An Investigation of Mysterious Oscillations in Black Hole
Accretion Disks
Adam Levine
Dept. of Physics, Stanford University, Stanford, CA 94305-4060
First Reader: Robert Wagoner Second Reader: Roger Blandford
We give an overview of the physics and observational signatures of accretion disks
surrounding black holes. By observing the hot, accreting matter falling into the
black hole, astronomers have discovered broad peaks in the X-ray power spectral
densities of these objects known as quasi-periodic oscillations (QPOs). We detail
this phenomenon and discuss various theoretical models proposed to explain it. We
also present results on the role of the magnetic fields on normal mode trapping in
the disk. Finally, while most QPOs have been observed in stellar mass black hole
binaries (M ∼ 1− 100M), we will also elaborate on the observational signatures of
super-massive (M ∼ 106 − 1010M) black holes in active galactic nuclei (AGN) and
the possibility of using optical power spectra to hunt for QPOs.
2
ACKNOWLEDGEMENTS
I would like to thank the friends and family that have made this thesis possible.
My research over the past two years has been fueled by their constant support. Thank
you Mom and Dad for letting me try to explain new physics concepts over the phone
(and for being the best parents!). Thanks to Amy, Luke, Lily, Sebastian, Benjy,
Johnny, Lucio and everyone else at TDX for helping me to lighten up and take my
mind off physics (at least for a little). Thank you especially to Lucio for the espresso.
Thanks to Jack, Ikshu, Conrad, Hart and all the other physics and math majors
that have taught me almost as much as my professors. I will never forget the late
nights on the fourth floor and in XΘX. I can’t wait to go to conferences and to keep
talking about math/physics for the rest of our lives.
I would like to thank the Stanford Physics Department and the Office of Un-
dergraduate Advising and Research for supporting my research over the past two
summers. Also, thank you to Roger Blandford for agreeing to read this thesis.
Last, but most certainly not least, I must thank Bob Wagoner for being my aca-
demic guide for the past two years. His constant support and patience has been
invaluable. I would be happy with even a small fraction of his intuition about black
holes. He stuck with me through thick and thin and successfully kept me posted
on every movement of Rory McIlroy. Bob is the reason undergraduates come to a
research university: to learn from and make a lasting connection with a leader in your
field.
This thesis is dedicated to my grandfather, who passed away this summer. I will
miss his Science magazine clippings and listening to jazz with him, but will never
forget the impact he had on my interest in science.
As we begin doing physics, it is worthwhile to remember one of Bob’s favorite
sayings: “Whatever you do, don’t panic!”
3
CONTENTS
Acknowledgements 2
I. Introduction 4
II. High Frequency Quasi-Periodic Oscillations Near Black Holes 5
A. Observations 5
B. Modeling HFQPOs 7
III. Dynamics around a Kerr Black Hole 9
IV. Accretion Disk Normal Modes 10
A. Modes without Magnetic Fields 10
B. The Effect of Magnetohydrodynamics on Normal Modes 14
V. Probing the Inner Accretion Disk of AGN 19
A. Radiative Transfer in Black Hole Accretion Disks 19
B. Observation of the g-Mode in AGN 23
C. Observing with Kepler II 28
VI. Conclusion 29
References 31
4
I. INTRODUCTION
Arguably the most profound consequence of Einstein’s general theory of relativity
(GR) is the existence of mysterious objects famously known as black holes. Roughly
20 X-ray sources have been declared black hole binaries (BHBs), with an estimated
mass that rules out the presence of neutron stars or white dwarfs. A list of these
sources can be found in Remillard & McClintock (2006). In this thesis, we give an
overview of the physics and observational signatures of black hole accretion disks and
their mysterious oscillations. We will also focus on the properties of Active Galactic
Nuclei (AGN), powered by the supermassive (∼ 106 − 109M) black holes present at
galactic centers.
Via x-ray and optical observations, astrophysicists can infer properties of the cen-
tral black holes such as mass - and more recently spin. The variability in these
objects can clue the observer into the dynamics of the disk and thus gravity in a
regime where GR effects are prominent. Through computation of the power density
spectrum (PDS) of certain black hole binaries, astronomers have gained more insight
into the inner region of the disk. In particular, some spectra contain broad peaks
known as Quasi-Periodic Oscillations (QPOs) whose origin as of today remains un-
known. Some plausible explanations relate these phenomena to normal modes of the
accretion disk. If this is true, QPOs provide a clear signature of relativistic dynamics.
Theoretically, we can model the disk much like a vibrating membrane, solving
linear wave equations similar to those found in quantum mechanics. In this thesis,
we will discuss the normal modes that arise out of these equations and the orbits of
both matter and light assuming a Kerr (rotating black hole) background. We will
also analytically model the radiative transfer of the accretion disk. For simplicity, we
partition the disk into three distinct regions, the sizes of which depend on M . In the
following table, we list what effects dominate the local pressure and opacity in each
region. In order to accurately model the spectrum of the disk, however, we employ
numerical ray tracing using the code of Dexter & Agol (2009) to find the fraction
of photons emitted from the inner region of the disk. These calculations could help
5
Inner Middle Outer
Pressure Radiation Gas Gas
Opacity Electron Scattering Electron Scattering Inverse Bremsstrahlung
inform future observations of BHBs as well as AGNs in the search for QPOs.
II. HIGH FREQUENCY QUASI-PERIODIC OSCILLATIONS NEAR
BLACK HOLES
A. Observations
Energy spectra of black-hole binaries (BHB) have portrayed a highly intricate
phenomenology. Different states with relatively few features have been observed,
termed Hard or Power Law and Soft or Thermal states, corresponding to higher
characteristic photon energies (Hard) versus lower energies (Soft). The hard states
generally show a characteristic power law dependence on frequency. Finally there
exists a very high, steep power law state (SPL) that will be of central importance in
our investigations (cf. Remillard & McClintock, 2006).
Over the years, two types of quasi-periodic oscillations - differentiated by frequency
- have emerged from observations: high frequency and low frequency. Low frequency
QPOs (LFQPO) have been observed in many BHBs in the intermediate (SPL) state.
For these systems, the QPO frequency is of order .1 - 30 Hz. In this paper, we focus
mainly on the high frequency quasi-periodic oscillations (HFQPOs) of BHBs since
their frequencies are more stable. LFQPOs on the other hand have been known to
vary over intervals of ∼ 10Hz (van der Klis 2000).
A table of some observed high-frequency QPOs in BHBs is presented in the Ap-
pendix. The spin for these objects was estimated either by observing gravitational
effects through the Iron-line or X-ray reflection (refl) method or through the contin-
uum fitting (cont) method which estimates the inner radius of the accretion disk (cf.
Reynolds (2013)). The masses of these BHBs were estimated through studying the
orbits of the companion star (cf. Ozel et al. (2010)).
6
FIG. 1: All known HFQPOs in BHB. The list in the Appendix also includes another
3:2 pair both with > 4σ significance discovered in an estimated ∼ 400M ULX.
