Institut d'astrophysique de Paris · Summary A fair fraction of observed galactic disks present a...

$: Institut d'astrophysique de Paris · Summary A fair fraction of observed galactic disks present a bar corresponding to elongated isophotes beyond the central core. Numerical simulations$
University of Cambridge May, 1994

Dynamics of

self-gravitating disks

Christophe Pichon

Clare College

and

Institute of Astronomy

A dissertation submitted to the University of Cambridgein accordance with the regulations for admission

to the degree of Doctor of Philosophy

ii

Summary

A fair fraction of observed galactic disks present a bar corresponding to elongated isophotes beyondthe central core. Numerical simulations also suggest that bar instabilities are quite common. Whyshould some galaxies have bars and others not? This question is addressed here by investigating thestability of a self gravitating disk with respect to instabilities induced by the adiabatic re-alignment ofresonant orbits. It is shown that the dynamical equations can be recast in terms of the interaction ofresonant flows. Concentrating on the stars belonging to the inner Lindblad resonance, it is argued thatthis interaction yields an instability of Jeans’ type. The Jeans instability traps stars moving in phasewith respect to a growing potential; here the azimuthal instability corresponds to a rotating growingpotential that captures the lobe of orbital streams of resonant stars. The precession rate of this growingpotential is identified with the pattern speed of the bar. The instability criterion puts constraints on therelative dispersion in angular frequency of the underlying distribution function. The outcome of orbitalinstability is investigated while constructing distribution functions induced by the adiabatic re-alignmentof inner Lindblad orbits. These distribution functions maximise entropy while preserving the underlyingaxisymmetric component of the galaxy and the constraints of angular momentum, total energy, anddetailed circulations conservation. Below a given temperature, the disk prefers a barred configurationwith a given pattern speed. A analysis of the thermodynamics of these systems is presented when theinteraction between the orbits is independent of their shapes.

Axisymmetric distribution functions characterise the equilibrium state of a galaxy. Their formalexpression in terms of the invariants of the motion is desirable both from a theoretical and observationalpoint of view: they constrain observed radial and azimuthal velocity distributions measured from theabsorption lines in order to asses the gravitational nature of the equilibrium; they correspond to thecore of all numerical and linear analyses studying non-axisymmetric features to describe the unperturbedaxisymmetric initial state. A general inversion method is presented here leading to the construction ofdistribution functions chosen to match either a given potential-density pair and a given temperatureprofile, or alternatively to account for detailed observed kinematics.

Complete sequences of new analytic solutions of Einstein’s equations which describe thin supermassive disks are constructed. These solutions are derived geometrically. The identification of pointsacross two symmetrical cuts through a vacuum solution of Einstein’s equations defines the gradientdiscontinuity from which the properties of the disk can be deduced. The subset of possible cuts whichlead to physical solutions is presented. At large distances, all these disks become Newtonian, but in theircentral regions they exhibit relativistic features such as velocities close that of light, and large redshifts.For static metrics, the vacuum solution is derived via line superposition of fictitious sources placedsymmetrically on each side of the disk by analogy with the classical method of images. The correspondingsolutions describe two counter-rotating stellar streams. Sections with zero extrinsic curvature yield colddisks. Curved sections may induce disks which are stable against radial instability. The general counterrotating flat disk with planar pressure tensor is found. Owing to gravomagnetic forces, there is nosystematic method of constructing vacuum stationary fields for which the non-diagonal component of themetric is a free parameter. However, all static vacuum solutions may be extended to fully stationary fieldsvia simple algebraic transformations. Such disks can generate a great variety of di!erent metrics includingKerr’s metric with any ratio of a to m. A simple inversion formula is given which yields all distributionfunctions compatible with the characteristics of the flow, providing formally a complete description ofthe stellar dynamics of flattened relativistic disks.

CambridgeFebruary 1994

iv Summary

This dissertation is the outcome of my own work, except where explicit reference has been madeto the work of others.

Section 2 of chapter 2 has already been published as Pichon, C. “Orbital instabilities ingalaxies”, contribution to the “Ecole de Physique des Houches” on: Transport phenomena inAstrophysics, Plasma Physics, and Nuclear Physics. In press. The idea of applying thermody-namics to study the evolution of bars within galaxies was suggested to me by Jim Collett.

Most of the material presented in chapter 4 was published as Bıcak, J., Lynden-Bell, D &Pichon C. “Relativistic discs and flat galaxy models”, Monthly Notices of the Royal AstronomicalSociety (1993) 265 126-144, where the basic method was suggested by Lynden-Bell, and carriedout independently Bıcak, Lynden-Bell and myself.

The material of chapter 5 has been submitted to Physical Review D for publication as:Pichon, C. & Lynden-Bell, D. “New sources of Kerr and other metrics: Rapidly rotating rela-tivistic disks with pressure support”, and the main ideas developped here are presented inPichon, C. & Lynden-Bell, D.“Quasar engines and relativistic disk models”, contribution to theconference: N-body problems and gravitational dynamics, CNRS Editions.

I hereby declare that this dissertation is not substantially the same as any that I havesubmitted for a degree or diploma or other qualification at any other University, and no part ofit has been or is submitted for any such degree, diploma or any other qualification.

This dissertation does not exceed 60,000 words in length.

Christophe PichonCambridge

February 1994

vi

vii

Dedicated tothe memory of John Treherne

viii

ix

Un de mes amis definissait l’observateur comme celuiqui ne comprend rien a la theorie, et le theoricien comme un

exalte qui ne comprend rien a rien.

L. Boltzmann (1890)

x

Contents

xii Contents

1

Prologue

Various aspects of the dynamics of self-gravitating disks are presented here. Chapter 2 ad-dresses the stability of thin axisymmetric galactic stellar disks with respect to non-axisymmetricperturbations. Models of such disks accounting for observed or parameterised kinematics areconstructed in chapter 3. Chapters 4 and 5 present a derivation of the relativistic stellar dy-namics of flattened systems applied to models of rotating and counter-rotating super-massivedisks.

The dynamics of disks has been the subject of many excellent reviews in recent years, ofwhich those of Sellwood (1993) [101] for bars, and Tremaine (1989) [109] are particularly worthyof mention for their scope and detail. In view of this, given that each chapter has its ownintroduction, a di!erent perspective is taken in this prologue: the main common features andhypotheses, such as that of an axisymmetric infinitely thin disk, are sketched. These assumptionsprovide the mathematical simplicity which allows more detailed investigation without resortingto sophisticated numerical analysis.

The infinitely thin disk approximation is used throughout as it reduces the number ofdimensions involved, and is essential in constructing solutions of Einstein’s equation geometri-cally. As in electromagnetism, the properties of the distribution of sources is deduced in thiscontext from the jump conditions at the boundary of the discontinuity. This approximation al-lows tremendous simplification in comparison with the fully three dimensional set of Einstein’sequations. This assumption together with that of axisymmetry makes the distribution functioninversion problem a degenerate case (two degrees of functional freedom in the distribution func-tion varying with energy and angular momentum, only one functional constraint while requiringthat the distribution adds up to the right surface density). In e!ect, the thin disk approxima-tion corresponds to an extra explicit integral of the motion which allows the inversion whilealso requiring that the solution satisfies all observed or prescribed kinematical properties. Theinfinitely thin disk approximation turns out to be useful also when studying the stability of thedisk as it reduces the number of phase space dimensions required to describe the dynamics totwice the number of available invariants for an axisymmetric disk. This in turn implies that themotion of stars is regular (not chaotic), and suggests labelling the trajectories by combinationsof these invariants called action variables. These variables have two attractive properties: theconjugate variable varies uniformly between zero and 2! as the star describes one oscillation ofthe corresponding degree of freedom1; actions may behave adiabatically, i.e. remain approxi-mately constant, as the slow perturbation is a!ecting the symmetry on which the conservationof these actions relies.

All disks are assumed to have initially reached a stationary equilibrium. The energy of each

1 The formalism of angle and action variables is presented in appendix A.

Chapter 1 Prologue 1

star, or fluid element, is therefore an invariant of the motion corresponding to the fact that theequilibrium is left unchanged by translations in time.

All disks are also assumed to be (at least initially) axisymmetric. This symmetry hypothesisyields a supplementary invariant corresponding to the conservation of angular momentum. Asmentioned above, this invariant is crucial both for the construction of distribution functionspresented in chapter 3 and to impose regular motions within the disk. In the context of relativity,axial symmetry reduces the number of unknown functions to be found from ten to three, andthe number of variables correspondingly.

Once all symmetry hypothesis have been made, it is instructive to analyse the behaviour ofthese disks while assuming that some combinations of the actions behave like quasi-invariantsas done in chapter 2, sections 3 and 4. At this level of description, it is then possible to analysethe instability in some detail, pointing out a clear analogy with the Jeans instability.

The above assumptions have a limited scope of validity which obviously depends on thetype of physical disks to which they are applied. Relativistic gaseous disks are, for instance,crudely described as infinitely thin systems or even as stationary objects given that they radiatetheir binding energy at a luminosity close to the Eddington luminosity. Some galactic disks,however, are more likely to be well described as flat and axisymmetric, though it remains tobe shown that all bars arise from initially axisymmetric disks. Adiabatic invariance assumesthat the instability occurs slowly compared with the typical period of a star describing its orbitwhich is not strictly in agreement compared to the results of some numerical simulations. Theasumption of adiabatic invariance yields nevertheless the correct criterion for the marginallyunstable modes, though it is unclear that non-axisymetric features observed in real galaxiescorrespond to such modes.

There are, however, good reasons for believing that most of the above assumptions aresatisfactory within the scope of this work. For instance, chaotic orbits are present in real galaxies,though the di!usion of chaotic orbits away from the periodic ones takes somewhat longer thanthe age of the universe. Similarly, it is a simple matter, using the linearity of Poisson’s equation,to thicken infinitely thin classical disks, and it is likely that a similar procedure is available forrelativistic disks.

2 Prologue Chapter 1

2

Orbital instabilitiesand bar formation

2.1 Introduction

Galactic bars were first mentioned in Astronomy by Curtis (1918) [22] who described “afairly common type of spiral nebula whose main characteristic is a band of matter extendingdiametrically across the nucleus and inner part of the spiral. [...] The general appearance isthat of the Greek letter "”. Later Hubble (1925 onwards) [48] introduced the term “barredgalaxy” and distinguished two parallel series of spirals: normal or ordinary (S), and barred(SB). Roughly one third of all spiral galaxies show developed bars (de Vaucouleur 1963) [24]while another third present weak bars. Barred galaxies do not di!er from non-barred ones asfar as integral properties, like luminosities, fractional HI content, colours etc, are concerned.Why some spiral galaxies should form bars, and others not is therefore a puzzling challenge forastronomers.

Bar formation or growth has been variously attributed to

(1) the gradual outward transfer of stellar orbital angular momentum by spiral density wavesand eccentricity pumping (Lynden-Bell & Kalnajs 1972 [76]),

(2) the density response of a stellar disk to a weak perturbation in the underlying potentialwhich is amplified into a standing wave (Toomre 1981 [105], Zang 1976 [114], Kalnajs[54,56,58,59]), and

(3) the trapping of elongated stellar orbits by a bar-like perturbation in the inner part ofgalaxies. (Lynden-Bell 1973 [77], 1979 [82]). Sellwood 1993 [101] gives a extensive reviewof the pro and cons of these theories. This chapter addresses the question of bar formationfrom the viewpoint of orbital instability.

2.1.1 Context

Consider a rotating dissipative protogalactic cloud collapsing into what will be a galaxy. Thecloud rapidly loses its energy of motion along the axis of rotation, but conserves its total angularmomentum. It therefore settles into a flat system. While the process of baryonic matter infallfrom the halo continues, it can be argued that the disk becomes thin and axisymmetric. Whilecooling, the gas produces stars. The disk surface density in stars increases and the velocitydispersion in stars decreases as more and more gas is injected. This process goes on until

4 Orbital instabilities and bar formation Chapter 2

eventually an azimuthal instability occurs which redistributes angular momentum. This scenariois compatible with the observations of galactic disks, and provides a symmetry necessary forthe analytical treatment. It does not rely on unknown initial conditions. The influence fromexternal fields arising from encounters with other galaxies undoubtedly amplifies the growthrate of unstable eigen-modes. This analysis is nevertheless restricted to the normal modes ofan isolated galaxy. The red colour of bar suggests neglecting the influence of the gas, thoughit is acknowledged that the inherent instability of the scarce gaseous component might triggerthe stellar instability. The self consistent gravitational field may conceptually be split into twocomponents: a mean field, satisfying Poisson equation for density, calculated by averaging locallyover a Dirac distribution corresponding to point-like stars and the fluctuation of this mean fielddue to neighbouring stars, which can be evaluated statistically in order to quantify its relevance.In fact, the long range interaction of gravity and the large numbers of stars to be found ingalaxies suggest that neglecting this latter component is a good approximation. The kinetictheory then provides a natural framework in which to follow statistically the behaviour of allthe stars as they describe their orbits. While the fluid description is attractive for describingthe global e!ect of the other stars on a given star, it is obviously insu#cient to account for self-gravity enhancement induced by the motion of two stars tracing orbits which have a constantrelative phase delay (i.e. orbits which resonate). Indeed, in stellar dynamics, as opposed tofluid mechanics, the mean free path of a given particle is quite large compared to the size ofthe system; therefore resonance will generally last until the weak but cumulative e!ect of thetorque has modified the nature of the two orbits. This reorganisation of resonant orbits mayrestructure the galaxy1.

The core of this analysis therefore relies on the two fundamental equations of galacticdynamics, namely

• Boltzmann’s equation, which describes the evolution of the distribution function in a givenpotential and characterises the dynamics,

• Poisson’s equation, which provides a prescription for the mean field induced by a givendistribution of stars.

These equations may be re-expressed exactly in order to identify the contribution from theresonances of orbits, independently of the contribution corresponding to the detailed phase ofthe stars on each orbit. The gravitational interactions of individual stars may be re-expressedin terms of the gravitational interactions of whole orbital streams, i.e. steady streams of starsthat follow the same orbit. The net e!ect of this interaction is to distort and re-orientate theorbits. Such instabilities restructure the galaxy only if the orbits co-operate by moving in thedirection of the torques on them. Orbits corresponding to the symmetry of bars present such acooperative behaviour. A collective re-alignment of quasi-resonant orbits is therefore expectedto produce the azimuthal analogue to Jeans gravitational instability. Jeans instability works bytrapping particles in a growing potential well. Azimuthal gravitational instability works herenot by trapping the stars but by trapping the lobes of orbital streams of stars azimuthally ina growing potential well. As with the classical instability the potential well may be rotatingrelative to inertial axes. The rotation velocity of this frame is to be identified with the patternspeed of the bar.

The actual instability criterion is derived by very classical means. The starting point is adistribution function specifying a given galaxy. For small perturbations away from equilibrium,

1 From an energy point of view, it can be argued that the disk was maintained far from its minimum energystate by the conservation of angular momentum. Hence the dominant factor controlling its dynamical evolution isthe way in which specific angular momentum may be redistributed within the disc. The mean field theory claimson the other hand that no angular momentum is exchanged between stars except on resonance. It follows thatthe interaction of resonant orbits provides a way to restructure the galaxy.

Section 2.2 The alignment instability model 5

the equations may be linearised to find a dispersion relation for the normal modes of the system.The condition for the existence of linear marginally-stable modes is derived and related to theconstraints the existence of these modes would impose on the underlying galaxy.

This analysis yield the following results:

• an azimuthal instability criterion for a given galaxy, and

• the angular frequency at which the instability corresponding to this criterion propagates.

2.1.2 Outline

The fully self consistent model of a real galaxy sketched in this introduction is postponed tosection 4. Section 2 introduces a toy model, the purpose of which is to give a general idea of themost relevant physical concepts; the behaviour of an assembly of tumbling ellipses as they gothrough bi-symmetric azimuthal instability is described in some detail. The idea of an e!ectiveinteraction potential between the ellipses is introduced. This leads to a Jeans-like instabilitycriterion. Section 3 focusses on a fictitious galaxy in which the only relevant resonance is theinner Lindblad one. Introducing an e!ective interaction potential similar to that of the firstmodel, together with some assumptions about its analytical expression leads to an analogouscriterion under less stringent conditions. It suggests that the orbits should be distorted asthey re-orient themselves with respect to the potential trough. The fate of this adiabatic orbitalinstability is analysed and illustrated in section 4. Below a critical temperature, the system findsa non-axisymmetric configuration which rotates at a temperature dependent pattern speed. Allthe resonances are taken into account in section 5, where the interaction potential is derivedexactly from Poisson’s equation. The exact integral equation accounting for the self consistencyof the field is given in terms of angles and actions. The corresponding formulation is contrastedwith that of Kalnajs (1971) [54] and Zang (1976) [114]. Various techniques to solve this integralequation are sketched and developed in the appendix.

· • !" ! • ·

2.2 The alignment instability model

The toy model described in the following section provides a simplified insight into themechanism that may be involved in the production of bars in galaxies. It is constructed aroundthree basic concepts introduced by Lynden-Bell (1979) [82] to account for bar formation viaorbital instability. These address the following questions

• Which orbits form a bar ?

If a particular star has an orbit with radial period 2!/" and an average period around thegalaxy 2!/$, then when viewed from axes that rotate at the rate $# "/2, it will close ina figure not unlike a centred ellipse. If this whole orbit is populated by an orbital stream,it will look stationary in the rotating frame but will tumble over and over at the angularrate $# "/2 in the fixed frame. Numerical simulations suggest that the stationary patternof such orbits form the skeleton of a bar.

• How do they interact ?

If two orbital streams tumble at the same rate, $!, but one has its lobes a little aheadof the other, then the gravitational interaction of the streams will produce a forwardstorque on the hindmost stream and a backwards torque on the foremost stream. Because


the two tumbling rates are equal (resonant), this torque will not reverse but will continueuntil it modifies the orbits. If however the tumbling rates are significantly di!erent, thenthe torques will reverse as the angle between the lobes changes, so the e!ects will cancel.Near-resonant orbits with a small di!erence of tumbling rates may nevertheless bring aboutsignificant changes before the di!erence can lead to any reversal of the torque.

• How does a backward torque a!ect a tumbling orbit ?

The orbits follow a remarkable generic behaviour as described by Lynden-Bell and Kalnajs(1972) [76]: a given orbit accelerates when pulled backwards adiabatically and slows downwhen pulled forwards. The underlying physical property is that their adiabatic momentof inertia is generally negative. Therefore no collective alignment is to be expected fromsuch “donkey” orbits.2 However, for the so called inner Lindblad resonance, the adiabaticmoment of inertia is positive in the central parts of galaxies. The lobes of two orbitalstreams may then attract and cooperate, vibrating around exact alignment and so forminga deeper potential well thats traps further orbits into that orientation, and so makes a bar.

The essence of these ideas are formalised in the following model.

2.2.1 Simplifying assumptions

Consider a set of ellipses or centred ovals rotating about their centre, which represents quasi-resonant inner Lindblad resonant orbits viewed from a rotating frame $p. They will all have thesame characteristics (i.e. same geometry and same mass). Each ellipse will therefore be entirelyspecified by the orientation of its semi-major axis with respect to the rotating frame, #, and theangular frequency $!.

The relative interaction of these ellipses will be assumed to derive from an e!ective align-ment potential:

$12 = G A2 cos 2 (#1 # #2) , (2.2.1)

where G is the gravitational constant, and #1 # #2 measures the relative azimuthal orientationof the two orbital lobes. In this model, A2 is taken to be constant. The torque that ellipse “two”applies on ellipse “one” may then be written:

d h1/dt = %$12/%#1 . (2.2.2)

The response in angular velocity to such a torque can be derived via the moment of inertia 1/&of the ellipse corresponding to the adiabatic moment of inertia of the inner Lindblad orbit thisellipse is supposed to represent, where & is given by

& =!%$!

%h

"

J

, (2.2.3)

so thatd$!/dt = &%$12/%#1 . (2.2.4)

Note that &may take both signs to account for the various orbital responses. Let F!(#,$!, t)d# d$!

be the number of ellipses with orientation between # and # + d# and with angular frequency

2 Collett, private communication nevertheless pointed out that a given orbit could possibly anti-align withthe potential trough created by the other orbits, and widely oscillate, therefore spending most of its time at ninetydegrees to the minimum of the potential, creating a pattern. Alternatively, he also pointed out that donkey orbitsmight be unstable with respect to the azimuthal equivalent of the two-stream instability in plasma physics.


Figure 2.2.1: The toy model problem: under what circumstances will an assembly of rotating centredovals rotating about their centre evolves spontaneously towards a barred configuration ?

between $! and $! + d$!. The conservation of orbits implies that this distribution functionsatisfies a continuity equation in (#,$!) space

%F !

%t+%

%#($!F

!) +%

%$!

!d$!

dtF !"

= 0 . (2.2.5)

Assuming that & is a constant of either sign, depending on the co-operative or “donkey” be-haviour of the orbits, d$!/dt becomes independent of $!, so F ! satisfies a Boltzmann equation:

%F !

%t+ $!

%F !

%#+%F !

%$!

!&%$

%t

"= 0 . (2.2.6)

The density of ellipses which have orientation # may then be defined by:

' (#, t) =#

F ! (#,$!, t) d$! . (2.2.7)

The interaction potential generated by the “two” ellipses may be obtained by multiplyingEq. (2.2.1) with F !

2 and integrating over their orientations

$ (#, t) = G A2#' (#2, t) cos 2 (## #2) d#2 . (2.2.8)

The stationary axisymmetric unperturbed state obeys F ! = F!0 ($!) and ' = '0 = constant.

Therefore $ is identically zero and Eq. (2.2.5), Eq. (2.2.6) and Eq. (2.2.7) are trivially satisfied.What is the stability of such a configuration to bi-symmetrical instabilities ?


Writting F ! = F!0 + f!, and linearising the equation with respect to the perturbation f! and $,

Eq. (2.2.6) becomes%f!

%t+ $!

%f!

%#+ &

%F !0

%$!

%$

%#= 0 . (2.2.9)

2.2.2 Instability criterion

The Fourier expansion of f! with respect to # reads

f! ="$

m=#"eim"f!

m ($!, t) , (2.2.10)

Similarly

' ="$

#"eim"'m , where 'm =

#f!

md$! . (2.2.11)

Substituting Eq. (2.2.11) into (2.2.8) gives:$ = !G A2

%'2 e2i" + '#2e

#2i"&

. (2.2.12)When the perturbation has no |m| = 2 component, Eq. (2.2.8) becomes

%f!/%t + $! %f!/%# = 0 . (2.2.13)

The general solution is f! = g(# # $!t) where g is an arbitrary function without an |m| = 2component. As Eq. (2.2.9) is linear, its complete solution corresponds to the sum of this solution(which propagates in ($,#) space) and a solution for the resonant modes |m| = 2. These modesare responsible for the gravitational instability. For growing modes 3, f!

m $ ei#t, where ( has anegative imaginary part. From Eqs. (2.2.9), (2.2.10) and (2.2.12), it follows that

i (( + m$!) f!m = #i m !G 'mA2&

%F !0

%$!, for |m| = 2 . (2.2.14)

Dividing by i(( + m$!)/A, and integrating over all $! so as to leave 'm on the r.h.s. leads tothe identity

1 = !G# #&A2 %F !

0 /%$!

$! # $pd$! , (2.2.15)

where $p = #(/m (which corresponds to the angular frequency at which the perturbation ispropagating). Recall that $p has a positive imaginary part (for m = 2) corresponding to thegrowth rate of the instability. Marginal stability is therefore reached as the imaginary part of$p vanishes. The integral in Eq. (2.2.15) may then be written as the sum of a Cauchy principalpart, and a half residue corresponding to the pole at $! = $p:

1 = !G# #&A2 %F !

0 /%$!

$! # $pd$! # i!2G

'&A2%F !

0 /%$!

$!

(

!p

, (2.2.16)

where $p is now real. Identifying real and imaginary parts, it follows that

1 = !G# #&A2 %F !

0 /%$!

$! # $pd$! , (2.2.17a)

%F !0

%$!= 0 for $! = $p . (2.2.17b)

Eq. (2.2.17b) fixes $p, the pattern speed of the bi-symmetrical instability.

3 Since this analysis concentrates on a dispersion relation for marginally stable modes, rather than transientbehaviour, there is no need to Laplace transform Eq. (2.2.12), and so get the instability criterion by avoiding thepoles, as is usually done for Landau damping.


Figure 2.2.2: phase space portrait of the azimuthal collapse; phase mixing occurs in any frame, whereasself-gravity is most e#cient in the frame corresponding to the maximum of F ($!).

2.2.3 Discussion: azimuthal Jeans instability

Candidates for the angular frequency of the perturbation are given by the extrema of F !0 .

Eq. (2.2.17a) tells which (if any) will be dominant. The pattern speed of the perturbationas well as of its stability is therefore a characteristic of the unperturbed galaxy, as illustratedby Fig. 2.2.24. When & is positive5, Eq. (2.2.16) may be re-arranged as

)2! % 1

2G M& , (2.2.18)

where )2! measures the weighted dispersion of the distribution function in the neighbourhood of

the resonance1)2

!

=# '

A2&%F !0 /%$!

$p # $!

(d$!

)#A2 &F !

0 d$! . (2.2.19)

Equation (2.2.18) is very similar to a Jeans instability criterion. It implies that the dispersionin $! around $p should be less than an e!ective G density, built from the adiabatic moment ofinertia &, and GA2, the amplitude of the e!ective interaction potential. The inverse adiabaticmoment of inertia, &, scales like [Length#2] while A2 scales like [Length#1]. As )2

! measuresthe dispersion in angular velocity in the frame of the maximum of F !

0 ($!), the criterion (2.2.19)implies that the lower the dispersion, the more e#cient the instability.

4 This description was suggested to me by Collett who derived independently the instability criterion (2.2.17)in 1987. Earn [30] also derives (2.2.17).

5 here, the equivalent to the two stream instability corresponds to a negative ! and a minimum of thedistribution function, "2F !

0 /"!2! ! 0, i.e. few repealing orbits widely librating so as to anti-align to the potential

valley.


· • !" ! • ·

2.3 The inner Lindblad resonance model

2.3.1 Orbital streams instability criterion

Consider a galaxy made of orbital streams of inner Lindblad resonances. Each orbital streamis characterised by its orientation, #, its specific angular momentum, h, and its circulation6

4! J = 4!(JR + 1/2 h) . As such streams interact gravitationally, # and h change, but J istaken to be adiabatically invariant. Indeed, it was shown in section 2.2 that the onset of anorbital instability required a narrow dispersion in the angular velocity of near resonant orbits.Under this assumption, the period corresponding to the relative libration of these orbits aroundtheir mean pattern speed, $p, should be large as they almost resonate. Many orbits are foundfor which the ratio of the orbital period to the oscillation period is very small. It is thereforeappropriate to construct the corresponding adiabatic invariant J. Strictly speaking, J is only aninvariant for the quasi-resonant orbits. On the other hand, non-resonant orbits are not expectedto play any significant role in restructuring the galaxy. It is also assumed in this model that anytwo orbital streams interact through their mutual potential energy, the angular dependent partof which is approximated by #%12 where

%12 = G A1A2 cos 2 (#1 # #2) . (2.3.1)

Here, #1 # #2 is the relative azimuth of the orbital lobes. The amplitude of the alignmentpotential must vanish if either orbit is circular, and for small ellipticities it should be proportionalto their product. The critical (and somewhat arbitrary) assumption in this section is that theamplitude depends on the orbits through a product A1 A2 with A1 depending on orbit one onlyand A2 on orbit two. While such a split is unlikely to be strictly satisfied – and indeed it is shownin the appendix not to be true – it should give a good qualitative account for the interactionbetween similar resonant orbits which are responsible for the type of orbital instability describedin section 2.2.The specific torque on stream “one” due to unit mass on stream “two” then reads

d h1/dt = %%/%#1 . (2.3.2)

Let F$(#, *, h, J, t)d#d* dh dJ be the mass weighted distribution function of stars belongingto an orbital stream with lobe azimuths in the range # to #+ d#, with orbital phase between* and * + d*, with angular momenta in the range h to h + dh and with circulation between Jand J + dJ.This distribution function obeys a Boltzmann equation corresponding to the conservation ofmass in phase space. Assuming that the only relevant resonance is the inner Lindblad one(i.e. all other resonances between orbits and in the relative orbital phases are disregarded), anaveraging principle may then be applied to this equation in order to write a Boltzmann equationfor the number of orbital streams. These are the relevant objects for this analysis according tothe above assumptions. Calling F =

*F $ d*, it follows that:

%F

%t+ $!

%F

%#+%%%#

%F

%h= 0 . (2.3.3)

6 See Appendix A & Lynden-Bell (1972) [76] for an introduction to angle and action variables.

Section 2.3 The inner Lindblad resonance model 11

With this definition, F is a distribution function in phase-space since (#, h) are canonicallyconjugate variables. The orientation potential, %, due to all the orbital streams is found byweighting Eq. (2.3.1) with F2 and integrating over all streams “two”:

% (#, h, J, t) = G A (h, J)#' (#2, t) cos 2 (## #2) d#2 , (2.3.4)

where ' is the “e!ective” density of streams at orientation #, and is defined by

' (#, t) =# #

A (h, J)F (#, h, J, t)dh dJ . (2.3.5)

Note that with this set of variables, % depends on h and J as well as orientation because thetorque applied on a given orbit depends on the shape and size of that orbit.Unperturbed axially-symmetric steady states have F = F0(h, J). Writing F = F0 + f andlinearising in the perturbed quantities f and $ equation Eq. (2.3.3) becomes

%f

%t+ $!

%f

%#+%%%#

%F0

%h= 0 . (2.3.6)

Note that%F0

%h=%F0

%$!· & , where & =

!%$!

%h

"

J

. (2.3.7)

Equation (2.3.6) may then be formally identified with Eq. (2.2.9). Equation (2.3.5) is the stellarequivalent of Eq. (2.2.7). The sum over d$ becomes here a double sum over dh dJ. Theassumption that the function A is separable with respect to the characteristics of orbits “one”and “two” leads to a solution which is derived following precisely the same steps as that ofsection 2.2 and reads

1 = !G# #

A2 (h, J) [#&%F0/%$!]$! # $p

dh dJ . (2.3.8)

2.3.2 Discussion: alignment and distortion

How does the criterion (2.3.8) relate to the toy model criterion, Eq. (2.2.17) ?Defining

F ! ($!, J) = F0 (J, h [$!, J]) , and A! ($!, J) = A (J, h [$!, J]) , (2.3.9)

the integrand in Eq. (2.3.8) may be integrated by parts##%%F !/%$!

&A2

!d J = # d

d$!

!#F ! A2

! d J"#+F ! A2

!

,+#

F !%%A2

!/%$!

&d J .

The square brackets correspond to the limits of integration, for which the argument vanishesbecause these are non-interacting orbits (circular orbits)

+F ! A2

!

,& 0. Calling

F !0 ($!) = 1/< A2& >

#F ! ($!, J) A2

! d J , (2.3.10a)

d

d$!+F !

0 ($!) = #1/< A2& >#

F ! ($!, J) %A2!/%$! d J , (2.3.10b)

equation (2.3.8) then becomes

1!G

=!# # < &A2 > d/d$! (F !

0 + +F!0 )

$! # $pd$!

", (2.3.11)


in direct analogy with Eq. (2.2.15). Here the constant < A2& > corresponds to the aver-age

*F !A2

!dJd$! in order to identify Eq. (2.3.8) with Eqs. (2.2.17). Note that F !d$!dJ =&F0 dh dJ. Therefore F ! d$! dJ is the distribution of orbital streams with angular frequency inthe range $! to $! + d$! and invariants between J and J + dJ weighted by the correspondingadiabatic moment of inertia. F !

0 ($!) is therefore the “e!ective” mass density of resonant orbitsat the frequency $!. As & generally changes sign, the above summation is algebraic. Recallthat negative & correspond to those inner Lindblad orbits which fail to cooperate. The inte-grand in Eq. (2.3.10b) may be rewritten as: F !

0 (%A2/%$!)J = F!0 (%A2/%h)J (%h/%$!)J . But

(%A2/%h)J measures the distortion – variation in size and eccentricity – of the resonant orbitwhen adiabatically pulled forward. and (%h/%$!)J cancels with the implicit weighting by & inEq. (2.3.10)7. Therefore +F !

0 is the correction to the e!ective mass density of orbits inducedby the distortion of the resonant orbits. In summary, for a galaxy made of orbital streams ofinner Lindblad resonant orbits, it was shown that as the orbits align, they are distorted by thegrowing potential.

2.4 Adiabatically relaxed bars

Once a bi-symmetric instability has occurred – triggered by an instability of orbital natureor indeed by other source of instability8– it is worth analysing the most likely path of evolution.Numerical simulations (e.g. Sellwood 1993 [101] and references enclosed) predict a characteristicsecular evolution of the amplitude and the pattern speed of the bar beyond the linear regimedescribed in the previous section. Such a trend is investigated here using the framework ofthermodynamics. The state of maximum entropy compatible with total energy and angularmomentum conservation corresponds to uniform rotation. In practise, this state is not reachedbecause of the preservation of other quasi-invariants such as the circulations of section 2.3 whichare kept adiabatically constant. The distribution function corresponding to the extremum ofentropy, given these supplementary constraints, is derived here. For that purpose, it is assumedthat the underlying axisymmetric component of the galaxy is left unchanged as a bar grows. Thishypothesis is only exact in the linear regime of perturbation theory. Here, it will neverthelessbe assumed to hold beyond that regime.

2.4.1 The coupling energy

The total energy, E, is the sum of the kinetic energy, K, and the gravitational energy of in-teraction, V . When only one resonance is relevant, E may be formally re-arranged as twocomponents:

• the energy arising from the orbits distributed axisymmetrically:

Eaxi = K + V0 =#, (h, J) F (J,#) dJ dh d## V0 , (2.4.1)

where ,(h, J) is the energy of the orbit (h, J) = J as defined by the axisymmetric componentof the potential, and V0 is the axisymmetric potential energy which is subtracted here toavoid counting its contribution twice;

7 this result is not surprising since a linear theory should include both the alignment and the distortionindependently

8 In any circumstance, the adiabatic re-alignment of elongated x1 orbits is believed to re-enforce existingbars.

Section 2.4 Adiabatically relaxed bars 13

• the non-axisymmetric component corresponding to the contribution from the interaction ofquasi-resonant inner Lindblad orbits:

V2 = #12

#F (J,#) % (J,#) d# d2J , (2.4.2)

where% (J,#) =

#A2 (J, J$) F (J$,#$) cos 2 (## #$) d#$ d2J$ . (2.4.3)

The function F (J,#) =*

F (J,#, *) d* is the distribution function of resonant orbitalstreams with adiabatic invariant J, angular momentum h, and orientation # which aretaking part in the instability. The expression &E, for the total energy measured withrespect to the axisymmetric potential energy follows

&E = E # V0 =# '

, (J)# 12% (J,#)

(F (J$,#$) d2J d# , (2.4.4)

Equation (2.4.4) together with Eq. (2.4.3) is assumed to represent an adequate descriptionof the energy of the system. Hereafter &E shall be referred to as the energy, and is assumedto be constant in this section. Its functional expression avoids the divergences usually foundin the statistical mechanics of gravitating systems. The energy of interaction remains finitefor overlapping orbits, and no orbit escapes the galaxy. In fact, the orbits become circular asthe star reaches its maximum specific momentum at a given circulation, and so the orbit stopscontributing to the instability.