. (Taken from Remillard and McClintock)
kHz QPOs have also been observed in accreting neutron star systems, but these
peaks are much less stable in frequency. High frequency QPOs in BHBs are stable
in frequency with an uncertainty of only ∼ 15% whereas the kHz QPOs have been
known to vary by a factor of 2 (cf. Remillard & Mclintock (2006)). The stability in
BHBs suggests an underlying mechanism related to intrinsic properties of the black
hole such as mass and spin. As can be seen from the QPO list in the Appendix as
well as Figure 1, the two highest frequency QPOs occur in a 3:2 ratio, sometimes
simultaneously but often not.
The data becomes sparser as we move to targets with larger black hole mass.
Yet, recently, two HFQPOs with frequency ratio of 3:2 were found in a possible
intermediate mass black hole with M ∼ 400M. They had large quality factors, Q,
and were identified with greater than 4σ confidence (cf. Pasham et al.).
So far only a few x-ray quasi-periodic oscillations have been observed in active
7
galactic nuclei. Most observed have long time scales - as expected from naive scaling
of the period with mass - and some are most likely the analogue of LFQPOs in stellar
mass BHBs. So far no confirmed HFQPOs have been observed in AGN, although
depending on mass estimates some of the declared LFQPOs may in fact be HFQPOs.
Indeed, Alston et. al found strong evidence for a 3:2 resonance in a 4 × 106M/M
mass AGN, suggesting that this QPO falls under the heading of high frequency. The
discover of HFQPOs in AGN motivates, in part, our research and would help to
differentiate the many models proposed to explain these phenomena.
B. Modeling HFQPOs
Many explanations for high frequency quasi-periodic oscillations have been pro-
posed. Although models were originally proposed to explain QPOs observed in accret-
ing neutron stars, black holes are theoretically simpler because they do not possess
an inner surface: the accretion disk has an inner edge at the innermost stable circular
orbit (ISCO). Furthermore, high frequency QPOs tend to have periods of order the
inner orbit and so are assumed to display the dynamics of the near horizon disk.
The three most well known models are the orbiting hotspot, epicyclic resonance
and normal mode models. The first - orbiting hotspot models (cf. Schnittmann and
Bertschinger, 2003) - propose that the variations come from regions of higher relative
emission in the disk. Such a region would normally be sheared out by the differential
rotation immediately [5]. Thus, the theoretical basis for the origin of these hotspots
is unknown and so we will not focus on them in this paper.
Epicyclic resonance models were proposed (cf. Abramowicz and Kluz’niak, 2001)
to help explain the observed 3:2 ratio in QPO frequencies. These models relied on
coupling between the radial and vertical epicyclic frequencies. For horizontal and
vertical perturbations of a torus, each obeys the following relations, respectively:
d2ξrdt2
+ κ2ξr = 0
d2ξzdt2
+ Ω⊥ξz = 0 (1)
8
These equations can be coupled by replacing κ2 → κ2(1 + χ1 cos(Ω⊥t)) and Ω2⊥ →
Ω2⊥(1+χ2 cos(κt). These modified ODEs are the Mathieu equations. According to the
methods of parametric resonance, one should then see 3:2 resonances at radii where
Ω⊥(r)/κ(r) ≈ 3/2. The downside of this model arises from the fact that it does
not provide an explanation for excitation of these resonances at the specified radius.
While some work has been done regarding excitation through stochastic driving terms
in the above equations, we will not pursue these explanations further in this paper.
Finally, the focus of this paper will be on models based on oscillations of accretion
disks (diskoseismic modes). These models have many nice theoretical advantages over
the others proposed. Disk modes, as will be fleshed out in detail, are trapped at certain
radii of the disk. These modes form a complete, orthonormal set and have relatively
simple characterizations. In a rotating hydrodynamical system, as around a Kerr
black hole, a fluid element that is displaced in the radial direction will feel a restoring
force due to competition between the gravitational pull of the central object and the
centrifugal force. There will be a vertical pressure and gravitational restoring force
that will lead to oscillations. Various combinations of radial and vertical restoring
forces produce the menagerie of disk modes known as the p-, g- and c-modes. These
different types of modes will be disambiguated below.
In particular, diskoseismic modes can become trapped in the inner disk due to
a form of the radial epicyclic frequency predicted general relativity: κ has a maxi-
mum κmax at a certain radius. The fundamental g-mode can be trapped below this
maximum frequency. In our work, the axisymmetric g-mode, which agrees nicely in
frequency with those of HFQPOs, will take on frequency values of ω2 ' κ2.
A final advantage of disk oscillations is that viscous driving provides a neat mech-
anism for exciting these modes. Furthermore, they can couple non-linearly between
each other and produce a frequency ratio of 3:2 as in Horak (2008). To gain fur-
ther insight into this compelling picture, we will need to mathematically explore the
dynamics of black hole accretion disks.
It is important to note that the observed QPOs have never been seen in simulations.
This indicates that some key physical ingredient is missing from the computer models.
9
On the other hand, the fundamental g-mode, which will be discussed in depth below,
has been seen along with p-waves propagating above κmax (Reynolds & Miller (2009)).
Some (Schnittman et al. (2006)) have suggested that the missing ingredient could be
related to a proper treatment of radiative transfer in the simulations.
III. DYNAMICS AROUND A KERR BLACK HOLE
In many ways, black holes are like large fundamental particles; Kerr black holes
are parameterized by two numbers only: their spin, a = cJ/GM2, and mass, M . In
this paper, we use Boyer-Lindquist coordinates with r measured in units of M . The
Kerr metric then takes the following form:
ds2 = −r2∆
Ad t2 +
A
r2(dφ− ω dt)2 +
r2
∆d r2 + d z2 (2)
with ∆ ≡ r2 − 2r + a2, A = r4 + r2a2 + 2ra2 and ω ≡ 2ar/A. For the remainder of
this paper, we also have set G = c = 1.
With this metric in hand, we readily solve the geodesic equation, uνuµ;ν = 0, to see
that
Ω ≡ ± 1
r3/2 + a(3)
where Ω = dφdt
is the dimensionless angular velocity and the dimensionless black
hole spin a can take on values from −1 to 1. The plus or minus sign in front of Ω
corresponds to a prograde or retrograde disk respectively.
For circular orbits, we assume the coordinate velocity is just dxµ
dt= (1, 0,Ω, 0).
Perturbing these velocities, we find vr satisfies [1] for ξr where
κ = Ω(r)(1− 6/r + 8a/r3/2 − 3a2/r2)1/2 (4)
and vz satisfies [1] for ξz where
Ω⊥ = Ω(r)(1− 4a/r3/2 + 3a2/r2)1/2 (5)
As is shown in Fig. 2, the radial epicyclic frequency has a maximum at rmax, around
which the most interesting disk oscillations are trapped. Most importantly for disk
10
4 6 8 10
0.01
0.02
0.03
0.04
Radial Epicyclic Frequency
a=.01
a=.5
a=.8
FIG. 2: The radial epicyclic frequency, κ for differing values of a. Note that κ has a
maximum - unlike in Newtonian gravity - and hits zero at some radius outside the
event horizon, which goes from r = 2− 1 for a = 0− 1.
dyanmics, however, is the fact that κ = 0 for some radius r > 1. This radius, we call
the Innermost Stable Circular Orbit (ISCO). For r < rISCO, any small perturbation
in the radial direction will be unstable and an orbiting object will fall to the horizon.
The shape of the radial epicyclic frequency determines much about the trapping of
various modes in the disk. We will now discuss this fact in detail and also examine
the effect that magnetic fields have on the trapping regions of various modes.
IV. ACCRETION DISK NORMAL MODES
A. Modes without Magnetic Fields
In this section, we will characterize the various modes that can propagate in the
inner accretion disk. As a rough first approximation, we will approximate the fluid
equations within a pseudo-Newtonian framework. For a fully relativistic treatment,
11
without use of the local WKB approximation, see Perez et al., 1997. We will follow
the basic derivation of the mode dispersion relation laid out in Fu & Lai (2008). The
main equations considered here will be the continuity equation and the momentum-
conservation equation:
∂ρ∂t
+∇ · (ρ~v) = 0 (6)
∂~v∂t
+ (v · ∇)~v = −1ρ∇Π−∇Φ + 1
ρ~T (7)
where Π is the gas and magnetic pressure of the disk, Φ is the gravitational potential
and ~T = 14π
( ~B · ∇) ~B is a magnetic stress-vector that will be needed later to describe
the effects of magnetic fields. For now, we set it to zero.
We assume a perfectly circular flow so that ~v = rΩ(r)φ and we perturb around
this solution. Following the notation of Fu & Lai (2008) and Ortega-Rodriguez et al.
(2015), the background configuration satisfies the equation
~G ≡ 1
ρ∇Π− ~T = rΩ2(r)r −∇Φ
Assuming the perturbations have the form δf ∝ eimφ−iωt, we then get the master
equations
− iωδρ+ 1r∂∂r
(ρrδvr) + imρrδvφ + ∂
∂z(ρδvz) = 0 (8)
− iωδvr − 2Ωδvφ = Grδρρ− 1
ρ∂∂rδΠ + 1
ρ(δ ~T )r (9)
− iωδvφ + κ2
2Ωδvr = − im
ρrδΠ + 1
ρ(δ ~T )φ (10)
− iωδvz = Gzδρρ− 1
ρ∂∂zδΠ + 1
ρ(δ ~T )z (11)
where the epicyclic frequency is defined as in Newtonian gravity as κ2 ≡ 2Ωrd(r2Ω)dr
and
ω ≡ ω −mΩ.
We will assume throughout this text that the disk fluid is barotropic so that
ρ = ρ(P ) and δρ = 1c2sδP . Here we have used the definition of the speed of sound as
c2s = ∂P
∂ρ. Imposing the WKB approximation, we postulate that the remaining spatial
dependence of the perturbations looks like eikrr+ikzz. Here we expand to first order in
the parameter (rkz)−1, (rkr)
−1 << 1. Plugging this ansatz into the above equations
12
and dropping negligible terms, we get
− iωρc2sδΠ + ikrδvr + ikφδvφ + ikzδvz = 0 (12)
− ikrρδΠ + iωδvr + 2Ωδvφ = 0 (13)
− ikφρ− κ2
2Ωδvr + iωδvφ = 0 (14)
− ikzρδΠ + iωδvz = 0 (15)
where kφ ≡ m/r. Furthermore, as will be done when we discuss the magneto-
hydrodynamic equations, we assume that Gz = −(∇Φ)z ∼ Ω2⊥z is negligible since
we are working near the z = 0 mid-plane. Furthermore, Gφ = 0 since we are assum-
ing an axisymmetric disk, and Gr is small since the force balance of the background
disk is assumed to be approximately Keplerian so that Ω2r ≈ (∇Φ)r.
To get the dispersion relation, we know that equations (13)-(16) define a coeffi-
cient matrix for the four unknown perturbations, δΠ, δvr, δvφ and δvz. Taking the
determinant of this matrix and setting it to zero, we get the following relation
(ω2 − κ2)(ω2 − k2zc
2s) = k2
rc2sω
2 (16)
As done more rigorously in Okazaki et al. (1987) and Nowak & Wagoner (1992), the
differential equations for the fluid perturbations can actually be separated. The ver-
tical dependence is not well described by eikzz, but, in fact, by Hermite polynomials.
Assuming an isothermal disk, the pseudo-Newtonian dispersion relation obtained in
both references is approximated by
(ω2 − κ2)(ω2 − nΩ2⊥) = k2
rc2sω
2 (17)
with n an integer. In this case, we can think of kz as quantized and write
kz ∼√nΩ⊥/cs (18)
This relation will be used below.
Equation [17] helps us to classify the various modes that propagate in the disk.
For n = 0, the modes are called fundamental p-modes. For m = 0, the dispersion
relation takes the simple form of (ω2 − κ2) = k2rc
2s. Thus, the p-mode only exists if
13
FIG. 3: Upper Left : The p-modes can propagate in the region indicated by the
green lines, where ω2 > κ2. Lower Left : The mode diagram for m > 0
non-axisymmetric p-modes. Upper Right : The mode diagram for m= 0 g-modes.
This agrees with the findings of Perez et al. (1996). The effect of magnetic fields on
these modes is discussed in the next section. Lower Right : The mode diagram for
non-axisymmetric g-modes and c-modes.
ω2 > κ2. As can be seen from the upper-left panel of Figure 3, these modes have
an important theoretically downfall; they require reflection at the ISCO in order to
remain trapped. This unlikely scenario is part of why we ignore the p-mode in the
discussion that follows.
For n = 1, the modes are called fundamental g-modes. We will consider mostly the
axisymmetric, m = 0 type. These modes satisfy the dispersion relation (ω2−κ2)(ω2−
Ω2⊥) = k2
rc2s. Thus, they can propagate in the region indicated in the upper-right
corner of Figure 3. The spectra of the g-mode was analyzed in a fully relativistic
background by Perez et al. (1996). They found the important relation that the
frequency of the fundamental g-mode obeys the equation
f(m = 0) ≈ 714(M/M)F (a) Hz (19)
where F (0) = 1 and F (.998) ≈ 3.443. This equation will be extremely important in
determining reasonable AGN targets for detailed observation. The g-mode trapping
14
FIG. 4: The mode trapping regions as a function of black hole spin. Note the
constancy of the g-mode region.
region also remains relatively stable as a function of spin as can be seen in Figure 4.
For n > 1 and m ≥ 1, the modes are called c-modes. These modes represent a pre-
cessing tilt of the inner disk. They can become trapped in the small region indicated
in the lower right panel of Figure 3. Again, they suffer from similar theoretical issues
as the p-modes. Thus, for the remainder of this thesis, we will focus our attention on
the g-mode and in particular the effect of magnetic fields on the trapping region of
Figure 3.
B. The Effect of Magnetohydrodynamics on Normal Modes
The following results were submitted to the Astrophysical Journal in Ortega-
Rodriguez et al. (2015). My contributions consisted of numerically analyzing the
effect of magnetic fields on the g-mode trapping region.
Accretion disks should contain magnetic fields accreting from the surrounding in-
terstellar medium. The study of magnetized fluids or magnetohydrodynamics (MHD)
15
has illuminated many new aspects of accreting flows. We extend the formalism of the
previous section in a simple way to include magnetic fields via the ideal (σ → ∞)
MHD equations. To equations (7) and (8), we include two more equations
∂ ~B∂t
= ∇× (~v × ~B) (20)
∇ · ~B = 0 (21)
Furthermore, we include the magnetic pressure in equation (8) so that Π = P + B2
8π.