2.4.2 States of extremum entropy

The Boltzmann entropy for an assembly of orbital streams reads

S = ##

F (J,#) log F (J,#) d2J d# , (2.4.5)

in units of Boltzmann’s constant set to unity. Angular momentum is exchanged between resonantorbits, but strict invariance is assumed for the adiabatic action corresponding to the motion ofa given star along its orbit9. All possible variations of F are explored, subject to the constraintthat $axi is left unchanged. The total energy, &E, the total angular momentum, H , and themass density of orbits, n(J) dJ, which have a circulation in the range J, J+dJ are all supposed tobe conserved in the process10. The latter hypothesis holds as long as the conditions of adiabaticinvariance hold. Realistic values of n(J) may be derived from the axisymmetric distributionfunctions, F0(,, h), constructed in the next chapter. Its behaviour is illustrated on Fig. 2.4.4for the Isochrone potential in the epicyclic approximation as described in the next subsection.Explicitly,

n (J) =#

F0 (, [J, h] , h) dh , (2.4.6)

where ,[J, h] is given implicitly by

J =12h +

1!

- .

2 ($0 (R) + ,)# h2

R2dR . (2.4.7)

9 for the Hamiltonian #(J)""(J,$), the canonical angle coupled to J is ignorable, therefore J is formally astrict invariant by construction.

10 In fact, for collisionless systems, the density in phase space described by F is also conserved following themotion; however, the coarse-grained distribution will evolve in time.


These conserved quantities put three constraints on the variation of F , namely:

n (J) =#

F (h, J,#) d# dh , (2.4.8a)

H =#

F (h, J,#) h d- , (2.4.8b)

&E =#

F (h, J,#)', (h, J)# 1

2% (h, J,#)

(d- , (2.4.8c)

with d- = d# dh dJ, the phase space volume element. The variation of V2 in Eq. (2.4.8c) viaEq. (2.4.3) may be carried out by switching indices in the double integration:

+V2 = #12

# #A2 (J, J$) (+F F $ + F +F $) cos [2 (## #$)] d- d- $ , (2.4.9a)

= ##+F (J,#) % (J,#) d- . (2.4.9b)

Note that the 1/2 factor of Eq. (2.4.9a) required to avoid counting all pairs of interaction twiceyields the correct potential in Eq. (2.4.9b). It follows that

+S = 0 =#+F [lnF (J,#) + 1# & (J) + ., (J)# .% (J,#)# .$ph] d- , (2.4.10)

where the Lagrange multipliers &(J) , .$p11 and . account for the constraints Eqs. (2.4.8).

Solving for all variations of +F gives

F (J,#) = C (J) exp [#. {, (J)#% (J,#)# $p h}] . (2.4.12)

Equation (2.4.12) is not a surprise; the distribution function is a function of the Jacobi invariant,,(J)#%(J,#)#$p h, which corresponds to the energy of an orbit measured in the frame whichrotates at the frequency $p.A solution must be found self-consistently as F is a functional of % which in turn is a functionalof F :

% (J,#) =#

A2 (J, J$) C (J$) exp (#. [, (J$)# $ph#% (J$,#$)]) cos [2 (## #$)] d- $ . (2.4.13)

11 Collett and Lynden-Bell (1989) [18] have shown that !p, the Lagrange multiplier associated with conser-vation of angular momentum can be identified with the pattern speed of the perturbation. The proof is derivedwhile requiring that the energy of a barotropic flow is stationary with respect to all displacements preservingcirculations, total angular momentum and number of particles. In fluid dynamics, the flow is characterised byits velocity field, u, its pressure, p, and density, %, responsible for the gravitational potential &. The first orderchange in the energy then reads

#E = !p#H +

# #'1 %v · d2r "

# # #'1 # · %v d3r

+

# # #'2

'"v $ ( + #

!1

2v2 +

#dp

%" & +

1

2(!p $ r)2

"(d3r , (2.4.11)

where the velocity, v = u " !p $ r, corresponds to the velocity of the flow measured in a frame which rotatesat the frequency !p, and ( = (# $ u) is the vorticity of the flow. The displacements, '1 and '2 may bevaried independently given the conservation of circulations. At constant total angular momentum, and constanttotal number of stars, the first three terms in Eq. (2.4.10) vanish for all displacements '1. The integrand of theremaining term corresponds to the equation of motion of a steady flow in a coordinate frame rotating with angularvelocity !p. Requiring that this term vanishes for all displacements '2 implies that the Lagrange multiplier !p

is the frequency of the frame in which the flow is stationary, i.e. the pattern speed. Note that Eq. (2.4.10) alsoprovides an value for the pattern speed directly from the adiabatic variation of E with respect to H , namely!p = ("E/"H )J . A similar proof holds for stellar dynamics.


This solution is easily found under the assumption that A2(J, J$) can be approximated as aproduct of A(J) by A(J$) as in section 2.3. Indeed Eq. (2.4.3) then becomes:

% (J,#) = A (J) cos (2#) K , (2.4.14)

where K, the “e!ective” bi-symmetric over-density, is defined to be

K =#

F (J,#) A (J) cos (2#) d- . (2.4.15)

All angles are measured with respect to the maximum of the over density which is assumed tobe symmetrical in #. Inserting Eq. (2.4.12) for F in Eq. (2.4.15) yields, self-consistently,

K =#

AC exp [#. (,# $ph# A cos (2#) K)] cos (2#) d2J d# , (2.4.16a)

= 2!#

A (J)C (J) I1 [.A (J)K] e#%&(J)+%!ph d2J . (2.4.16b)

where In is the Bessel function of order n. Equation (2.4.16) may be re-written formally as

K

N=/

A (J)I1 [.A (J)K]I0 [.A (J)K]

0& G (.K) . (2.4.17)

The bracket ' ( denotes the normalised average over phase space weighted by the distribu-tion function F (J,#). Non-trivial solutions of this implicit equation correspond to the relationbetween the amplitude of the bar and the temperature of the system. A critical temperature,1/.crit, may be defined in terms of G and its derivatives below which Eq. (2.4.17) has morethan one trivial root (K & 0 is always a solution given that I1[0+]/I0[0+] = 0+; I1[x]/I0[x]tends asymptotically to 1 as x) *; therefore the two curves y = N#1x/. and y = G(x) mustintersect for su#ciently high values of .).

The actual values for the Lagrange multipliers C(J), . and $p may be derived from theconstraints, Eqs. (2.4.8). These may be re-arranged as

n (J) = 2!C (J)#

I0 (.A (J)K) e#% &(J)+%!phdh , (2.4.18a)

'h( & H

N=% logN

%/, where / = $p. , (2.4.18b)

'e( & &E

N= #% log N

%.+

K2

2, (2.4.18c)

where the auxiliary function, N , corresponds to the total number of orbits:

N =#

F (J,#)d2J d# = 2!#

C (J) I0 [.A (J) K] e#% &(J)+%!phd2J . (2.4.19)

In Eqs. (2.4.18), the partial derivation is done holding constant the other Lagrange multipli-ers, the boundaries of integration and K, the e!ective number of orbits caught in the bar.Eqs. (2.4.16) and (2.4.18) yield the relationship between the pattern speed of the bar and itsamplitude as a function of the temperature, which follows from the macroscopic properties ofthe disk (mean energy and angular momentum, initial distribution of orbital streams with agiven circulation).


0 0.1 0.2 0.3 0.4 0.5 0.6

1/beta

0

0.2

0.4

0.6

0.8

1

K

-1.5 -1 -0.5 0 0.5 1 1.5

phi

-1

-0.5

0

0.5

1

gamma

Figure 2.4.1: the amplitude of the bar K/AN (left panel) as a function of the temperature (.NA2)#1.The width of the bar – defined here by the separation between the zeros of the reduced autocorrelation/(*) &< '(*+#)'(#) > / < '(#)2 > and plotted on the right panel as a function of * – decreases slowlyas a function of .AK = 0.5, 1, ..5 .

2.4.3 A simple example: rigid rods

Consider first the thermodynamics of identical rigid rods. For these objects, both the mo-ments of inertia, I , and the amplitude of interaction, A2, are assumed to be constant. ConditionEq. (2.4.17) then becomes

K =I1

%.AK

&

I0

%.AK

& AN . (2.4.20)

Equation (2.4.20) will have non-trivial solutions if

.A2N + 1(I1/I0)

$ (0)= 2 . (2.4.21)

This equation gives the critical temperature, A2N/2, below which the rods start aligning asillustrated on Fig. 2.4.1. The kinetic energy of each rod reads I#1h2/2, which implies that

N = C

.2! I

.exp

!12I$2

p

"I0

%.AK

&, (2.4.22)

therefore Eqs. (2.4.34) become $p = 'h(/I and 'e( = I$2p/2 + .#1/2# N#1K2/2. The pattern

speed of the aligned rods corresponds to the mean angular frequency. The energy versus tem-perature variations are illustrated on Fig. 2.4.2. The specific heat at constant total momentum,CH = #.2%'e(/%., is given by CH = 1/2 + .2K%K/%.. It is positive everywhere, which is aproperty of one dimensional self-gravitating systems12. Note that CH is discontinuous at the

12 This property is best understood by contrasting the qualitatively di$erent behaviour of gravity in one andtwo dimensions; consider a test particle moving in the potential well & of a central mass concentration. In twodimensions, the test particle describes a rosette while oscillating radially between perigee and apogee. In onedimension, the particle oscillates symmetrically between "xmax and xmax = &#1("#). If energy is retrieved fromthat particle, it readjusts its trajectory to a state of narrower oscillation, corresponding to lower mean kineticenergy. In two dimensions, the particle re-adjust its trajectory to a more eccentric Lissajous figure which, contraryto the one dimensional case, has a higher mean kinetic energy. If this increase in energy more than counter balancesthe energy loss induced by the self-consistent change in potential energy, an ensemble of self-gravitating particlesdisplays negative specific heat.


0 0.1 0.2 0.3 0.4 0.5 0.6

T=1/beta

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

energy

0.1 0.2 0.3 0.4 0.5 0.6

T=1/beta

0.5

1

1.5

2

2.5

specific heat

Figure 2.4.2: the energy of the bar, (left panel), and its specific heat, (right panel), as a function of thetemperature (.NA2)#1. Note the discontinuity of CH – inducing a kink in the energy – at the criticaltemperature.

critical temperature as shown on Fig. 2.4.2, which is characteristic of a phase transition describedin the mean field approximation.

2.4.4 Low temperature expansion of the general case

The temperature, 1/., measures the relative dispersion in the distribution of orbits withangular momentum h. In the low temperature limit, the regime in which the bar forms, thisdispersion is small and the integration over h is well estimated using a steepest descent approx-imation (cf Fig. 2.4.3). This in e!ect describes the integrand of Eq. (2.4.19) as a truncatedGaussian centred on the maximum of this integrand. Equation (2.4.19) then reads

N = 2!#

C (J) e#%(&#!ph) I0

%.AK

&.

2!I.

dJ , (2.4.23)

where ¯( ) corresponds to ( ) evaluated at h, and h(J,$p, .) is the angular momentum corre-sponding to the extremum of F (h, J) at fixed J. In Eq. (2.4.23), I corresponds to an e!ective– temperature dependent – moment of inertia given by

I = I$

1

2erf

3

4..

2I$

%hmax (J)# h

&5

6

7

82

, where I$ ='%2,

%h2# 1.

%2 log I0

%h2

(#1

, (2.4.24)

and erf(x) = 1/,!* x0 exp(#t2)dt. The erf function in Eq. (2.4.24) accounts for the correction

of the moment of inertia I$ induced by a maximum for the angular momentum reached by thecircular orbits rotating at the frequency $p with adiabatic action J. In Eq. (2.4.24), hmax = 2Jis the angular momentum of these circular orbits with invariant J. Note that A vanishes forthese orbits (characterised by JR = 0 = J # hmax/2). In Eq. (2.4.24), 1/I$ is the sum of twocontributions: &, the donkey parameter, and .#1%2 log I0/%h2 which accounts for the fact that Avaries with h; this contribution scales like . at low temperature which implies that the adiabatic


-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

h

1.25

1.5

1.75

2

2.25

2.5

F(h,J)

Figure 2.4.3: the schematic variation of F (J,# = !/4) given by Eq. (2.4.12) for di!erent values of Jand $p. An analogous function, the integrant of Eq. (2.4.19), is represented by a truncated Gaussian inthe steepest descent approximation. Here the potential was assumed to be that of the Isochrone disk.The widths of these curves vary like 1/..

distortion of the orbits dominates the last stage of the collapse. The value of h($, J) is givenby the smaller13 root of

$!

%h&&!%,

%h

"= $p +

1.

!% log I0

%h

"= $p +

!%A

%h

"I1

I0K , (2.4.25)

which fixes the pattern of the bar, $p, for a given h. Note the similarity between Eq. (2.4.25)and the value of $p predicted by the linear theory of section 3: both analyses yield a patterncorresponding to the maximum of the e!ective number of orbits rotating at the frequency $!,once a correction has been made to account for the orbit adiabatic distortion by the bar. Infact, (%A/%h) is generally negative which implies that patterns speed above the maximum of $!

may be reached while the orbits are adiabatically distorted.Putting Eq. (2.4.23) into Eqs. (2.4.18) yields after some algebra14

'h( =1N

#n (J) h

9

1 +1

2.$ph

% log I% log$p

:

dJ , (2.4.26a)

'e( =1N

#n (J)

9

,+12.

;

1# % log I% log.

<

# K2

2N

:

dJ . (2.4.26b)

13 Equation (2.4.25) has generally two roots corresponding to the inner-inner Lindblad resonance and theouter-inner Lindblad resonance respectively. The latter is ignored in this analysis as it corresponds to donkeyorbits which anti-align.

14 Equation (2.4.17) was used to write*

(" log I0/")) n(J) dJ =*

AK n(J) I1/I0 dJ = K2/N . Equation

Eq. (2.4.25) is also taken in to account.


Equations (2.4.26), together with Eq. (2.4.25) fix the temperature and the pattern speed of a barembedded in a galaxy with specific angular momentum 'h(, specific energy 'e( and circulationdistribution n(J). All thermodynamic properties of the bar follow from Eqs. (2.4.26). Forinstance, the adiabatic specific heat at constant total momentum, CH = #.2%'e(/%., is givenby

CH =1

2N

#n (J)

9;

1 +% log I% log .

<

#;%2 log I% (log.)2

<

# 2.2 %,

%.+ 2.2 K

N

%K

%.

:

dJ , (2.4.27)

where %K/%. + 0 is given implicitly by Eq. (2.4.17). Here , is a function of . via $p =$p[., K(.)]. The variation of the e!ective moment of inertia of the orbits is crucial in fixing thesign of CH , and the values of . and $p. Now

% log I% log$p

=% log I!

% log$p(1# L [X ]) + 2L [X ]

% log+hmax # h

,

% log$p, (2.4.28a)

where L (x) =% log erf (x)% logx

, and X =

3

4..

2I$

%hmax # h

&5

6 . (2.4.28b)

The sign of % log I/% log$p therefore depends on the detailed behaviour of A(J) in Eq. (2.4.24),but will certainly be negative when L(X)) 1, i.e. when Eq. (2.4.25) fixes h su#ciently close tohmax in units of

=.crit/I$. For a set of initial conditions (n(J), H, E) yielding such a sub-critical

temperature, Eq. (2.4.26a) implies that h should, on the average, be larger than 'h(, inducinga bar which rotates faster than the mean angular frequency. This result corresponds formallyto the intuitive fact that a skewed distribution function biased towards the escape angularmomentum has a maximum higher than its mean. As this distribution becomes narrower, themean and the maximum converge. In the zero temperature limit, Eq. (2.4.26a) suggests indeedthat $p converges towards this mean rotation rate like15 1/..

Equations (2.4.26), (2.4.27) and (2.4.28) are compatible with a scenario in which a galaxyforms with a mean energy, a mean angular momentum and a circulation distribution such thatthe corresponding temperature is below the critical temperature for bi-symmetric instabilityand such that Eqs. (2.4.28) yield a negative specific heat for some temperature range. Thecollective alignment of inner Lindblad orbits would then act as an energy sink; these orbitswould exchange energy and momentum via other resonances, but without inducing any collectivecollapse. Weakly trapped orbits could then be thought of as a heat bath coupled to a bardisplaying negative specific heat. If this specific heat were larger in modulus to that of the bath,the equilibrium would be secularly unstable. In e!ect, the heat flow would drive energy awayfrom the bar, yielding a stronger, colder bar which would rotate at slower and slower angularspeed until it reaches temperatures for which L[X ] ) 0. The bar may then stop displayingnegative specific heat – leading the system, bar plus galaxy, towards a secular equilibrium, orthe system may continue to cool down.

15 The low temperature value might not be reached, as X scales like ), therefore L(X) % 0 when ) % &. Inthis regime, as mentioned earlier, the adiabatic distortion of the orbit dominates the variation of its moment ofinertia, fixing the sign of " log I/" log!p.


0 0.1 0.2 0.3 0.4 0.5

J

0

0.1

0.2

0.3

0.4

n(J)

Q=0.8

Q=1

Q=1.2

Q=2

Figure 2.4.4: behaviour of n(J) =*

FQ(h, J) dh for di!erent values of the Toomre Q number in thetepid Isochrone disk described by Eqs. (2.4.6).

2.4.5 Illustration: the Isochrone disk

The above analysis is now applied to the Isochrone disk. An analytic expression for theenergy as a function of actions is available for this disk (in units of GM = b = 1):

, (h, J) = #12

!J +

>h2/4 + 1

"#2

. (2.4.29)

The angular frequency, $! = %,/%h, and the inverse moment of inertia, & = %2,/%h2, followreadily from Eq. (2.4.29) and are illustrated on Fig. 2.4.5. The collective instability describedin this chapter requires that & be positive, which in turn implies that only these orbits shouldbe counted in Eqs. (2.4.8), yielding

'e( =1N

Jmax#

0

2J#

0

, (h, J) F0 (h, J) dh dJ , 'h( =1N

Jmax#

0

2J#

0

F0 (h, J) h dh dJ , (2.4.30)

where N =* Jmax

0 n(J)dJ, and Jmax - 0.75. Distribution functions for the Isochrone are derivedin the next chapter. Here, for simplicity, tepid disks are constructed by the ansatz

FQ (h, JR) =' )2

R

"exp

!#"JR

)2R

" !%R

%h

", where )R =

!Q'"

. (2.4.31)

The function )R measures the radial velocity dispersion corresponding to a disk with Toomrenumber Q. The epicyclic frequency, ", the surface density, ', and )R are all functions of


0 0.2 0.4 0.6 0.8 1 1.2 1.4

h

0

0.2

0.4

0.6

0.8

1J

Figure 2.4.5: the isocontours of $! and & in (h, J) space. The section of the plot below the lineJ = 1/2h is unphysical. Therefore for each value of J , h may vary between 0 and hmax = 2J . From rightto left, the vertical contours correspond to & = 0, 0.01...0.25, while the horizontal contours correspondfrom bottom to top to $! = 0.11, 0.15, ..., 0.01. Note the & = 0 = 1/I$ contour, normal to the maximaof $!, which corresponds to orbits which are indi!erent to torques. In this analysis, only orbits on theleft of that contour have positive & and contribute to the collective instability. This yields a maximumvalue for J - 0.75.

R. the angular momentum of circular orbits. In the epicyclic approximation, h is identifiedto hc(R), the angular momentum of circular orbits so that h becomes a function of R and* hc(R)0

*fQ(h, JR)dJR dh =

* R ' dR. The function fQ(h, JR) presents an exponential cuto! inthe number of eccentric orbits with a characteristic energy scaling like )2

R $ Q2. Then

n (J) =#

FQ

!h, J # h

2

"dh =

# ' )2R

"exp

!#" (J # hc (R) /2)

)2R

"dR . (2.4.32)

The distribution of the number of orbits with adiabatic circulation J given by Eq. (2.4.32) isillustrated on Fig. 2.4.4.

In the spirit of section 2.3, consider here an “idealised” Isochrone galaxy, for which a smallsubset of resonant orbits is responsible for the onset of the instability, in e!ect describing thedistribution n(J) shown on Fig. 2.4.4 as a delta function centred on J $ Q. Let Eq. (2.4.30) fix'e( and 'h( for the corresponding Q number. For simplicity, assume also that the adiabatic dis-tortion of the orbits can be neglected. Therefore A(h, J) = Const. & A. Condition Eq. (2.4.17)then reads again

K =I1

%.AK

&

I0

%.AK

& AN , (2.4.33)

and Eq. (2.4.21) gives the critical temperature, A2N/2, below which a bi-symmetrical pattern isspontaneously generated within the disk, which corresponds essentially to a Jeans-like criterion.


0.5 1 1.5 2 2.5

h

0

0.02

0.04

0.06

0.08

0.1

0.12omega

Figure 2.4.6: pattern speed, $p, versus momentum, 'h(, for di!erent temperatures measured by(.A2N)#1 and J in the tepid Isochrone disk. This figure illustrates the “temperature” behaviour of(.A2N)#1. The di!erent bundles, from top to bottom, correspond to J = 0, 0.2, 0.4. The lines in eachbundle correspond to 5, 7, ..., for .A2N moving up the page.

Note that K/A, which represents the number density of orbits trapped in the bar is boundedby N , the total number of orbits, as shown on Fig. 2.4.1. The pattern speed of the bar, $p(.)corresponds to the solution of the system

h = 'h( # 12.$p

;% log I% log$p

<

and $p = $!

%h&

, (2.4.34)

while the temperature, 1/., follows from $p(.), Eq. (2.4.33) and Eq. (2.4.26b) which now reads

'e( = ,+12.

;

1# % log I% log .

<

# K2

2N. (2.4.35)

The e!ective moment of inertia obeys

% log I% log$p

= $p%&#1

%h(1# L [X ])# 2

$p &#1

hmax # hL [X ] , (2.4.36)

where L[X ] is given by Eq. (2.4.28b). For realistic galaxies, & & %2,/%h2, is a decreasing functionof h. Its isocontours are illustrated on Fig. 2.4.5 for the Isochrone disk.

In practise, a given model is best implemented by choosing $p and . > .crit. Equation(2.4.33) yields the amplitude of the bar, while h is given by h = $#1

! ($p). The mean momentumand energy follow in turn from Eqs. (2.4.34) and (2.4.35) via Eq. (2.4.36). Fixing 'h( yields$p = $p(.) and therefore 'e( as a function of .; its derivative corresponds to CH . The aboveprescription is implemented on Fig. 2.4.7 for two disks with di!erent global temperature J $ Q.Collett and Lynden-Bell (1989) [18] have shown that, on these diagrams, the bar will evolvealong the lines obeying +'e( = $p +'h(.


0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

T=1/beta

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Omega

0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

T=1/beta

0

0.01

0.02

0.03

0.04

Omega

Figure 2.4.7: isocontours of 'h( and 'e( in temperature (horizontal axis) and pattern speed (vericalaxis) space for a cold (right panel),J = 0.15, and a warm (left panel),J = 0.5, Isochrone disk.On the right panel from bottom to top, the horizontal contours correspond to 'h( = 0.1, 0.2..., 2, andfrom left to right, the vertical contours correspond to 'e( = #0.825,#0.8125, ..# 0.50. Similarly, on theleft panel, from bottom to top, horizontal contours correspond to 'h( = 0.5, 1, ..., 10, while from left toright the vertical contours correspond to 'e( = #0.65,#0.60, ...0. The intersection of the correspondingcontours yields the temperature and the pattern speed of the instability. If this temperature is below thecritical temperature, Eq. (2.4.33) yields the amplitude of that perturbation.

In the low temperature limit, % log I/% log . = #% log I/% log$p, and Eq. (2.4.27) gives for thespecific heat

CH =12

;

1 +% log I% log$p

<

# 12

;%2 log I

% log . % log$p

<

# .2 %,

%.+ .2 K

N

%K

%., (2.4.37a)

=12

+ .2 K

N

%K

%.. (2.4.37b)

When . )*, the orbits start behaving similarly to rigid rods, the moment of inertia convergestowards &#1, and CH is positive as illustrated on Fig. 2.4.2. Recall that for simplicity, theself-consistent distortion of orbits has not here been fully taken into account while taking A as aconstant. A more detailed analysis accounting for both the dispersion in n(J), the intermediatetemperature regime, and the variations of A(J) is required in order to be conclusive about thefate of real galaxies.

2.4.6 Discussion

The equations (2.4.17) and (2.4.33) may be understood both as a bar formation and alsobar dissolution criteria. Indeed, numerical evidence suggests that the process of galaxy for-mation occurs in a highly non-symmetrical environment which is likely to trigger transientnon-axisymmetric features within the forming disks. The viability of these grand design struc-tures depends on the intrinsic properties of the disk such as its kinetic temperature. Criterion


Eq. (2.4.17) provides an estimate of the temperature above which a bar is not viable and willdisappear adiabatically.

The formation of initially fast bars that slow down as they capture adiabatically more orbitscorresponds clearly to the results of numerical simulations, and is compatible with observations.The mechanism presented in this section should therefore be implemented in details on realisticmodels such as those constructed in the next chapter to account for other observable trends suchas aspect ratio evolution, etc

· • !" ! • ·

2.5 Global azimuthal instabilities

The previous models were introduced with simplifications to present some physical insightin the phenomenon of bar formation. The instability is now investigated within the frameworkof the Boltzmann and the Poisson equations.

2.5.1 Boltzmann’s Equation: the dynamics

Boltzmann’s equation reads%F/%t + [H, F ] = 0 , (2.5.1)

where H is the Hamiltonian for the motion of one star, F is the mass-weighted distributionfunction in phase space, and the square bracket denotes the Poisson bracket

[H, F ] =%H

%p· %F%q

# %H%q

· %F%p

. (2.5.2)

Writing F = F0 + f , and % = $0 + $ and linearising Eq. (2.5.1) in f and $ yields

%f/%t + [H0, f ]# [$, F0] = 0 . (2.5.3)

According to Jeans theorem, the unperturbed equation, [H0, F0] = 0, is solved if F0 is anyfunction of the energy, E, and the angular momentum, h (= Rv').

Following Lynden-Bell and Kalnajs (1972) [76], angle and action variables of the unper-turbed Hamiltonian H0 are chosen here as canonical coordinates in phase-space. Indeed, theunperturbed Hamilton equations are quite trivial in these variables, which makes them suitablefor perturbation theory in order to study quasi-resonant orbits16. In polar coordinates (R, *),the unperturbed Hamiltonian reads

H0 = 12

?p2

R + p2'

@# $0 (R) , where pR = vR and p' = h/R . (2.5.4)

The actions are defined in terms of these variables by J = (JR, J') = (JR, h), where

JR =12!

- ARBdR, J' = h = R2* . (2.5.5)

Here [R] is a function of the radius R, the specific energy E, and the specific angular momentum,h, given by A

RB

=>

2E + 2$0 (R)# h2/R2 . (2.5.6)

16 See Appendix A for more details on angle and action variables

Section 2.5 Global azimuthal instabilities 25

The phase-angles conjugate to JR and h are ! = (#R,#'), where

#R = "# R A

RB#1

dR , (2.5.7a)

#' = *+# R %

h/R2 # $& A

RB#1

dR . (2.5.7b)

The generalised epicyclic frequency, ", obeys

2!/" =- A

RB#1

dR = PR , (2.5.8)

where PR is the (radial) period between apocentres. The angle between apocentres is

" =-

h/R2ARB#1

dR and $ = "/PR . (2.5.9)

Hamilton’s equation for the unperturbed orbits reads

! = (%H0/%J) = ! (J) , (2.5.10)

where ! stands for (",$), while the J are kept constant as the actions were designed so that H0

is independent of the phases ! 17. The stationary unperturbed Boltzmann equation Eq. (2.5.1)is solved by F = F0(J), since [H0, J] = 0. When expanded in Fourier series with respect to theangles !, Eq. (2.5.2) becomes

%fm/%t + i m ·! fm + i m · %F0/%J $m = 0 , (2.5.11)

where m is a integer vector with components (0, m), and $m and fm are the Fourier transformsof $ and f with respect to (#R,#').For growing instabilities, fm and $m are taken to both be proportional to ei#t, ( having anegative imaginary part. Eq. (2.5.11) then becomes

fm =#m#1m · %F0/%J

$! # $p$m , (2.5.12)

where $! = m#1m ·! = $+ 0"/m and $p = #(/m. When 0 = #1 this coincides with the lobetumbling rate of the orbits discussed in section 2.3. When 0 = 1 it gives the lobe tumbling rateof the type of orbits encountered at outer-Lindblad-resonance while 0 = 0 gives $! = $, thetumbling rate of co-rotating epicycles.

There is evidently a resonant response from all those parts of phase-space for which $!(J)is close to Re{$p}. For di!erent 0 values these are distinct parts of phase-space because $! #$!! = (0$ # 0)"/m .= 0, for 0 .= 0$. In dealing with each resonant term, new angle and actionvariables are introduced which are more appropriate and make the particular resonance labelledby (0, m) especially simple. These “fast” and “slow” coordinates are obtained from the canonicaltransformation generated by the generating function

S (Js, Jf ,#R,#') = Js (#' # $pt# 0#R/m) + Jf#R . (2.5.13)

The transformation to the new variables is given by

%S/%#' = h gives Js = h , (2.5.14a)%S/%#R = JR gives Jf = JR # 0 h/m & J! , (2.5.14b)%S/%Js = #s gives #s = #' + 0/m#R # $pt , (2.5.14c)%S/%Jf = #f gives #f = #R . (2.5.14d)

17 the actions are a combination of integrals of the unperturbed motion by construction. See the appendix fordetails


Note that$s = #s = $# 0 "/m# $p = $! # $p . (2.5.15)

In axes that rotate at $ + 0"/m the orbit closes with m lobes after |0| turns about the centre.From Eq. (2.5.15), $s is zero on resonance, and for all near-resonant orbits #s varies slowly. Infact #s is the azimuth of the lobe of an orbit in the frame that rotates with angular velocity $p.It is convenient to define

#! & #' + 0 #R/m , (2.5.16)

which is the lobe’s direction in absolute space. Then #s = #!#$pt. It is interesting to see that(%H0/%h)Jf = $! = #! for every 0.Returning to Eq. (2.5.12) and evaluating it using Eq. (2.5.14)

m#1m · %F0/%J =!%F0

%h

"

JR

+0

m

%F0

%JR,

=!%F0

%h

"

JR

+!%F0

%JR

"

h

!%JR

%h

"

Jf

,

=!%F0

%h

"

Jf

. (2.5.17)

Thus Eq. (2.5.12) reads

f! =[#%F0/%h]J!

$! # $p

$! . (2.5.18)

The result (2.5.18) shows how strikingly simple the dynamics is in angle action variables.

2.5.2 Poisson’s equation

The Poisson equation relates the potential, $, to the density perturbation:

$ (R$, *$) =# #

f (R, v)|R#R$| dRd* dvR dv' . (2.5.19)

This equation may be rewritten in order to make explicit the contribution from the interactionof orbits. Here again angle-action variables are useful, as a given unperturbed orbit is entirelyspecified by its actions. It is straightforward to identify in Poisson’s equation the contributioncorresponding to the interaction of given orbits. Re-expressing this equation in terms of angleand actions (!, J) and taking its Fourier transform with respect to ! implies:

$m (J) = !G$

m!

#fm! (J$)Amm! (J, J$) d2J $ , (2.5.20)

whereAmm! =

14!3

# # exp i (m$ · !$ #m · !)|R#R$| d2#$ d2# . (2.5.21)

The (double) sum in Eq. (2.5.20) extends for both 0 and m going from minus infinity to infinity.R = R(!, J) and R$ = R(!$, J $) are radii re-expressed as functions of these variables. Note thata m$- fold symmetry family of orbits may contribute to the m$ the component of the potentialaccording to Eq. (2.5.20). Now |R#R$| depends on #' and #$

' in the combination #$'##' = &#

only. Thus(m$ · !$ #m · !) = m$&## (m#m$)#' # 0$#$

R + 0#R . (2.5.22)


As |%#$' %#'/%&# %#'| = 1, the order of integration may then be reversed doing the #' inte-

gration with &# fixed. This yields 2!+mm! , so m$ becomes (0$, m) in the surviving terms. Thisgives for Eq. (2.5.22)

Amm! =+mm!

2!2

# # # exp i (m&## 0$#$R + 0#R)

|R#R$| d&# d#$R d#R . (2.5.23)

Now #! = #' + 0#R/m and #f = #R, so Eq. (2.5.23) becomes

Amm! =+mm!

2!2

# # # exp im (#$!! # #!)

|R#R$| d (#$!! # #!) d#$

f d#f ,

=+mm!

2!2

# # # d#f d#$f

|R#R$| exp im (&#!) d (&#!) , (2.5.24)

where |R # R$| is thought of as a function of J, J$, the angle between the lobes of the twoorbits, &#!, and the two phases around those orbits #f and #$

f . Equation (2.5.24) states aninteresting result; it implies that two orbits with di!erent m-fold azimuthal symmetry do notinteract. Eq. (2.5.24) has a simple physical interpretation in terms of potential energy. Considerthe energy of gravitational interaction, V , given by

V = #12

#' (R,#) $ (R,#) dRd# , (2.5.25)

where the surface density reads in terms of the distribution function

' (R, *) =#

f (R, *, vR, v') dvR dv' . (2.5.26)

ThereforeV = #1

2

# #f (R, *, vR, v') $ (R, *) dvR dv' dRd* , (2.5.27)

Angle and action variables of the axisymmetric Hamiltonian are introduced again to enlight thecontribution of each orbit to the overall potential energy. The potential $ expressed in terms of(!, J) is given by Eq. (2.5.20)

$ (J, !) =$

m,m!

#Am,m! (J, J$)ei(m·!#m!·!!) f (J$, !$) d1$ , (2.5.28)

with d1 = d2J$ d2#$, the elementary phase space volume. Inserting Eq. (2.5.28) into Eq. (2.5.27)leads to the formal expression for V

V =$

m,!,!!

Vm,!,!! =$

m,!,!!

12

#f (J, !) Am,!,!! (J, J$) ei(m·!#m!·!!) f (J$, !$) d1 d1$ . (2.5.29)

Switching the fast and slow angles, Eq. (2.5.29) becomes

Vm,!,!! = 2!2

# # #f! (J,#!) Am,!,!! (J, J$) eim""! f!! (J$,#!!) d2J d2J$ d&#! , (2.5.30)

wheref! (J,#!) =

12!

#f (J,#!,#R) ei"R d#R (2.5.31)

Imagine a unit mass orbital stream on the orbit specified by m, 0$, J$,#$!!. In the gravitational

potential of that stream, place another stream specified by m, 0, J,#!; then their mutual gravi-tational potential energy is some function

V" (m, 0, 0$, J$, J,&#!) .


If V" is expanded as a Fourier series in &#!, the angle between the lobes of the two orbits, thenthe first non-zero angular-dependent term is

Am,!,!! exp (m&#!) .

Formally this follows from Eq. (2.5.30) where delta functions are substituted for f! and f!! . V"

is therefore the exact analogue of the orientation potential % introduced in section 2.3.

2.5.3 Instability criterion

Putting Eq. (2.5.18) into Eq. (2.5.20) leads to the integral equation

$!1 (J1) = 2!2G$

!2

# #Amm! (J1, J2)

[#%F0/%h]J!2

$!2 # $p$!2 (J2) d2J2 . (2.5.32)

Equation (2.5.32) is formally identical to the integral equation found in section 2.3 except forthe summation over all 0 resonances.

• The integral equation Eq. (2.5.32) could be tackled analytically under the assumption thatthere exists a way to split Amm!(J1, J2) = A!1(J1)A!2(J2). Eq. (2.5.32) then becomes

1 = 2!2G$

!2

# #A2

!2 (J2)[#%F0/%h]J!2

$!2 # $pd2J2 , (2.5.33)

which is similar to equation Eq. (2.3.8) apart for the summation over all orbits. Identifyingboth real and imaginary part of Eq. (2.5.33) yields

2!2G$

!2

# #A2

!2 (J2)[#%F0/%h]J!2

($!2 # $R)2 + 22d2J2 = 0 , (2.5.34a)

2!2G$

!2

# #[$!2 # $R]A2

!2 (J2)[#%F0/%h]J!2

($!2 # $R)2 + 22d2J2 = 1 , (2.5.34b)

where $p = $R+i2. When the main resonant contribution for bars is assumed to come fromthe inner Lindblad and co-rotation, [#%F0/%h]Jf is generally positive for Jf = JR, 02 = 0 atco-rotation, and negative at the inner Lindblad resonance (Jf = JR + h/2, 02 = #1). As aconsequence, Eq. (2.5.34a) may be satisfied, while both resonance contribute to Eq. (2.5.34b)positively, when $0 > $p while $#1 < $p.