Fu & Lai (2009) (FL) perturb these equations around the background magnetic
field ~B = (0, Bφ, Bz). They found that for Bφ = 0 the p-modes obey a modified
dispersion relation
ω2 = κ2 + k2r(c
2s + v2
Az) (22)
where vAi = Bi/√
4πρ is the Alfven velocity. Solving the basic magnetohydrody-
namical equations reveals modes that propagate in the direction of the background
magnetic fields. These are called Alfven waves, and they propagate with the Alfven
velocity, which can be thought of as speed associated to a vibrating string with tension
B2. Equation [22] shows that the magnetic field just shifts the p-mode propagation
speed to√c2s + v2
Az and does not affect its existence.
FL, however, did find that the g-modes are affected by the introduction of Bz. We
will follow this treatment as done in Ortega-Rodriguez et al. (2015) and examine
more generally the effect of magnetic fields on the these modes.
In order to do so, we will try to see the effect of introducing a small Br. Unfortu-
nately, Br spoils the stationarity of the background magnetic field configuration. For
simplicity, we assume Br = Cr
, Bφ = Bφ(r) ∼ rq and Bz = constant. Then ∇ · ~B = 0
is automatically satisfied. Equation (22), however, is not. This form of the magnetic
field introduces a term in the MHD equations such that
Bnewφ ≡ Bold
φ (r) + rBrdΩ
drt (23)
where we assume that Br << Bφ as confirmed by simulation. Furthermore, in order to
maintain axisymmetry of the background configuration, we need to introduce a small
radial velocity vr ∼ ε2α∗rΩ. Here, α∗ is the viscosity parameter and ε = H/r is the
width of the disk divided by the radius. In our thin disk approximation, H/r << 1.
16
In order to examine the modes, we now perturb equations (7)-(8) and equations
(21) and (22) around this modified background. In order to focus on the axisymmetric
g-mode, we set m = 0. Imposing the perturbation form eikrr+ikzz, we get the master
equations:
− iωρc2sδΠ + ikrδvr + ikzδvz + iω
4πρc2s(BφδBφ +BzδBz +BrδBr) = 0 (24)
− ikrρδΠ + iωδvr + 2Ωδvφ + 1
4πρ(ikzBz + ikrBr)δBr − Bφ
2πρrδBφ = 0 (25)
iωδvφ − κ2
2Ωδvr + 1
4πρ(ikzBz + ikrBr)δBφ +
(1+q)Bφ4πρr
δBr = 0 (26)
− ikzρδΠ + iωδvz + 1
4πρ(ikzBz + ikrBr)δBz = 0 (27)
ikzBzδvr + iωδBr − ikzBrδvz = 0 (28)
ikrBφδvr − (ikzBz + ikrBr)δvφ + ikzBφδvz − iωδBφ − pΩδBr = 0 (29)
ikrBzδvr − ikrBrδvz − iωδBz = 0 (30)
We apply the same process for finding the dispersion relation as in the ~B = 0 case.
We then get a very large polynomial equation from the determinant of this 7 × 7
matrix, which takes the form:
P (ω, kr, kz, a, r, ~B...) = 0 (31)
where P is a 7th order complex polynomial in ω. In order to see the effect of the
various magnetic fields, we need to isolate the branch of this polynomial that reduces
to κ2 in the ~B → 0 limit. In this limit, Re(P ) = 0, so we focus on Im(P ). As a
crude first approximation, and following equation (30) of FL, we set kr = 0. With
kr = 0, this equation can be solved numerically as a function of kz, spin, radius and
magnetic field. The results are shown in Figure 3.
In the Br = Bφ = 0 limit, as taken in FL, the branch corresponding to the m = 0
g-mode takes the form
ω2 =1
2[κ2 + 2η(Ω2
⊥b2 +
√κ4 + 16η(Ω⊥Ω)2b2)] ≡ (κ′)2/2 (32)
where the approximation is made - as discussed in equation [18] - that kz ∼√ηΩ⊥/cs
with η ∼ 1. From this relation and from Fig. 5, it is easy to see what happens
to the trapping region: as bz is increased, the innermost value of κ′ moves up until
17
FIG. 5: The g-mode trapping region for various field configurations taken from the
simulations listed in the figure in the Appendix. The radius is listed in units of
GM/c2. The top panel corresponds to a spin of a = 0 and the bottom panel to near
maximal or a = .95. Furthermore, bi ≡ vAi/cs
κ′(rISCO) > κmax. In order to be trapped, the g-mode then would require reflection
at the ISCO - the same hypothetical effect that we chose to disregard earlier. What
our approximate results show, as an addendum to FL, is that for reasonable magnetic
fields, this trapping region is actually still present.
By expanding perturbatively in the parameters ε = H/r and bibj << 1, we can
also get more detail about the modification of the g-mode behavior. Expanding the
18
branch frequency as
ω = ω0 + Λbφbz + λbφbr + Γb2z + γb2
r + βbzbr (33)
we find order by order that each coefficient has a real and imaginary part. This
suggests some instability in this mode. Interestingly, the leading order imaginary
term comes from the bzbφ term, explaining why FL did not find it. The imaginary
contribution to this term is
Im(Λ) =kzkrc
2s[2(2 + p)Ω2 − κ2]
4Ω(k2zc
2s − κ2)
(34)
Interestingly, for Newtonian mechanics, where κ2 = 2Ωrd(r2Ω)dr
, Im(Λ) = 0 and so this
instability (or damping) is purely relativistic. The effects of it are minimal as they are
suppressed by order ε1/2, but more work investigating the potential for this imaginary
term as a mode excitation mechanism or otherwise is needed.
Finally, we note that the g-mode appears to be safe from the famous magneto-
rotational instability (MRI) of Balbus & Hawley (1998). We found perturbatively
that the timescale of the g-mode, τg, can be related to the MRI timescale, τMRI by
the relationτg
τMRI
∼ Ωb1/2z
κ(35)
This relation allows roughly 3 g-mode oscillations for typical bz values as listed in
the Appendix. This means that the g-mode will oscillate a few times before being
destroyed by MRI (cf. Ortega-Rodriguez et al. (2015)).
In summary, we find no reason to discard the g-mode based upon destruction of
the κ trapping effect, as FL suggested in their work. For the magnetic fields listed in
the Appendix, and those used to generate Fig. 5, the g-mode trapping region remains
intact. Furthermore, the magnetic field needs to only take on acceptable values for
relatively short amounts of time since QPO duty cycles are quite small (Rodriguez et
al. (2015)).
These results were focused more on QPOs observed in stellar mass black hole bina-
ries. With few new X-ray telescopes launching in the next decade, optical telescopes
may provide a new avenue to observe black hole accretion disks. Since the tempera-
ture of the disk scales inversely with the mass, as will be discussed in the next section,
19
the plausibility of observing the inner disk with optical telescopes is higher for targets
with large masses. We now explore the possibility of using super-massive black holes
that power active galactic nuclei (AGN) to continue the hunt for QPOs and other
strong-gravity phenomena.
V. PROBING THE INNER ACCRETION DISK OF AGN
We now discuss using observations of AGN to access the inner accretion disk
via lower energy spectra. This could increase the number of targets with observed
QPOs and give us more observational evidence for the scalings of basic properties of
this phenomena as a function of mass. We examine the properties of AGN spectra
in order to inform a selection of adequate targets. We have submitted a proposal
for observation of selected targets in the Kepler II mission. The advantages and
disadvantages of this particular search will be discussed below. The use of data from
LSST will also be considered.
A. Radiative Transfer in Black Hole Accretion Disks
We begin by studying the basics of accretion around black holes and how modes
will modulate observable properties of this disk. Following the treatment in Shapiro
and Teukolsky (1983), we note that the main energy generation mechanism occurs
from viscous stresses slowing down and heating up the in-falling gas. The hottest
part of the disk will occur near the inner edge of the disk. The main mechanism of
accretion will occur through angular momentum transfer to larger radii with mass
transfer inward. Quantitatively, we can see this phenomenon in the approximation
of a Keplerian disk around a black hole. For illustration purposes, we will derive the
luminosity of a black hole accretion disk in Newtonian gravity.
First, we assume as in the previous section that, H(r) << r. Thus we will integrate
out any dependence on z. In this spirit, we define
Σ =
∫ H
−Hρd z ≈ ρ2H (36)
20
is the surface density of the disk. We also assume that vφ = rΩ = (GMr
)1/2. Using the
Navier-Stokes equations in combination with force balance, we find that the azimuthal
stress due to two radially separated fluid elements (r → r + dr) is just
fφ = −trφ = −3
2η
(GM
r3
)1/2
(37)
Here η = αρHcs is the dynamic viscosity. The viscosity parameter α <∼ 1 appears in
the disk equations discussed below.
Then the angular momentum transport inward across a cylinder of radius r and
heigh 2h will just be J+ = M√GMr where
√GMr is just the specific angular
momentum of a fluid element at radius r and M is the mass accretion rate of the
central black hole. The amount of angular momentum swallowed by the black hole
is just βM√GMrISCO, with β ≤ 1, and so, by angular momentum conservation, we
find that the total angular momentum added to the section of the disk between rISCO
and r is just
trφ · 2πr · 2h = M[β√GMrISCO −
√GMr
](38)
Note, trφ < 0 indicating that angular momentum is being carried out to larger radii.
Then the basic equations of energy transport in a fluid state that the heat (or
entropy by the Second Law of Thermodynamics) are given by the square of the stress
tensor. Thus,
Q = ρT s ≈t2rφη
= −fφtrφη
(39)
Assuming this heat is totally radiated away, the flux per unit area of heat energy is
then just F (r) = 1/2× 2HQ. The 1/2 is for each side of the disk, and the 2H comes
from integrating out the z-dependence. Thus, using [37] and [38], we get
F (r) =3M
8πr2
GM
r
[1− β
(rISCOr
)1/2]
(40)
Integrating over r to get the total disk luminosity, we finally get
L =
∫ ∞rISCO
2F × 2πrd r =
(3
2− β
)GMM
rISCO(41)
This final equation is how we estimate accretion rate and mass of the central black
hole of AGN. While it is not relativistic, it provides many heuristics for the radiation
properties of black hole accretion disks.
21
In our work, we also took into account the effects of opacity and radiative transfer,
which simulations so far have not been able to do successfully. Proper radiative trans-
fer prevents disks from puffing up into their thick, radiatively inefficient counterpart.
In order to include these effects, we must discuss the overall structure of an accretion
disk.
As discussed in the introduction, there are three regions of the disk: outer, middle
and inner. In the outer region, gas pressure dominates radiation pressure. Further-
more, free-free (e− → e−) absorption contributes most significantly to the opacity.
The middle-outer transition occurs when the contribution to the total opacity from
free-free absorption is roughly that from electron scattering (e− + γ → e− + γ) or
κff ≈ κes. Finally the middle - inner transition occurs where the radiation pressure
and gas pressure are comparable or where Pgas ∼ Prad.
In order to assess the effects of opacity on the emergent disk spectrum, we first
need to clear up some terminology regarding what temperature we are actually cal-
culating. In 1973, Shakura & Sunyaev solved for the disk temperature T involved in
the radiative transport equation. They found that, for the inner region of the disk,
T = (5× 107K)(αM)−1/4(2r)−3/8 (42)
with r in units of M , M in units of M and α <∼ 1 is the viscosity parameter mentioned
above. This T corresponds to the temperature at the midplane of the disk. We can
also define a surface temperature Ts. For optically thick regions of the disk, the
surface temperature will be roughly the same as the effective blackbody temperature
Teff =(
4F (r)a
)1/4
. Using equation [40] that we derived above, we can solve for this
temperature as a function of radius.
This approximation will only hold, however, when free-free absorption dominates
the disk opacity. In the inner and middle disk, where κff <∼ κes, we have to take
into account that photons in the disk zig-zag before arriving at the surface; they
originate at larger z values than in the outer disk. Since fewer photons will arrive
at the surface, the emitted photons will have higher characteristic frequency so as to
radiate the same total energy as for a black-body spectrum. We expect then that this
modified black-body spectrum will be skewed toward higher photon energies.
22
Quantitatively the modified black-body spectrum can be modeled by the equation
Iν ∼ Bν(Ts)
(κffνκes
)1/2
(43)
A detailed derivation of this modification can be found in Shapiro & Teukolsky (1983).
Equation [43] will be essential for our results.
Using the formulae derived in Novikov & Thorne (1973) for
κffν ∼ (1.5× 1025cm2g−1)ρT−7/2gff1− e−x
x3(44)
and the well-known value of κes ∼ .40cm2g−1 from Thomson scattering calculations,
we can explicitly write down an expression for Iν using (43). Integrating over the
upper half-sphere and over frequency, we find an expression for F :
F =∫ π/2
0
∫∞0Iν cos θdΩ dν
∼ C 2.54×10−3kBh
ρ1/2T9/4s [erg/cm2/s] (45)
where C =∫∞
0x3/2e−x/2√
ex−1. This is the modified black-body analogue of the famous
relation F ∼ T 4. The use of equation (45), (40) and the relation for ρ found in
Novikov and Thorne then allows us to solve for Ts(r). For the inner region, we find
that the temperature is
Ts = 8.47× 1018 · L/(M · η · r3) · ρ−1/2 · Q(r, a)/(B(r, a) ·√C(r, a))4/9[K] (46)
where Q,B and C are given in Novikov & Thorne (1973), M is in units of M and r
is in units of GM/c2.
The temperature of the disk as a function of the radial coordinate can be seen in
Fig. 6. Equation (46) can then be plugged back into (45) and integrated over r to
find an expression for Lν and L respectively. With these equations in hand, we can
investigate how the observational properties of the g-Mode change with varying spin,
mass, luminosity and inclination angle of a target AGN. We take up this subject in
the next section.
23
5 10 15 20r @MD
5´104
1´105
2´105
5´105
Ts @KDa = .8M, L=.3 LEdd
FIG. 6: The temperature of an accretion disk with central object mass of 109M,
luminosity L ∼ .3LEdd and spin a = .8. Note that the maximum occurs near the
maximum of κ (r = blahblahblah PUT THIS IN) and so g-modes are modulating a
region of high output flux. For this reason, the g-mode should be more likely
observable.