• More accurately, solutions to a given integral equation may be found either by expand-ing the unknowns over a set of orthonormal functions that is likely to describe the typeof perturbation correctly, or by searching for the eigen modes of the Kernel. An alterna-tive prescription would, for instance, involve constructing a complete set of orthonormaldistribution functions on which the perturbation is projected

f! (J) =$

n

1n,! fn (J) . (2.5.35)

The system of equations which follows from inserting Eq. (2.5.35) into Eq. (2.5.32) has anon-trivial solution for the 1n,!’s only if

D ($p) = det |1 + M ($p) | = 0 , (2.5.36)


where 1 is the identity matrix, and M is a matrix given by

!M

(n,n!)

(!,!!)

"= !G

# #fn (J)

9(#%F0/%h)J!

$! # $p

:#1

A!,!! (J, J$)[#%F0/%h]J!!

$!! # $pfn! (J$) d2J d2J$ .

(2.5.37)Equation (2.5.36) gives the criterion for the existence of exponentially growing unstablemodes. These are combinations of the eigenvectors of the null space of 1 + M($p). Thecorresponding perturbed distribution function characterises the orbits which are first caughtby the instability. If this distribution function is peaked in the neighbourhood of a givenorbit caught in a given resonance, the instability is orbital in nature. If not, the methodof potential density pairs developed by Kalnajs (1971) [54] gives a more direct route tothe standing wave generated by the instability. Note that this wave is available fromEq. (2.5.35) by simple integration over velocity space. For the purposes of this method, anumerical evaluation of the coupling coe#cients Am

!,!! is given in the appendix. Distributionsfunctions parameterised by their temperature are also constructed in the next chapter. Inpractise, the choice of an appropriate basis function, Eq. (2.5.35), is critical to ensure a fastconvergence of the determinant in Eq. (2.5.36). Indeed, a good description of the naturalmode of the system must be chosen di!erently for each galaxy.

• An alternative systematic inversion method for Eq. (2.5.32) is presented in the appendix.The eigen-modes (3n, un) of the Kernel A!,!!(J, J) are constructed in the epicyclic approxi-mation with softened gravity. Within the framework of that approximation, the integrationover eccentricity and over the angles may be performed in Eq. (2.5.24), yielding a one di-mensional integral equation for which simple analytic modes are derived. The projectionof Eq. (2.5.32) onto these modes leads to the dispersion relation

D ($p) = det |"+ H ($p) | = 0 , (2.5.38)

where

H(n,n!)

(!,!!)= /

(!,!!)!G

# #An

! (J) An!

!! (J)[#%F0/%h]Jf

$! # $pd2J , (2.5.39)

and the e!ective An! are defined by:

An! (J) = a! [e (J)]un

+R (J)

,. (2.5.40)

The eccentricity dependence, a!(e), and the eigen-functions, un, are given in the appendix.These functions are characteristic of the law of gravitation, and as such need not be tailoredto each specific underlying galaxy.

The dispersion relations Eqs. (2.5.36) and (2.5.38) give the criterion for the existence ofexponentially growing unstable modes. They are functions of a free parameter $p correspondingto the pattern velocity of the growing bar. Their interpretation is as follows: consider thecomplex $p plane and a contour that traverses along the real axis and then closes around thecircle at * with Im($p) positive. The determinant D is a continuous function of the complexvariable $p; so, as $p traces out that closed contour, D($p) traces out a closed contour in thecomplex D plane. If the D($p) contour encircles the origin, then the system is unstable. Anintuitive picture of how the criterion operates on Eq. (2.5.38) is provided with the followingthought experience. Imagine turning up the strength of G for the perturbation (i.e. turn onself-gravity, which is physically equivalent to allowing for the gravitational interaction to playits role). Starting with small G, all the Hnn!

!!! are small, so for all $p’s the determinant D($p)remains on a small contour close to

Ci 3i .= 0. As G is increased to its full value, either the D

contour crosses the origin to give a marginal instability or it does not. If it does not, then by


continuous change of G does not modify the stability, and therefore the self-gravitating systemhas the same stability as in the zero G case, i.e. is stable. If, however, it crosses, and remainscircling the origin then it has passed beyond the marginally stable case and is unstable. Asimilar argument holds for Eq. (2.5.36).

2.5.4 Overview

The simple picture invoking the alignment of inner Lindblad orbits described in section 2.3as the triggering mechanism for the instability in stellar disk is somewhat endangered both bythe results of numerical N-body simulations (see Sellwood 1993 [101] for a review) and morecrucially by linear analysis of the natural density modes of the disk carried by Kalnajs (1976) [58]Sawamura (1988) [99] and Hunter (1992) [50]. Indeed the orbital analysis should predict thesame pattern speeds and growth rates and reproduce, via integration over velocity space, allfeatures of the natural density modes. But these analyses predict pattern speeds which aresignificantly (by a factor of two) above the maximum pattern speed reached by inner Lindbladorbits. In fact the fastest inner Lindblad orbits correspond to circular orbits. This factor oftwo discrepancy is therefore quite di#cult to reconcile with a model in which the root of theinstability corresponds to the alignment of such orbits via relative torques! In other words, thealignment of inner Lindblad orbits described in section 2.3 does not seem to trigger the instabilitywith the fastest growth rate in the models studied by Kalnajs [54] and others. Merritt [86]Polyachenko [94] and Allen 1992 [2] have shown that a similar type of instability involving thealignment of radial orbits may occurs in the core of elliptical galaxies.The adiabatic alignment of inner Lindblad orbits may nevertheless play some role later in theevolution of the structure of the bar beyond the linear regime as described in section 2.4. Thecorresponding results are in e!ect more likely to account for the observations than those ofthe linear theories described in sections 2.2, 2.3 and 2.5. Section 2.4 demonstrated that thealignment of resonant orbits could not be treated independently of their distortion, which ine!ect reconciled the mechanisms of eccentricity forcing [76] and alignment instability [82]. Theformalism presented in section 2.5 is likely to re-unify all theories as it involves both the waveand the orbital description.

It may well turn out that the type of standing (fluid) waves described by Toomre [105]and others corresponds to the fastest linearly growing modes of stellar disks, and that thisdescription is indeed still accurate for the evolved structures observed in real galaxies. Yetthis picture is unsatisfactory as the swing amplification mechanism – a leading density wavewhich is gravitationally enhanced at co-rotation while becoming trailing – does not really makeany prediction for the non-existence of bars in a given model. Moreover, this scenario doesn’taccount for the nature of the flow in the bars found in numerical simulations, in which the bulkof the bar appears to be made of closed elongated orbits. There is a significant gap betweenthe physical mechanism of wave amplification on the one hand, and the corresponding growthrates and pattern speed found by the matrix projection method of Kalnajs [58] on the otherhand. This gap is di#cult to fill while resorting to a partially numerical inversion method whichrequires di!erent implementation for each galaxy.

A detailed implementation of the analysis of section 2.5 is required to address these draw-backs and make any final conclusion on the relevance of the orbital description for galacticbars.

In the final analysis, the problem of bar formation remains an open question!

· • !" ! • ·

A

Aspects of orbital

instabilities

A.1 Numerical evaluation of the A m/m

For numerical purposes, it is desirable to re-express the the coupling factor Am!m introducedin section 5 in terms of integration over positions rather than angles. It may be re-arrangedfrom Eq. (2.5.24) as

Am!m =1

4!3

# # # exp (im&#') exp (i&{0#R})|R# R$| d&#' d#R d#R! , (A.1.1)

where the & symbol stands for the di!erence of its argument primed minus unprimed. Thedenominator in Eq. (A.1.1) may be expanded as

|R # R$| =?R2 + R$2

@1/2

[1 + x cos (&*)]1/2 , (A.1.2)

where x = 2RR$/(R2 + R$2). Given that

#' = *# "#1#

h

R2d#R + "#1#R

2!

-h

R2d#R , (A.1.3)

Am!m may be re-written as

Am!m =1

4!3

# # exp (i&{0#R + m#' #m*})?R2 + R$2

@1/2

9# exp (im&*)[1 + x cos (&*)]1/2

d&*:

d#R d#R! .

(A.1.4)Calling

Cm (x) =12!

# exp (im&*)[1 + x cos (&*)]1/2

d&* , (A.1.5)

and

#!$ [R] = 0#R + m#' #m* = 0#R #m"#1

#h

R2d#R + m"#1#R

2!

-h

R2d#R , (A.1.6)

yields

32 Aspects of orbital instabilities Appendix A

-20 -10 0 10 20

-4

-2

0

2

4

h: 1.5 J: 1.62

omega: 0.0111

-7.5-5-2.502.557.5-1.5

-1-0.5

00.5

11.5

h: 1.5 J: 0.673

omega: 0.031

-4 -2 0 2 4

-1.5-1

-0.50

0.51

1.5

h: 1.5 J: 0.252

omega: 0.0559

-10 -5 0 5 10-2

-1

0

1

2

h: 1. J: 1.25

omega: 0.0181

-4 -2 0 2 4

-0.75-0.5

-0.250

0.250.50.75

h: 1. J: 0.526

omega: 0.0486

-3-2-1 0 1 2 3-1

-0.5

0

0.5

1

h: 1. J: 0.199

omega: 0.0846

-6-4-2 0 2 4 6-0.4

-0.2

0

0.2

0.4

h: 0.5 J: 0.991

omega: 0.0208

-3 -2 -1 0 1 2 3-0.4

-0.2

0

0.2

0.4

h: 0.5 J: 0.427

omega: 0.0521

-1.5-1-0.500.511.5

-0.4

-0.2

0

0.2

0.4

h: 0.5 J: 0.165

omega: 0.0849

-20 -10 0 10 20

-15-10-505

1015

h: 1.5 J: 1.62

omega: 0.041

-7.5-5-2.502.557.5-7.5-5

-2.50

2.55

7.5

h: 1.5 J: 0.673

omega: 0.115

-4 -2 0 2 4-4

-2

0

2

4

h: 1.5 J: 0.252

omega: 0.21

-10 -5 0 5 10-10

-5

0

5

10

h: 1. J: 1.25

omega: 0.086

-4 -2 0 2 4

-4

-2

0

2

4

h: 1. J: 0.526

omega: 0.238

-3 -2 -1 0 1 2 3

-2

-1

0

1

2

h: 1. J: 0.199

omega: 0.426

-6 -4 -2 0 2 4 6

-4

-2

0

2

4

h: 0.5 J: 0.991

omega: 0.177

-3 -2 -1 0 1 2 3

-2

-1

0

1

2

h: 0.5 J: 0.427

omega: 0.474

-1.5-1-0.500.511.5-1.5

-1

-0.5

0

0.5

1

1.5

h: 0.5 J: 0.165

omega: 0.821

Figure A.1.1: The angular frequency, $!, and shape of a resonant orbit as a function of angularmomentum, h, radial action, JR, and resonance m – Inner Lindblad on the left panel, outer Lindbladon the right panel. Note the departure from simple elongated ellipses for the eccentric orbits (JR large).Here the axis are re-scaled for each plot.

Am!m =1

2!2

Ra#

Rp

R!a#

R!p

expAi#!

$ [R]# i#!!

$ [R$]B

?R2 + R$2

@1/2Cm

! 2RR$

R2 + R$2

"dRARB dR$AR$B . (A.1.7)

Here [R] is a function of R and of the actions (J, h) via

ARB

=,

2', (J, h) + $ (R)# h2

2R2

(1/2

; (A.1.8)

the radial angle #R is in turn a function of R via the identity d#R = dR/[R]; consequently#!m$ [R] is also a function of R, J and h. The integral Eq. (A.1.7) may therefore be numerically

evaluated over R and R$ space. Note the the real part of C2 reads

Re [C2 (x)] = # 83 ! x2

,1 + x

'(1 + x) E

! 2 x

1 + x

"#K

! 2 x

1 + x

"(# 2

3 !,

1 + xK! 2 x

1 + x

".

The Kernel, Gm, in Eq. (A.1.7) presents an essential singularity near the line R = R$. Physically,it corresponds to the contribution from tangential orbits for which the energy of interactionis infinite in the infinitely thin disk description. This artifact of this description is removedby softening gravity. In e!ect, softening restores the third dimension by putting all orbits in‘di!erent’ planes. Fig. A.1.2 illustrates this calculation of the interaction of inner Lindbladorbits.

Equation (A.1.7) requires the knowledge of the radial action JR and its derivatives. Ageneral semi-algebraic algorithm was implemented to construct these functions for any givenpotential as shown on Fig. A.1.1 for the Kuzmin disk.

Section A.2 Epicyclic Criterion 33

0.2 0.4 0.6 0.8

f

0.6

0.8

1

1.2

1.4

h

Figure A.1.2: The iso-contours of the coupling coe#cient A!,!! between two orbits trapped in the innerLindblad resonance of a Kuzmin disk as a function of angular momentum, h, and fraction of the circularenergy, f = ,/,h of the second orbit. The first orbit corresponds to f = 0.55, h = 0.85, and softeningwas taken to be equal to 0.1. Note the elongation along h $ f , which accounts for the similar perigeeof these orbits, when the overlapping of the two trajectories corresponds to the largest contribution inEq. (A.1.7).

A.2 Epicyclic Criterion

In this section, (A.1.7) is expanded using epicycles: this allows one to carry the integrationover geometry and eccentricity, reducing the integration over the full dimensional phase-spacein Eq. (2.5.32) to a simple one dimensional integral equation. This integral equation is in turninverted by a simple ansatz which e!ectively soften gravity for quasi-identical orbits. A stabilitycriterion similar to that of section two is recovered, with a detailed prescription for the couplingfactor A!(J).

A.2.1 Epicyclic expansion and phase space reduction

In the epicyclic approximation, the coordinates of a star i describing its orbit read

Ri = Ri + ai cos (#Ri) , (A.2.1a)

*i = #'i # 2ai

Ri

$i

"isin (#Ri) , (A.2.1b)

where the indices refer to orbit “one” and “two”. Here ai is the epicyclic radius, Ri the guidingcentre radius, and *i the orientation of the guiding centre of star i = 1, 2 measured from therotating frame $pi . Recall that "i and $i are respectively the epicyclic frequency and the angular


frequency of the star i. In the epicyclic approximation, Eq. (A.1.6) for #!i$ becomes

#!i$ [#Ri ] = 0i

'#Ri # 22i

ai

Ri

sin (#Ri)(

where 2i = 1 +$pi

"i

m

0i. (A.2.2)

Here $pi , i = 1, 2, are intermediate variables obeying $i # $pi = 0i"i/m; their purpose isdescribed below. Substituting Eq. (A.2.2) and Eq. (A.2.1a) into Eq. (A.1.7) allow Am!m" to bere-arranged as

Am!m" =+m2

m1

2!2

# #

DEEEEEEEEF

EEEEEEEEG

Gm

'R1

!1 +

a1

R1

cos (#R1)"

, R2

!1 +

a2

R2

cos (#R2)"(

exp

1

HHH2

i02

'#R2 # 222

a2

R2

sin (#R2)(#

i01

'#R1 # 221

a1

R1

sin (#R1)(

7

III8

JEEEEEEEEK

EEEEEEEEL

d#R1 d#R2 ,

(A.2.3)where the function Gm stands for

Gm (R1, R2) =1

(R21 + R2

2)1/2

Cm

' 2R1R2

R21 + R2

2

(, (A.2.4)

and Cm is given by Eq. (A.1.5). Equation (A.2.3) is now Taylor-expanded in power of ei &ai/Ri

1. The zeroth order term of the expansion is

A(0)m!m"

=+m2

m1

2!2

# #Gm exp [i02#R2 # i01#R1 ] d#R2 d#R1 = 2 +m2

m1+!10 +

!20 Gm , (A.2.5)

and Gm stands for Gm(R1, R2). Equation (A.2.5) corresponds to the interaction of two orbits atco-rotation. It does not involve any eccentricity, as the orbits need not be eccentric to interactat co-rotation.The first order term in ei of the expansion reads

A(1)m!m"

=+m2

m1

2!2

# #exp i (02#R2 # 01#R1)

3

M4Gm (i sin(#R1) e12101 # i sin (#R2) e22202)+

cos (#R2) R2e2%Gm

%R2

+ cos (#R1) R1e1%Gm

%R1

5

N6 d#R1 d#R2

(A.2.6)where Gm and its derivatives are to be evaluated at Ri = Ri, i = 1, 2. Performing the doubleintegration over the angles d#1d#2 leads to

A(1)m!m"

=+m2

m1

2

DEEEEF

EEEEG

+!1#1 +!20

9

Gm 2101 + R1%Gm

%R1

:

e1 + +!11 +!20

9

R1%Gm

%R1

# Gm 2101

:

e1+

+!2#1 +!10

9

Gm 2202 + R2%Gm

%R2

:

e2 + +!21 +!10

9

R2%Gm

%R2

# Gm 2202

:

e2

JEEEEK

EEEEL

(A.2.7)

These contributions correspond to the cross-interaction of orbits trapped in the di!erent reso-nances 0 = #1, 1 with orbits at co-rotation. It only requires the inner or the outer Lindbladorbit to be eccentric. The expression corresponding to the interaction between inner Lindblad

1 In the epicyclic approximation, the expansion is expected to converge fast only if ai = o(R2 " R1) becauseGm diverges when R1 = R2.


orbits described in the previous sections comes into the expansion as a second order term in ei

which is derived similarly:

A(2)m!m"

=+m2

m1

4e1e2

DEEEEF

EEEEG

+!1#1+!2#1

9

Gm21012202 + 2101R2%Gm

%R2+ 2202R1

%Gm

%R1+ R1R2

%2Gm

%R1%R2

:

+

+!11 +!21

9

Gm21012202 # 2101R2%Gm

%R2# 2202R1

%Gm

%R1+ R1R2

%2Gm

%R1%R2

:

JEEEEK

EEEEL

,

(A.2.8)where all corrections to the lower order terms have been dropped. The first group in Eq. (A.2.8)corresponds to the interaction of inner Lindblad orbits and the second to outer Lindblad orbits.It requires both orbits to be eccentric.Recognising in the integrand of Eq. (A.2.4) the generator for Legendre polynomials, it is a simplematter to expand Gm as

Gm

%R1, R2

&=

DEEEEF

EEEEG

$

n

(nR2n

1

R2n+12

if R2 + R1 ,

$

n

(nR2n

2

R2n+12

if R1 + R2 ,

(A.2.9)

But R2n1 /R2n+1

2 and ¯R2n2 /R2n+1

2 are eigen-functions for the operators Ri%/%Ri, i = 1, 2. It istherefore straightforward to find new /!1,!2

n ’s corresponding to expressions like the brackets inEq. (A.2.7) and Eq. (A.2.8) so that

A!1!2 (e1, e2, R1, R2) = a!1 (e1)a!2 (e2)$

n

/!1,!2n gn

%R1, R2

&, (A.2.10)

where the coupling term of this equation obeys

gn (R1, R2) =

DEEEF

EEEG

g1n (R1) g2

n (R2) =R2n

1

R2n+12

if R2 + R1 ,

g1n (R2) g2

n (R1) =R2n

2

R2n+11

if R1 + R2 ,

(A.2.11)

and the dependency on the eccentricity is – to second order in e –

a!1 (e1) =%+!1#1 + +!11

&e1 + +!10 . (A.2.12)

In principle, the /!1,!2n are here functions of Ri via 2i. In practice, it is desirable to restrict the

emphasis on the quasi-resonant orbits which contribute significantly to Eq. (2.5.32) as it wasargued in section 3. Under these circumstances, $p1 0 $p2 0 $p. In simulations, the patternspeed of the bar, $p, is usually found well below2 "/2. Therefore, the variations of 2i above andbelow 1 will be neglected here-after.Equation (A.2.10) is then directly separable in eccentricities and reference radii, which pro-vides a unique opportunity to simplify the integral equation (2.5.32). Putting Eq. (A.2.10) intoEq. (2.5.32) yields

$!1 (J1) = a!1 (e1) 2!2G$

!2,n

/n,!1,!2

# #a!2 (e2) gn

%R1, R2

& [#%F0/%h]J!2

$!2 # $p$!2d

2J2 . (A.2.13)

2 The maximum of !" */2 certainly satisfies this constaint in the inner parts of galaxies.


Switching to the new variables eccentricity and mean radius, (e, R), and calling the core of thekernel

K!2

%R2, e2

&= 2!2G

OOOO%J!2 %h

%e %R2

OOOO[#%F0/%h]J!2

$!2 # $p, (A.2.14)

equation (A.2.13) becomes

$!1

%R1, e1

&= a!1 (e1)

$

!2,n

/!1,!2n

#gn

%R1, R2

& '#a!2 (e2) K!2

%R2, e2

&$!2

%R2, e2

&de2

(dR2 .

(A.2.15)Multiplying Eq. (A.2.15) by a!1(e1) K!1(R1, e1) and integrating over e1 yields

%!1

%R1

&= H!1

%R1

& $

!2,n

/!1,!2n

#gn

%R1, R2

&%!2

%R2

&dR2 , (A.2.16)

where the new unknown vector reads

%!i

%Ri

&=#

a!i (ei) K!i

%Ri, ei

&$!i

%Ri, ei

&dei , i = 1, 2 , (A.2.17)

and the characteristics of the underlying galaxy are all taken care of in H!1(R1), given by

H!1

%R1

&=#

a2!1 (e1) K!1

%R1, e1

&de1 ,

=#

a2!1 (e1)

OOOO%Jf %h

%e %R1

OOOO[#%F0/%h]J!1

$!1 # $pde1 . (A.2.18)

A.2.2 Softened gravity and Green kernels

At this stage, a supplementary simplifying assumption is taken by assuming that the sum-mation in Eq. (A.2.10) reduces to one term

A!1!2 (e1, e2, R1, R2) = a!1 (e1)a!2 (e2) /!1,!2n gn

%R1, R2

&, (A.2.19)

where the coupling term, gn(R1, R2), is given by Eq. (A.2.11). This assumption in e!ect softengravity for near resonant orbits of similar size. The precise exponent n chosen does not constrainwhat follows. In fact, the only compulsory feature of a model for the coupling factor g is a jumpin the derivative at R1 = R2 which is preserved by ansatz Eq. (A.2.19). This ansatz yieldssimple analytic eigen-functions corresponding to a “natural” set of functions (only dependenton the laws of gravity), on which the criteria Eq. (A.2.13) is expanded. A more realistic modelfor the core of gravity would lead to more complex sets of eigen-functions.Eq. (A.2.16) may then be re-written more elegantly as:

% (R1) =#

gn

%R1, R2

&H%R2

&·%

%R2

&dR2 , (A.2.20)

where % is an unknown vector of dimension m, where m is the order of the resonance, and H isa (2m)1 (2m) matrix, given by

H =

1

HHH2

/(#m,#m)H(#m) /(#m,#m+1)H(#m+1) . . . /(#m,m)H(m)

/(#m+1,#m)H(#m) /(#m+1,#m+1)H(#m+1) . . . /(#m+1,m)H(m)

...... . . . ...

/(m,#m)H(#m) /(m,#m+1)H(#m+1) . . . /(m,m)H(m)

7

III8 . (A.2.21)


The / coe#cients in Eq. (A.2.21) are constants depending on the geometry of the orbit whichcan be calculated for each resonance, and H!(R) is given by Eq. (A.2.18). The function gn inEq. (A.2.19) has the properties of a Green function because of the discontinuity in its derivative.Hence there exists a di!erential operator, L, corresponding to the integral equation Eq. (A.2.20)so that: L G(R1, R2) = +(R1 # R2). Applying L to Eq. (A.2.20) yields:

L % =#+%R1 # R2

&H%R2

&·%

%R2

&dR2 = H ·% . (A.2.22)

The operator L is calculated later. The existence of a complete set of orthonormal eigen-functions, un(R), and eigenvalues, 3n, compatible with the integral equation is also discussedlater. These eigen-functions do not depend on the resonance 0 under consideration. %! maytherefore expanded over the same set of eigen-functions as follows

%! =$

n

x!,nun . (A.2.23)

Inserting Eq. (A.2.23) into Eq. (A.2.22) leads to 2m relations of the form:$

n!

x!,n! 3n! un! +$

n!

$

!!

H!,!! x!!,n! un! = 0 . (A.2.24)

Multiplying Eq. (A.2.24) by un and integrating over R (given that the un functions form anorthonormal set), implies

x!,n 3n +$

n!

$

!!

x!!,n!

#un

%R&H!,!!

%R&un!

%R&dR = 0 . (A.2.25)

Calling

x = ((x1,#m, · · · , x1,m) , · · · , (xn,#m, · · · , xn,m) , · · ·) , (A.2.26a)" = Diag (Diag (31, · · · , 31) , · · · , Diag (3n, · · · , 3n) , · · ·) , (A.2.26b)

H ($p) =?Hn,n!

@=!#

un

%R&H%R&un!

%R&dR

"=!

H(n,n!)

(!,!!)

", (A.2.26c)

yields for Eq. (A.2.25)" · x + H ($p) · x = 0 . (A.2.27)

Here H is given by Eq. (A.2.22). The coe#cients H(n,n!)

(!,!!)are functions of $p given by

H(n,n!)

(!,!!)= /

(!,!!)!G

#un

%R&un!

%R&P#

a2! (e)

OOOO%J! %h

%e %R

OOOO[#%F0/%h]J!

$! # $pde

Q

dR . (A.2.28)

Equation (A.2.28) may be rearranged as

H(n,n!)

(!,!!)= /

(!,!!)!G

# #An

! (J) An!

!! (J)[#%F0/%h]Jf

$! # $pd2J , (A.2.29)

where the e!ective An! are defined by:

An! (J) = a! [e (J)]un

+R (J)

,. (A.2.30)

The system of equations Eq. (A.2.27) has a non-trivial solution for the xn,!’s only if

D ($p) = det |"+ H ($p) | = 0 . (A.2.31)

Eq. (A.2.31) gives the criterion for the existence of exponentially growing unstable modes in theepicyclic approximation. The corresponding results are discussed in the main text.


A.2.3 Reduced eigenvalue problem

The eigen-functions and eigenvalues of the Sturm Liouville equation corresponding to the integralequation Eq. (A.2.20) are now derived. The r.h.s. of Eq. (A.2.20) reduces to the integral equation

u%R1

&=#

gn

%R1, R2

&u%R2

&dR2 , (A.2.32)

where u(R) is the unknown. Given Eq. (A.2.11), Eq. (A.2.32) may be re-written as:

u%R1

&= g2

n

%R1

&U1

%R1

&+ g1

n

%R1

&U2

%R1

&, (A.2.33)

where

U1

%R&

=# R

g1n

%R&u%R&dR , (A.2.34a)

U2

%R&

=#

Rg2

n

%R&u%R&dR . (A.2.34b)

The gin’s are given by Eq. (A.2.11). Double di!erentiation of this equation leads to the system

u = g2nU1 + g1

nU2 , (A.2.35a)u$ = g2

n$U1 + g1

n$U2 , (A.2.35b)

u$$ = g2n$$U1 + g1

n$$U2 +

%g2

n$g1

n # g1n$g2

n

&u . (A.2.35c)

Eliminating U1 and U2 between the first two equations yields the Sturm Liouville eigenvalueproblem

Lu & (pu$)$ + qu = 3u , (A.2.36)

with

p%R&

= g1n$g2

n # g2n$g1

n =1 + 4 n

R2, (A.2.37a)

q (R) = g1n$$g2

n$ # g2

n$$g1

n$ # p

%R&2 = #1 + 10 n + 28 n2 + 16 n3

R4. (A.2.37b)

This di!erential equation is equivalent to the integral equation Eq. (A.2.32), when associatedto the initial conditions which follows from Eq. (A.2.32), namely that U1 = [ug1

n$ # u$g1

n], andU2 = [ug2

n$ # u$g2

n] vanish at the endpoints. The integration of Eq. (A.2.36) yields

u(

%R&

= AR 54 J#) (2R) + B R5

4 J) (2R) , (A.2.38)

where R =

.#3

16 (1 + 4 n)R2 and & =

(5 + 24 n + 16 n2)1/2

4. (A.2.39)

Given the mixed boundary conditions+ug1

n$ # u$g1

n

, %Rmin

&= 0 and

+ug2

n$ # u$g2

n

, %Rmax

&= 0 , (A.2.40)

and requiring that Eq. (A.2.38) is normalised reduces the variation in 3 to a discrete set ofpossible eigenvalues. As L is hermitian (by construction), the eigenvalues are real. It is thereforelegitimate to expand H over the eigenfunctions ofL. It can be checked that Eq. (A.2.36) does notallow for a solution compatible with the initial conditions Eq. (A.2.40) when 3 = 0. ThereforeC

i 3i .= 0. The convergence of the determinant Eq. (A.2.31) should follow from the asymptoticbehaviour of the solution to the di!erential equation. The asymptotic solutions vary like ncos(nR2). Therefore Hnn!

!!! = o(n#1n$#1) according to Lebesgue-Riemann Lemma.

Section A.3 A brief introduction to angle action formalism 39

1 2 3 4 5

R

-20

-10

0

10

20u

Eigen modes

Figure A.2.1: Eigen-functions of the Green kernel (Eq. (A.2.32)) corresponding to the eigen-values3n = #3.065,#5.313,#11.62,#20.29,#31.33. The boundary of the disk are chosen to be Rmin = 1 andRmax = 5. The Green Kernel corresponds to n = 1.

A.3 A brief introduction to angle action formalism

A fair fraction of the content of this subsection was inspired from D. Earn’s [30] PhD. Its purposeis to introduce Angle Action variables and generating functions, and their relevance to galacticdynamics (See Born (27) [16] for references).

Let q and p be some known coordinates and momentum for Hamilton’s equation. Theyobey the equation of motion:

q =%H

%p(A.3.1a)

p = #%H%q

(A.3.1b)

where H is the Hamiltonian for this set of variables. This can be expressed as a variationalprinciple, writing

+# t2

t1

(p · q#H) dt =# t2

t1

R+p

!q# %H

%p

"# +q

!p +

%H

%q

"Sdt + [p · +q]t2t1 . (A.3.2)

The r.h.s of Eq. (A.3.2) vanishes because the variation are assumed to vanish at the end points,and q and p obey Hamilton’s equation.

A new set of variables, say P and Q with H the new Hamiltonian expressed in thesevariables, will satisfy formally the same variational principle.

+# t2

t1

?P · Q#H

@dt = 0 . (A.3.3)


This will have the same physical content as A1 only if there exists a function S so that

P · Q#H = p · q#H +dS

dt, (A.3.4)

where d/dt is a total derivative with respect to time.Restricting possible change of variables to new sets for which S(q, P) = Sq(q)+ SP (P)#P ·Q,Eq. (A.3.4) is satisfied only if

p =%S

%q, Q =

%S

%Pand H = H , (A.3.5)

which can be written in a more compact form as:

H!q,%Sq

%q

"= H

!%SP

%P, P

". (A.3.6)

Notice that H is a function of P only, by choice of S. The new equations of motion thereforeread:

P = #%H%Q

& 0 and Q =%H%P

, (A.3.7)

which leads to the immediate solution:

P = constant , and Q =%H%P

t + Q0 . (A.3.8)

Therefore S is a generating function to a set of variables for which the momenta are integral ofthe motions.

• Angle and action in galactic dynamics.

Restricting this analysis to multiply periodic bound motion, some new momenta for whichthe associated coordinates are the phase of the oscillation of each degree of freedom maybe constructed. Calling this new set of variables (#, J), it follows

2! =-

d#i =- !

%#i

%q· dq +

%#i

%p· dp

"=- !

%

%q%S

%Ji· dq +

%

%p%S

%Ji· dp

"

=%

%Ji

-%S

%q· dq =

%

%Ji

-p · dq , (A.3.9)

where the loop integral refers to integration over one complete period of oscillation for thecorresponding degree of freedom. This relation will be satisfied if

Ji =12!

-pidqi (A.3.10)

These momenta are called actions. They correspond to the circulation of each degree offreedom. In section 3, new momenta were built in the context of galactic dynamics forthe radial and azimuthal motion of a star in the disc of the axisymmetric galaxy whosepotential is %0. Namely J = (JR, J') = (JR, h), where

JR =12!

- ARBdR, J

¯'= h = R2* .

Here [R] is the function of R, the specific energy E, and the specific angular momentum hof the star given by A

RB

=>

2E + 2%0 (R)# h2/R2

Section A.4 Numerical N-body and restricted 3-body simulations 41

Actions are relevant to galactic dynamics for the following reasons

(i) the equations of motion are trivial in these variables as the unperturbed actions areby construction integrals of the motion. The perturbation theory is therefore easy toimplement;

(ii) the actions label unperturbed orbits;

(iii) the actions can be adiabatically invariant.

Two other concepts central to the analysis of the main sections are the averaging principleand the adiabatic invariance

• The averaging principle

Let F be a distribution function in phase space parameterised by angles and actions. Ex-pressing this function as the sum over its Fourier components with respect to the phase#

F (J,#, t) =$

m

Fm (J, t) eim" (A.3.11)

it follows thatdF

dt=%F

%t+%H%J

· %F%#

# %H%#

· %F%J

= 0 (A.3.12)

This implies in turn that for each m

Fm (J, t) = Fm (J) exp!#im · %H

%Jt"

(A.3.13)

If m · %H/%J .= 0 the time average of Fm(J, t) will be zero. If there exists J so thatm ·%H/%J & m ·! = 0 for a given m, i.e. if there is a resonance between the two degrees offreedom, the time average of the corresponding Fm will not vanish. The averaging principleas applied in Section 3 of chapter 1 assumed that the distribution function does not haveany resonant component but the inner Lindblad, for all J. Strictly speaking, this statementimplies that F & 0. Nevertheless, it may in practise be assumed that for a su#ciently longtime (of the order of 1/|!m|), the contribution from the higher m’s is negligible. For finitetime, F (J,#, t) is therefore well described by F0 =< F (J,#, t) > when restricting to theother low order resonances.

• Adiabatic Invariance

The adiabatic invariance implies that when a system is slowly perturbed, its actions stillbehave as a first integral to second order in the ratio of the natural frequency of thecorresponding phase to the characteristic time scale of the perturbation. This relies on theaveraging principle, and therefore assumes no low order resonances are involved betweenthese motions. In the main text, the libration of a given orbit in the potential well of thebar is taken to be an external slowly varying perturbation for the stream motion of a staralong its orbit. The circulation corresponding to the latter motion is therefore assumed tobe adiabatically invariant. This reads:

J

J= o

9!$! # $p

"

"2:

· • !" ! • ·


-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

"Particle"

Figure A.4.1: Trajectories of four initially resonant inner Lindblad orbits in an Isochrone disk whileself gravity is slowly switched on. Four particles are required to fix the centre of mass frame, but do notprevent the mean angular drift of that frame.

A.4 Numerical N-body and restricted 3-body simulations

Two numerical experiments where carried out to illustrate and clarify the analysis of chap-ter 2.

The purpose of the first experiment was to verify the donkey or cooperative behaviour of resonantinner Lindblad orbits. A restricted four bodies code was written to account for the interactionbetween stars describing resonant orbits as prescribed by the mean field. Fig. A.4.1 gives thetrajectory of two initially resonant orbits – represented by four stars – in the mean field de-scription, when self gravity is switched on smoothly. The generic behaviour of the orientationof resonant orbits given by #2 # #1 $ #(&1 + &2) sin(2#2# 2#1) is recovered qualitatively, untila close encounter invalidates the description of the orbit in terms of a unique star.The other experiment corresponds to an N-body simulation of a fully self gravitating disk. Theintegrator is a tree code developed by Richardson. The qualitative evolution is given on Fig. A.4.2and Fig. A.4.3. The disk displays spontaneously non-axisymmetric features and rapidly forma strong bar, once some fraction of the angular momentum has been carried away by transientspiral waves.The stars describe elongated x1 orbits oriented along the direction of the bar, in agreement withthe models of section 3-5. The pattern speed is found below the maximum pattern speed of innerLindblad resonances, but this apparent disagreement with Sellwood’s results probably accountsfor the di!erent regime investigated and the crudeness of this model3. A more quantitative anal-ysis would undoubtedly require using a better Poisson kernel with symmetrised initial conditions

3 This model explores the non-linear regime of bar formation with a relatively noisy code – a direct forceintegrator, starting with a set of initial conditions which does not correspond to an equilibrium.

Section A.4 Numerical N-body and restricted 3-body simulations 43

Figure A.4.2: Iso-density contours in log scale of a self-gravitating Kuzmin disk made of 32 000 particles.The initial conditions correspond to an isotropic pressure disk with a Toomre number, Q = 3.36G'0/)R",equal to unity.

(so called quiet stars as introduced by Sellwood [101]), along the lines of Earn’s [30] work, whosenumerical integrator allows a natural extension of the linear analysis in the non-linear regime.

· • !" ! • ·


Figure A.4.3: Velocity field of the same self-gravitating Kuzmin disk for the time step displayed onthe top right corner of Fig. A.4.2 (Only 8 000 particles are shown here). The stars in the bar describeelongated trajectories along the major axis of the bar.

3

Stellar dynamics

of round galactic disks

3.1 Introduction

Over the next decade, ample and accurate observational data on the detailed kinemat-ics of nearby disks galaxies will become available. It would be of great interest to link theseobservations with theoretical models for the underlying dynamics. Independent measurementof the radial and azimuthal velocity distribution functions could, for instance, be contrastedwith predictions arising from the gravitational nature of the interaction. Indeed the laws ofthe motion and the associated conserved quantities together with the assumption that the sys-tem is stationary put strong constraints on the possible velocity distributions. This is formallyexpressed by the existence of an underlying distribution function which characterises the dy-namics completely. The determination of realistic distribution functions which could accountfor observed line profiles is therefore an important project vis a vis the understanding of galacticstructure. Producing theoretical models that accounts globally for the observed line profiles ofa given galaxy provides a unique opportunity to inspect the current understanding of the dy-namics of S0 galaxies. It should then be possible to study quantitatively all departures from theflat axisymmetric stellar models. Indeed, axisymmetric distribution functions are the buildingblocks of all sophisticated stability analyses, and a good phase space portrait of the unperturbedconfiguration is often needed in order to asses the stability of a given equilibrium state as shownin chapter II. Numerical N-body simulations also require sets of initial conditions which shouldreflect the nature of the equilibrium.

In this chapter inversion methods are constructed and implemented. These lead to classicaldistribution functions compatible with a given surface density (section 2), or a given surfacedensity and a given pressure profile (section 3). This latter technique yields distribution functionsthat correspond to given line profiles (or rather, to the dispersion in the line profiles inducedby the velocity distributions which will loosely be called line profiles in this chapter) which mayeither be postulated or chosen to match the observations. In fact, only a sub-sample of theobservation is required to construct a fully self-consistent model which accounts, in turn, forall the observed kinematics. The corresponding redundancy in the data may be exploited (assketched in section 4) to address the limitations of the description.

· • !" ! • ·

46 Stellar dynamics of round galactic disks Chapter 3

3.2 Distribution functions for a given surface density

The problem of finding the distribution function for an axially symmetrical system mayformally be solved by using Laplace transforms (Lynden-Bell 1960 [70]) or by using power series(Fricke 1952 [37]). These methods have been further developed for applications to flat galaxiesby Kalnajs (1976) [58] and by Miyamoto (1974) [88] and Hunter & Qian (1992) [96] who chosespecific forms of distribution function because the flat problem has no unique solution. Infact there is a whole functional freedom left in the distribution function. Here this freedom isexploited to choose special forms of distribution function which make the solution of the resultingintegral equation remarkably simple. Explicit inversions are given for distribution functionswhose even parts are of the form f+(,, h) = (#,)n+ 1

2 Gn(h2/2) where h is the specific angularmomentum and , the specific energy of a given star. Inversions are also provided when f+(,, h) =h2nFn(,). Examples are given and illustrated graphically . The part of f which is antisymmetricin h = Rv' cannot be determined from the surface density alone unless it is assumed that all thestars rotate in the same sense about the galaxy, in which case f# = Sgn(h)f+. More generally,if the mean velocity 'v'( is specified – say by the azimuthal drift prescription – a similar ansatzfor f# allows its explicit determination too. The inversion formulae require expressing powersof R times the surface density '(R) as functions of either the potential $(R) or of the relatedfunction Z(R) = R2$. These formulae can only be applied to those density distributions forwhich the potential in the plane of the matter is known (either self-consistently or not). Theother characteristics of the disk follow from the knowledge of f and cannot be specified a-priori.

3.2.1 Derivation

For a flat galaxy all the stellar orbits are confined to a plane and by Jeans’ Theorem the steadystate mass-weighted distribution function must be of the form f = f(,, h), where the specificenergy , is given by1

, = 12

?v2

R + v2'

@# $, (3.2.1)

and the specific angular momentum, h, reads:

h = R v' . (3.2.2)

The surface density, '(R), arises from this distribution of stellar orbits provided that the integralof f overall bound velocities is '(R), i.e.

' (R) =# #

f (,, h) dvRdv' , (3.2.3)

where the integral is over the region 12 (v2

R + v2') < $.

The following inversion method assumes that '(R) is given and its potential on the plane,$(R) is known. This is achieved either by requiring self-consistency via Poisson’s equation

22$ = #4 ! + (z) ' (R) , (3.2.4)

or alternatively by assuming that the disk is embedded in a halo and choosing both ' and $independently. At constant v' and R, vR dvR = d, and hence, using Eq. (3.2.1) and Eq. (3.2.2)

dvR = d,+2 (, + $)# h2R#2

,#1/2. (3.2.5)

1 The convention throughout this chapter the minus sign in the definition of the potential.

Section 3.2 Distribution functions for a given surface density 47

Furthermore, at constant R, dh = R dv', and therefore Eq. (3.2.3) reads

' (R) =# +,

2R2*

#,

2R2*2# 0

h22R2 #*

f (,, h) d, dh=2 (, + $)R2 # h2

. (3.2.6)

The factor of 2 arises because vR takes both positive and negative values which are mapped intothe same range of , which depends only on v2

R.

Splitting f into its odd and even parts:

f+ (,, h) = 12[f (,, h) + f (,,#h)]

f# (,, h) = 12[f (,, h)# f (,,#h)]

Q

, (3.2.7)

leads to

' (R) = 4# ,2R2*

0

# 0

h22R2 #*

f+ (,, h) d, dh=2 (,+ $)R2 # h2

, (3.2.8)

as only the even part of f contributes to Eq. (3.2.6).

• Even component of f : first Ansatz

A first ansatz is to look for solutions for f+ which take the form

I f+ (,, h) = (#,)n+ 12 |h|Gn

!h2

2

", (3.2.9)

where n is first assumed to be an integer. Here the |h| is taken out for convenience; its appearancein no way implies that f+(,, 0) = 0 because Gn(h2/2) may behave as |h|#1 for small h. Thequantity n measures the level of anisotropy in the disk. Large n correspond to cold, centrifugallysupported disks; small n correspond to isotropic, pressure supported disks. Inserting the ansatzinto Eq. (3.2.8) gives

' (R) =2 3

2

R

# R2*

0Gn

!h2

2

"9# Y

0

(#,)n+ 12 d (#,)

=Y # (#,)

:

d!

h2

2

", (3.2.10)

where the e!ective potential, Y , is given by

Y = $ # h2

2R2. (3.2.11)

The inner integral can be transformed to Y n+1In where

In =# 1

0

xn+ 12 dx,

1# x=

(2n + 1) (2n# 1) ...12n+2 (n + 1)!

! , (3.2.12)

and x has been written for (#,/Y ).Re-expressing Y n+1 via Eq. (3.2.11),Eq. (3.2.10) becomes

' (R) = 232 In R#1

# R2*

0

!$ # h2

2R#2

"n+1

Gn

!h2

2

"d!

h2

2

". (3.2.13)

Writing H for h2/2 and Z for R2$, this expression becomes on multiplication by R2n+3

Sn & R2n+3' (R) = 232 In

# Z

0(Z #H)n+1 Gn (H) dH . (3.2.14)


Z & R2$ is known, so R2n+3'(R) can be re-expressed as a function Sn(Z). Di!erentiatingEq. (3.2.14) n + 2 times with respect to Z, gives:

!d

dZ

"n+2

Sn (Z) = 232 In (n + 1)! Gn (Z) . (3.2.15)

Hence, substituting H for Z and using Eq. (3.2.12) yields

Gn (H) =1,

2+%

n + 12

& %n# 1

2

&.... 1

2

,!

!d

dH

"n+2

Sn (H) . (3.2.16)

The ansatz Eq. (3.2.9) therefore leads to the solution of the integral equation Eq. (3.2.8) in theform

f+ = (#,)n+ 12 |h|Gn

!h2

2

", (3.2.17)

where Gn is given by Eq. (3.2.16). Eq. (3.2.16) specifies the part of the distribution functionwhich is even in h. Demanding no counter-rotating stars would imply, for instance, f = f++f# =0 for h < 0. In that case f# = #f+ for h < 0 which leads to f# = Sgn(h) f+(,, h). Hence

f = f+ + f# =

DEF

EG

2f+ (,, h) h > 0 ,

f+ (,, h) h = 0 ,

0 h < 0 .

(3.2.18)

These solutions are called maximally rotating disks. The formal solution, Eq. (3.2.16), must benon-negative for all h so that Eq. (3.2.17) corresponds to a realizable distribution of stars. Withinthat constraint, the choice of n is, in principle, free. There are many distribution functions whichgive the same surface density distribution because there are two functional freedoms in f(,, h)and only one function '(R) is given. Here it has been shown that even within the specialfunctional form of ansatz I there is potentially a solution for any chosen integer n. When n isnot an integer, n = n0 # &, 0 < & < 1, Eq. (3.2.10) may still be inverted using Abel transforms(see appendix B for details), though the final solution involves an integral which might requirenumerical evaluation

Gn (H) =sin (!&) 2#3/2!#1I#1

)

[(n0 + 1# &) (n0 # &) ... (1# &)]

3

MMMM4

# H

0

;d

dZ

n0+2

Sn

<dZ

(H # Z)1#)

+;

d

dZ

n0+1

Sn

<

Z=0

1H1#)

5

NNNN6, (3.2.19)

where

I) =# 1

0

xn+ 12 dx,

1# x=,! (

%n + 3

2

&

2( (n + 3). (3.2.20)

Alternatively, there are many ways in which it is possible to re-express '(R) in the general form

' (R) =$

n

R#2n#3S!n

%R2$

&. (3.2.21)

For any particular sum, a distribution function G!n(H) can be found which gives rise to that

part of the density given by the nth term S!n. Indeed the relationship between G!

n and S!n is just

that given by Eq. (3.2.16) between Gn and Sn. Since the original integral equation is linear, itfollows that each expression for '(R) in the form Eq. (3.2.21) yields a distribution function inthe form

f+ (,, h) =$

n

(#,)n+ 12 |h|G!

n

!h2

2

". (3.2.22)


There is no requirement that each G!n should be positive provided that the sum f+ is positive

for all , and h. Thus the expression for the density Eq. (3.2.21) with the distribution functionEq. (3.2.22) gives a very considerable extension of the original ansatz Eq. (3.2.9).

• Odd Component of f and azimuthal drift

When the average azimuthal velocity field is known, a similar inversion procedure is avail-able in order to specify the odd part in h of the distribution function. This field may be assumedto be given by the azimuthal drift equation which takes the form

' 'v'(2 = p' # pR

!"2R2

4V 2c

", (3.2.23)

where the azimuthal pressure, p', and the radial pressure, pR, are derived from the even com-ponent of f#, and the epicyclic frequency and the circular velocity " and Vc follow from $.Formally, the only di!erence from the previous analysis is that in place of Eq. (3.2.3), theintegral equation relating f# and 'v'( becomes

' 'v'( =# #

f (,, h) v' dvR dv' . (3.2.24)

In place of Eq. (3.2.8), this leads to the equation

R' 'v'( = 4# ,2R2*

0

# 0

h22R2 #*

f# (,, h)=

2 (, + $)R2 # h2h d, dh . (3.2.25)

Replacing ansatz I by ansatz I$:

I $ f# (,, h) = (#,)n+ 12 TGn

!h2

2

"Sgn (h) , (3.2.26)

yields a solution for TGn which is formally equivalent to Eq. (3.2.16), but with TSn defined by

TSn (Z) = R2n+4' 'v'( . (3.2.27)

The extension to solutions for f# when

' 'v'( =$

n

R#2n#4 TS!n (Z) , (3.2.28)

is, of course,

f# (,, h) =$

n

(#,)n+ 12 TG!

n

!h2

2

"Sgn (h) . (3.2.29)

Equation (3.2.29) yields that part of the distribution function which is antisymmetric in h andis compatible with the mean azimuthal flow 'v'( given ansatz I’.

• Alternative Ansatz

In place of ansatz I, consider a di!erent ansatz:

II f+ (,, h) = h2nFn (,) , (3.2.30)

where n is also first assumed to be an integer. Inserting Eq. (3.2.30) into Eq. (3.2.8), andreversing the order of the integrations, yields

' (R) = 4# 0

#*Fn (,)

9# X1/2

0

h2ndh,X # h2

:

d, , (3.2.31)


whereX = 2 (,+ $) R2 . (3.2.32)

The inner integral is Xn Jn, where

Jn =# 1

0

x2ndx,1# x2

=+%

n# 12

& %n# 3

2

&... 1

2

,!/ (2n!) , (3.2.33)

and x = h/X12 . Thus Eq. (3.2.31) becomes, using Eq. (3.2.32) for Xn

' (R) = 2n+2JnR2n

# 0

#*(,+ $)n Fn (,) d, . (3.2.34)

Re-expressing R#2n'(R) & Sn($) yields!

d

d$

"n+1

Sn ($) = 2n+2n! Jn Fn (#$) . (3.2.35)

Writing , for #$ leads to a solution for Fn(,)

Fn (,) =2#n#1

+%n# 1

2

& %n# 3

2

&... 1

2

,!

!d

d (#,)

"n+1

[Sn (#,)] ; (3.2.36)

hencef+ = h2nFn . (3.2.37)

Again, Eq. (3.2.18) is used to determine f# when no retrograde orbits are allowed. EquationEq. (3.2.37) was derived independently by Sawamura (1987) [99].When n is not an integer, the solution to Eq. (3.2.34) can still be found, though with a supple-mentary integration (with n = n0 # &, n0 the closest upper integer), to give

Fn (#,) =2#n#2!#1J#1

n sin!&[(n0 + 1# &) (n0 # &) ... (1# &)]

3

MMM4

# &

0

!dn0+1

d$Sn

"d$

[,# $]1#)

+1,1#)

!dn0

d$Sn

"

*=0

5

NNN6 , (3.2.38)

where

Jn =# 1

0

x2n

,1# x2

dx =,! (

%12

+ n&

2( (1 + n). (3.2.39)

The ansatz in Eq. (3.2.30) may be also generalised when '(R) is expressed in the form

' (R) =$

n

R2nS!n ($) ; (3.2.40)

each such decomposition corresponds to a distribution function

f+ (,, h) =$

n

h2n F !n (,) , (3.2.41)

where the F !n are given in terms of the S!

n by Eq. (3.2.36) with stars inserted on Fn and Sn.The above procedure is also straightforward to implement in order to constrain the antisymmet-ric component of the distribution function. Indeed ansatz II$:

II $ f# (,, h) = h2n#1 TFn (,) , (3.2.42)


may be chosen in place of ansatz I$. Then Eq. (3.2.31) becomes

R' 'v'( = 4# 0

#*

TFn (,)# X1/2

0

h2ndh,X # h2

d, . (3.2.43)

Hence the solution for Fn is formally identical to Eq. (3.2.36) but with TSn substituted for Sn,where

TSn ($) = R1#2n' 'v'( . (3.2.44)

Again, the integral equation is linear, so if ' 'v'( is expressed as

' 'v'( =$

n

R2n#1 TS!n ($) , (3.2.45)

thenf# (,, h) =

$h2n#1 TF !

n (,) , (3.2.46)

where the relationship of TF !n to TS!

n is the same as in Eq. (3.2.36).

3.2.2 Disk properties

• Radial velocity dispersion

The average radial velocity dispersion, )2R, is an important quantity for the local stability

of the disk. It satisfies

')2R =

# #f (,, h) v2

R dvRdv' , (3.2.47a)

=4R

,2R2*#

0

0#

h22R2 #*

f+ (,, h)

.

2 (,+ $)# h2

R2d, dh , (3.2.47b)

where the integration is carried out over the region 12(v2

R + v2') < $. Given Eq. (3.2.9) and using

ansatz I, Eq. (3.2.47) may be rearranged as

')2R =

2 52

R

# R2*

0Gn

!h2

2

"9# Y

0(#,)n+ 1

2

>Y # (#,) d (#,)

:

d!

h2

2

", (3.2.48)

where the e!ective potential Y = $ # h2R#2/2. The inner integral is

# Y

0(#,)n+ 1

2

>Y # (#,) d (#,) = Yn+2In , (3.2.49)

whereIn =

# 1

0xn+ 1

2,

1# xdx =! (2n + 1)!!2n+2 (n + 2)!

. (3.2.50)

Re-expressing Y n+2, Eq. (3.2.48) then becomes

')2R =

2 52In

R2n+5

# Z

0(Z # H)n+2 Gn (H) dH , (3.2.51)


where H = h2/2 and Z = R2$. The calculation of )2R(R) is therefore straightforward once the

function Gn has been found. Note the similarity between Eq. (3.2.51) and Eq. (3.2.14). In factthis similarity provides a check for self-consistency since one should have

%

%Z

%)2

R'R2n+5&

= 2 (n + 2)In/InR2n+3' = R2n+3' . (3.2.52)

Putting Eq. (3.2.30) into Eq. (3.2.47) yields, for Ansatz II:

R2 ' )2R = 4

# 0

#*Fn (,)

9# X1/2

0h2n,

X # h2 dh

:

d, , (3.2.53)

where X = 2 (,+ $)R2 . The inner integral is Xn+1 Jn, where

Jn =# 1

0x2n,

1# x2 dx =! (2n# 1)!!2n+2 (n + 1)!

, (3.2.54)

and x = h/X1/2. Thus Eq. (3.2.53) becomes, using Eq. (3.2.32) for Xn

' )2R = 2n+3JnR2n

# 0

#*(,+ $)n+1 Fn (,) d, . (3.2.55)

The similarity between Eqs. (3.2.34) and (3.2.55) yields the identity:

%

%$

%')2

RR#2n& = 2 (n + 1)'R#2nJn/Jn = 'R#2n . (3.2.56)

• Azimuthal velocity dispersion

The average azimuthal velocity dispersion, )', is an observable constraint on models whichsatisfies

'?)2' + 'v'(2

@= '

Uv2'

V=# #

f (,, h) v2' dvRdv' , (3.2.57a)

=4R3

,2R2*#

0

0#

h22R2 #*

f+ (,, h) h2 d, dh>

2 (,+ $)# h2

R2

, (3.2.57b)

where the mean azimuthal velocity 'v'( is given in terms of f# by linear combinations ofEqs. (3.2.25) and (3.2.43).Putting Eq. (3.2.9) into Eq. (3.2.57b) and following the substitutions Eq. (3.2.6)-(3.2.16) yields,for the first Ansatz,

R2n+5 '?)2' + 'v'(2

@= 2

52 In

# Z

0(Z #H)n+1 HGn (H) dH , (3.2.58)

where H = h2/2 and Z = R2$. In and Gn are given by Eq. (3.2.12) and (3.2.16).Putting Eq. (3.2.30) into Eq. (3.2.57b) gives, for the second ansatz,

'Uv2'

V=

4R2

# 0

#*Fn (,)

9# X1/2

0

h2n+2dh,X # h2

:

d, , (3.2.59)


where X = 2 (,+$)R2 , and Fn is given by Eq. (3.2.36). The inner integral is Xn+1 Jn+1, whereJn is given by Eq. (3.2.33). Therefore

'Uv2'

V= 2n+4Jn+1R

2n

# 0

#*(,+ $)n+1 Fn (,) d, . (3.2.60)

Note that )R, $, ' and 'v2'( are related via the equation of radial support:

Uv2'

V# R

%$

%R=% (R')2

R)' %R

. (3.2.61)

The properties of the disks constructed with ansatz I & II are illustrated in Figs. 3.2.2 andFig. 3.2.1 for the examples described below.

3.2.3 Examples

• Isochrone disks

Distribution functions for the flat Isochrone are simplest with the assumptions of AnsatzII. In the equatorial plane, the Isochrone potential reads

$ = GM/ (b + rb) where rb2 = R2 + b2 . (3.2.62)

The corresponding surface density is

' =Mb

2!R3{ln [(R + rb) /b]# R/rb} =

Mb

2!R3L . (3.2.63)

But R2$ = Z = (rb2 # b2)GM/(rb + b) = GMb(s # 1), where s is the dimensionless variable

rb/b. Therefore

'R2n+3 =Mb

2!R2nL (s) =

Mb2n+1

2!%s2 # 1

&nL (s) , (3.2.64)

where L(s) = ln(,

s2 # 1+s)+,

s2 # 1/s. Note that L$(s) is quite simple: L$(s) =,

s2 # 1/s2.With this notation,

!%

%Z

"n+2 %R2n+3'

&=

Mb2n+1

2! (Gmb)n+2

!%

%s

"n+2 A%s2 # 1

&nL (s)

B, (3.2.65)

which implies, in turn, that

Gn

;h

2

2<

=M#n#1bn#1

25/2!GIn (n + 1)

!%

%s

"n+2 +%s2 # 1

&L (s)

,OOOOOs=1+h2/2

. (3.2.66)

Therefore, the final distribution function (or rather its symmetric part, f+) therefore reads

f+ (,, h) =[#,/(GM/b)]n+1/2

4!2Gb (n + 1/2)!!

!h2

2GMb

"1/2 !%

%s

"n+2 A%s2 # 1

&nL (s)

B. (3.2.67)

with s = 1+h2/(2GMb). For example, the first two distribution functions are given by (in unitsof GM = b = 1):

f+,0 (,, h) =,

2 (#,)1/2 (#4 + 4 h2 + h4)(2 + h2)3,4 + h2 !2

, (3.2.68a)

f+,1 (,, h) =2,

2 (#,)3/2 (40 + 8 h2 + 18 h4 + 8 h6 + h8)3(2 + h2)4

,4 + h2 !2

. (3.2.68b)


0 0.5 1 1.5 2 2.5 3

R

1.6

1.8

2

2.2

2.4

2.6

2.8

Toomre’s Q n=1

n=2

n=3

n=4

Figure 3.2.1: the “Toomre Q number” Q = )R"/(3.36'0) against radius R for the Isochrone modeldescribed by Eqs. (3.2.68) and (3.2.70). The parameter n in Eq. (3.2.67) corresponds to a measure of thetemperature of the disk.

Using these distributions functions, Toomre’s Q criterion for radial stability (Q = )R"/3.36G'0)may be calculated via Eq. (3.2.51). Note that for the Isochrone potential

)2R (R) =

25/2In

R2n+5' (R)

# Z

0(Z # H)n+2

9!%

%s

"n+2

L (s)%s2 # 1

&n

:

s=H+1

d H . (3.2.69)

For instance

' )2R,0 =

12 !R5

[#2 R + arctan (R) + r log (r + R)] , (3.2.70a)

' )2R,1 =

112 !R7

'10 R# 8 R3

3# 6 arctan (R) + 2 r

%#2 + R2

&log (r + R)

(, (3.2.70b)

where r =,

1 + R2.

• Toomre-Kuzmin disks

Ansatz II is also appropriate when seeking distribution functions for the Toomre-Kuzmindisks. In the equatorial plane, the potential of the disk reads

$ = #GM

rb, with rb =

,b2 + R2 , (3.2.71)

while the surface density is

' =12!

GM b

r3b

. (3.2.72)


Ansatz II yields the distribution (in units of b and GM/b )

fn+ (,, h) =

2n#2 h2n

+%n# 1

2

& %n# 3

2

&... 1

2

,!2

's2n+3

(1# s2)n

((n+1)

s%#&

. (3.2.73)

For instance,

f+,0 (,, h) =3 (#,)2

4 !2, (3.2.74a)

f+,1 (,, h) = h2(#,)3

A10# 9 (#,)2 + 3 (#,)4

B

2A1# (#,)2

B3!2

, (3.2.74b)

f+,2 (,, h) = h4(#,)4

A#5 + (#,)2

B A7 + (#,)4

B

2A(#,)2 # 1

B5!2

, (3.2.74c)

This example illustrates the drawbacks of Ansatz II. The factor h2n in Eq. (3.2.30) removes allzero angular momentum orbits. Self-consistency yields a solution with circular orbits near theangular momentum origin (f scales like powers of h2/[1 # (#,)2]3/2). In position space, thisimplies that the inner core of the galaxy is made of circular orbits! This qualitative featureinduces rising radial pressure curves given by

' )2R,0 =

$4

8 !, (3.2.75a)

' )2R,1 =

#2 + $2 + $4

8 !+

(#1 + $2) log (1# $2)4 !$2

, (3.2.75b)

' )2R,2 =

6# 9$2 + 2$4 + $6

8 ! $2+

3 (#1 + $2)2 log (1# $2)4 !$4

, (3.2.75c)

for the examples above.

Realistic distribution functions will therefore require superpositions of solutions such as(3.2.74), in the spirit of Eqs. (3.2.41) or (3.2.22) as illustrated by Fig. 3.2.2. It is also clear fromEqs. (3.2.75) that ' )2

R does not fall o! fast enough to yield cold disks (it ought to scale like $6

to lead to constant Q profiles).

• Power law disks

The simplicity of power law disks leads to solutions for the inversion problem which maybe extended to models with a continuous free “temperature” parameter as described above.Recall that this parameter is a measure of the “temperature” of the disk. This temperature maytherefore be varied continuously between the two extremes corresponding to the isotropic andcold disk.

The potential and surface density for the power law disk read

$ = R#% and ' =S%

2!R#%#1 , (0 < . < 1) , (3.2.76)

where S% is given by :

S% = .!( [1/2 + ./2] ( [1# ./2]( [1/2# ./2] ( [1 + ./2]

". (3.2.77)


0 0.2 0.4 0.6 0.8 1

R

0

0.01

0.02

0.03

0.04

P

Figure 3.2.2: radial (solid line) and azimuthal (dotted line) pressure for a Kuzmin disk constructed bylinear superposition of Eqs. (3.2.75) corresponding to the distribution functions given by Eqs. (3.2.74).

It is assumed here that distances are expressed in terms of R0, the reference radius, and energiesin terms of $0 = $(R0). (in units of G = 1). The inversion method corresponding to Ansatz Imay be carried out and, using (3.2.17), yields the distribution function

f+ (,, h) = S% (#,)n+1/2 |h|9

2#5/2 !#3/2( [&+ 1]( [n + 3/2]( [& # n# 1]

:!h2

2

")#n#2

(3.2.78)

where & = 2n + 2# ./(2# .). The parameter n is varied continuously in the range [(3. #2)/(4# 2.),*[ to account for the relative amount of pressure support in the disk. The velocitydispersions follow from Eqs. (3.2.19) and Eq. (3.2.39):

)2' + 'v'(2 =

n.

(2# . + n)R#% , (3.2.79a)

)2R =

2# .2 (2# . + n)

R#% . (3.2.79b)

By construction, these dispersions satisfy the equation of radial equilibrium:

Uv2'

V# R

%$

%R= # . (2# .)

(2# . + n)R#% =

% (R')2R)

' %R. (3.2.80)

Toomre’s number follows from Eqs. (3.2.76) and (3.2.79b)

Q =)R"

!G'=

2# .S%

' 2.2# . + n

(1/2

, (3.2.81)

Section 3.3 Distribution functions for given kinematics. 57

where S% is given by Eq. (3.2.77). Here, the e!ect of the parameter n on temperature is obvious.These results for the power-law disks were also derived by Evans [33] who used the ansatz II forthe inversion.

• Generalised Mestel disk

The generalised Mestel disk has more realistic features such as a very flat rotation curve.Its potential and surface density read (in natural length and energy units):

$ (R) =1R

log [r + R] and ' (R) =12!

1r (r + 1)

, (3.2.82)

where r =,

1 + R2. Ansatz I yields the following distribution:

f+,0 (,, h) =(#,)1/2 |h|

23/2!2

3

MMMM4

3 R (4 + 4 r + 3 R2 + r R2)(1 + r)3 (1 + R2) (R + r log [r + R])3

log [r + R]

# R4 (9 + 9 r + 12 R2 + 8 r R2 + 3 R4 + r R4)(1 + r)3 (1 + R2)2 (R + r log [r + R])3

5

NNNN6

R log[R+r]=h2/2

(3.2.83)Note that for these disks, Gn is a function of h via the implicit relation R log[R + r] = h2/2.

0 1 2 3 4 5

h

0

0.02

0.04

0.06

0.08

G(h)

Figure 3.2.3: the function Gn(h) – defined by Eq. (3.2.16), n = 0, 1, ..4 from bottom to top, for thegeneralised Mestel disk defined by Eq. (3.2.82). The n = 0 distribution function is given by Eq. (3.2.83).

· • !" ! • ·


3.3 Distribution functions for given kinematics.

The above Ansatz I & II lead to recursive expressions for f which start at the hot diskend. This artifact of the construction scheme is unfortunate because observed disks are fairlycold. Mathematical simplicity was achieved at the expense of a priori realism. For instance itwas unclear in advance that any of the above inversions would lead to a positive distributionfunction. It is desirable to use the functional freedom left in f in order to construct a distributionfunction which accounts for all the kinematics, either observed or desired (i.e. which accountsfor the line profiles observed, or which are marginally stable to radial modes). Here a di!erenttechnique is suggested which addresses most of these drawbacks.The philosophy behind this global inversion method is to introduce an intermediate observable,F'(R, v'), the distribution function for the number of stars which have azimuthal velocity v' atradius R (or alternatively G&(,, R) the distribution function for the number of stars which havespecific energy , at radius R). This technique may be applied to the reconstruction of distributionfunctions accounting for observed line profiles. In fact, given some symmetry assumptions aboutthe shape of an observed galaxy, it is shown that only a subset of the available line profiles – saythe azimuthal velocity distribution – is required to re-derive the complete kinematics. Thereforethis method provides a general procedure for constructing self-consistent models for the completedynamics of disk galaxies. These models predict radial velocity distributions which may, in turn,be compared with observations as discussed in section 4.Alternatively, a ‘natural’ functional form for F' or G& may be postulated and parameterisedso that it is compatible with imposing the surface density, the average azimuthal velocity andthe azimuthal pressure profiles (or equivalently the Toomre number Q, as Q follows from theequation of radial support). The distribution function f(,, h) of this disk follows in turn fromF' or G& via simple Abel transforms. This prescription is very general and especially usefulwhen setting the initial conditions of numerical N-body simulations such as those described inthe appendix of the previous chapter.

3.3.1 Inversion via line profiles

The number of stars which have azimuthal velocity v' within dv' at radius R reads

F' (R, v') =#

f (,, h) d vR =,

20#

#Y

f (,, h),, + Y

d, , (3.3.1)

where the e!ective potential, Y , is given by Y = $ # h2 R#2/2 = $ # v2'/2. The line profile

F'(R, v') may be expressed in terms of (h, Y ). Indeed the identity Y = $ # h2 R#2/2 may besolved for R which yields R(h, Y ). The azimuthal velocity v' becomes in turn a function of h, Ydefined by v' = h/R(h, Y ).

Calling F'(h, Y ) & F'(R, v') allows the inversion of Eq. (3.3.1) by an Abel transform (cfappendix B):

f (,, h) =1,2!

#&#

0

%F'

%Y

dY=(#,)# Y

, (3.3.2)

where the partial derivative is taken at constant h. It was assumed here that the distributionF'(R, v') vanishes at the escape velocity. Note that F yields both the symmetric and theantisymmetric parts of the distribution function. The r.h.s of Eq. (3.3.2) is advantageouslyre-expressed in terms of the variables (R, h) given that (for monotonic integration)

;%F'

%Y

<

h

dY =!%F'

%R

"

h

!%R

%Y

"

h

dY =!%F'

%R

"

h

Sgn'!%R

%Y

"

h

(dR . (3.3.3)


0 2 4 6 8

R

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0Y

Figure 3.3.1: the relationship between integration in terms of the e!ective potential Y and R integration.The equation Y = #, has two roots corresponding to the perigee and the apogee of the star, while Y = 0has two roots corresponding to infinity and Re, the inner bound of an orbit with zero energy and angularmomentum h. The sign of the slope of Y (R) gives the sign of the contribution of each branch.

This yields two contributions for Eq. (3.3.2)

f (,, h) =1!

Rp#

Re

(%F'/%R)h dR=

h2/R2 # 2$ (R)# 2,# 1!

"#

Ra

(%F'/%R)h dR=

h2/R2 # 2$ (R)# 2,, (3.3.4)

where Rp(h, ,), and Ra(h, ,) are, respectively, the apogee and perigee of the star with invariantsh and ,, and Re(h) is the inner radius of a star on a “parabolic” (zero energy) orbit withmomentum h. Note that the derivative in Eq. (3.3.4) is performed keeping h = Rv' constant.Equation (3.3.4) yields the unique distribution function compatible with an observed azimuthalvelocity distribution. All macroscopic properties of the flow follow from f . For instance thesurface density reads

' &# '#

f+ (,, h) d vR

(dv' = 2

ve#

0

F' (R, v') dv' , (3.3.5)

while the radial velocity distribution, FR(R, vR), obeys

FR (R, vR) =2R

R(2*#v2R)1/2

#

0

f!

v2R

2+

h2

2R2# $ (R) , h

"dh . (3.3.6)

Both FR and ' may in turn be compared with the corresponding observed quantities.


3.3.2 Disks with given Q profiles

From a theoretical point of view, F' may be chosen to match given constraints such as aspecified potential and temperature profile of a disk. The azimuthal pressure p' is then fixedvia the equation of radial support and by the temperature of the disk defined by Toomre’s Qnumber:

p' = 'V 2c +

% (RpR)%R

, and pR = Q2 !2'3

"2, (3.3.7)

where the circular velocity, the epicyclic frequency and the surface density2 are given by

V 2c = #

' 1R

%$

%R

(, "2 =

1R3

9% (RVc)

2

%R

:

, and ' =12!

'%$

%z

(. (3.3.8)

The surface density, ', and the azimuthal pressure, ''v2'(, may in turn be expressed in terms

of F'(R, v') via

' &# '#

f+ (,, h) d vR

(dv' = 2

ve#

0

F' (R, v') dv' , (3.3.9a)

''v2'( &

# '#f+ (,, h) d vR

(v2' dv' = 2

ve#

0

F' (R, v') v2' dv' , (3.3.9b)

where the circular escape velocity, ve, is equal to,

2$. The function f+ stands for the evencomponent of the distribution function in h. Any function F' satisfying these moment equationscorresponds to a state of equilibrium stable against ring formation when Q > 1. Realistic choicesfor F' are presented in the next sections.