B. Observation of the g-Mode in AGN
Since the g-mode should occupy the region near the hottest part of the disk -
as can be seen from comparison of Fig. 4 and 6 - they should be the most easily
observable candidate mode. In this section, then, we focus on what exactly should
be observed in AGN with a characteristic g-mode. We will follow and expand upon
the treatment given in Nowak & Wagoner 1993. Most of the following work was done
with the intention of using the upcoming Large Synoptic Survey Telescope (LSST)
to monitor AGN in optical bands. We will then use band widths specific to LSST,
but the results can be easily generalized to any other search.
In order to estimate the effect of the mode on the object luminosity, we need to
first find how the fractional luminosity varies with spin. Assuming that the fractional
modulation at radius r of the g-mode is given by fg(r), we can find the fractional
modulation by integrating:
δLν/Lν =
∫ ∞r=rISCO
fg(r)Fν′(r)2πr dr/
∫ ∞r=rISCO
Fν′(r)2πr dr (47)
where ν ′ is the frequency of light at observer. For our work, we were interested mostly
24
in scaling of these observational signatures and so set fg(r) to be a step function
comprising the mode region. As indicated in Fig. 4, we set boundaries of this step
function to [1.3rISCO, 1.3rISCO + 2].
The problem with trying to extract information from optical power spectra comes
from the fact that the characteristic photon frequency of the g-mode region is ν ∼
1015Hz, which lies well into the UV. In order to compensate for this, we focused on
AGN candidates whose redshift was large enough to move the Lyman limit to the low
frequency end of the LSST u-band. This corresponds to a redshift of z ∼ 3.42.
The work of Nowak & Wagoner was done non-relativistically and did not account
for the effects of gravitational redshift and doppler shifts. Per the discussion of the
previous paragraph, the gravitational redshift could actually help bring the higher
energy photons of the inner disk into the observation band. Yet, as we increase a,
the disk will be pulled in closer to the event horizon, increasing the characteristic
temperature of the disk. A priori, we do not know which effect will dominate if any.
We carried out an expanded analysis to examine this interplay. The frequency shift
of a photon originating near a Kerr black hole is
ν0
ν∞=
r3/2 + a+ b sinα cosϕ
r3/4(r3/2 − 3r1/2 + 2a)1/2(48)
where b is the impact parameter at which the photon crosses the observers image plane
at infinity, ϕ is the angular coordinate in the image plane and α is the inclination
angle of the disk. Furthermore, we accounted for the angular dependence of a rotating,
scattering dominated atmosphere given in (Perez, 1993). This gives
Iν0 =2π
c2MBF (r)
as + bsγ
2π(as2
+ bs2
)(49)
where as ≈ 0.36 and bs ≈ 0.64 as found by (Perez, 1993) through numerical fitting to
theoretical results in (Schneider & Wagoner, 1987). MBF is the modified blackbody
function as discussed above. Here also we have included the cosine, γ, of the angle of
the emergent photon with respect to the disk normal. The relativistic generalization
25
FIG. 7: The image of a black hole accretion disk generated using the code detailed
in Dexter & Agol (2009). This image represents a black hole accretion disk with
central spin of a = .8M , M = 108M, L = .3LEdd and α = 80
of this function is
γ = − ~p·n~p·~U
= 1r[b2(sin2 ϕ+ cos2 α cos2 ϕ) + a2 cos2 α]1/2
× r3/4(r3/2−3r1/2+2a)
r3/2+a+b sinα cosϕ(50)
which accounts for relativistic beaming effects of the accretion disk.
Actually incorporating this redshift into equation (47), however, is difficult because
it depends non-trivially on the initial conditions of the photon. To overcome this
difficulty, we use the Geokerr code detailed in Dexter & Agol (2009). We construct
an image plane and raytrace back to the disk. We then calculate the local intensity
at the final point of the geodesic, account for the redshift via Iν = Iν0/(1 + z)3 and
then sum over all the geodesics. For a given inclination angle, mass and spin, this
produces an image of the accretion disk as seen in Fig. 7.
Accounting for all these competing effects, we can also plot the spectra of our
26
ò
ò
ò
ò
ò
ò
òò
òò ò ò ò ò ò ò ò ò
ò
ò
ò
ò
ò
ò
à
à
à
à
à
à
à
àà à
à
à
à
à
à
Lyman
Limit
G-Band
14 15 16 17 18 19
Log10
Ν
36
38
40
42
44
Log10
HΝdLΝ
dW
L
L=.3LEdd,M=108M, i=30°
ò a=.98
à a=.20
(a) The spectrum for an M = 108M,
L = .3LEdd and α = 30 accretion disk
with two differing spins (a=.98 on top
and a=.20 on bottom). Note the
modified black body behavior where the
a = 0.98 spectrum takes the form of νp
for ν ∼ 1015.5 − 1017.5. Our estimation
puts p ≈ 1/6.
ò
ò
ò
ò
ò
ò
òò
ò ò ò ò ò òò
ò
ò
ò
ò
ò
òò
à
à
à
à
à
à
à
àà
àà
àà à à
à
à
à
à
à
Lyman Limit
G-Band
15 16 17 18
Log10
Ν
34
36
38
40
42
44
Log10
HΝdLΝ
dW
L
L=.3LEdd,M=108M, a=.8
ò i = 0°
à i = 85°
(b) The spectrum for an M = 108M,
L = .3LEdd and a = .8M accretion disk
with differing inclination angles of
α = 0 on top and α = 85 on bottom.
Note the effect of inclination angle.
Higher inclination angles will contribute
more relativistically-beamed photons to
the spectrum.
FIG. 8: Accretion disk spectra accounting for gravitatoinal redshift
accretion disk for differing inclination angles and spins as in Fig. 6. The effect of
spin is obvious; the accretion disk is pulled in closer to the horizon and so increases
in temperature. The flux thus increases overall and gets harder contributions from
emergent disk photons. Note that inclination angle appears to have a much smaller
effect on the spectrum than spin of the black hole. We are now in a tricky position.
In order to optimize flux from the g-mode region (inner disk), we should be observing
high spin AGN. Conversely, this will pull more intensity out of the optical bands and
into the UV. Thus, we need to examine more closely how the fractional modulation
looks for various disk parameters.
Indeed, we find a very interesting result, that Nowak & Wagoner (1993) did not
find in their work. As can be see from Fig. 9, the fractional modulation due to the g-
mode appears to minimize at a given mass of roughly M ∼ 107M. Interestingly, very
small mass AGN (105M) at a high inclination angle are about as good a candidate
as high mass AGN (109M).
27
òò
ò
ò
ò
ò
à à à à àà
5 6 7 8 9 10
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Log10
MM
Fra
cti
onalL
um
inosi
ty
L=.3LEdd, Ν=2.84*1015
Hz, i=30°
ò a=.2
à a=.98
(a) The fractional modulation due to a
g-mode for differing spins. Note that
higher spin seems to pull light out of the
optical bands as expected.
òò ò ò
ò
ò
à
à
à
à
à
à
5 6 7 8 9 10
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.085 0.85 8.5 85 850 8500
Log10
MM
Fra
cti
onalL
um
inosi
ty
G-Mode Period in Hours
L=.3LEdd, Ν=2.84*1015
Hz, a=.8
ò i=0°
à i=85°
(b) The fractional modulation due to a
g-mode for differing inclination angles
(α = 0 and α = 85.) Note how the
higher inclination angle surprisingly
increases the overall intensity in the
optical band.