3.3.3 Alternative inversion method

Another intermediate observable, G&(R, ,), the distribution function for the number of starswhich have specific energy ,, may also be parameterised to fix the surface density and the averageenergy density profiles. For external galaxies G& is not directly observed. For our Galaxy, G&

is measurable indirectly – via the distribution of stars with given radial and azimuthal velocitygiven by spectroscopy and proper motions – but only yields the even component in h of thedistribution function. However, the corresponding inversion method still provides a route forthe construction of models with specified temperature profiles.The number of stars which have specific energy , within d, at radius R reads

G& (R, ,) =#

f+ (,, h)vR

dv" =2,2

X#

0

g (,, h2/2)=X # h2/2

d!

h2

2

", (3.3.10)

where X was defined in Eq. (3.2.32) as

X = R2 ($ + ,) . (3.3.11)

Here the auxiliary function g is given by3

g%,, h2/2

&= f+ (,, h) /|h| . (3.3.12)

2 It is assumed here that the field is self-consistent and obeys Eq. (3.2.4); alternatively % may be givenindependently of & and the method described here still applies.

3 Note that only the symmetric component of the distribution function follows from Eq. (3.3.12).


The surface density, and mean energy density, ',(, may be expressed in terms of G&(R, ,) via

' &# #

f+ (,, h) d v' d vR =0#

#*

G& (R, ,) d, , (3.3.13a)

' ',( &# #

f+ (,, h)9v2'

2+

v2R

2# $

:

dv' dvR =0#

#*

G& (R, ,) , d, . (3.3.13b)

The local mean energy density ',( = '#1(pR/2 + p'/2) # $(R) is fixed by the temperature ofthe disk defined by Toomre’s Q number:

',( =12

''V 2

c +% (RpR)%R

+ pR

(# $ (R) given pR = Q2 !

2'3

"2, (3.3.14)

where " and Vc are given by Eq. (3.3.8). The function G(R, ,) may be expressed in terms of,, X via Eq. (3.3.11) and R = R($). Calling G&(R, X) & G&(R, ,) leads to the inversion ofEq. (3.3.10) by an Abel transform:

g%,, h2/2

&=

1,2!

h2/2#

0

%G&

%X

dX=h2/2# X

. (3.3.15)

f+(,, h) follows from g(,, h2/2) via Eq. (3.3.12).Any function G& satisfying the moment constraint Eqs. (3.3.13) yields, via Eq. (3.3.15) and(3.3.14), a maximally rotating disk stable against ring formation.

3.3.4 Implementation: Gaussian line profiles

Gaussian velocity distributions are desirable both as building blocks to fit measured line pro-files and as “realistic” choices for F' in the construction scheme of disks parameterised by theirtemperature. The former point is discussed in the next section.The construction of Gaussian line profiles compatible with a given temperature requires a sup-plementary assumption for the mean azimuthal velocity of the flow, 'v'(, on which the Gaussianshould be centred. In additions to the the two constraints, Eqs. (3.3.9), this puts a third con-straint on F', namely

' 'v'( &# '#

f# (,, h) d vR

(v' dv' = 2

ve#

0

F' (R, v') v' dv' , (3.3.16)

where f# is the odd component in h of the distribution function. Suppose the following functionalform for F

F' (R, v') = S (R) Wn (R, v') exp;

# [v' # v (R)]2

2)2 (R)

<

, (3.3.17)

where the window function Wn is intended to damp F'(R, v') near the escape velocity ve. Apossible expression for Wn is

Wn (R, v') =

DEEEEF

EEEEG

exp

3

M4#v2'

n2?v2

e (R)# v2'

@2

5

N6 |v'| < ve ,

0 elsewhere .

(3.3.18)


Self-consistency requires that the unknown functions (v, ), S) of Eq. (3.3.17) are solved in termsof 'v'( , ', and p' via Eqs. (3.3.9) and Eq. (3.3.16). For practical purposes, the temperaturerange explored in realistic disk models is such that (ve # v)2/2)2 is quite large at all radii; if nis also chosen so that at temperature )

n3 ve 'v'()?v2

e # 'v'(2@

(ve + 'v'(), (3.3.19)

then the tail of the Gaussian function need not be taken explicitly into account. The line profileF then reads directly in terms of 'v'( , ', and p':

F' (R, v') =' (R),2! )'

exp;

# [v' # 'v'(]2

2)2'

<

, (3.3.20)

where )' is the the azimuthal velocity dispersion

)2' = p'/'# 'v'(2 . (3.3.21)

The azimuthal pressure p' follows from the equation of radial support and the temperatureof the disk defined by Toomre’s Q number given by Eq. (3.3.7). The expression of the averageazimuthal velocity, 'v'(, may be taken to be that which leads to an exact asymmetric driftequation:

' 'v'(2 = p' # pR

!"2R2

4V 2c

". (3.3.22)

Equations (3.3.20)-(3.3.22), together with Eqs. (3.3.7) provide a prescription for the Gaussianazimuthal line profile F' which yield the distribution function f(,, h) of a disk at given temper-ature profile via Eq. (3.3.4).

3.3.5 Example: constant temperature Gaussian Kuzmin disks

For the Kuzmin disk, the above prescription for the parameters of the line profile yields

)' =Q

4 (1 + R2)3/4and 'v'( =

R (256# 60 Q2 + 128 R2# 21 Q2 R2 + 16 R4)1/2

4 (1 + R2)3/4 (4 + R2),

where Q is Toomre’s number. From Eq. (3.3.17), F' becomes as a function of h

F [h, R] ==

2/!3Q#1

(4 + R2)exp

3

4#8 (1 + R2)3/2

Q2

;h

R# R (256# 60 Q2 + 128 R2# 21 Q2 R2 + 16 R4)1/2

4 (1 + R2)3/4 (4 + R2)

<25

6 .

Di!erentiating with respect to R gives

% log F

%R=%4 + R2

&

3

MMMM4

8 h2 (2# R2)Q2 R3 (1 + R2)1/4

+4 RP6

Q2 (1 + R2)3/4 (4 + R2)3

# 3 R

2 (1 + R2)7/4# 6 h F6 R

Q2 (1 + R2) (4 + R2)2

5

NNNN6, (3.3.23)

with

F6 =(#1024 + 216 Q2 # 768 R2 + 106 Q2 R2 # 192 R4 + 7 Q2 R4 # 16 R6)

(256# 60 Q2 + 128 R2 # 21 Q2 R2 + 16 R4)1/2, (3.3.24a)

and P6 = #256 + 60 Q2 # 192 R2 + 27 Q2 R2 # 48 R4 # 4 R6 . (3.3.24b)The equations (3.3.23) and (3.3.24) together with Eq. (3.3.9) fully characterise a complete familyof distribution functions parameterised by their temperature via Toomre’s number.


Figure 3.3.2: isocontours of the Gaussian distribution functions for the Kuzmin disk with Q = 1.5(bottom panel), Q = 2 (top panel). A hotter component is apparent at larger momentum for the Q = 2disk where the contours are more widely spaced.

3.3.6 Example : Algebraic Kuzmin Disks

A polynomial in v' for the line profile F' yields analytic distribution functions; the line profilemust be parameterised in order to satisfy given constraints on its first two moments. A possibleparameterisation reads, as a function of h = R v' and R,

Fn (R, h) =;

&+ .!

h

R

"2n+2<;!

h

R

"2

# v2e (R)

<n

, (3.3.25)

where & and . are parameters, and the integer n is a free index which should be varied to reachdisks of di!erent temperature. Putting Eq. (3.3.25) into Eqs. (3.3.9) and solving for & and .leads to a formal solution which is a function of h/R, ve(R) and p'/(' v2

e). Now, for the Kuzmindisks with Q < 2, p'/(' v2

e) varies between Q2/32 and 1/2. Indeed p' and ' v2e read, in terms

of $,

p' ($) =$4 (1# $2)

2 !+

Q2 $6 (#5# 3$2 + 12$4)8 ! (1 + 3$2)2 and ' v2

e = !#1 $4 . (3.3.26)

Requiring the line profile Fn to remain positive in that interval puts constraints the minimumpower n compatible with a given Q. The previous section showed that distribution functionsconstructed from polynomials in h lead to disks with temperatures scaling like the inverse ofthe degree of the polynomial. Indeed, low order polynomials are not likely to produce a good fitto 'v'(. Here too, high values of n are required to reach low values of Q. For instance, Q = 1requires n > 14. For example, the line profile Fn corresponding to n = 6 (

=32/11 < Q < 2)


when written as a function of $ and h reads

F6 ($, h) =/,

2! $13/2

!h2$2

1# $2# 2$

"4

3

MMMM4

1056$5# 4199 h10$10

(1# $2)5 +;

46189 h10$6

2 (1# $2)5 # 1008$<

P

$4%1# $2

&+

Q2$6 (#5# 3$2 + 12$4)4 (1 + 3$2)2

Q

5

NNNN6

where / = 231/5242880. The integrand of Eq. (3.3.4) becomes in terms of $!%F6

%R

"dR

=h2/R2 # 2 ($ + ,)

=(%F6/%$) d$

=h2$2/ (1# $2)# 2 ($ + ,)

, (3.3.27)

Integrating Eq. (3.3.27) along the path described in Fig. 3.3.1 yields the analytic, thoughcomplicated, distribution function corresponding to a disk with Toomre number in the range=

32/11 < Q < 2 (roughly the value measured in the Galaxy).

· • !" ! • ·

3.4 Observational implications

The inversion method described in section 3 may be applied on observational data. The obser-vations may be carried out as follows: consider a disk galaxy seen almost edge on. It should bechosen so that it looks approximately axisymmetric and flat. It may contain a fraction of gasin order to derive the mean potential $ from the observed H II velocity curve (Alternatively,the potential may be derived directly from the kinematics if the asymmetric drift assumptionholds). Putting a slit on its major axis and cross correlating the derived absorption lines withtemplate stellar spectra yields an estimate of the projected velocity distribution which shouldessentially correspond to the azimuthal line profile F'(R, v')4. An estimate of the radial lineprofile FR(R, vR) arises when the slit is placed along the the minor axis. But FR(R, vR), givenby Eq. (3.3.6), reads

FR (R, vR) =2R

R(2*#v2R)1/2

#

0

f!

v2R

2+

h2

2R2# $ (R) , h

"dh , (3.4.1)

while Eq. (3.3.4) gave f expressed as a function of F'(v', R), namely:

f (,, h) =1!

Rp#

Re

(%F'/%R)h dR=

h2/R2 # 2$ (R)# 2,# 1!

"#

Ra

(%F'/%R)h dR=

h2/R2 # 2$ (R)# 2,, (3.4.2)

Therefore, putting Eq. (3.4.2) into Eq. (3.4.1) provides a simple way of predicting the radial lineprofiles if the azimuthal line profiles are given. The surface density ' follows from both FR(vR, R)and F'(v', R) and could also be compared with the photometry of that galaxy. The likelydiscrepancy between the predicted and the residual data may be used to asses the limitationsof the reconstruction scheme. Indeed, the above prescription for the inversion relies on a set ofhypotheses for the nature of the flow (axial symmetry, thin disk approximation etc..) and the

4 in practise, projection e$ects ought to be taken into account

Section 3.4 Observational implications 65

quality of measurements. The following features of an observed galaxy may constrain the scopeof this analysis.

• Thick disk, with random motions perpendicular to the plane of the galaxy.

The measured line profile yields a mean value corresponding to the integrated emissionthrough the width of the disk; as the galaxy is not edge on, the result of the cross correlationof the spectra does not give F' and FR exactly, but rather gives the projection onto theline of sight of the full velocity distributions F',z(v', vz, R) and FR,z(vR, vz, R). However,it is consistent with the thin disk approximation to assume that the motion in the planeis decoupled from that perpendicular to the plane. The observed line profile, Fv(R, v),then corresponds to the convolution of these functions with the component of the velocitydistribution perpendicular to the plane of the galaxy, Fz(R, vz). Formally, for the line profilemeasured along the major axis, this reads

Fv (R, v) =# "

#"Fz [R, v sin (i) + u cos (i)] F' [R, v cos (i)# u sin (i)] du (3.4.3)

where v and u are the velocities along the line of sight, and perpendicular to the line ofsight respectively. A similar expression for the line profile measured along the minor axisinvolving FR follows. Here, the angle i measures the inclination of the plane of the galaxywith respect to the line of sight. At this level of approximation the velocity distributionnormal to the plane is accurately described by a centred Gaussian distribution; its variancefollows roughly from the equation of radial support and is used to deconvolve Eq. (3.4.3)(or practically to convolve the predicted line profile pairs and compare them to the data).Note that a rough estimate of the error induced is )2

z/)2' tan2(i) 0 1% in our Galaxy when

i = 20 degrees.

• Non-axisymmetric halo or disk, with dust and absorption.

The departure from axisymmetry reduces the number of invariants – angular momentumis not conserved, which invalidates this method. Some fraction of the population of starsmay then follow chaotic orbits which cannot be characterised by a distribution function.Some real galaxies either present intrinsic non axisymmetric features such as bars, spiralor lopsided arms. Triaxial halos may induce warps within the disk, hence compromis-ing the evaluation of the kinematics. Non-axisymmetrically distributed dust or molecularclouds may also a!ect measurement of the light from the stellar disk. On the positive side,non-axisymmetric features often appear as small perturbations which may only contributeweakly to the overall mass distribution. Under such circumstances, an estimate of the un-derlying distribution function for the axisymmetric component of the disk may be extractedfrom the data. Sets of initial conditions for an N-body simulation may also be extractedfrom the distribution function. The time evolution of the code would yield constraints onthe stability of the observed galaxy. The above analysis could be carried out on an isolatedS0 galaxy and on an Sb galaxy showing a grand design spiral pattern (or alternatively onan Sa presenting an apparent lopsided HI component to isolate plausible di!erent instabil-ity modes). The relative properties of the two galaxies and their supposedly distinct fateswhen evolved forward in time should provide an unprecedented link between the theory ofgalactic disks and the detailed observation of the kinematics of these objects.

• Data quality.

The method described in section 3 involves a deconvolution of the measured line profileswhich yields the velocity distributions which, in turns, are related by Abel transforms. Bothtransformations are very sensitive to the unavoidable noise in the data. Sources of noise arepoor seeing, photon noise, detector noise, thermal and mechanical drift of the spectrograph,


poor telescope tracking, etc... It is therefore desirable to construct from Eqs. (3.4.2), (3.4.1)and (3.3.17) a set of pairs of Gaussian velocity distributions (F', FR) on which to projectthe observed quantities since this will help to assess the errors induced by this noise. Indeed,Eq. (3.4.2) is linear; therefore any superposition of Gaussian profiles corresponding to a goodfit of the observed line profile will lead to a distribution function expressed in terms of sumsof solutions of Eq. (3.4.2). These Gaussians may in turn be identified with populations atdi!erent temperatures – say corresponding to a young component and an older populationof stars. Formally this translates as

F' (R, v') =$

i

Si (R) exp;

# [v' # vi (R)]2

2)2i (R)

<

, (3.4.4)

where the functions Si(R),)i(R) and vi(R) should be fitted to the observed azimuthal lineprofiles. Each component in Eq. (3.4.4) may then be identified with those in Eq. (3.3.20),yielding the temperature profile of that population. This approach should yield the bestcompromise between fitting the radial and the azimuthal profile while avoiding the decon-volution of noisy data. It also provides directly a least square estimate of the errors.

The observations described in this section may be achieved with today’s technology, asillustrated in Fig. 3.4.1 and Fig. 3.5.1.

Figure 3.4.1: Contour plot of 300 s Rc band CCD image of NGC 7332 (Fisher et al. 1994 [35]). Lowestcontour level is 22 Rc mag arcsec#2 with successive contours at 1 mag intervals. Orientation of the sixslit illustrated in Fig. 3.5.1 positions are shown.

· • !" ! • ·

Section 3.5 Conclusions 67

3.5 Conclusions

The ansatz presented in section 2 and 3 yield direct and general methods for the constructionof distribution functions compatible with a given surface density for the purpose of theoreticalmodeling. These distribution functions describe stable models with realistic velocity distribu-tions for power law disks, and also the Isochrone and the Kuzmin disks. Simple general inversionformulae to construct distribution functions for flat systems whose surface density and Toomre’sQ number profiles are given in section 4 and illustrated. The purpose of these functions isto provide plausible galactic models and assess their critical stability with respect to globalnon-axisymmetric perturbations. The inversion is carried out for a given azimuthal velocitydistribution (or a given specific energy distribution) which may either be observed or chosenaccordingly. When the azimuthal velocity distribution is measured from data, only a subset ofthe observationally available line profiles, namely the line profiles measured on the major axis,is required to re-derive the complete kinematics from which the line profile measured along theminor axis is predicted. This prediction may then be compared with the observed minor axisline profile.

· • !" ! • ·


Figure 3.5.1: NGC 7332 stellar velocity dispersions ) (upper panel) and derived velocities (lower panel)plotted against radius in arcsseconds along the position angles indicated.

B

B.1 Distribution functions for counter-rotating relativistic disks

In the next chapters, static solutions of Einstein’s equations corresponding to flat disks made oftwo equal counter rotating streams of stars are presented. The pressures and energy densitiesof the disks are those induced by a known vacuum field; these may therefore not be chosenindependently. The inversion method presented in the previous section – which produces dis-tribution function for a given surface density and pressure profile – deserves a generalisation tothe relativistic regime for these objects.

The relativistic flow is characterised by its stress energy tensor, T ()%), which, in Vierbeincomponents, reads

T ()%) =# #

f (,, h) P ()) P (%)

P (t)dP (R)dP (') . (B.1.1)

From the geodesic equation it follows that

PµPµ = #1 4 P(R) =1,/tt

+,2 # h2 /tt//'' # gtt

,1/2, (B.1.2)

where the line element in the disk is taken to be

ds2 = #/tt dt2 + /RR dR2 + /'' d*2 . (B.1.3)

Di!erentiating Eq. (B.1.2) implies

dP (R)

P (t)=

d,

[,2 # h2 /tt//'' # /tt]1/2

, while dP (') =dh,/''

, (B.1.4)

where the di!erentiation is done at constant R and h , and R and , respectively. Combinationsof Eqs. (B.1.1) lead to

) = /''1/2

?T (tt) # T ('') # T (RR)

@=# #

f (,, h) dh d,>,2 # h2/tt//'' # /tt

, (B.1.5a)

& = /''3/2 T ('') =

# #h2 f (,, h) dhd,>,2 # h2/tt//'' # /tt

. (B.1.5b)

Note that Eq. (B.1.5b) is precisely Tolmann’s formula corresponding to the total gravitatingmass of the disk. Introducing

R2 = /''//tt , e =12%,2 # 1

&, % =

12

(1# /tt) and f (e, h) =f (,, h),

, (B.1.6)

70 Appendix B

gives, for Eqs. (B.1.5):

) =# #

f (e, h) dhde

[2 (e + $)# h2/R2]1/2and & =

# #h2 f (e, h) dhde

[2 (e + $)# h2/R2]1/2. (B.1.7)

Eq. (B.1.7) is formally identical to Eqs. (3.3.9), replacing (),&, f,R, e,%) by (R', R3 p',f, R, ,,$). In fact (),&, f,R, e,%) tends to (R', R3 p', f, R, ,,$) in the classical regime.Introducing the intermediate functions F (h,R), which is chosen so that its moments satisfyEqs. (B.1.5) and following the steps of Eqs. (3.3.1)-(3.3.4) yields

f (e, h) =1!

Rp#

Re

?%F/%R

@

hdR

=h2/R2 # 2%# 2e

# 1!

"#

Ra

?%F/%R

@

hdR

=h2/R2 # 2%# 2e

, (B.1.8)

where Rp(h, e), and Ra(h, e) are respectively the apogee and perigee of the star with invariants(h, e), and Re(h) is the inner radius of a star on a “parabolic” (zero energy) orbit with momentumh. Eq. (B.1.8) provides direct and systematic means to construct families of distribution functionfor counter-rotating disks characterised by their stress-energy tensor in the plane. In chapter 4,essentially all static solutions of Einstein equations describing these disks are derived. The aboveinversion method is also generalised to disks with mean rotation, where the dragging of inertialframes requires to fix three moments of the velocity distribution to account for the given energydensity, pressure law, and rotation law.

B.2 Abel transform

Suppose that a function g(t) obeys the integral equation

x#

0

g (t) dt

(x# t))= f (x) , where 0 < & < 1 , (B.2.1)

then g may be expressed formally in terms of f by

g (t) =sin (!&)!

d

dt

3

4t#

0

f (x) dx

(t# x)1#)

5

6 =sin (!&)!

3

4t#

0

df

dx

dx

(t# x)1#) +f (0)t1#)

5

6 , (B.2.2)

This classical result is demonstrated in Binney & Tremaine [14] while substituting Eq. (B.2.1) inthe first part of Eq. (B.2.2) and integrating by part having interchanged the order of integrationand given the identity

!

sin (!&)=

1#

0

du

u) (1# u)1#) . (B.2.3)

The transformations Eqs. (B.2.1) and (B.2.2) are called Abel transforms and are usedthroughout chapter 3 and 5, and in the above section.

· • !" ! • ·

4

New solutionsof Einstein’s equations:

Relativistic disks

4.1 Introduction

Thin relativistic disks are sources of axisymmetric metrics with gradient discontinuitiesacross the disk. Recent progress in Newtonian potential theory enables one to generate thepotential-density pairs of classical axisymmetric disks in closed forms. These results are trans-posed to construct complete sequences of new solutions of Einstein’s equations which describesuper massive disks. To avoid the general-relativistic complications caused by inertial framedragging (gravo-magnetic forces), the relativistic counterparts of Newtonian disks must havevanishing net angular momentum. Following Morgan & Morgan(1969) [90], one may considerdisks to be made of two equal streams of collisionless particles (stars) that circulate in oppositedirections around the barycentre. Those disks are described as counter-rotating. Counter ro-tating disks have recently become more attractive since the observations of NGC 4550 by V.CRubin et al. (1992) [98], Rix et al. (1992) [97] and NGC 7217 by Merrifield & Kuijken (1993) [85]where the kinematics suggest the existence of such streams.

4.1.1 Background

In Newtonian theory, once a solution for the vacuum field of Laplace’s equation is known, itis straightforward to build solutions to Poisson’s equation for disks by using the method ofmirror images. Indeed Kuzmin constructed his potential by considering a point mass placed ata distance b below the centre R = 0 of the plane z = 0 and it gravitational field above thatplane. By reflecting this potential in the plane z = 0, he obtained a symmetrical solution ofPoisson’s equation everywhere, which has a discontinuous normal derivative in the plane. Thisjump he identified with the surface density of the disk.

Indeed, the potential of the Kuzmin disks in cylindrical coordinates (R, z) is given by

4 = #M/rb, where r2b = R2 + (|z|+ b)2 , (4.1.1)

and M is the total mass of the disk. The corresponding classical surface density of mass is

'0 (R) = (4!)#1'%4

%z

(0+

0#= (2!)#1 Mb/

%R2 + b2& 3

2 . (4.1.2)

72 New solutions of Einstein’s equations: Relativistic disks Chapter 4

Figure 4.1.1: geometrical construction of solutions of Poisson’s equation. Consider a point mass placedat a distance b below the centre R = 0 of the plane z = 0 and its gravitational field above that plane.By reflecting this potential in the plane z = 0, a symmetrical solution of Poisson’s equation everywhereis constructed which has a discontinuous normal derivative in the plane.

Evans & de Zeeuw [33] pointed out that the linearity of Poisson’s equation allows one togenerate general potential density pairs for disks in a closed and compact form via fictitious linedensity distributions of mass along the negative z-axis. The potential is then given by

4 = ## b2

b1

W (b) dbAR2 + (|z|+ b)2

B 12

, (4.1.3)

where the limits of integration are determined by the interval < b1, b2 > along the negative z-axis, in which the line density W (b) – or “the weight function” – is non-zero. The correspondingsurface density of mass on the disk becomes

'0 (R) = (2!)#1# b2

b1

W (b) b

(R2 + b2)32db , (4.1.4)

The total mass is* b2

b1W (b)db.

For Kuzmin’s disk, W is a delta function, and for the family of Kuzmin-Toomre’s disks it isgiven by combinations of derivatives of the delta function. In Kuzmin’s picture this correspondsto placing multipoles at a distance b below the centre R = 0. In the case of the Kalnajs-Mestelfamily (Mestel 1963 [87], Kalnajs 1976 [58]),

W (b) = M b#m , for b + b1, (4.1.5)

M = const, and m = 0, 1, ... in the intervals < b1, b2 = * >. The resulting potential(Eq. (4.1.3)) diverges logarithmically at infinity for m = 0, and the total mass of the disk

Section 4.2 General characteristics of counter-rotating disks 73

is infinite for both m = 0 and m = 1. However, “truncated” Kalnajs-Mestel disks may beconstructed by considering line densities (weight functions) of the form Eq. (4.1.5) only in theintervals b1 < b2 < * yielding well-behaved potentials and masses even when m = 0 or 11.

In the present chapter, static axisymmetric metrics which are solutions of Einstein’s equa-tions are constructed representing the relativistic generalisation of the Kuzmin-Toomre and theKalnajs-Mestel families of disks. These solutions provide physical sources (interior solutions)for static axisymmetric gravitational fields. The general properties of counter-rotating disks arepresented in section 2. The Kalnajs-Mestel disks are analysed in Section 3. The generating func-tion for the second constituent that determines the complete relativistic metric for any memberof the Kalnajs-Mestel family is constructed. The explicit forms of the metrics are written downfor the truncated Kalnajs-Mestel disks with m = 0, m = 1 and for the relativistic version ofthe Isochrone disks (m = 2). Section 4 gives solutions representing relativistic Kuzmin-Toomredisks. In Section 5, the physical properties of the disks, their velocity curves, the graphs of thespecific angular momentum, and both surface mass-energy and rest-mass densities are discussedfor typical cases.

· • !" ! • ·

4.2 General characteristics of counter-rotating disks

Weyl (1917) [111] has shown that axially symmetric solutions of Laplace’s equation in flat spacemay be used to construct static solutions of Einstein’s equations (cf. Synge (1971) [100], foran extensive exposition). If the energy-momentum tensor of a source satisfies the conditionT R

R + T zz = 0, which, hereafter, will be assumed, then the metric in the ‘cylindrical’ coordinates

(t, R, z, *) can be written in the form

ds2 = #e2+dt2 + e25#2+%dR2 + dz2

&+ R2e#2+d*2 , (4.2.1)

where, as a consequence of Einstein’s equations, the functions 4(R, z) and 5(R, z) satisfy

224 = 4! e25#2+?T '' # T t

t

@, (4.2.2)

Here 22 is the standard flat space Laplace operator in cylindrical coordinates. Hence Eq. (4.2.2)implies that outside the matter, 4 satisfies the flat-space Laplace equation. The potential ofany classical disk will therefore serve as a possible solution for 4. Thus to each classical diskthere corresponds a relativistic disk for which 4 has exactly the same mathematical form whenexpressed in Weyl’s coordinates. When the parameters are such that the potential is much lessthan c2, the physical properties of the classical and relativistic disks will coincide. However,when the parameters are such that the disks are truly relativistic, this correspondence generatesone of many possible relativistic generalisations of the classical disk. Others would be obtainedby demanding that the circular velocities should have the same functional form when expressedin terms of some suitably defined radial coordinate (such as circumferential radius) or that theangular velocity or surface density profiles should coincide. However this generalisation leadsto exactly tractable mathematics. The matter tensor of these objects are confined to an axiallysymmetric surface which divides space into two reflectionally symmetric parts. The metricfunctions are finite everywhere; only their second derivatives have the required singularity on

1 In fact, in Einstein’s theory, the vacuum solution generated by the potential of a rod of length b2 " b1 withconstant line density equal to 1/2 is the standard Schwarzschild metric with mass M = 1

2 (b2 " b1) (Kramer et. al.1980) [63]; corresponding disks are thus of special interest in general relativity.


the disk.The remaining Einstein equations yield

%5

%R= R

9!%4

%R

"2

#!%4

%z

"2:

, (4.2.3)

%5

%z= 2R

%4

%R

%4

%z, (4.2.4)

and%25

%R2+%25

%z2#224 +

!%4

%R

"2

+!%4

%z

"2

= 4! e25#2+?T t

t + T''

@. (4.2.5)

The Laplace equation (4.2.2) is the integrability condition for Eq. (4.2.3) and (4.2.4). WhenR = 0 and |24| has no singularities above z = 0, then 5 is constant along the axis by Eq. (4.2.4).Hence the condition that 5 = 0 at infinity ensures that 5 = 0 on the axis. Furthermore Eq. (4.2.3)then implies that 5 is 0(R2) close to the axis which ensures elementary flatness of the metric onthe axis.For infinitesimally thin disks the stress energy tensor components along z = 0 are of the formS(R)+(z). Integration through the disk gives surface stress tensor

-'' =#

T '' e5#+dz = ,V 2 , (4.2.6)

- tt =

#T t

t e5#+dz = #, , (4.2.7)

where V is the velocity of the stream as measured by a static observer (V = ,g""d#/d-LOC)

and, = )/

%1# V 2

& 12 = 2,0/

%1# V 2

&; (4.2.8)

here , and ) are the surface densities of energy and rest mass measured by the static observer,,0 is the surface proper rest mass density of one stream measured by a co-moving observer.Integrating Eq. (4.2.1) - (4.2.5) across z = 0 yields

'%4

%z

(0+

0#= 4! e

5#+?-'' # - t

t

@& 4! e

5#+,%1 + V 2

&, (4.2.9)

9%5

%z

:0+

0#

= 2R%4

%R

'%4

%z

(0+

0#, (4.2.10)

andR%4

%R=

V 2

1 + V 2. (4.2.11)

The disk will only be physically realizable in terms of counter-rotating streams of particles if themass-energy surface density , is positive and if the dominant energy condition is satisfied; fromEq. (4.2.6) and Eq. (4.2.7), this reduces to ,(1 # V 2) + 0, i.e. |V | % 1. From Eq. (4.2.11) thisimplies the condition R %4/%R % 1/2 on z = 0. Not every mathematical potential on the diskwill give a realistic counter-rotating disk; the potential must not be too concentrated. In practisethis condition gives a limit on how compact one can make the mass distribution correspondingto a given form of disk potential of which total mass is known. From Eq. (4.2.9) the conditionthat , is not negative is satisfied provided that %4/%z at z = 0+ is not negative at any R.This condition is automatically fulfilled for all the disks considered here because they come frompositive mass density Newtonian disk solutions. Indeed, in general this condition is equivalentto the condition that the corresponding Newtonian disk have a non-negative surface density. A

Section 4.3 Relativistic Kalnajs-Mestel disks 75

su#cient but not necessary condition for that is that W (b) be non-negative. In summary, allthe disks considered here will have physical sources provided V < 1; for certain types of diskthis places a limit on their compactness.The magnitude of the velocity of counter-rotating streams, as measured by the local staticobservers, follows from Eq. (4.2.11):

V ='R%4

%R/!1# R

%4

%R

"(1/2

. (4.2.12)

The angular velocity of counter-rotation, as measured at infinity, $ = d#/dt, is

$ = V e2+/R , (4.2.13)

and the specific angular momentum of a particle rotating at radius R (defined as h = p"/µ0 =g""d#/d- where µ0 and - are the particle’s rest mass and proper time) is

h = RV e#+/%1# V 2

& 12 , (4.2.14)

where V is given by Eq. (4.2.12). The surface mass-energy density, ,, and the surface rest massdensity, ), follow from Eqs. (4.2.8), (4.2.9) and (4.2.12) in the form

, (R) = '0 e+#5 (1# R%4/%R) , and (4.2.15)

) (R) = '0 e+#5 [(1# 2R%4/%R) (1# r%4/%R)]

12 , (4.2.16)

where '0(R) = (4!)#1[%4/%z]0+0# is the classical density. Finally, for static spacetimes, it is of

interest to know the redshift z of a photon emitted by an atom at rest and received by a staticobserver at infinity. This redshift factor reads

1 + z = (|g00|)#12 = e#+ . (4.2.17)

· • !" ! • ·

4.3 Relativistic Kalnajs-Mestel disks

The classical potentials of the Kalnajs-Mestel disks are given by Eq. (4.1.3), where the weightfunctions are chosen as inverse powers of b, i.e. , according to Eq. (4.1.5).

4.3.1 Derivation

Substituting Eq. (4.1.5) into Eq. (4.1.3), yields the potential of the mth order Kalnajs-Mesteldisks in the form

4m = #M# b2

b1

db

bmAR2 + (|z|+ b)2

B1/2. (4.3.1)

Introducing the new variables u and v by

u = R#1

R|z| + b +

AR2 + (b + |z|)2

B 12S

, and v = 2|z|/R , (4.3.2)


so that2R

(b + |z|) = u# 1u

, (4.3.3)

Eq. (4.3.1) takes the form

4m = #M! 2

R

"m # ub

ua

1(u# 1/u# v)m

du

u, (4.3.4)

where the integration limits are determined by Eq. (4.3.2),

ua = R#1

R|z| + b1 +

AR2 + (b1 + |z|)2

B1/2S

, (4.3.5)

and a similar expression for ub involving b2. Writing

Im =# ub

ua

1(u # 1/u# v)m

du

u, (4.3.6)

leads to the recursion relationdIm

dv= m Im+1 , (4.3.7)

which implies

Im =1

(m# 1)!

!d

dv

"m#1

I1 . (4.3.8)

Finally, introducing w by

w = (r + |z|)/R , so that v = w # 1w

, (4.3.9)

yields an easy integral for I1

I1 =# ub

ua

du

(u # w) (u + 1/w)=

w

w2 + 1

'lnWWW

u# w

u + (1/w)

WWW(ub

ua

. (4.3.10)

In terms of w the recurrence relation Eq. (4.3.8) reads

Im =1

(m# 1)!

!w2

w2 + 1d

dw

"m#1

I1 . (4.3.11)

The classical potential 4 of the mth-order truncated Kalnajs-Mestel disks Eq. (4.3.4) thereforereads

4m = #M! 2

R

"m

Im , (4.3.12)

where Im is given by Eq. (4.3.8) or Eq. (4.3.11), and the generating function I1 is given byEq. (4.3.10), with u1, u2 and w given by Eq. (4.3.5) and (4.3.9) in terms of R and z.Integrating Eq. (4.1.4) by parts, leads to a classical surface density for the mth order truncatedKalnajs-Mestel disks in the form

'm (R) =M4!

DF

G

9

# 1bm (R2 + b2)1/2

:b2

b1

#m! 2

R

"m+1

Im+1 (w = 1)

JK

L . (4.3.13)

These expressions for 4 and ' in the limit b1 = a, b2 = * are in exact agreement with thosegiven by Evans & de Zeeuw (1992) [33] for the first four members of the Kalnajs-Mestel family.

Section 4.3 Relativistic Kalnajs-Mestel disks 77

The function 5 is then required in order to obtain the complete relativistic metrics. It may befound by integrating Eq. (4.4.6) with functions 4 given by Eq. (4.3.12). Because Eq. (4.4.6) andEq. (4.2.10) are quadratic in 4, the formal solution for 5 should read

5 =# #

W (b1)W (b2)Z (b1, b2, r1, r2) db1db2 (4.3.14)

where Z was found by Bicak, Lynden-Bell & Katz(1993) [11] to be

Z = # 12r1r2

P

1#!

r2 # r1

b2 # b1

"2Q

+ 0 . (4.3.15)

Here b1 and b2 are the distances of two points below the disk’s centre and r1 and r2 are thedistances measured from these points to a point above the disk. Below the disk, Z is givenby reflection with respect to the plane z = 0. On the z-axis ri = bi + |z| and near thereri = bi + |z| + 1

2R2/(bi + |z|). Thus, near the axis, Z = #1

4R2(b1 + |z|)#2(b2 + |z|)#2, which

clearly makes 5 of order R2 near the axis. This ensures regularity on and near the axis for allsuch solutions. Eq. (4.3.15) may be rearranged using the variables ui (cf. Eq. (4.3.2)) as

5 = #2# ub

ua

# ub

ua

W1W2

(u1u2 + 1)2du1du2 (4.3.16)

whereWi = W

'12R!

ui #1ui

"# |z|

(, i = 1, 2 . (4.3.17)

The limits of integration are given by Eq. (4.3.5). In the Kalnajs-Mestel disks, the function Wis just proportional to the inverse power of its argument according to Eq. (4.1.5). Hence, for themth order truncated Kalnajs-Mestel disks 5 reads

5m = #2M2

! 2R

"2m

Jm (ua, ub; v1 = v, v2 = v) , (4.3.18)

where the function Jm of four variables is defined by

Jm (ua, ub; v1, v2) =# ub

ua

# ub

ua

um1

(u21 # v1u1 # 1)m (u1u2 + 1)#2 du1 du2

(u22 # v2u2 # 1)m (4.3.19)

Recall that v1 = v2 = v = 2|z|/R; so 5m are indeed functions of R and |z|. Inspecting Eq. (4.3.19)leads immediately to the recursion relation

%2Jm

%v1%v2= m2Jm+1 , (4.3.20)

which implies

Jm =1

[(m# 2)!]2!

%2

%v1%v2

"m#1

J1 . (4.3.21)

Hence, all 5m’s will be determined once the generating function,

J1 (ua, ub; v1, v2) =# ub

ua

# ub

ua

u1u2 du1du2

(u21 # v1u1 # 1) (u2

2 # v2u2 # 1) (u1u2 + 1)2, (4.3.22)

is known. Since the calculation of J1 is lengthy and not straightforward – indeed, it is surprisingthat it can be expressed in terms of elementary functions – its details are given in the Appendix.The final form of J1 for the general, truncated Kalnajs-Mestel disks is given in Eqs. (C.15)-(C.16).


4.3.2 Examples

• Kalnajs-Mestel m = 0 truncated disk

All the recursion formula above are valid for m + 1. The metric for the m = 0 disk musttherefore be directly derived from the original expressions: Eq. (4.3.1) for 4, and Eq. (4.3.18)for 5. The resulting family of disks was first derived by Bicak, Lynden-Bell & Katz(1993) [11]and called the generalised Schwarzschild disks since they are generated by the potential of arod of length b2 # b1 with constant line density M; the disk with M = 1/2 corresponds to thestandard Schwarzschild metric with mass M = (b2 # b1)/2.Consider here truncated m = 0 Kalnajs-Mestel disks which have well-behaved potentials andfinite masses. The metric functions are

40 = N lnWWW

r1 + |z|+ b1

r2 + |z|+ b2

WWW, and (4.3.23a)

50 = 2N2 lnWWW

(r1 + r2)2 # (b2 # b1)

2

4r1r2

WWW , (4.3.23b)

wherer2

i = R2 + (|z|+ bi)2 , N = M/ (b2 # b1) . (4.3.24)

• Kalnajs-Mestel m = 1 disk

When m = 1, the potential 4 is well-defined even in the un-truncated case. Let b1 = a andb2 )*. The final expressions for 4 and 5 are simpler in terms of r, µ which are related to R, zby

R = r sin 6 , z = r cos 6 , | cos 6| = µ . (4.3.25)

Introducing the constant K1, the dimensionless radial coordinate r, and the function P by

K1 = 4! a'0 , r = r/a , P =+r2 + 2µr + 1

, 12 , (4.3.26)

where '0 = M!#1 a#2/4 is the classical central surface density of the disks (cf. Eq. (4.3.13)),yields

41 = #K1

rln'r + µ + P

1 + µ

(, and (4.3.27a)

51 = K21

R 1r2

(1 + µr # P ) + ln'1 + µr + P

P 2

(+

1r2

[µ + P (r # µ)] ln'r + µ + P

1 + µ

(

+µ2 # 1

2r2

'ln!

r + µ + P

1 + µ

"(2

# ln 2Q

. (4.3.27b)

The potential 41 and 51 were derived by direct integration and also independently by usingEq. (4.3.18) with the generating function Eq. (C.18), (C.19) and substitutions. Note that 51 ) 0as r ) * and 51 = 0 on the z-axis (µ = 1). Hence, the spacetime is asymptotically flat andregular on the z-axis. (The condition 5 = 0 for R = 0 guarantees that the metric Eq. (4.2.1) isregular on the axis – cf. e.g. Synge 1971).

• Kalnajs-Mestel m = 1 truncated disk

Although the spacetimes of the un-truncated Kalnajs-Mestel disks with m = 1 have rea-sonable properties, the total mass of each disk is infinite, as can easily be seen by integratingW (b) or '0 given in Eq. (4.3.13). This defect is avoided with truncated disks, as it was done

Section 4.4 Relativistic Kuzmin-Toomre disks 79

for the m = 0 case. Let 0 < b1 < b2 < * be two given constants. Starting from the generalexpressions Eq. (4.3.1), Eq. (4.3.18), the potential 4$1 reads

4$1 = # M

r ln (b2/b1)ln'!

b2

b1

"r + b1µ + P1

r + b2µ + P2

(, (4.3.28)

where r and µ are given by Eq. (4.3.25), and

Pi =%r2 + 2birµ + b2

i

&1/2, (4.3.29)

and M = M ln(b2/b1) is the total mass of a disk. The second metric function, 5$

1, is slightlymore complicated:

5$1 = #

' 2M

ln (b2/b1)

(2 2R2

J $1 , (4.3.30)

where J $1 is given in Appendix (Eq. (C.20) ).

• Kalnajs-Mestel m = 2 Isochrone disk

In contrast to the disks with m = 0 and 1, the Isochrone disks have finite mass even in thestandard, un-truncated form. In terms of the same variables as Eqs. (4.3.25),and Eq. (4.3.26),the metric functions read

42 = #M

a

'r + 2µ

r (1 + P )# µ

r2ln!

r + µ + P

1 + µ

"(, (4.3.31)

and52 =

!M

a

"2 Rµ

r+

14r2

%1# 3µ2 # 4P

&+

µ

2r3

%#7 + 9µ2 + 3P

&

+1

2r4

%#4 + 9µ2

&(1# P ) + ln

!1 + µr + P

P 2

"

+1

2r4

+µ%#7 + 9µ2

&(1# P ) +

%2r2 # 2µr + 3µ2 # 1

&rP

,ln!

r + µ + P

1 + µ

"

+1

4r4

%1# 10µ2 + 9µ4

& 'ln!

r + µ + P

1 + µ

"(2

# ln 2Q

.

(4.3.32)

The potential 42, after changing variables (r, µ) back to (r, z), as stated in Eq. (4.3.26) is equalto the expression given first by Evans & de Zeeuw (1992) [33]. Again, note that 52 ) 0 asr ) * and 52 = 0 on the z-axis so that the axis is regular and the spacetime with the metricEq. (4.2.1), in which 4 and 5 are given by Eq. (4.3.31) and (4.3.32), is asymptotically flat. Directcalculations confirm that the functions 42 and 52 satisfy the field equations Eq. (4.2.3)-(4.2.5).

· • !" ! • ·

4.4 Relativistic Kuzmin-Toomre disks

The general nth order classical Kuzmin-Toomre disks are described by the potential (Toomre1963 [107], Nagai & Miyamoto 1976, Evans & de Zeeuw 1992)

4n+3/2 = # M

(2n# 1)!!

n$

k=0

(2n# k)!bk

2n#k (n# k)!1

rk+1b

Pk (| cos6|) , (4.4.1)

where Pk is the kth Legendre polynomial, | cos 6| = (b + |z|)/rb. The simple potential of theKuzmin disk is obtained while putting n = 0 in Eq. (4.1.1).


4.4.1 Derivation

Equation (4.4.1) can be re-arranged in the form:

4n+ 32

= #n$

k=0

(#1)k

k!Uk,n

%k

%bk

DEF

EG1

AR2 + (|z| + b)2

B1/2

JEK

EL, (4.4.2)

whereUk,n =

(2n# k)!2n#k (2n# 1)!! (n# k)!

Mbk . (4.4.3)

The potential of the nth order Kuzmin-Toomre disks at z > 0 therefore arises in the Kuzminpicture from the sum of the fields of elementary multipoles of the order k, 0 % k % n, placed at adistance b below the centre of a plane z = 0, the multipole moments being given by Eq. (4.4.3).Correspondingly, the weight function W entering Eqs. (4.1.3) and (4.1.4) becomes

W (b, b$) =n$

k=0

(#1)k

k!Uk,n+

(k) (b# b$) , (4.4.4)

where +(k) denotes the kth derivative of a +-function and the integration in Eq. (4.1.3) is over b$.From Eq. (4.4.3), it is easy to prove the recurrence relation

4n+1+3/2 = 4n+3/2 #b

2n + 1%

%b4n+3/2. (4.4.5)

The function 5 is to be extracted from Eqs. (4.2.3) and (4.2.4) to determine the complete metricoutside the matter. For this purpose, the two equations can be combined to give

D5 = R (D4)2 , (4.4.6)

whereD = %/%R# i %/%z (4.4.7)

is an operator invariant under the translations z ) z + b. Introducing spherical coordinateswith the origin anywhere on the z-axis and putting

D, = %/%6# ir%/%r, (4.4.8)

gives, for Eq. (4.4.6),D,5 = sin 6ei, (D,4)

2 . (4.4.9)By centring the spherical coordinates at the distance b below the disk, Eq. (4.4.9) takes the form

!%

%6# irb

%

%rb

"5 = sin 6 ei,

9!%4

%6

"2

# 2irb%4

%6

%4

%r b# r2

b

!%4

%r b

"2:

, (4.4.10)

where r2b = R2 +(|z|+ b)2 and sin 6 = R/rb. Substituting 4 given by Eq. (4.4.1) into Eq. (4.4.10)

and integrating yields after some manipulation

5 =# sin2 6

[(2n# 1)!!]2

3

4n$

k,!=0

Bk,!,nbk+!Pk,! (6)1

rk+!+2b

5

6 , (4.4.11)

where

Bk,!,n =(2n# k)! (2n# 0)!

22n#l#! (n# k)! (n# 0)! (k + 0+ 2),

Pk,! (6) = (k + 1) (0+ 1)PkP! + 2 (k + 1) | cos 6|PkP$! # sin2 6P $

kP$! ,

(4.4.12)

and P $k = (d/d| cos6|)Pk.

Section 4.5 Discussion 81

4.4.2 Examples

• Kuzmin-Toomre n = 0 disk

For the simple Kuzmin disk, 4 is given by Eq. (4.1.1) while 5 follows from Eq. (4.4.12) orfrom direct integration of the real component of Eq. (4.4.10):

43/2 = #M/rb, (4.4.13)

and53/2 = #1

2M2R2/r4

b . (4.4.14)

The 6 integration gives the same result since Eqs. (4.2.3) and (4.2.4) are consistent as a conse-quence of the condition 224 = 0. The metric Eq. (4.2.1) with 4 and 5 given by Eq. (4.1.1) andEq. (4.4.14) represents the spacetime of the Kuzmin-Curzon disks2.

• Kuzmin-Toomre n = 1 disk

The explicit form of the metric functions in the original cylindrical coordinates is the fol-lowing:

45/2 = #M (R2 + z2 + 3b|z|+ 2b2)AR2 + (b + |z|)2

B3/2, (4.4.15)

and5

5/2 = # M2R2

2AR2 + (b + |z|)2

B4

'%R2 + z2 + 4b|z|+ 3b2

&2 # 12b2R2

(. (4.4.16)

Substituting these into Eq. (4.2.1) yields the explicit form of a new metric. Notice that for thesetwo solutions 5 $ R2 near R = 0 which ensures that the space is regular with no cusp at R = 0.

· • !" ! • ·

4.5 Discussion

The disk models may be compared by exhibiting the physical quantities introduced insection 2 as functions of the “circumferential” radius3 TR = Re#+(R,0).The velocity curves for two families of the truncated m = 0 Kalnajs-Mestel disks are shownin Fig. 4.5.1. The Schwarzschild disks with N = 1/2 (full lines), and the disks with N = 1/5(dashed lines). The velocities (measured in units of the velocity of light) are determined byEq. (4.2.11); 4 is given by Eq. (4.3.24). Here b1 = a and b2 = b. Note that with a decreasing(i.e. with the upper end of the rod approaching R = z = 0), the velocities increase close toR = 0. For N % 1/2, a may approach zero and the value of the maximum velocity is given by

Vmax = [N/ (1# N )]1/2 , N % 12.

In Fig. 4.5.1 the parameter a is fixed, but the e!ect of increasing b, i.e. of increasing the lengthof the rod or, correspondingly, the mass M (cf. Eq. (4.3.24)) at fixed N is presented. All thevelocity curves rapidly increase at small R but with b increasing they remain flat over largerand larger intervals of R. By sending the lower end of the rod to infinity (b)*, N fixed), one

2 Bicak, Lynden-Bell & Katz [11] used this terminology since Eq. (4.1.1) and (4.4.14), with rb being replacedby a standard radial coordinate, give Curzon’s vacuum solution of Einstein’s equations (Kramer et. al. 1980) [63].

3 The physical circumference of the circle R = const is given by 2+ TR.


Figure 4.5.1: the velocity curves for two families of the truncated m = 0 relativistic Kalnajs-Mesteldisks: N = 1/2 (full lines) and N = 1/5 (dashed lines). The numbers on the individual curves denote thevalues of the dimensionless parameter b/a. As this parameter increases, the velocity curves become flat.When N = 1/2, the velocities approach the velocity of light, when N = 1/5, one half of this velocity.

arrives at the infinite mass, constant velocity disks of Lynden-Bell & Pineault (1978) [79,80].Note that the curves in Fig. 4.5.1 approach the maximal velocities according to the formulae :Vmax = 1 for N = 1/2 (full lines), while Vmax = 1/2 for N = 1/5 ( dashed lines).The velocity curves, the curves of the specific angular momentum, the surface mass-energydensity, and the surface rest mass density in the dimensionless form V , h/M , M, and M) aregiven in Fig. 4.5.2,Fig. 4.5.3 and Fig. 4.5.4 as functions of the dimensionless radius TR/M . Thesecurves were calculated starting from Eqs. (4.2.12), (4.2.14), (4.2.15) and (4.2.16), with functions4 and 5 given in Sections 3 and 4. Fig. 4.5.2 corresponds to the curves for the relativisticKuzmin-Curzon disks with n = 1, described by metric functions Eq. (4.4.15) and (4.4.16).Fig. 4.5.3 corresponds to the same quantities for the truncated Kalnajs-Mestel disks with m = 1,starting out from Eqs. (4.3.28) and (C.21). In Fig. 4.5.4 are illustrated the properties of the(un-truncated) Isochrone disks. In all the plots the total mass M of the disks is fixed but severalcurves are plotted for di!erent values of the parameters b (for the Kuzmin-Toomre disks), b1

and a (for the Kalnajs-Mestel disks). In all cases the disks become more relativistic the smallerthese parameters are, i.e. the closer the distribution of mass along the negative z-axis is to thecentre of the plane z = 0. In the plots of V and h the parameters decrease from bottom to thetop of the data. For , and ) this is the case only at small TR; at larger TR’s the curves beginto cross. As it is easy to see from Eqs. (4.2.6) and (4.2.7), the condition that the velocity ofcounter-rotating streams does not exceed the velocity of light reduces to the dominant energycondition, #- t

t > -'' (Hawking & Ellis (1973) [43]). The maximal relativistic velocities appearat larger TR/M - 3 # 4 for the Kuzmin-Toomre disks with n = 1, i.e. at larger radii than forthe Kalnajs-Mestel disks. In the highly relativistic disks, V approaches the velocity of light as

Section 4.5 Discussion 83

Figure 4.5.2: The same quantities as in Fig. 4.5.1 for nine relativistic Kuzmin-Toomre disks of ordern = 1 corresponding to M/b = 0.30, 0.33, ..., 0.54. In the highly relativistic disks, as TR decreases, Vapproaches the velocity of light and the specific angular momenta start to increase, indicating a relativisticinstability. At those radii where V ) 1, the surface densities, in particular the surface rest mass density,decrease rapidly with TR.

TR decreases, and the specific angular momenta cease to decrease with decreasing TR but startincreasing. The angular momenta of circular test-particle orbits in Schwarzschild spacetime alsoshow such a rise for r < 6M which is associated with their relativistic instability. This e!ectis observed in all three families of disks (Fig. 4.5.2b, Fig. 4.5.3b, Fig. 4.5.4b). For the Kuzmin-Toomre disks it is more pronounced than for the Kalnajs-Mestel disks. (Compare, for example,the Isochrone disk with Vmax 0 0.8 with the analogous Kuzmin-Toomre disk).As the disks become more relativistic, the surface mass densities rise rapidly towards the centre.This e!ect is most distinct in the truncated Kalnajs-Mestel disks with m = 1 (Fig. 4.5.3c,d).Nevertheless, these disks have long tails of surface density falling as TR#3 at large TR, as isderived from Eq. (4.3.13) with m = 1 At large TR, , - ) go over to the classical density'0. In fact, the most compact disks are the Kuzmin-Toomre disks with large n. In the limitn ) * the classical Kuzmin-Toomre disks approach the circular Gaussian disk in which ' ='0 exp(#R2/R2

c) (Toomre (1963) [107]), where Rc

,ln 2 is the half-mass radius. An interesting

property of the rest mass surface density may be observed in the highly relativistic disks. Atthose radii where the velocities of the streams approach the velocity of light, the rest mass densitydecreases rapidly with TR, as appears in all three Fig. 4.5.2(d), Fig. 4.5.3(d) and Fig. 4.5.4(d). Inextreme cases, such as in the Isochrone disk in Fig. 4.5.4(d), it even reaches a local minimum.Finally, the redshifts Eq. (4.2.17) of these objects are largest in the centre. In the case of thenth-order Kuzmin-Toomre disks, Eq. (4.2.17) and Eq. (4.4.1) imply

1 + z0 = exp+#4m+3/2 (0, 0)

,,


Figure 4.5.3: The same quantities as in Fig. 4.5.1 for eleven relativistic truncated Kalnajs-Mestel disksof order m = 1 with b2 = e2b1, b1/M = 2.3, 2.1, ..., 0.3. The angular momenta of the first ten disks keepdecreasing with decreasing TR. In the most relativistic disk the rest mass density decreases very rapidlywith TR for those TR where V approaches the velocity of light.

where

#4m+3/2 (0, 0) =M

bAn , and An =

m$

k=0

(2n# k)!2n#k (n# k)!

. (4.5.1)

In the first family,A0 = 1, A1 = 2, A2 =

83, A3 =

165

.

As mentioned above, with n)* these disks approach the circular Gaussian disk. In this case(cf. Evans & de Zeeuw 1992 [33]), one finds

#4" (0, 0) =M

Rc

,! = !

=M'0 ,

where M = !'0 and '0 is the classical central density. However, these disks can not producearbitrarily large central redshifts by decreasing Rc because V must not exceed 1. Indeed themaximum value of R d4/dR is 0.527

,!M/Rc which is attained at R = 1.257Rc. Now by

Eq. (4.2.11) this quantity can not exceed 1/2 for V % 1. Thus the maximum central redshift, zc

for a disk of this type is given by 1 + zc = 2.58.The central redshifts can be arbitrarily large in the Kalnajs-Mestel disks with m = 0 whenb1 ) 0, as shown by Bicak, Lynden-Bell & Katz [11].

In the Isochrone disks (cf. Eq. (4.3.31))

1 + zc = exp!

M

2a

",

Section 4.6 Conclusion 85

Figure 4.5.4: The same quantities as in Fig. 4.5.1 for thirteen Isochrone disks corresponding to M/a =0.5, 0.7, ..., 2.9. In the highly relativistic disks the behaviour of the angular momentum again indicatesa relativistic instability. At those radii for which V ) 1, the surface rest mass density decreases rapidlywith TR , and it even reaches a local minimum in the most relativistic case.

so the central redshift zc cannot be made arbitrarily large by putting a ) 0 as the dominantenergy condition, where Vmax = 1, occurs for M/a = 2.92 leading to the limit

1 + zc % 4.30.

· • !" ! • ·

4.6 Conclusion

Recent results in Newtonian potential theory of axisymmetric infinitesimally thin diskshave been used to construct infinite sequences of new solutions of Einstein’s equations whichdescribe static, counter-rotating disks. The disks are infinite but, except for exceptional casessuch as the un-truncated Kalnajs-Mestel disks with m = 1, their mass is finite. Relativisticgeneralisations of all Kuzmin-Toomre disks and Kalnajs-Mestel disks have been constructed.Einstein’s equations can be integrated starting from the corresponding classical potentials andthe resulting metrics, though in general complicated, can all be expressed explicitly in closedforms.

Although the disks become Newtonian at large distances, in their central regions interestingrelativistic features arise such as velocities close to the velocity of light, large redshifts, and


peculiar behaviour of the surface rest mass density. Some of these solutions correspond to non-spherical sources which lead to arbitrarily large redshifts. The disks discussed satisfy reasonablephysical requirements such as a dominant (positive) energy condition, and the spacetimes theyproduce are asymptotically flat.

Besides the relativistic instability exhibited in the behaviour of the specific angular momen-tum in the most relativistic disks, the disks described in this chapter are unstable with respectto the formation of rings since no radial pressure is available to prevent the collapse. In spiteof the recent observations of classical stellar counter-rotating disks mentioned in section 1, thevery existence of relativistic Counter-rotating disks needs yet to be clarified. In the followingchapter, these drawbacks will be addressed by constructing massive rotating disks with partialpressure support.

· • !" ! • ·

C

Generating function

for the Kalnajs-Mestel disks

In this appendix, the generating function for the potentials 4m of the mth-order Kalnajs-Mestel disks is derived explicitly. In the main text this generating function was defined as

J1 (ua, ub; v1, v2) =# ub

ua

# ub

ua

u1u2 du1du2

(u21 # v1u1 # 1) (u2

2 # v2u2 # 1) (u1u2 + 1)2. (C.1)

Defining

vi = wi #1wi

, (i = 1, 2) , (C.2)

J1 can be cast into the form:

J1 =# ub

ua

dQ

du2

u2 du2

(u2 # w2)?u2 + 1

w2

@ , (C.3)

whereQ =

# ub

ua

u1 du1

(u1 # w1)?u1 + 1

w1

@?u1 + 1

u2

@ . (C.4)

The last integral can be calculated via decomposition into partial fractions, giving

Q = [Q1 + Q2]ub

ua,

where

Q1 =1

w21 + 1

3

4 w21?

w1 + 1u2

@ ln |u1 # w1| +1?

1u2# 1

w1

@ ln |u1 +1w1

|

5

6 ,

andQ2 =

u2?u2 + 1

w1

@(u2 # w1)

ln |u1 +1u2

| . (C.5)

After substituting for Q into Eq. (C.3) and performing minor manipulations, J1 becomes thesum of three integrals

J1 = R1 + R2 + S, (C.6)

88 Generating function for the Kalnajs-Mestel disks Appendix C

where, given Eq. (C.2),

R (w1, w2) = R1 =

3

M4ln |u1 # w1|(w2

1 + 1)

# ub

ua

u2 du2?u2 + 1

w1

@2

(u22 # v2u2 # 1)

5

N6

ub

ua

, (C.7)

R2 = R!# 1

w1, w2

", (C.8)

andS (v1, v2) =

'# ub

ua

dQ2

du2

u2 du2

(u22 # v2u2 # 1)

(ub

ua

, (C.9)

with Q2 given by Eq. (C.5). Di!erentiating the log term in Q2 and then integrating by parts inthe last integral gives

S (v1, v2) = #T +9

u22 ln |u1 + 1

u2|

(u22 # v2u1 # 1) (u2

2 # v1u2 # 1)

:ub,ub

ua,ua

#9# ub

ua

d

du2

9u2 ln |u1 + 1

u2|

u22 # v2u2 # 1

:u2du2

(u22 # v1u2 # 1)

:ub

ua

,

(C.10)

where this last integral is S(v2, v1) by comparison with Eq. (C.9). T is

T ='# ub

ua

u2 du2

(u22 # v1u2 # 1) (u2

2 # v2u2 # 1) (u1u2 + 1)

(ub

ua

. (C.11)

Note that T is symmetrical in v1 and v2. From the symmetry of J1 with respect to variablesv1, v2 it is clear that

J1 (ua, ub; v1, v2) = J1 (ua, ub; v2, v1) . (C.12)

ThusJ1 (v1, v2) = 1

2[J1 (v1, v2) + J1 (v2, v1)] ,

so that2J1 (v1, v2) = R (w1, w2) + R (w2, w1) + R

!# 1

w1

, w2

"+ R

!# 1

w2

, w1

"

+S (v1, v2) + S (v2, v1) . (C.13)

From Eq. (C.10) the last two terms are simply given by

S (v1, v2) + S (v2, v1) = #T +9

u22 ln |u1 + 1

u2|

(u22 # v1u2 # 1) (u2

2 # v2u2 # 1)

:ub,ub

ua,ua

. (C.14)

The symmetrisation avoids the last integral in Eq. (C.10) which cannot be expressed as a finitecombination of elementary functions.

The function J1(ua, ub; v1, v2) is therefore given by Eq. (C.13), where R1 and R2 are givenby Eq. (C.7) and (C.8) and S(v1, v2)+S(v2, v1) by Eq. (C.14). Evaluating Eq. (C.7) and (C.11)which are readily calculated by decomposition into partial fractions, leads, after some algebra,to the final form for the primitive function J1:

J1 (ua, ub; w1, w2) = 12

ATJ (u1, u2; w1, w2) + TJ (u1, u2; w2, w1) + j (u1, u2; w1, w2)

Bub,ub

ua,ua

,

Appendix C Generating function for the Kalnajs-Mestel disks 89

where

j =u2

2 lnA(u1u2 + 1)2 /u2

B

(u22 # v1u2 # 1) (u2

2 # v2u2 # 1), (C.15)

and TJ is given byTJ (u1, u2; w1, w2) =

w21w

22 ln |u1 # w1|

(w21 + 1) (w2

2 + 1)

9

#(w2

1 + 1) (w22 + 1) ln |u2 + 1

w1|

(w1w2 + 1)2 (w1 # w2)2 +

ln |u2 # w2|(w1w2 + 1)2

+ln |u2 + 1

w2|

(w1 # w2)2

:

+w2

1w22 ln |u1 + 1

w1|

(w21 + 1) (w2

2 + 1)

9

#(w21 + 1) (w2

2 + 1) ln |u2 # w1|(w1w2 + 1)2 (w1 # w2)

2 +ln |u2 # w2|(w1w2 + 1)2

+ln |u2 + 1

w2|

(w1 # w2)2

:

+w1w2

(w21 + 1) (w1 # w2) (w1w2 + 1)

9w2

1 ln |u1 + 1w1|

u2 # w1+

ln |u1 # w1|u2 + 1

w1

+

w22 ln |u2 + 1

w2|

u1 # w2+

ln |u2 # w2|u1 + 1

w1

: . (C.16)

This expression is the generating function for the second constituent 5 required for the completerelativistic metric of any member of the Kalnajs-Mestel family.Note that in terms of the wi’s the recursion relation Eq. (4.3.21) reads

Jm =1

[(m# 1)!]2'

w21w

22

(w21 + 1) (w2

2 + 1)%2

%w1%w2

(m#1

J2 . (C.17)

The explicit form of J1 for the untruncated Kalnajs-Mestel disks, follows from Eq. (C.15) and(C.16) after putting ua = u and ub )*:

J1 (ua = u, ub = *; w1, w2) = 2w1w2

XTJ1 (u; w1, w2) + TJ1 (u; w2, w1)

+u2

(u# w1) (u# w2) (1 + uw1) (1 + uw2)ln |u +

1u|

+w1w2 (1 + w1w2)

#2

(1 + w21) (1 + w2

2)

'ln |u# w1| ln |u# w2| + ln |u +

1w1

| ln |u +1w2

|(Q

.

, (C.18)

where

TJ1 (u; w1, w2) =w1

(w2 # w1) (1 + w1w2) (1 + w21)

9ln |u# w1|1 + uw1

+w1 ln |u + 1

w1|

u# w1

:

+w1w2

(w2 # w1)2 ln |u +

1w1

|9

ln |u# w2|(1 + w2

1) (1 + w22)# ln |u# w1|

(1 + w1w2)2

:

. (C.19)

This generating function gives 5m for all un-truncated Kalnajs-Mestel disks which, together with4m given by Eq. (4.3.10), then determines the metrics completely.Another asymptotic limit is readily derived from Eq. (C.15) and (C.16) by setting w1 and w2 tow. This yields after some algebra

J $1 = J1 (ua, ub; w1 = w, w2 = w) = K +

9

L +4w4

(1 + w2)4lnWWW

ua + 1w

ub + 1w

WWW

:

lnWWW

ub # w

ua # w

WWW

90 Generating function for the Kalnajs-Mestel disks Appendix C

+9

M +2w4

(1 + w2)4 lnWWW

ua + 1w

ub + 1w

WWW

:

lnWWW

ua + 1w

ub + 1w

WWW+2w2u2

a (1 + uaw)#2

(ua # w)2lnWWW

1 + u2a

1 + uaub

WWW

+2w2u2

b (1 + ubw)#2

(ub # w)2lnWWW

1 + u2b

1 + uaub

WWW+2w4

(1 + w2)4

'lnWWW

ua # w

ub # w

WWW(2

, (C.20)

where

K =2w4 (ua # ub)

2

(1 + w2)2 (ua # w) (ub # w) (1 + uaw) (1 + ubw),

L =2w3 (ua # ub) [#2# w (ua + ub) + w3 (ua + ub) + 2w4uaub]

(1 + w2)3 (a + uaw)2 (1 + ubw)2,

M =2w3 (ub # ua) [2w4 + w (ua + ub)# w3 (ua + ub)# 2uaub]

(1 + w2)3 (ua # w)2 (ub # w)2.

(C.21)

This expression characterises 51 for the m = 1 truncated Kalnajs-Mestel disk.

· • !" ! • ·

5

Dynamics of rotating

super-massive disks

5.1 Introduction

The general-relativistic theory of rapidly rotating objects is of great intrinsic interest andhas potentially important applications in astrophysics. Following Einstein’s early work on rela-tivistic collisionless spherical shells, Fackerel (1968) [34], Ipser (1969) [51] and Ipser & Thornes(1968) [52] have developed the general theory of relativistic collisionless spherical equilibria. Thestability of such bodies has been further expanded by Katz and Horowitz (1975) [60]. Here thecomplete dynamics of flat disk equilibria is developed and illustrated, extending the results ofthe previous chapter to di!erentially rotating configurations with partial pressure support. Ageneral inversion method for the corresponding distribution functions is presented, yielding acoherent model for the stellar dynamics of these disks. Di!erentially rotating flat disks in whichcentrifugal force almost balances gravity can also give rise to relatively long lived configurationswith large binding energies. Such objects may therefore correspond to possible models for thelatest stage of the collapse of a proto-quasar. Analytic calculations of their structure have beencarried into the post Newtonian regime by Chandrasekhar. Strongly relativistic bodies have beenstudied numerically following the pioneering work on uniformly rotating cold disks by Bardeen& Wagoner [6]. An analytic vacuum solution to the Einstein equations, the Kerr metric [61],which is asymptotically flat and has the general properties expected of an exterior metric of arotating object, has been known for thirty years, but attempts to fit an interior solution to itsexterior metric have been unsatisfactory. A method for generating families of self-gravitatingrapidly rotating disks purely by geometrical methods is presented. Known vacuum solutionsof Einstein’s equations are used to produce the corresponding relativistic disks. One familyincludes interior solutions for the Kerr metric.

5.1.1 Background

In 1919 Weyl [111] and Levi-Civita gave a method for finding all solutions of the axially sym-metric Einstein field equations after imposing the simplifying conditions that they describe astatic vacuum field. In the previous chapter, complete solutions corresponding to counter ro-tating pressure-less axisymmetric disks were constructed by matching exterior solutions of theWeyl- Levi-Civita type on each side of the disk. This method will here be generalised, allowing

92 Dynamics of rotating super-massive disks Chapter 5

true rotation and radial pressure support for solutions describing self-gravitating disks. Thesesolutions provide physical sources (interior solutions) for stationary axisymmetric gravitationalfields.

5.1.2 Pressure Support via Curvature

In Newtonian theory, once a solution for the vacuum field of Laplace’s equation is known, it isstraightforward to build solutions to Poisson’s equation for disks by using the method of mirrorimages. In general relativity, the overall picture is similar. In fact, it was pointed out by Morgan& Morgan (1969) [90] that for pressureless counter rotating disks in Weyl’s co-ordinates, theEinstein equation reduces essentially to solving Laplace’s equation (cf. previous chapter). Whenpressure is present, no global set of Weyl co-ordinates exists. The condition for the existence ofsuch a set of co-ordinates is that TRR + Tzz = 0. On the disk itself, this is satisfied only if theradial eigenvalue of the pressure tensor, pR, vanishes (Tzz vanishes provided the disk is thin).When pR is non-zero, vacuum co-ordinates of Weyl’s type still apply both above and belowthe disk, but as two separate co-ordinate patches. Consider global axisymmetric co-ordinates(R$, *$, z$) with z$ = 0 on the disk itself. For z$ > 0, Weyl’s co-ordinates R, *, z may be used.In terms of these variables, the upper boundary of the disk z$ = 0+ is mapped onto a givensurface z = f(R). On the lower patch, by symmetry, the disk will be located on the surfacez = #f(R). Thus in Weyl co-ordinates, the points z = f(R) on the upper surface of the diskmust be identified with the points z = #f(R) on the lower surface. The intrinsic curvature ofthe two surfaces that are identified match by symmetry. The jump in the extrinsic curvaturegives the surface distribution of stress-energy on the disk. A given metric of Weyl’s form isa solution of the empty space Einstein equations, and does not give a complete specificationof its sources (for example, a Schwarzschild metric of any given mass can be generated by astatic spherical shell of any radius). With this method, all physical properties of the sourceare entirely characterised once it is specified that the source lie on the surface z = ±f(R) andthe corresponding extrinsic curvature (i.e. its relative layout within the embedding vacuumspacetime) is known. The freedom of choice of 4 in the Weyl metric corresponds to the freedomto choose the density of the disk as a function of radius. The supplementary degree of freedominvolved in choosing the surface of section z = f(R) corresponds classically to the choice of theradial pressure profile.

5.1.3 Rotating Disks

For rotating disks, the dragging of inertial frames induces strongly non-linear fields outside thedisk which prohibit the construction of a vacuum solution by superposition of the line densitiesof fictitious sources. nevertheless, in practise quite a few vacuum stationary solutions have beengiven in the literature, the most famous being the Kerr metric. Hoenselaers, Kinnersley &Xanthopoulos (1979) [46,47](HKX hereafter) have also given a discrete method of generatingrotating solutions from known static Weyl solutions. The disks are derived by taking a cutthrough such a field above all singularities or sources and identifying it with a symmetrical cutthrough a reflection of the source field. This method applies directly to the non-linear fieldssuch as those generated by a rotating metric of the Weyl-Papapetrou type describing stationaryaxisymmetric vacuum solutions. The analogy with electromagnetism is then to consider the fieldassociated with a known azimuthal vector potential A', the analog of gt'/gtt. The jump in thetangential component of the corresponding B-field gives the electric current in the plane, justas the jump in the t, * component of the extrinsic curvature gives the matter current.

The procedure is then the following: for a given surface embedded in curved space-timeand a given field outside the disk, the jump in the extrinsic curvature on each side of the mirror

Section 5.1 Introduction 93

images of that surface is determined in order to specify the matter distribution within the disk.The formal relationship between extrinsic curvatures and surface energy tensor per unit areawas introduced by Israel and is described in details by Misner Thorne Wheeler (1973) p 552. [89]Plane surfaces and non-rotating vacuum fields (i.e. the direct counterpart of the classical case)were investigated in the previous chapter and led to pressureless counter-rotating disks. Anycurved surface will therefore produce disks with some radial pressure support. Any vacuummetric with gt' .= 0 outside the disk will induce rotating disks.