FIG. 9: The fractional modulation due to a g-mode with 100% modulation at the
frequency ν = 2.84× 1015Hz. Note, we used equation 19 to estimate the g-mode
period in hours in the upper horizontal axis.
On the other hand, as expected from Fig. 8, the fractional modulation at the
specified frequency of ν = 2.84 × 1015Hz actually goes down for higher spins. This
qualitatively makes sense as higher spin AGN emit more strongly in the UV/X-
ray. The motivating factor for these results was to investigate the interplay between
heating and local redshift for increasing spin. These diagrams seem to show that
heating wins despite more exaggerated gravitational effects. Note, the frequency of
ν = 2.84 × 1015 Hz was chosen as so that the observed frequency ν ′ = ν/(1 + z)
is in the middle of the LSST g-band for z = 3.42. This redshift selects for u-band
dropouts.
As a comparison of different possible observing bands, we also computed the frac-
tional modulation for various bands without the gravitational redshift accounted for.
These are plotted in Fig. 10. Note, as expected, the fractional modulation increases
as we increase the band which we are looking in. Note for the U band we would
28
ææ
æ
æ
8.0 ´10141.0 ´1015 1.2 ´1015 1.4 ´1015 1.6 ´1015
Ν @HzD0.011
0.012
0.013
0.014
0.015
0.016
∆LΝLΝ
FIG. 10: The fractional modulation in different bands. The horizontal axis
corresponds to the middle frequency of the band; I band (leftmost), R band
(left-middle), G band (right-middle), U band (rightmost). This plot corresponds to
a black hole with spin a = .8M , L = .3LEdd and M = 108M and z = 3.42
need to look for ∼ 1.6% modulations in the light curve. Indeed, we explore the ob-
servational viability of using power spectra of AGN as a tool for studying the inner
accretion disk. This resulted in a proposal sent to NASA for their Kepler II mission
as described in the following section.
C. Observing with Kepler II
Although originally designed with the intent of finding exoplanets (planets out-
side our own solar system), the loss of a gyroscope has forced NASA to repurpose
the Kepler satellite telescope. What follows is taken from a proposal to NASA to
participate in Kepler II (K2) Campaign 6.
Campaign 6 will be optimal for extragalactic observation. The above calculations
were used to estimate the best candidates. Indeed, the QPO period for an AGN is
estimated to be
P = CP (1 + z)(M/107M)hours (51)
where z is the cosmological redshift and CP is a numerical factor that ranges from
1.1 for a = M to 4.0 for a = 0. K2 will have a maximum cadence of 30 minutes and
29
duration of 80 days. Using equation (51), we know that the AGN mass - and thus
luminosity via the Eddington relation - will range from
1.7× 1011L << (CP/CL)(1 + z)L << 7× 1014L (52)
with CL = L/LEdd < 1. We then used the redshift-magnitude relation for an FRW
universe
mi = Ai − 2.5 log(L/L)− 2.5 log(Qi(z)) + 5 log(D(z)) (53)
for a given band i. Ai here is a constant, D(z) is the luminosity distance and Qi(z) is
the luminosity emitted into the observed band divided by the total luminosity. With
our calculations above, Qi(z) was calculated for a characteristic mass of M ∼ 108M
and a = .8M . The redshifts were only allowed to be in the range 0.7 < z < 2.9
to avoid both the Lyman-α forest as well as contamination from the host galaxy.
STEM students from the high school Phillips Academy then used mi and z values
from the Million Quasar (MILLIQUAS) catalog to compute estimates on L and thus
M . Luminosities outside of the above range were rejected. A list of these targets is
presented in the Appendix.
Note that even if this search does not produce any QPOs, useful information can
be obtained from the PSDs. Break frequencies where the power-law slope changes
discontinuously at some characteristic frequency or the slope values themselves can
contribute to a deeper understanding of the dynamics around the central object.
VI. CONCLUSION
In summary, we have discussed the effect magnetic fields have on normal mode
trapping in a black hole accretion disk. We showed that, despite the claims of Fu &
Lai (2009), the g-mode trapping region is relatively stable under the magnetic field
configurations indicated by simulations. Furthermore, we showed that the vertical
and azimuthal magnetic field couple to give a purely general relativistic instability.
Although this instability arises at higher order in perturbation theory, its existence
indicates the need for further examination of basic MHD in accretion disks.
30
We have also expanded on the work of Nowak & Wagoner (1992) in including the
effects of gravitational redshift and relativistic Doppler shift. In order to provide
a full treatment, we used the code of Dexter & Agol (2009) to raytrace from the
image plane. We found interesting behavior in the fractional luminosity due to g-
mode modulation. In particular, competing effects due to decreased temperature but
increased overall luminosity seem to create a mass of minimum fractional modulation
at M ∼ 107M. We also found that the spectrum could be well approximated by
ν dLνdΩ∼ ν0.6. This calculation allowed us to compute Qi(z) and compile a target list
of AGN. We proposed this list for Kepler 2 Campaign 6.
We now await observation time with K2. If awarded guest observer status, we will
begin data analysis this Fall after data collection over the Summer. In the future, we
would like to take this work in two other directions:
• Investigate the use of the deep drilling LSST fields to obtain more optical power
spectra of AGN. The relative dearth of high cadence X-ray telescopes launching
within the next 10 years has required different techniques for finding QPOs.
LSST has a maximum cadence similar to that of K2. This requires creating a
metric to evaluate the large scale simulations of future LSST runs. Our metric
will detail the feasibility of using LSST to monitor AGN for observing QPOs.
• Explore the effect of turbulent driving of modes and the coupling between modes
and turbulence. This could include a detailed analysis of the instability discov-
ered in our investigations of magnetic disk modes.
• Finally, we hope to further characterize the various dependencies of accretion
disk spectral features on black hole mass, luminosity and spin. This would
aid observers in understanding and identifying the signatures of accretion disk
dynamics.
This work was supported through the Stanford Physics Department as well as
through an Undergraduate Advising and Research Grant.
31
[1] Alston et al., 2015, MNRAS, 449, 467.
[2] Balbus, S. and J. Hawley, 1998, Rev. Mod. Phys., 70, 1.
[3] Bardeen et al., 1972, ApJ, 178, 347.
[4] Belloni, T. and L. Stella, 2014, arXiv:1407.7373v1.
[5] Blandford, Roger. In conversation, 25, May, 2015.
[6] Dexter, J. and E. Agol, 2009, 696, 1616.
[7] Fu, W. and Dong Lai, 2009, ApJ, 690, 1386.
[8] Horak, J., 2008, A&A, 486,1.
[9] Kato, S., Fukue, J., & Mineshige, S. (1998). Black-hole accretion disks. Kyoto, Japan:
Kyoto University Press.
[10] Lehr et al., arXiv:astro-ph/0004211.
[11] Lin et al., 2013, arXiv:1309.4440v2.
[12] Mohan, P. and A. Mangalam, 2014, ApJ, 791, 74.
[13] I.D. Novikov and K.S. Thorne, ”Astrophysics of Black Holes,” in Black Holes, eds. C.