Figure 5.1.1: Symbolic representation of a section z = ±f(R) through the gt' field representing theupper and lower surface of the disk embedded in two Weyl-Papapetrou fields. The isocontours of ((R, z)correspond to a measure of the amount of rotation in the disk.

In section 2, the construction scheme for rotating disks with pressure support is presented.In section 3, the properties of these disks are analysed, and a reasonable cut, z = f(R), issuggested. This method is then first applied to warm counter rotating disks in section 4; thedominant energy condition given for Curzon disks by Chamorow et. al.(1987) [17], and Lemos ’solution for the self - similar Mestel disk with pressure support are recovered. In section 5, theKerr disk is studied and the method for producing general HKX disks is sketched.

· • !" ! • ·


5.2 Derivation

The jump in extrinsic curvature of a given profile is calculated and related to the stressenergy tensor of the matter distribution in the corresponding self gravitating relativistic disk.

5.2.1 The Extrinsic Curvature

The line element corresponding to an axisymmetric stationary vacuum gravitational field is givenby the Weyl-Papapetrou (WP) metric

ds2 = # e2 + (dt # ( d*)2 + e2 5#2 +?dR2 + dz2

@+ R2e#2 + d*2, (5.2.1)

where (R, *, z) are standard cylindrical co-ordinates, and 4, 5 and ( are functions of (R, z)only1. Note that this form is explicitly symmetric under simultaneous change of * and t. Thevacuum field induces a natural metric on a 3-space-like z = f(R) surface

d )2 = g)%%x)

%xa

%x%

%xbdxadxb & g)% h)

ah%b dxadxb,

& /ab dxa dxb,(5.2.2)

where {xa} are the co-ordinates on the embedded hypersurface z = f(R), {xa} = (t, R, *),and {x)} = (t, R, z, *) are Weyl’s co-ordinates for the embedding spacetime. /ab stands for theembedded metric, and h)

a is given by

h)a =

%x)

%xa=

3

41 0 0 00 1 0 f $

0 0 1 0

5

6 , (5.2.3)

where hereafter a prime, ( )$, represents the derivative with respect to R. The line element isthen

d)2 = # e2 + (dt # ( d*)2 + R2e#2 +d*2 + e25#2 +%1 + f $2& dR2. (5.2.4)

Let the upward pointing normal Nµ to the surface z = f(R) be

Nµ =%

%xµ[z # f (R)] = (0,#f$, 0, 1) . (5.2.5)

The normalised normal is therefore

nµ =Nµ,N)N)

=Nµ=

g)%N%N)

=Nµ,

1 + f $2 e5#+. (5.2.6)

The extrinsic curvature tensor K is given in its covariant form by

Kab = nµ

?%ah

µb + (µ

) % h)a h%

b

@, (5.2.7)

where %ahµb = %hµ

b /%xa. Eq. (5.2.6) and (5.2.3) together with the Christo!els given in theAppendix lead to the computation of K , the extrinsic curvature of that surface embedded in

1 geometric units G = c = 1 are used throughout this chapter

Section 5.2 Derivation 95

the Weyl Papapetrou metric.

Ktt = # e+#5

,1 + f $2

!%4

%N

"e4+#25 , (5.2.8a)

KRR = # e+#5

,1 + f $2

1

2 f $$

1 + f $2 +%?4 # 5

@

%N

7

8%1 + f $2& , (5.2.8b)

K't = # e+#5

,1 + f $2

'% [4 + 1

2 log (()]%N

((e4+#25 , (5.2.8c)

K'' = # e+#5,1 + f $2

'!f $

R+%4

%N

"R2e#25 + e4+#25(2% [4 # log (()]

%N

(, (5.2.8d)

where the notation %/%N & (%/%z # f$%/%R) has been used.

5.2.2 The Stress Energy Tensor

The stress energy tensor per unit surface -ab is the integral of of the stress energy tensor carried

along the normal to the surface z = f(R). In a locally Minkowskian frame co-moving with themean flow of the disk, the corresponding orthonormal tetrad is

e(0)i = e+ (1, 0,#() , (5.2.9a)

e(1)i = e

5#+=

1 + f $2 (0, 1, 0) , (5.2.9b)e(2)

i = Re#+ (0, 0, 1) , (5.2.9c)

so thatd s2 = 2(a)(b)

?e(a)

i d xi@?

e(b)j d xj

@, (5.2.10)

with 2(a)(b) = Diag(#1, 1, 1). In that frame,

A- (a)(b)

B

0=

3

4, 0 00 pR 00 0 p'

5

6 . (5.2.11)

After a Lorentz transformation to a more general frame in which the flow is rotating with relativevelocity V in the * direction, this stress becomes

- (a)(b) =1

1# V 2

3

4, + p' V 2 0 (p' + ,) V

0 (1# V 2) pR 0(p' + ,) V 0 p' + ,V 2

5

6 . (5.2.12)

5.2.3 The Discontinuity Equations

Israel (1964) [64] has shown that Einstein’s equations integrated through a given surface ofdiscontinuity can be re-written in terms of the jump in extrinsic curvature, namely

-ab =

18!

[Kab #Ka

a +ab ]+# & La

b . (5.2.13)


where [ ]+# stands for ( ) taken on z = f(R) minus ( ) taken on z = #f(R). Lab is known as

the Lanczos tensor.

In the tetrad frame (5.2.9), the Lanczos tensor given by Eqs. (5.2.13) and (5.2.8) reads

L(a)(b) =e+#5

4!,

1 + f $2

3

MMMM4

f !!

1+f !2 + f !

R +-

?2+#5

@

-N 0 -#-N

e2"

2R

0 #f !

R 0-#-N

e2"

2R0 # f !!

1+f !2 + -5-N

5

NNNN6. (5.2.14)

Identifying the stress energy tensor - (a)(b) with the tetrad Lanczos tensor L(a)(b) according toEq. (5.2.13), and solving for pR , p' , , and V gives

V =Re#2+

%(/%N

'!f $

R+ 2

%4

%N

"#Q

(, (5.2.15a)

, =e+#5

4!,

1 + f $2

3

4Q

2+

1

2 f $$

1 + f $2 +f $

2R+%?4 # 5

@

%N

7

8

5

6 , (5.2.15b)

p' =e+#5

4!,

1 + f $2

3

4Q

2#

1

2 f $$

1 + f $2 +f $

2R+%?4 # 5

@

%N

7

8

5

6 , (5.2.15c)

pR =e+#5

4!,

1 + f $2

!#f $

R

", (5.2.15d)

where

Q &

.!f $

R+ 2

%4

%N

"2

# e4+

R2

!%(

%N

"2

. (5.2.16)

All quantities are to be evaluated along z = f(R). Eq. (5.2.15) gives the form of the most generalsolution to the relativistic rotating thin disk problem provided the expressions for ,, p', pR arephysically acceptable.

· • !" ! • ·

5.3 Physical properties of the warm disks

Physical properties of interest for these disks are derived and related to the choice of profilez = f(R) compatible with their dynamical stability.

Defining the circumferential radius Rc, proper radial length R, and a synchronised propertime -! by

Rc = R e#+ , (5.3.1a)

R =# =

1 + f $2e5#+ dR , (5.3.1b)

-! =#

(T )

e+!

1# (d*

dt

"dt , (5.3.1c)

where the integral over dR is performed at z = f(R) and that over dt is performed along a giventrajectory (T ), the line element on the disk (5.2.4) reads

d )2 = #/d-2! + dR2 + R2

cd*2. (5.3.2)

Section 5.3 Physical properties of the warm disks 97

For circular flows (/d-! = e+ [1# (d*/dt] dt), V may be re-written as

V = Rcd*

/d-!. (5.3.3)

Equation (5.3.3) is inverted to yield the angular velocity of the flow as measured at infinity:

$ & d*

dt=

V e+/Rc

1 + (V e+/Rc. (5.3.4)

Similarly, the covariant specific angular momentum h of a given particle reads in terms of thesevariables

h & p'µ

= /t'ut + /''u

' =1,

1# V 2(RcV + ( e+) , (5.3.5)

where µ is the rest mass of the particle. The covariant specific energy 7 of that particle reads

7 & #pt

µ= #/t'u

' # /t tut =

e+,1# V 2

. (5.3.6)

which gives the central redshift, 1 + zc = exp (#4). The velocity of zero angular momentumobservers (so called ZAMOs) follows from Eq. (5.3.5):

Vz = #(e+

Rc. (5.3.7)

With these observers, the cuts z = f(R) may be extended inside ergoregions where the drag-ging of inertial frames induces apparent superluminous motions as measured by locally staticobservers. The circumferential radius Rz as measured by ZAMOs is Lorentz contracted withrespect to Rc, becoming

Rz &,

g'' = Rc

+1# V 2

z

,1/2, (5.3.8)

while the velocity flow measured by ZAMOs is

V/z =V # Vz

1# V Vz. (5.3.9)

The total angular momentum of the disk H follows from the asymptotic behaviour of thevacuum field ( at large radii, ( ) #2H/r. Define the binding energy of the disk &E asthe di!erence between the baryonic rest mass M0 and the total mass-energy of the disk M asmeasured from infinity given by 4 ) #M/r:

&E = M0 #M. (5.3.10)

M is also given by the Tolmann formula, but corresponds, by construction, to the mass of theline source for the vacuum field. The baryonic rest mass can be expressed as

M0 &#

',#g UµdSµ =

# [1# ($]#1

,1# V 2

' 2!RcdR , (5.3.11)

where the baryonic energy density ' can be related to the energy density 7 through some yetunspecified equation of state or the knowledge of a distribution function. Possible equations ofstate are derived in the appendix and will be presented in sections 4 and 6 for concrete examples;a general method to derive distribution functions corresponding to a given flow is presented insection 5.


5.3.1 Constraints on the internal solutions

The choice of the profile z = f(R) is open, provided that it leads to meaningful physical quan-tities. Indeed it must lead to solutions which satisfy , + 0, pR + 0, p' + 0, andV % 1 whilethe dominant energy condition implies also that , + p' and , + pR. These translate into thefollowing constraints on f

, + 0 4,N 2 # O2 + (N # 2Z) + 0 ; (5.3.12a)

0 % V % 1 4 N + O and O + 0 ; (5.3.12b)0 % p' % , 4 4Z N +

%O2 + 4Z2

&and N + 2Z ; (5.3.12c)

0 % pR % , 4 f$ % 0 and # 2f$/R %,N 2 # O2 + (N # 2Z) ; (5.3.12d)

given

N & f $

R+ 2

%4

%N, O & e2+

R

%(

%N, and Z &

;%5

%N# f $$

1 + f $2

<

. (5.3.13)

The existence of a solution satisfying this set of constraints can be demonstrated as follows:in the limit of zero pressure and counter-rotation (i.e. O) 0,N ) 2%4/%z, and Z ) %5/%z),any cut z = Const & b 3 m satisfies Eqs. (5.3.12). By continuity, there exist solutions withproper rotation and partial pressure support. In practise, all solutions given in sections 4 and5 satisfy the constraints (5.3.12). Note that in the limit of zero radial pressure, Eq. (5.3.12b)implies %/%z [(/R + exp 24] % 0.

5.3.2 Ansatz for the profile

In the following discussion, the section z = f(R) is chosen so that the corresponding radialpressure gradient balances a given fraction of the gravitational field which would have occurredhad there been no radial pressure within the disk. This choice ‘bootstraps’ calculations for allrelevant physical quantities in terms of a single degree of freedom (i.e. this fraction 2), ratherthan a complete functional. This gives

%pR

%R= # 2

(1# V 2)%4

%R

+,+ pR + V 2 (p' # pR)

,. (5.3.14)

On the r.h.s. of Eq. (5.3.14), pR is put to zero and V, p', , are re-expressed in terms of 5, 4,(via Eqs. (5.2.15) with f $ = f $$ = 0 and z = Const. On the l.h.s. of Eq. (5.3.14), pR is chosenaccording to Eq. (5.2.15d). In practise, this ansatz for f is a convenient way to investigatea parameter space which is likely to be stable with respect to ring formation as discussed inthe next subsection. In principle, a cut, f , could be chosen so as to provide a closed boundedsymmetrical surface containing all fictitious sources. Indeed, by symmetry no flux then crossesthe z = 0 plane beyond the cut, and the energy distribution is therefore bound to the edge ofthat surface: this corresponds to a finite pressure supported disk. It turns out that in generalthis choice is not compatible with all the constraints enumerated in the previous section. Morespecifically, the positivity of p' fails for all finite disk models constructed. The ansatz describedabove gives for instance an upper bound on the height of that surface when assuming that allthe support is provided by radial pressure. This height is in turn not compatible with positiveazimuthal pressure at all radii.

Section 5.3 Physical properties of the warm disks 99

5.3.3 Stability

To what extent are the equilibria studied in the previous sections likely to be stable under theaction of disturbances? The basic instabilities can be categorised as follows:

• dynamical instabilities, which are intrinsic to the dynamical parameters of the disks, growon an orbital time scale, and typically have drastic e!ects on the structure of the system.

• secular instabilities, which arise owing to dissipative mechanisms such as viscosity or grav-itational radiation, grow on a time scale which depends on the strength of the dissipativemechanism involved, and slowly drive the system along a sequence of dynamical equilibria.

Amongst dynamical instabilities, kinematical instabilities correspond to the instability ofcircular orbits to small perturbations, and collective instabilities arises from the formation ofgrowing waves triggered by the self-gravity of the disk. Rings, for instance, will be generatedspontaneously in the disk if the local radial pressure is insu#cient to counteract the self-gravityof small density enhancements. Even dynamically stable non-axisymmetric modes may drive thesystem away from its equilibrium by radiation of gravitational waves which will slowly removeangular momentum from the disk. For gaseous disks, viscosity and photon pressure will a!ectthe equilibria. Radiative emission may disrupt or broaden the disk if the radiation pressureexceeds the Eddington limit. The energy loss by viscosity may induce a radial flow in the disk.However, the disks discussed in this chapter have anisotropic pressures inappropriate for gaseousdisks and the accurate description of the latter two processes requires some prescription for thedissipative processes in the gas. The scope of this section is therefore restricted to a simpleanalysis of the dynamical instabilities.

Turning briefly to the corresponding Newtonian problem, Toomre (1963) [107] gave thelocal criterion for radial collective instability of stellar disks,

)R +3.36G'0

", (5.3.15)

where '0 is the local surface density, )2R the radial velocity dispersion, and " the epicyclic

frequency of the unperturbed stars. This criterion is derived from the first critical growingmode of the dispersion relation for radial waves. The corresponding critical wavelength is Jeanslength 0 2)2

R/G'0.

For the relativistic disks described in this chapter, spacetime is locally flat, which suggestsa direct translation of Eq. (5.3.15) term by term. The constraints that stability against ringformation places on these models will then be addressed at least qualitatively via the Newtonianapproach. The proper relativistic analysis is left to further investigation. The relativistic surfacedensity generalising '0 is taken to be the co-moving energy density , given by Eq. (5.2.15b).The radial velocity dispersion )2

Ris approximated by pR/,. The epicyclic frequency is calculated

in the appendix. Putting Eqs. (D.2.4) and (D.2.5) into Eq. (5.3.15) give another constraint onf for local radial stability of these disks. Note that the kinematical stability of circular orbitsfollows from Eq. (D.2.3) by requiring "2 to be positive.

· • !" ! • ·


5.4 Application: warm counter rotating disks

For simplicity, warm counter-rotating solutions are presented first, avoiding the non-linearitiesinduced by the dragging of inertial frames. These solutions generalise those of chapter III, whileimplementing partial pressure support within the disk. Formally this is achieved by putting((R, z) identically to zero in Eq. (5.2.1); the metric for the axisymmetric static vacuum solu-tions given by Weyl is then recovered:

ds2 = #e2 + dt2 + e25#2 +?dR2 + dz 2

@+ R2 e#2 +d*2, (5.4.1)

where the functions 5(R, z) and 4(R, z) are generally of the form (cf. Eq. (4.1.3), Eq. (4.3.14))

4 = ##

W (b) db>R2 + (|z|+ b)2

, (5.4.2a)

5 =# #

W (b1)W (b2)Z (b1, b2) db1 db2, (5.4.2b)

with Z given by Eq. (4.3.15). When the line density of fictitious sources is of the form W (b) $b#m (Kuzmin-Toomre), or W (b) $ +(m) (b) (Kalnajs-Mestel), the corresponding functions 5 , 4have explicitly been given in the previous chapter. For the metric of Eq. (5.4.1), the Lanczostensor (5.2.14) reads

3

4L(t)(t)

L(R)(R)

L(')(')

5

6 =e+#5

4!,

1 + f $2

3

MMMM4

f !!

1+f !2 + f !

R +-

?2+#5

@

-N#f !

R

#f !!

1+f !2 + -5-N

5

NNNN6. (5.4.3)

Consider again counter-rotating disks made of two equal streams of stars circulating inopposite directions around the disk centre. The stress energy tensor is then the sum of thestress energy tensor of each stream:

- (a)(b) =

3

4, 0 00 pR 00 0 p'

5

6 =2

1# V 20

3

4,0 + p0 V 2

0 0 00 (1# V 2

0 ) p0 00 0 p0 + ,0V 2

0

5

6 , (5.4.4)

the pressure p0 in each stream being chosen to be isotropic in the plane. Subscript ( )0 repre-sents quantities measured for one stream. Expressions for ,, p' and pR follow from identifyingEqs. (5.4.4) and (5.4.3) according to Eq. (5.2.13). Solving for ,0, V0, and p0 in Eq. (5.4.4), givenEq. (5.4.3) and (5.2.13) gives

,0 =e+#5

4!,

1 + f $2

3

4 f $$

1 + f $2 +%?4 # 5

@

%N

5

6 , (5.4.5a)

p0 =e+#5

8!,

1 + f $2

'#f $

R

(, (5.4.5b)

V 20 =

;%5

%N# f $$

1 + f $2 +f $

R

<1

2 f $$

1 + f $2 +%?24 # 5

@

%N

7

8

#1

. (5.4.5c)

The above solution provides the most general counter rotating disk model with pressure support.Indeed, any physical static disk will be characterised entirely by its vacuum field 4 and its radial

Section 5.4 Application: warm counter rotating disks 101

pressure profile pR, which in turn defines W (b) and f(R) uniquely according to Eqs. (5.4.5b)and (4.3.15). The other properties of the disk are then readily derived.

The angular frequency and angular momentum of these disks follow from Eq. (5.3.5), (5.3.4)on putting ( to zero and V to V0. The epicyclic frequency " given by Eq. (D.2.4) may be recastas

"2 =e+

R3c

dh2

dR, (5.4.6)

which relates closely to the classical expression "2 = R#3dh2/dR.The ansatz given by Eq. (5.3.14) for z = f(R) becomes, after the substitutions ,) ,0, V )

V0, p' ) 0,

(f $/R)$ = #2 %4%R

9%4

%z# 1

2%5

%z

:

, (5.4.7a)

= 2%4

%R

%4

%z

'R%4

%R# 1

(, (5.4.7b)

where z is to be evaluated at b. Equation (5.4.7b) follows from Eq. (5.4.7a) given the (Rz )

component of the Einstein equation outside the disk.The equation of state for a relativistic isentropic 2D flow of counter-rotating identical par-

ticles is derived in the appendix and reads

,0 = 2 p0 +'

1 + p0/'. (5.4.8)

Solving for ' gives' =

12

A,0 # 2 p0 +

=,0 # 2 p0

=,0 + 2 p0

B, (5.4.9)

which in the classical limit gives ' ) ,0 # p0. The binding energy of these counter rotatingdisks is computed here from Eq. (5.4.9) together with Eqs. (5.3.11) and (5.4.5). Alternatively,it could be computed via the distribution functions presented in the appendix B.

5.4.1 Example 1 : The Self-similar disk

The symmetry of the self-similar disk allows one to reduce the partial di!erential equationscorresponding to Einstein’s field equations to an ordinary di!erential equation with respect tothe only free parameter 6 = arctan(R/z) (Lemos, 89 [65]). Lemos’ solution may be recoveredand corrected by the method presented in section 2. Weyl’s metric in spherical polar co-ordinates(',1, *) defined in terms of (R, z, *) by ' =

=(R2 + z2) and cot(1) = z/R is

d)2 = #e2+d- 2 + '2

9

e#2+ sin2 (1) d*2 + e25#2+

;d'

'2

2

+ d12

<:

. (5.4.10)

A self-similarity argument led Lemos to the metric

ds2 = #r2neNdt2 + r2k

9

e2P#N d*2 + eZ#N

;dr

r2

2

+ d62<:

, (5.4.11)

where P , N , and Z are given by

P (6) = log [sin (k + n) 6] , (5.4.12a)

N (6) =4n

k + nlog [cos (k + n) 6/2] , (5.4.12b)

Z (6) =8n2

(k + n)2log [cos (k + n) 6/2] + 2 log (k + n) . (5.4.12c)


Here the *’s in both metrics have already been identified since they should in both case varyuniformly in the range [0, 2![. The rest of the identification between the two metrics followsfrom the transformation

1 = (n + k) 6, (5.4.13a)' = rn+k, (5.4.13b)- = t, (5.4.13c)

with the result that 4 and 5 take the form:

24 = N + 2n log (r) , (5.4.14a)25 = Z # 2 log (n + k) . (5.4.14b)

This can be written in terms of the new variables given in Eqs. (5.4.13) as

5 =4n2

(k + n)2log

!12

", (5.4.15a)

4 =2n

(k + n)2log

!12

"+

n

k + nlog (') . (5.4.15b)

The procedure described earlier may now be applied. In order to preserve the self-similarity ofthe solution and match Lemos’ solution, f(R) is chosen so that

f $ (R) = const. = cot (2) say. (5.4.16)

Then%

%N&!%

%z# f $ %

%R

"= # 1

' sin (2)%

%2. (5.4.17)

Putting Eqs. (5.4.15) and Eq. (5.4.17) into Eq. (5.2.15) gives3

4,pR

p'

5

6 =r#k

4!

'cos

!2

2

"( 2n(k#n)(k+n)2

3

M4cot (2) + 2n k

(k+n)2tan (2/2)

# cot (2)2n2

(k+n)2tan (2/2)

5

N6 , (5.4.18)

with 2 to be evaluated at (n + k) !/2. The above equations are in accordance with the solutionfound by Lemos by directly integrating the Einstein field equations.

5.4.2 Example 2: The Curzon disk

The Curzon disk is characterised by the following pair of Weyl functions:

4 = #&/r, 5 = #&2R2/%2 r4

&, (5.4.19)

with r =,

R2 + f2, given the dimensionless parameter & = M/b (recall that G & c & 1), alllengths being expressed in units of b. This disk is the simplest example of the Kuzmin-Toomrefamily, where W (b) $ +(b). It also corresponds to the building block of the expansion given inEqs. (5.4.2). Equations (5.4.3) and (5.4.4) then imply

3

4,pR

p'

5

6 =e#

#r

%1## R2

2r3

&

4!b,

1 + f $2

3

M4

f !!

1+f !2 + f !

R + )r6 {2f (r3 # &R2) + [& (R2 # f2)# 2r3] Rf $}

#f !

R

# f !!

1+f !2 + )2

r6 [2R2f # (R2 # f2) Rf $]

5

N6 . (5.4.20)

Section 5.4 Application: warm counter rotating disks 103

0 2 4 6 8 10

R

1

2

3

4

5Toomre’s Q

Figure 5.4.1: the “Toomre Q number” Q = )R"/(3.36'0) is plotted here as a function of circumferentialradius Rc for di!erent compactness parameter & = b/M when 2 = 1/25.

The weak energy conditions , + 0, , + pR + 0, and , + p' + 0 which follow are inagreement with those found by Chamorow et al.(1987) [17]. Their solution was derived by directintegration of Einstein’s static equations using the harmonic properties of the supplementaryunknown needed to avoid the patch of Weyl metric above and below the disk. The cut z = f(R)corresponds to the imaginary part of the complex analytic function, the existence of whichfollows from the harmonicity of that new function.

The ansatz (5.3.14) leads here to the cut

f (R) = b +2

2

9b m2

4 (b2 + R2)# 2b m3 (4b2 + 7R2)

105 (b2 + R2)5/2

:

. (5.4.21)

In Fig. 5.4.1, the ratio of the binding energy of these Curzon disks (which is derived fromEqs. (5.3.11), (5.4.9),(5.4.20) and (5.4.21)) over the corresponding rest mass M0 is plotted withrespect to the compactness parameter b/M0 for di!erent ratios p0/,0 measured at the origin.The relative binding energy decreases in the most compact configurations because these containtoo many unstable orbits ("2 < 0). A maximum ratio of about 5 % is reached.

· • !" ! • ·


1 1.5 2 2.5

b/M

0.015

0.02

0.025

0.03

0.035

0.04

0.045

dM/M

Figure 5.4.2: the binding energy over the rest mass &E/M0 for Curzon disks, plotted against thecompactness parameter & = b/M for di!erent ratios p0/,0 fixed by 2 = 0.01, 0.16, · · · , 0.7.

5.5 Distribution functions for rotating super-massive disks

The method described in section 2 will generally induce rotating disks with anisotropicpressures (p' .= pR). These objects may therefore be described in terms of stellar dynamics.It should be then checked that there exist a stellar equilibria compatible with a given vacuumfield and a given cut z = f(R). Here a general procedure to derive all distribution functionscorresponding to a specified stress energy tensor is presented, generalising the results of AppendixB to disks with non-zero mean rotation. The detailed description of the dynamics of the diskfollows.In Vierbein components, the stress energy tensor reads

T ()%) =# #

f (,, h) P ()) P (%)

P (t)dP (R)dP (') , (5.5.1)

where f(,, h) is the distribution of stars at position R, * with momentum P (R), P ('). For astationary disk, it is a function of the invariant of the motion ,, h. Now for the line element(5.2.4):

d)2 =# e2 + (dt # ( d*)2 + e25#2 +?1 + f $2

@dR2 + R2e#2 + d*2, (5.5.2)

the Vierbein momenta read

P (') = e+ R#1 (h # ( ,) , (5.5.3a)P (t) = e#+ , , (5.5.3b)

P (R) =A,2e#2+ # e2+ R#2 (h# ( ,)2 # 1

B1/2. (5.5.3c)

Section 5.5 Distribution functions for rotating super-massive disks 105

Calling 1 = h/, and 8 = 1/72, Eqs. (5.5.3) becomes

P (') = e+ R#1 8#1/2 (1# () , (5.5.4a)P (t) = e#+ 8#1/2 , (5.5.4b)

P (R) = 8#1/2Ae#2+ # e2+ R#2 (1# ()2 # 8

B1/2. (5.5.4c)

In terms of these new variables, the integral element dP (')dP (R) becomes

dP (')dP (R) =

OOOOO%P (') %P (R)

%1 %8

OOOOO d1 d8 =

e#+R#1 8#2 d1 d8

2>

e#2+ # e2+ R#2 (1# ()2 # 8. (5.5.5)

Given Eq. (5.5.5) and Eq. (5.5.3b), Eq. (5.5.1) may be rewritten as

T ()%) =# #

P ())P (%) f (,, h) R#1 8#3/2 d1 d8

2>

e#2+ # e2+ R#2 (1# ()2 # 8. (5.5.6)

In particular,

RT (tt) e2+ =# #

f (,, h) 8#5/2 d1d8

2>

e#2+ # e2+ R#2 (1# ()2 # 8, (5.5.7a)

R2 T (t') =# #

(1# ()f (,, h) 8#5/2 d1 d8

2>

e#2+ # e2+ R#2 (1# ()2 # 8, (5.5.7b)

R3 T ('') e#2+ =# #

(1# ()2f (,, h) 8#5/2 d1 d8

2>

e#2+ # e2+ R#2 (1# ()2 # 8. (5.5.7c)

Note that given Eqs. (5.5.7), T (RR) follows from the equation of radial support.Let

F (1, R) =12

#f (,, h)8#5/2 d8

>e#2+ # e2+ R#2 (1# ()2 # 8

=Y#

1

f$ (1, 8) d8

2,Y # 8

, (5.5.8)

where f$ (1, 8) = f (,, h)8#5/2 , and Y (R,1) = e#2+ # e2+ R#2 (1# ()2 . (5.5.9)

The set of Eqs. (5.5.7) becomes

RT (tt) e2+ =#

F (1, R) d1 , (5.5.10a)

R2 T (t') =#

F (1, R) (1# () d1 , (5.5.10b)

R3 T ('') e#2+ =#

F (1, R) (1# ()2 d1 . (5.5.10c)

Any positive function F (1, R) satisfying the moment constraints Eqs. (5.5.10) corresponds toa possible choice2. Note that the dragging of inertial frames requires to fix three moments ofthe velocity distribution to account for the given energy density, pressures and rotation law, incontrast with the Newtonian inversion described in chapter III, or the relativistic static inversion

2 For instance, a truncated Gaussian parameterised in width, mean, and amplitude similar to that constructedin section 3 of chapter 3.


described in appendix B. Let F (1,Y) = F (1, R), where Y(R,1) is given by Eq. (5.5.9). Writtenin terms of F , the integral equation Eq. (5.5.8) is solved for f$ by an Abel transform

f$ (1, 8) =2!

.#

1

;%F

%Y

<

1

dY,8# Y

. (5.5.11)

The latter integration may be carried in R space and yields

f (,, h) =2,4!

# !%F

%R

"

1

dR># (P (R))2

, (5.5.12)

where P (R) is given by Eq. (5.5.3c) as a function of R, ,, and h. Note the minus sign in front of%P (R)

&2. Here F is chosen to satisfy Eq. (5.5.10) which specifies the stress energy componentsT ()%). All properties of the flow follow in turn from f(,, h). For instance, the rest mass surfacedensity, ', may be evaluated from the detailed “microscopic” behaviour of the stars, leading toan estimate of the binding energy of the stellar cluster.

An equation of state for these rotating disks with planar anisotropic pressure tensors isalternatively found directly while assuming (somewhat arbitrarily) that the fluid correspondsto the superposition of two isotropic flows going in opposite directions. In other words, theanisotropy of the pressures measured in the frame co-moving with the mean flow V is itselfinduced by the counter rotation of two isotropic streams such as those described in section 4.This prescription gives for ' :

' =,# p' # pR +

>(, # p' # pR) (,# p' + 3pR)

2=

1# V 20

, (5.5.13)

where V0 =>

(p' # pR)/(,+ pR) is the counter rotating velocity measured in the frame co-moving with V which induces p' .= pR. In the classical limit, ' ) (, # p')/

=1# V 2

0 . Thebinding energy of these rotating disks is computed from Eq. (5.5.13) together with Eqs. (5.3.11)and (5.2.15).

· • !" ! • ·

5.6 Application: rotating disks

The problem of constructing disks with proper rotation and partial pressure support ismore complicated but physically more appealing than that of counter rotating solutions. It isnow illustrated on an internal solution for the Kerr metric, and a generalisation of the solutionsconstructed in the previous chapter to warm rotating solutions is sketched.

Section 5.6 Application: rotating disks 107

5.6.1 A Kerr internal solution

The functions (4, 5,() for the Kerr metric in Weyl-Papapetrou form expressed in terms ofspheroidal co-ordinates (R = s

=(x2 5 1)(1# y2), z = sxy) are given by

e2+ =#m2 + s2 x2 + a2 y2

(m + s x)2 + a2 y2, (5.6.1)

e25 =#m2 + s2 x2 + a2 y2

s2 (x2 5 y2 ), (5.6.2)

and( = #2 a m (m + s x) (1# y2)

#m2 + s2 x2 + a2 y2, (5.6.3)

where s = (±(m2 # a2))1/2. Here ± corresponds to the choice of prolate (+), or oblate (#)spheroidal co-ordinates corresponding to the cases a < m and a > m respectively. In this setof co-ordinates, the prescription given in section 2 leads to completely algebraic solutions. Thenormal derivative to the surface z = f(R) reads (when a > m)

(%/%N) = N (x, y) (%/%x) + N (y, x) (%/%y) , (5.6.4)

whereN (x, y) =

%x2 # 1

& +y/s + x

%y2 # 1

&f $/R

,/%x2 # y2

&.

Di!erentiating Eqs. (5.6.1) through (5.6.3) with respect to x and y together with Eq. (5.6.4)leads via Eqs. (5.2.15) to all physical characteristics of the Kerr disk3 in terms of the Kerr metricparameters m and a, and the function z = f(R) which must be chosen so as to provide relevantpressures and energy distributions according to Eqs. (5.3.12).

On the axis R = 0, f & b, and f$$ & #c, while Eqs. (5.3.12) imply f$ = 0 at (x = b/s, y =1). This in turn implies

(p')0 = (pR)0 =c

4!

.a2 + b2 #m2

a2 + (b + m)2, (5.6.5a)

(,)0 =14!

.a2 + b2 #m2

a2 + (b + m)2

3

42m

A(b + m)2 # a2

B

?a2 + (b + m)2

@(a2 + b2 #m2)

# c

5

6 , (5.6.5b)

where ( )0 stands for ( ) taken at R = 0. The constraint that all physical quantities shouldremain positive implies

b +,

m2 # a2, (5.6.6a)

c % 2m(b + m)2 # a2

?a2 + (b + m)2

@(a2 + b2 #m2)

. (5.6.6b)

Equation (5.6.6a) is the obvious requirement that the surface of section should not enter thehorizon of the fictitious source. Similarly, ergoregions will arise when the cut z = f(R) enterthe torus !

z

m

"2

+!

R

m# a

m+

12

m

a

"2

=!1

2m

a

"2

. (5.6.7)

3 Bicak & Ledvinka (1993) [12] have constructed independantly a cold Kerr solution when a < m.


R

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

Azimuthal Velocity

0 1 2 3 4 50

0.001

0.002

0.003

0.004

0.005

Radial Pressure

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

Radial/Azimuthal Pressure

0 1 2 3 4 50

0.025

0.05

0.075

0.1

0.125

0.15

0.175

Relative Kinetic Support

(b)(a)

(c) (d)

Figure 5.6.1: the azimuthal velocity (a), the radial pressure (b), the ratio of radial over azimuthalpressure and the relative fraction of Kinetic Support (d) namely ,V 2/(,V 2 + p' + pR) for the a/m = 0.5Kerr disk as a function of circumferential radius Rc, for a class of solutions with decreasing potentialcompactness, b/m = 1.1, 1.2 · · ·1.7, and relative radial pressure support 2/m = 1/5 + 2(b/m# 1)/5 .

The analysis may be extended beyond the ergoregion via the ZAMO frame. The characteristicsof these frames are given by Eq. (5.3.7), (5.3.8) and (5.3.9). The central redshift

1 + zc = e#+ =(m + b)2 + a2

b2 + a2 #m2, (5.6.8)

can be made very large for such models constructed when b),

m2 # a2.

On Fig. 5.6.1, Fig. 5.6.2 and Fig. 5.6.3, some characteristics of these Kerr disks with botha > m and a < m are illustrated. For simplicity, pressure is implemented via the cut (5.4.21).These disks present anisotropy of the planar pressure tensor (p' .= pR).

The anisotropy follows from the properties of the ( vacuum field which gives rise to thecircular velocity curve. Indeed, in the outer part of the disk, the specific angular momentum htends asymptotically to a constant. The corresponding centrifugal force is therefore insu#cientto counter-balance the field generated by the 4 function. As this construction scheme generatessuch equilibria, the system compensates by increasing its azimuthal pressure away from isotropy.When a > m, and 2 = 0 the height of the critical cut corresponding to the last disk with positivepressures everywhere scales like a (a2 #m2)1/2 in the range 2 % a/m % 20. Note that this limitis above that of the highly relativistic motions which only occurs when b tends asymptoticallyto the Kerr horizon (a2 #m2)1/2. The binding energy which follows Eq. (5.5.13) reaches valuesas high as one tenth of the rest mass for the most compact configurations.


R

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

Azimuthal Velocity

0 1 2 3 4 50

0.0005

0.001

0.0015

0.002

Radial Pressure

0 1 2 3 4 50

0.5

1

1.5

2Radial/Azimuthal Pressure

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6


(b)(a)

(c) (d)

Figure 5.6.2: as in Fig. 5.6.1 with a/m = 0.95, 2/m = 1/15 + (b# 1)/5. Note the large relative kineticsupport.

5.6.2 Other rotating solutions

The rotating disk models given in section 2 require prior knowledge of the corresponding vacuumsolutions. However, while studying the symmetry group of the stationary axially symmetricEinstein Maxwell equations, Hoenselaers, Kinnersley & Xanthopoulos (1979) [46,47] found amethod of generating systematically complete families of stationary Weyl-Papapetrou vacuummetrics from known static Weyl solutions. Besides, the decomposition of the Weyl potential40 into line densities provides a direct and compact method of finding solutions to the staticfield. The N+1 rank zero HKX transform is defined as follows: let 40 and 50 be the seed Weylfunctions given by Eqs. (5.4.2). HKX define the N 1 N matrix # parameterised by N twistparameters &k and N poles, ak, k = 1, · · ·N as

%(±&

k,k! = i&ke%k

rk

'rk # rk!

ak # ak!± 1

(, (5.6.9)

with rk = r(ak) =>

R2 + (z # ak)2 and .k = .(ak) , . being a function satisfying (5.6.11a)

below. Defining r = Diag(rk), they introduce the auxiliary functions

D± =OO1 + #±OO , and L± = 2D± tr

A%1 + #±&#1 #± r

B,

(± = D+ ± e2+0D#, and M± = 9(± + 2%L+ 5 e2+0L#

&,

which lead to the Weyl-Papapetrou potentials

e2+ = e2+0 6'D#

D+

(, (5.6.10a)


R

0 200 400 600 800 1000 12000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Azimuthal Velocity

0 200 400 600 800 1000 1200

0

-101. 10

-102. 10

-103. 10

-104. 10

-105. 10

Azimutal Pressure

0 200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1


0 200 400 600 800 1000 12000

2

4

6

8

10

12

14

Angular Momentum

(b)(a)

(c) (d)

Figure 5.6.3: the azimuthal velocity (a), the azimuthal pressure (b), the relative fraction of KineticSupport (c), namely ,V 2/(,V 2 + p') and the angular momentum (d) for the a/m = 10 zero radialpressure Kerr disk as a function of circumferential radius Rc, for a class of solutions with decreasingpotential compactness b/m = 210, 260 · · ·510. The first curve corresponds to a cut which induces negativeazimuthal pressure (tensions).


e25 = ke250 6AD#D+!B

, (5.6.10b)

( = 279M!

+ ((+ + (#)#M# ((+ + (#)!

|(+|2 # |(#|2

:

, (5.6.10c)

where 6 and 7 stand respectively for the real part and the imaginary part of the argument and( )! represents the complex conjugate of ( ). k is a constant of integration which is fixed bythe boundary conditions at infinity. The two yet unspecified function . and 9 satisfy:

2. (ak) =1rk

([z # ak]2+ R e' 12) 40, (5.6.11a)

29 = R e' 1240, (5.6.11b)

where 2 = (%/%R, %/%z) and e' is the unit vector in the * direction. The linearity of theseequations suggests again that solutions should be sought in terms of the line density, W (b),characterising 40, namely

. (ak) =#

W (b)ak

9R2 + (z # b# ak)

2

R2 + (z # b)2

: 12

db, (5.6.12a)

9 =#

W (b) (b# z)>R2 + (z # b)2

db. (5.6.12b)

It is therefore a matter of algebraic substitution to apply the above procedure and construct allnon-linear stationary vacuum fields of the Papapetrou form from static Weyl fields. Following theprescription described in section 2, the corresponding disk in real rotation is then constructed.The requirement for the physical source to be a disk is in e!ect a less stringent condition onthe regularity of the vacuum metric in the neighbourhood of its singularity because only thehalf space which does not contain these singularities is physically meaningful. A suitable Ehlerstransformation on D#/D+ ensures asymptotic flatness at large distance from the source.

To illustrate this prescription consider the vacuum field given by Yamazaki (1981) [113].This field arises from 2 HKX rank zero transformation on the Zipoy-Voorhess static metric givenby

40 = 12+ log

(x# 1)(x + 1)

, (5.6.13a)

50 = 1

2+2 log

(x# 1)(x2 # y2)

, (5.6.13b)

in prolate spheroidal co-ordinates. It corresponds to a uniform line density between ±+. Thefunctions . and 9 follow from Eqs. (5.6.12). Choosing poles ak, k = 1, 2 at the end of the rod±+, leads to

. (a±) = 12+ log

x2 # 1[x5 y]2

, (5.6.14a)

9 = 2+y. (5.6.14b)

Equations (5.6.10) together with (5.6.13) and (5.6.11) characterize completely the three Weyl-Papapetrou metric functions 4, 5, and(; all physical properties of the corresponding disks follow.For instance, Fig. 5.6.4 gives the zero radial pressure velocity curve of the Yamazaki disk spanningfrom the Schwarzschild (+ = 1/2) metric when &1 = &2.


0 5 10 15 20R

0

0.025

0.05

0.075

0.1

0.125

0.15

Azimuthal Velocity

Figure 5.6.4: the azimuthal velocity for the a/"Y zero radial pressure Yamazaki disk as a functionof circumferential radius Rc, for a class of solutions with decreasing potential compactness b/"Y =3.6, 3.8, · · ·5. "Y is the natural unit of distance in these disks as defined by Yamazaki’s Eq. (Y-2) [113].

More generally, the extension of this work to the construction of disks with planar isotropicpressures should be possible by requiring that Eq. (5.2.15c) is identically equal to Eq. (5.2.15d)given Eq. (5.6.10).

· • !" ! • ·

5.7 Conclusion

The general counter rotating disk with partial pressure support has been presented. Thesuggested method for implementing radial pressure support also applies to the construction ofstationary axisymmetric disk with rotation but requires prior knowledge of the correspondingvacuum solution. In this manner, a disk-like source for the Kerr field has been constructed. Thecorresponding distribution reduces to a Keplerian flow in the outer part of the disk and presentsstrong relativistic features in the inner regions such as azimuthal velocities close to that of light,large central redshift and ergoregions. The disk itself is likely to be stable against ring formationand the ratio of its binding energy to rest mass can be as large as 1:10. The broad lines of how toconstruct all rotating disks arising from HKX-transforming the corresponding counter rotatingmodel into a fully self-consistent model with proper rotation and partial pressure support hasbeen sketched. It should be a simple matter to implement this method with the additionalrequirement that the pressure remains isotropic with a sensible polytropic index. One couldthen analyse the fate of a sequence of gaseous disks of increasing compactness and relate it tothe formation and evolution of quasars at high redshift. Alternatively, the inversion methoddescribed in section 5 yields a consistent description of all disks in terms of stellar dynamics.In those circumstances, the inversion method for distribution functions presented in section 5yields a complete description of the detailed dynamics of these objects. It would be worthwhileto construct distribution functions for the Kerr disk presented in section 6.

· • !" ! • ·

D

Aspects of the dynamics

of relativistic disks

D.1 Equation of state for a relativistic 2D adiabatic flow

The equation of state of a planar adiabatic flow is constructed here by relating the perfect fluidstress energy tensor of the flow to the most probable distribution function which maximisesthe Boltzmann entropy.1 These properties are local characteristics of the flow. It is thereforeassumed that all tensorial quantities introduced in this section are expressed in the local Vierbeinframe.

Consider a infinitesimal volume element d8 defined in the neighbourhood of a given event,and assume that the particles entering this volume element are subject to molecular chaos.The distribution function F of particles is defined so that F (R, P) d8d2P gives the number ofparticles in the volume 8 centred on R with 2-momentum pointing to P within d2P .

The most probable distribution function F $ for these particles is then given by that whichmaximises Boltzmann entropy S subject to the constraints imposed by the conservation of thetotal energy momentum and by the conservation of the number of particles within that volume.This entropy reads in terms of the distribution F

S = #d8#

F logF d2P . (D.1.1)

The stress energy tensor corresponds to the instantaneous flux density of energy momentumthrough the elementary volume d8 (here a surface)

T)b =#

F $P)P % d2P

|E| , (D.1.2)

where the 1/|E| factor accounts for the integration over energy-momentum space to be restrictedto the pseudo-sphere P)P) = m2. Indeed the detailed energy-momentum conservation requiresthe integration to be carried along the volume element

#+%P)P) #m2

&d3P =

#+?P 02 # E2

@d2PdP 0 =

d2P

|E|, (D.1.3)

1 Synge [100] gives an extensive derivation of the corresponding equation of state for a 3D flow.

114 Aspects of the dynamics of relativistic disks Appendix D

where E =,

m2 + P2. It is assumed here that momentum space is locally flat. Similarly thenumerical flux vector reads

*)$ =#

F $P) d2P

|E| . (D.1.4)

The conservation of energy-momentum of the volume element d8 implies that the flux of energymomentum across that volume should be conserved; this flux reads

T)% n% d8 = d8#

F $P) d2P = Const. , (D.1.5)

where n% is the unit time axis vector. Keeping the population number constant provides thelast constraint on the possible variations of S

*)$n)d8 = d8#

F $d2P = Const. . (D.1.6)

Varying Eq. (D.1.1) subject to the constraints (D.1.5) (D.1.6), and putting +S to zero leadsto

(logF $ + 1) +F$ = a+F$ + 3)P)+F $ , (D.1.7)

where a and 3) are Lagrangian multipliers corresponding resp. to (D.1.5) (D.1.6). Thesemultipliers are independent of P a but in general will be a function of position. The distributionwhich extremises S therefore reads

F $ = C exp (3)P)) . (D.1.8)

The 4 constants C and 3a are in principle fixed by Eqs. (D.1.5) and (D.1.6). The requirement forF $ to be Lorentz invariant – that is independent of the choice of normal n) – and self consistent,is met instead by demanding that F $ obeys Eq. (D.1.2) and (D.1.4) , namely

C#

P) exp (3µPµ)d2P

|E| = *)$ , (D.1.9)

C#

P)P % exp (3µPµ)d2P

|E| = T)% , (D.1.10)

where *)$ and T)% satisfy in turn the covariant constraints

%T)%

%x%= 0 and (D.1.11a)

%*)$%x)

= 0 . (D.1.11b)

Equations (D.1.9)-(D.1.11) provide a set of thirteen equations to constraint the thirteen func-tions C, 3), *)$ , and T)%. Equations (D.1.9) and (D.1.10) may be rearranged as

*)$ = C%"%3)

, T)% = C%2"

%3)%3%, (D.1.12)

where the auxiliary function " is defined as " =*

exp(3)P)) d2P/|E|; " is best evaluated usingpseudo polar coordinates corresponding to the symmetry imposed by the energy momentumconservation P)P) #m2 = 0; these are

P 1 = m sinh1 cos* ,

P 2 = m sinh1 sin* ,

P 0 = im cosh1 ,

(D.1.13)

Section D.2 The relativistic epicyclic frequency 115

where the coordinate 1 is chosen so that the normal n) lies along 1 = 0. In terms of thesevariables, d2P/|E| then reads

d2P/|E| = dP1dP 2/m cosh1 = m sinh1d1d* . (D.1.14)

Moving to a temporary frame in which 30 is the time axis ( 30 = i3 = i(#3)3))1/2, 3k = 0 k =1, 2), " becomes

" = 2!m# "

0exp (#m3 cosh1) sinh1d1 =

2!3

exp (#3) . (D.1.15)

From Eq. (D.1.9)-(D.1.10) and (D.1.15), it follows that

T)% =2!C

33exp (#3)

9Y32 + 33+ 3

Z 3)

3

3%

3+ {3+ 1} +)%

:

, (D.1.16)

*)$ =2!C

33exp (#3) {3+ 1}3) . (D.1.17)

On the other hand, the stress energy tensor of a perfect fluid is

T)% = (,0 + p0)U)U% + p0 +)% , (D.1.18)

while the numerical surface density of particles measured in the rest frame of the fluid ' isrelated to *)$ via

' = #*)$U) . (D.1.19)Eq. (D.1.16) has clearly the form of Eq. (D.1.18) when identifying

U) = 3)/3 , (D.1.20a)

,0 + p0 =2!C

33exp (#3)

Y32 + 33+ 3

Z, (D.1.20b)

p0 =2!C

33exp (#3) {3+ 1} . (D.1.20c)

Eliminating 3,C between Eqs. (D.1.17)–(D.1.20c) yields the sought after equation of state

,0 = 2 p0 +'

1 + p0/'. (D.1.21)

where p0 = '/3. This is the familiar gas law given that 1/3 is the absolute temperature.The equation of state Eq. (D.1.21) corresponds by construction to an isentropic flow. Indeed,using the conservation equations Eq. (D.1.11a) dotted with U), and Eq. (D.1.11b) together withU)U) = #1 yields after some algebra to the identity

3d

ds

'(32 + 33+ 3)3 (3+ 1)

(+

13

= #%U)

%x)=

1'

d'ds

, (D.1.22)

where d/ds = (dx)/dx))(%/%x)) is the covariant derivative following the stream lines. Writingthe r.h.s of Eq. (D.1.22) as an exact covariant derivative leads to the first integral

d

ds

' 11 + 3

# 2 log (3) + log (1 + 3)# log (')(

= 0 . (D.1.23)

Given that p0 = '/3, it follows!1 +

p0

'

"p0

'exp

! 11 + '/p0

" 1'

= Const. (D.1.24)

which in the low temperature limit gives '#2p0 = Const. This corresponds to the correct adia-batic index / = (2 + 2)/2.


D.2 The relativistic epicyclic frequency

The equation for the radial oscillations follows from the Lagrangian

L = #[

e2+?t# (*

@2

# e25#2+ (1 + f $2) R2 # e#2+ *2R2, (D.2.1)

where·

( ) stands here for derivation with respect to the proper time - for the star describingits orbit. In Eq. (D.2.1), * and t are ignorable which leads to the invariants:

%L%*

= h%L% t

= #7. (D.2.2)

The integral of the motion for the radial motion follows from Eqs. (D.2.2) and UµUµ = #1,namely:

e25#2+ %1 + f $2& R2 +e2+

R2(h# 7 ()2 # 72e#2+ = #1

This equation, together with its total derivative with respect to proper time provides the radialequation of motion, having solved for the angular momentum h and the energy 7 of circular orbitsas a function of radius when equating both R and R to zero. the relativistic generalisation ofthe classical epicyclic frequency is defined here to be the frequency of the oscillator calling backlinear departure from circular orbits. Hence the equation for radial perturbation reads

+R + "2+R = 0, (D.2.3)

which gives for "2 :

"2 =e#2Z%Z$$ # 2Z$2& # 72e#2L

%L$$ # 2L$2&#

e2KA

(h# 7()2%K $$ + 2K$2& # 7 (h# 7() (4K$($ + ($$) + 72($2

B,

(D.2.4)

where ( )$ stands in this Appendix for d/dR & %/%R + f$ %/%z and L, K, andZ are given interms of the potential 5 and 4 by:

L = 5 + log=

1 + f $2, (D.2.5a)Z = 5 + log

=1 + f $2 # 4, (D.2.5b)

K = 24 # 5 # log=

1 + f $2 # logR. (D.2.5c)

Here 7 and h are known functions of R given by the roots of:

R = 0 4 e#2L72 = (h # 7 ()2 e2N + e#2Z, (D.2.6a)R = 0 4 e#2ZZ$ = e#2LL$72+ (D.2.6b)

(h # 7() e2N [(h# 7 () N $ # 7 ($] .

Section D.3 Christoffel Coefficients for the WP metric 117

D.3 Christo!el Coe"cients for the WP metric

The connection coe#cients of the Weyl-Papapetrou metric are given by:

(tRt =

%4

%R+

12 R2

e4 + (%(

%R, (D.3.1)

(tzt =

%4

%z+

12 R2

e4 + (%(

%z, (D.3.2)

(t'R =

(

R# 2(

%4

%R# 1

2%(

%R# 1

2 R2e4 + (2 %(

%R, (D.3.3)

(t'z = #2(

%4

%z# 1

2%(

%z# 1

2 R2e4 + (2 %(

%z, (D.3.4)

(Rtt = e4 +#25 %4

%R, (R

RR = # %4%R

+%5

%R, (D.3.5)

(RzR = #%4

%z+%5

%z, (R

zz =%4

%R# %5

%R, (D.3.6)

(R't = e4 +#25

!#!(%4

%R

"# 1

2%(

%R

", (D.3.7)

(R'' =

!#R + R2 %4

%R

"e#25 + e4 +#25

!(2 %4

%R+ (

%(

%R

", (D.3.8)

(ztt = e4 +#25 %4

%z, (z

RR =%4

%z# %

5

%z, (D.3.9)

(zzR = # %4

%R+%5

%R, (z

zz = #%4%z

+%5

%z, (D.3.10)

(z't = e4 +#25

!#!(%4

%z

"# 1

2%(

%z

", (D.3.11)

(z'' =

!R2 %4

%z

"e#2 5 + e4 +#25

!(2 %4

%z+ (

%(

%z

", (D.3.12)

('Rt =

12 R2

e4 + %(

%R, ('

zt =1

2 R2e4 + %(

%z, (D.3.13)

(''R =

1R# %4

%R# e4 + (

%(

%R

12 R2

, (D.3.14)

(''z = #%4

%z# e4 + (

%(

%z

12 R2

. (D.3.15)

· • !" ! • ·

6

Epilogue

The work described in the previous four chapters presented new aspects of the dynamics offlattened self gravitating systems. It has covered the following aspects:

• Orbital instabilities in galaxies

Instabilities of stellar systems are worth studying because they may account for the struc-ture of observed and numerical data on disks. Unstable configurations are also possible output ofgalaxy formation and stability analysis puts constraints on the viability of models which are sup-posed to account for observed galaxies. Considerable e!orts have been devoted to understandingthe stability of bars within galaxies, but the problem of their formation remains to a large extenta puzzle. This question was addressed here by considering the linear stability of self-gravitatingstellar systems from the viewpoint of instabilities triggered by interactions between orbits. Itwas shown that the set of equations describing the linear instability can be exactly written interms of the interaction of orbital streams, i.e. resonant flows of stars. For a given resonancethis interaction leads to an instability of Jeans’s type. Jeans’s gravitational instability can bethought of as a process in which more and more particles are trapped in a growing potentialwell which may move relative to an inertial frame. In galaxies azimuthal gravitational instabilityworks by trapping the lobes of orbital streams of stars moving azimuthally in a growing potentialwell which may be rotating. Such instabilities can only occur if the orbital lobes respond bymoving in the direction of the torques applied on them. This co-operation occurs at the innerLindblad resonance in the central parts of galaxies. The general picture combines the e!ect ofall resonances compatible with the symmetry of the mode under consideration. This treatmentis formally related to the potential density pair method of Kalnajs, but treats both Boltzmannand Poisson’s equations directly in action space. The basis functions on which the perturbationis expanded then correspond to known families of orbits for which the exact coupling factors de-scribing the interaction have been computed. These correspond to the core of Poisson’s equationwritten in angle and action variables. Identifying the fastest growing mode and its eigen-spaceyields the roots of the orbital instability. The fate of the above orbital instabilities has beeninvestigated and the type of distribution functions expected from the adiabatic re-alignment ofresonant orbits under the assumption of a unique inner Lindblad resonance was derived. It wasassumed that the star trajectories would evolve towards their state of maximum entropy, giventhe constraints fixed by conservation of circulation, overall angular momentum and energy. Thisanalysis yields a critical temperature below which the preferred state of equilibria is that of abarred galaxy which may display negative specific heat. The amplitude and pattern of a bar isthen related to the other macroscopic properties of the flow. Conversely the criteria states thatbarred configurations are unstable to orbital dilution for some range of energies and momenta.

• Distribution functions for flat galaxies: observational implications

120 Epilogue Chapter 6

Simple inversion formulae were given to construct distribution functions for flat systemswhose surface density and Toomre’s Q number profile. These distributions describe stablemodels with realistic velocity distributions for the power law, the Isochrone and the Kuzmindisks. The purpose of these functions is to provide plausible galactic models and assesstheir critical stability with respect to global non-axisymmetric modes. The inversion mayalso be carried out for a given azimuthal velocity distribution (or a given specific energydistribution) which may either be observed or chosen accordingly. This method should beapplied to real galaxies and provides a unique opportunity to confront predictions on theradial velocity distributions with the same observed quantity. The likely residual could beused to assess the departures from axisymmetry or other limiting aspects of the model.Sets of initial conditions for an N-body simulation may be extracted from the distributionfunction. A numerical stability analysis may then be carried out to address the intrinsictendency for the disk to form such non-axisymmetric structures. The properties of thegalaxy when evolved forward in time should provide an unprecedented link between thetheory of galactic disks and the detailed observation of the kinematics of these objectswhich may be achieved with today’s technology.

• Relativistic stellar dynamics of flattened systems and quasar formation

Complete families of new exact analytical solutions of Einstein’s equations have been con-structed in closed form for thin axisymmetric rotating disks derived from new metrics withgradient discontinuities. A general geometrical procedure is presented to implement partialradial pressure support within these disks. These solutions are relevant to astronomy forthe following reasons: firstly, one of these families represents a possible internal solutionfor the Kerr metric which is ultimately the field generated by a rotating black hole. Thisinternal solution had been sought for thirty years; secondly, these models correspond todynamically (locally) stable, di!erentially rotating, relativistic flattened stellar clusters forwhich distribution functions are available. When the pressure is isotropic, these configu-rations describe super-massive disks corresponding to possible models for the latest stageof the collapse of a proto-quasar. The binding energy of these thin disks can reach valuesas high as ten percent of the rest mass, suggesting that the corresponding configurationmight be su#ciently long lived to have observational implications. A natural follow-up ofthis work would be to study the global dynamical stability of these disks. It would alsobe worthwhile to address the secular stability of super-massive disks by studying the e!ectof viscosity and radial angular momentum transport. These would undoubtedly induce arelative broadening of the disk which should be assessed. The corresponding model couldthen be contrasted with that of the spherical collapse of a non-rotating super-massive star.The detailed dynamical properties of massive flattened stellar clusters also deserve furtherwork.

E

Resume

Une fraction importante des disques galactiques observes presentent une structure barreequi se caracterise par des isophotes allonges au dela du bulbe central. L’analyse photometriquede ces isophotes suggere que la barre est constituee d’etoiles agees; la barre ne semble donc pasetre le fruit d’un phenomene transitoire, et la description du systeme en termes de dynamiquestellaire parait appropriee pour comprendre sa formation et son evolution. De gros e!orts ontete consacres ces dernieres annees a l’etude de la stabilite de ces objets mais leur formation restea ce jour un probleme ouvert. S’il est vrai que les simulations numeriques sont sujette a de fortesinstabilites bi-symetriques, seulement quarante pour cent des disques observes sont constituesd’une forte barre. Pourquoi certaines galaxies forment-elles une barre et d’autres pas? J’ai tentede repondre a cette question en menant l’etude de la stabilite d’un systeme stellaire autogravi-tant avec pour point de vue les instabilites induites par l’alignement d’orbites resonnantes. Defait, j’ai montre qu’il est possible de re-ecrire les equations de la dynamique exclusivement en ter-mes de couplage entre des flots d’etoiles decrivant une meme orbite resonante. En portant monattention sur les etoiles appartenant a un flots en resonnance interne de Lindblad, j’ai montreque ce couplage conduisait a une instabilite de type instabilite de Jeans. L’instabilite de Jeanspeut etre interprete comme un processus au cours duquel de plus en plus d’etoiles sont captureespar un potentiel d’amplitude croissante. De la meme maniere, ici l’ instabilite azimutale procedepar capture du lobe des orbites resonnantes par un potentiel tournant. Le taux de precessionde ce potentiel d’amplitude croissante correspond alors au taux de precession de la barre. Cetype d’instabilite n’est possible que si le moment d’inertie adiabatique des orbites pieges estpositif. Dans ce cas, l’orbite resonnante repond dynamiquement a un couple attracteur par undeplacement dans la direction de la source de ce couple. Ce mouvement cooperatif a lieu a laresonance interne de Lindblad dans les parties centrales des galaxies. Le critere d’instabiliteglobal est obtenu en considerant l’e!et de toutes les resonances compatibles avec la symetrie del’instabilite considere. Une relation de dispersion pour la frequence et le taux de croisance dela barre se deduit par projection, soit sur la base des fonctions propres du noyau de couplageentre les orbites dans l’approximation epicyclique, soit sur une base complete de fonctions dedistribution susceptible de bien decrire la geometrie de la barre dans le regime lineaire. Dans cedernier cas, mon analyse s’apparente a la methode des paires densite-potentiel de Kalnajs, maispresente l’avantage de traiter le couplage directement dans l’espace des phases, ce qui permetd’identifier les orbites responsables de l’instabilite. Les caracteristiques de l’onde stationaire dedensite de deduisent par simple integration sur l’espace des vitesses.J’ai par ailleurs etudie le devenir de ce type d’instabilite en construisant la fonction de distri-bution induite par l’alignement adiabatique d’orbites en resonance interne de Lindblad. Cettefonction de distribution est obtenue em maximisant l’entropie du systeme compte tenu des con-traintes de conservation du nombre d’orbites, du moment angulaire et de l’ energie totale dusysteme, et de la conservation adiabatique detaillee de la circulation de chaque etoile decrivant

122 Resume Appendix E

son orbite. On montre alors qu’il existe une temperature critique en deca de laquelle le systemeevolue spontanement vers un etat barre avec une composante bi-symetrique naissante tournanta la frequence predite par l’analyse ci-dessus. Le systeme peut presenter dans ce cas une chaleurspecifique negative; en considerant l’ensemble des etoiles non resonnantes comme une sourcethermique a chaleur specifique positive, on consoit qu’il se produise l’equivalent d’une catas-trophe gravothermale. La barre s’amplifie alors et son taux de precession decroıt pendant quela temperature diminue encore, en accord avec les resultats des simulations numeriques. Il estimportant de noter que ces resultats peuvent aussi bien rendre compte du devenir d’une barregeneree spontanement par instabilite orbitale par exemple, mais aussi expliquer la disparitionadiabatique d’une structure barree qui possederait une temperature d’equilibre en deca de latemperature critique.Il convient maintenant de mettre en oeuvre ce type de formalisme sur des exemples concrets degalaxies realistes, afin d’evaluer la pertinence de ce type d’instabilites pour la formation ou ladestruction des barres galactiques.

L’etat d’equilibre dynamique d’une galaxie est caracterise par une fonction de distribu-tion qui s’exprime en termes des invariants du mouvement de chaque etoile. Cette fonctionpresente un double interet sur le plan theorique et observationnel. Elle permet de contraindreles distributions en vitesse radiales et azimuthales obtenues par deconvolution des profils de raiesd’absorbtion pour rendre compte de la nature gravitationnelle de l’equilibre. Elle constitue lapremiere pierre d’une analyse lineaire ou numerique telle que celle decrite ci dessus pour etudierla stabilite globale des disques galactiques vis-a-vis de modes non-axisymetriques.Dans ce domaine, j’ai trouve une methode d’inversion tres generale qui permet de constru-ire les fonctions de distribution correspondant a une densite surfacique donnee et un profil detemperature choisi. Ces fonctions de distribution conduisent a des modeles localement stablesavec des distributions de vitesse realistes. L’inversion peut aussi etre menee directement a partirde donnees observationnelles comme la distribution en vitesse azimutale ou la distribution enenergie specifique. La mise en oeuvre d’une telle methode d’inversion est prometteuse pour lesraisons suivantes: elle constitue une opportunite unique de confronter la distribution en vitesseradiale observee a la meme quantite deduite independamment via la fonction de distribution,de la distribution en vitesse azimutale. Les residus permettent alors de contraindre le modeleet d’evaluer ses faiblesses. Avec cette methode, il est d’autre part aise de generer les conditionsinitiales d’une simulation numerique correspondant exactement d’un point de vue dynamiqueaux caracteristiques de la galaxie observee. Il serait interessant de proceder a ce type d’analysesimultanement pour une galaxie S0 et pour une galaxie spirale presentant une structure non ax-isymetrique apparente. Les proprietes relatives de ces deux types morphologiques et l’existencepresumee de modes propres d’instabilite azimutale dans le second cas consituraient un lien sansprecedent entre la theorie dynamique des disques galactiques et les observations detaillees deleur cinematique rendues possible par la technologie actuelle. En particulier, cette observationpermettrait d’apporter des elements de reponse pour le probleme du caractere transitoire ouintrinseque de la formation des spirales dans les galaxies.

Lors de l’e!ondrement d’un nuage baryonique de plus 108M&, la conservation du momentangulaire conduit a son applatissement. La contraction se poursuit jusqu’a ce qu’une descriptionrelativiste du coeur devienne necessaire. S’il semble etabli que la phase ultime de l’e!ondrementcorresponde a un trou noir supermassif, il convient d’etudier plus en detail les etapes conduisantsa la formation d’un tel objet.Pour se faire, j’ai construit des familles de solutions analytiques exactes aux equations d’Einsteincorrespondant a des disques supermassifs minces. Ces solutions sont etablis de maniere geome-trique en identifiant de part et d’autre d’un plan de symetrie les points decrivant deux sur-faces de sections traversant un champ du vide donne. Cette identification definit le saut desderivees normales du champ a partir desquelles les proprietes du disque se deduisent. J’ai etudiela nature des sections conduisant a une solution physique. Pour une metrique statique, j’ai

Appendix E Resume 123

montre que le champ du vide peut etre construit par superposition lineaire de sources fictivesde part et d’autre du plan de symetrie en analogie avec la demarche newtonienne. La solutioncorrespondante peut alors etre interpretee en termes de deux flots stellaires en contre rotation.Le choix d’une section a courbure extrinseque nulle conduit a des disques denues de pressionradiale. Tout profil courbe induira au contraire un disque susceptible d’etre stable vis a vis desinstabilites radiales. En raison de la complexite induite par les forces gravomagnetiques (quientrainent les reperes inertiels), il n’existe pas de solution stationnaire generale pour laquelle lacomposante non diagonale de la metrique peut etre choisie independamment. Neanmoins, il estpossible de generer par transformation algebrique des familles completes de metriques station-naires a partir de solutions statiques connues. On construit de cette maniere un disque avecune distribution d’energie et de pression et une courbe de vitesse qui se deduit du choix de lametrique du vide dans laquelle le disque est plonge. A grande distance, ces disques deviennentnewtoniens mais presentent dans leur region centrale des proprietes relativistes (decalage versle rouge important, ecoulement luminique, ergoregions etc ...). J’ai transpose dans ce contextela methode d’inversion newtonienne presentee ci-dessus pour construire les fonctions de distri-bution compatibles avec un ecoulement relativiste donne.Ce type de solutions presente un interet a la fois astrophysique et theorique. La generalisation duformalisme de la dynamique stellaire des disques minces au cadre de la relativite d’Einstein, et ladescription statistique detaillee de l’ecoulement pour un ensemble tres general de solutions rela-tivistes constitue un progres theorique important, en particulier au vu du nombre tres restreintde solutions physiques aux equations d’Einstein a symetrie non spherique. A titre d’exemple,une de ces solutions represente une source physique interne pour la metrique de Kerr qui corre-spond ultimement au champ genere par un trou noir en rotation. Cette solution interne a faitl’objet d’une recherche continue depuis trente ans. Ces solutions sont plus generalement suscep-tibles de decrire des amas stellaires relativistes en rotation di!erentielle localement stables. Lesconfigurations a pression isotrope correspondent eventuellement a des disques supermassifs quel’on peut associer a l’etape d’e!ondrement final d’un proto-quasar. J’ai calcule la fraction del’energie de masse qui est rayonnee lors de l’e!ondrement de ces objets. Celle-ci peut atteindredix pour cent de la masse au repos du disque, ce qui suggere que ces derniers ont une duree devie su#sante pour avoir des consequences observationnelles.Il convient maintenant d’etudier la stabilite dynamique globale de ces disques. Il serait aussisouhaitable d’etudier leur stabilite seculaire en ajoutant une description de la viscosite et dutransport radial du moment angulaire. Ce type d’instabilite devrait induire un epaississementdu disque dont il faudrait tenir compte.

124 Resume Appendix E

Acknowledgements

It is a great pleasure to thank my supervisor, Donald Lynden Bell, for his enthusiasm and constantavailability. It has been an overwhelming experience to learn so much from such an extraordinary person.I am also very grateful to Francoise Combes for her e!cient supervision from abroad.

I would like to thank Robert Cannon warmly for his kind friendship and constant help on so manyaspects of my life in the last three years, ranging from relativity to the repairing of hoovers. I am verygrateful to Jim Collett for his precious advice, physical insight and kindness. Let me thank also GuinevereKau"man for her enlightening feminine touch and friendship, Rodrigo Ibata for his curious mind andgenerous attention, and Martin Haehnelt for his patience and continental solidarity. Thanks to Cookyfor countless discussions about “the revolution” and to Ravi Sheth for his peaceful approach and concern.Also worthy of special recognition are David Earn for his positive criticism, Robin Williams for enlightingconversations, Derek Richardson for his skilful help with Unix “en version francaise”, Andrew Dunn for hisrefreshing solipist philosophy, Simon White for his support, advice and understanding in both languages,Isobel Hook for her good mood, Erik Stengler for his sunny accent, Geraint Lewis for the Welsh touch,Angel Lopez-Sainz for Thesis, Mike Hudson and Caleb Scharf for their even tempered approach to beingfourth year students of Donald, Martin Becket for all his help with fixing PCs and macs, Blane Little forlate night discussions and Dave Syer for introducing me to Mathematica.

My work at the Institute of Astronomy has been greatly facilitated by the kindness and smoothe!ciency of Michael Ingham and Judith Moss and by Alice Duncan’s helpful availability.

Special thanks are due to Jiri Bicak for his energetic introduction to the the relativity of counter-rotating disk, and to Martin Rees and Gerry Gilmore who have given me much valuable advice.

Je souhaite aussi remercier Christian Boily pour son soutient de francophone face aux “envahisseurs”,Ian Bonnell et Gisele pour leur humeur joviale et leur patriotisme quebequois plus vrai que nature, Jean-Francois Sygnet pour son flegme tres parisien, Michel Tagger pour ses conseils de carriere et son approchepolemique de la theorie des ondes spirales, et Pierre Encrenaz pour ses encouragement a pousuivre danscette voie. Un grand merci a Stephane Colombi pour son language corosif et les journees passees a meconvaincre des merites de la recherche et a Frederique Dizier pour les heures de telephone pour me per-suader du contraire. Merci aussi a Jonathan pour son entetement a vouloir reconstruire le monde surdes bases plus saines apres chaque repas. et a Laurent dont l’amitie et les longues conversations parcourrier electronique m’ ont ete d’un grand secours. Je tiens a remercier Michel Tallon pour sa patiencea m’enseigner quelques rudiments de physique dans la bonne humeur, et Eric Thiebaut dont l’entraintforce l’admiration. Un grand merci a Renaud et Francoise pour leur gentillesse et leur hospitalite maintefois mise a l’epreuve.

It has been a real pleasure to spend time with with Steve, Robert, Ted, Karl, Richard, Kristian,Christopher, Ariane, Jamila, Francine, Patrick, Guy, Juliet, Maryvonnes, Fabienne and Cathy. I amgrateful to Margaret for making England just like home, Marcus and Debby for their hospitality andkindness, Mark for his brilliant skills on the deck, the pitch, and the pub, and Greg for his enthusiasmfor football and the French revolution. From all these people I have leant a lot about friendship, kindnessand sometimes even physics!

My research was generously supported by the Ecole Normale Superieure de Lyon, the Ministere dela Recherche, the British Council and the Isaac Newton Studentship.

Let me express my gratitude to my parents: they have supported and encouraged me to follow myinterests at every stage of my life.

Finally, a very special thank thank to Heloıse, Jean and Brigit for constantly reminding me whatreal life is all about.

CambridgeFebruary 1994

126 Acknowledgements

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