DeWitt and B. DeWitt (Gordon and Breach, Paris, 1973), pp. 343-450
[14] Nowak, M. and R. Wagoner, 1991, ApJ, 378, 656.
[15] Nowak, M. and R. Wagoner, 1993, ApJ, 418, 187.
[16] Okazaki et al., 1986, Pub. Astron. Soc. Japan, 39, 457.
[17] Ortega-Rodriguez, 2014, 440, 3011.
[18] Ortega-Rodriguez et al., 2015, submitted to ApJ.
[19] Ozel et al., 2010, ApJ, 725, 1918.
[20] Pasham et al., 2014, Nature, 74, 514.
[21] Perez, C.A., 1993, Ph.D. Thesis, Stanford University
[22] Perez et al., 1996, 476, 589.
[23] Remillard, R. and J. McClintock, 2006, Annu. Rev. Astron. Astrophys., 44, 49.
[24] Reynolds, C. and M. Miller, 2008, ApJ, 692, 869.
[25] Reynolds, C. S. 2014, Space Sci. Rev., 183, 277.
32
[26] Shapiro, S. L, & Teukolsky, S. A. (1983). Black holes, white dwarfs, and neutron stars
: the physics of compact objects. New York: Wiley.
[27] Silbergleit et al., 2000, arXiv:astro-ph/0004114.
[28] Wagoner et al., 2015, “Proposal to Kepler 2.”
[29] Wagoner, 2008, New Astron. Rev., 51, 828.
[30] Schnittman, J. and E. Bertschinger, 2003, arXiv:astro-ph/0312406.
[31] Schnittman, J. et al., 2006, ApJ, 651, 1031.
[32] Shu, F. H. (1991). The physics of astrophysics. Mill Valley, Calif.: University Science
Books.
[33] Wagoner, 2012, ApJ, 752, L18.
33
HFQPOs in Stellar Mass Black Holes
Source M/M QPO f(Hz) BH Spin
GRS1915+105
(i = 60± 5 deg)
12.5± 1.9 (KK)
41± 0.4 (Q,R)
67± 0.4(Q,R,S)
113±?
168± 5 (T,U)
0.54 <a <.58 (refl,F)
0.97 <a <0.99 (refl, F, EE)
0.98 <a <1.00 (cont, D)
XTEJ1550-564
(i = 75± 4 deg)
9.1± 0.6 (I)
92±? (V)
184± 5 (V)
276± 2 (V)
-0.11 <a <0.71 (cont, C)
0.33 <a <0.70 (refl, C)
0.75 <a <0.77 (refl, A)
GROJ1655-40 6.3± 0.5 (J)300± 9 (V,W,X)
450± 5 (V,W,X)
0.65 <a <0.75 (cont, E)
0.90 <a <1.00 (refl, B)
0.97 <a <0.99 (refl, A)
Cyg X-1 14.8± 1.0 (K) 135 (P) ?
0.60 <a <0.99 (refl, GG)
0.98 <a <1.00 (cont, G)
0.95 <a <0.98 (refl, CC)
0.77 <a <0.95 (refl, DD)
0.83 <a <1.00 (refl, FF)
XTEJ1650-500 5± 2 (L) 250± 5 (Y) 0.78 <a <0.80 (refl, A)
XTEJ1859+226 >5.4 (M) 190± 12 (Z)
H1743-322165± 6 (AA,BB)
241± 3 (AA, BB)-0.3 <a <0.7 (cont, N)
4U 1630-47
49.3± 3.0 (HH)
164.3± 9.8 (HH)
187.4± 6.5 (HH)
262.2± 45.5 (HH)
38.1± 7.3 ?
179.3± 5.7 ?
0.97 <a <0.99 (refl, II)
M82 X-1 400? (JJ)3.32± 0.06 (JJ)
5.07± 0.06 (JJ)
TABLE I: The QPOs in a 3:2 ratio are in bold
34
HFQPOs in AGN
Source M/M Period References
REJ1034+396 1.07 hours Alston, W.N. et al. 2014, MNRASL 445, L16
2XMM J12103.2 +110648 ∼ 3.8 hours Lin, D. et al. 2013, Ap. J. Lett. 776: L10
MS 2254.9-3712 ∼ 4× 106 1.9 hours Alson, W.N. et al. 2015, MNRAS 449, 467
Swift J1644 49.3+573451 0.058 hours Reis, R.C. et al. 2012, Science 337, 949.
PSO J334.20
28+01.40751010±0.5 542± 15 days Liu, T. et al. 2015, Ap. J. Lett. 803: L16
Various Magnetic Fields Produced in Simulations
Taken from Ortega-Rodriguez et al. (2015)
A) Miller, J.M. et al., 2009, Ap.J. 697, 900-912.
B) REis, R.C. et al., 2009, MNRAS, 395, 1257.
C) Steiner, J.F. et a., 2011, MNRAS, 416, 941.
D) McClintock, J.E. et al., 2006, Ap.J., 652, 518.
35
E) Shafee, R. et al., 2006, Ap.J. 626, L113.
F) Blum, J.L. et al., 2009, Ap.J., 706, 60.
G) Gou, L. et al., 2009, Ap.J. 742, 85.
H) Steeghs, J. et al., 2013, arXiv:1034.1808.
I) Orosz, J.A. et al., 2011, Ap.J. 730, 75.
J) Greene, J., Bailyn, C.D. & Orosz J.A., 2001, Ap.J., 554, 1290.
K) Orosz, J.A. et al., 2011, Ap.J., 742, 84.
L) Orosz, J.A. et al., 2004, Ap.J., 616, 376.
M) Corral-Santana, J.M. et al., 2011, MNRAS, 413, L15.
N) Steiner, J.F., McClintock, J.E. & Reid, M.J., 2012, Ap.J., 745, L7
P) Remillard, R. & McClintock, J.E., 2011, private communication with R.V.
Wagoner.
Q) Strohmayer, T.E., 2001, Ap.J., 554, L169.
R) Morgan, E.H., Remillar, R.A. & Greiner, J., 1997, Ap.J., 482, 993.
S) Belloni, T., Mendez, M. & C, Sanchez-Fernandez, 2001, A&A, 372, 551.
T) Belloni, T. et al., 2006, MNRAS, 369, 305.
U) Remillard et al., 2002 in Durouchoux Ph., Fuchs Y. & Rodriguez J., eds, New
View on Microquasars, Vol. 49 Center for Space Physics, Kolkata, India.
V) Remillard, R. et al., 2002, Ap.J., 580, 1030.
W) Remillard, R. et al., 1999, Ap.J., 522, 397.
X)Strohmayer, T.E., 2001, Ap.J., 552, L49.
Y) Homan, J. et al., 2003, Ap.J., 586, 1262.
Z) Cui, W. et al., 2000, Ap.J., 535, L123.
AA) Remillard, R. et al., 2006, Ap. J., 637, 1002.
BB) Homan, J. et al., 2005, Ap.J., 623, 383.
CC) Fabian, A.C. et al., 2012, MNRAS, 424, 217.
DD) Duro, R. et al., 2011, A&A, 533, L3.
EE) Miller, J.M. et al., 2013, Ap.J., 775, L45.
FF) Tomsick, J.A. et al., 2013, arXiv:1310:3830.
GG) Miller, J.M. et al., 2012, Ap.J. 757, 11.
Top Related