IMPRS-LECTURE - Fall 2003
Transcript of IMPRS-LECTURE - Fall 2003
The Interstellar Medium
IMPRS-LECTURE
- Fall 2003 -Dieter Breitschwerdt
Max-Planck-Institut für Extraterrestische Physik
Prologue...it would be better for the true physics if there were no mathematicians on earth. Daniel Bernoulli
A THEORY that agrees with all the data at any given timeis necessarily wrong, as at any given time not all the data
are correct. Francis Crick
TELESCOPE, n. A device, having a relation to the eye similar to that of the telephone to the ear, enabling distant objects to plague us with a multitude of needless details.Luckily it is unprovided with a bell ... Ambrose Bierce
For those who want some proof that physicists are human, the proof is in the idiocy of all the different units which theyuse for measuring energy. Richard P. Feynman
OverviewLECTURE 1: Non-thermal ISM Components• Basic Plasma Physics (Intro)
– Debye length, plasma frequency, plasma criteria• Magnetic Fields
– Basic MHD– Alfvén Waves
• Cosmic Rays (CRs)– CR spectrum, CR clocks, grammage– Interaction with ISM, propagation, acceleration
LECTURE 2: Dynamical ISM Processes• Gas Dynamics & Applications
– Shocks – HII Regions– Stellar Winds – Superbubbles
• Instabilities– Kelvin-Helmholtz Instability– Parker Instability
LECTURE 1Non-Thermal ISM Components
I.1 Basic Plasma Physics• Almost all baryonic matter in the universe
is in the form of a plasma (> 99 %)• Earth is an exception• Terrestrial Phenomena: lightning, polar
lights, neon & candle light• Properties: collective behaviour, wave
propagation, dispersion, diagmagneticbehaviour, Faraday rotation
• Plasma generation:– Temperature increase
However: no phase transition! Continuous increase of ionization (e.g. flame)
– Photoionisation (e.g. ionosphere)– Electric Field („cold“ plasma, e.g. gas discharge)
• Plasma radiation:– Lines (emission + absorption)– Recombination (free-bound transition)– Bremsstrahlung (free-free transition)– Black Body radiation (thermodyn. equilibrium)– Cyclotron & Synchrotron emission
1. Definition:• System of electrically charged particles
(electrons + ions) & neutrals• Collective behaviour long range
Coulomb forces2. Criteria & Parameters for Plasmas:i. Macroscopic neutrality: Net Charge Q=0
L >> ( ... Debye length) for collective behaviourii. Many particles within Debye sphere iii. Many plasma oscillations between ion-neutral
damping collisions
Dλ Dλ13 >>≈ DeD nN λ
1 >>enτω
Quasi-neutral Plasma• Physical volume: L• Test volume: l• Mean free path: λ
• Requirement:
Net Charge: Q = 0
L
lλ Ll <<<<λ
• To violate charge neutrality within radius r requires electric potential:
( ) ( )
( )
K107or V106.03 of voltagea need we
01.0 ,cm 10 cm, 1 ands g cm 1 esu 1 esu, 10803.4
For
34
34)(
34
73
311
1-1/23/210
2
33
×≈×=Φ
≈−===×=
−==Φ⇒
−=−+=
−
−
T
nnnnre
nnerrq
nnerenenrq
iei
ei
eiei
π
ππ
Debye Shielding• Violation of Q=0 only within Debye sphere
e e e
eee
+ +q x
yz
r
Plasmaq ... positive test charge
Equilibrium is disturbed
Is there a new equil. state?Look for steady state!
)(exp )( , )(exp )( 00
Φ−=
Φ=Tk
ennTk
ennB
iB
errrrr
rr
r
)(rErrr
Φ∇−=• Electrostatics: • Equilibrium: Boltzmann distribution
• Total charge density:
)( )(exp )(exp
)())()(()(
0 rrr
rrrr
rrr
rrrr
δ
δρ
QTk
eTk
een
Qnne
BB
ie
+
Φ−−
Φ−=
+−−=
)(4 rErrr
πρ=∇• Relation between charge distribution and
charge density (Maxwell):
• Disturbed potential thus given by diff.eq.:
)( 4 )(exp )(exp 4)( 02 rrrr
rrr
r δππ QTk
eTk
eneBB
−=
Φ−−
Φ−Φ∇
• Test charge with small electrostatic potential energy:
Tke B<<Φ )( rr
• Thus the transcendental equation can be approximatedTkTk
eBB
)r(1)r(exprr
Φ±≈
Φ±
)( 4 )( 2 4)( 02 rrr
rr
r δππ QTk
eneB
−=
Φ−Φ∇
• Defining the Debye length:
024 neTkB
D πλ =
)( 4)(2)( 22 rrr rrr δπ
λQ
D
−=Φ−Φ∇
• Electrostatic forces are central forces: In spherical coordinates:
0)( :
)()( :0
:conditionsboundary with
0)(2)(12
22
→Φ∞→
=Φ→Φ→
=Φ−
Φ
rrrQrrr
rrdrdr
drd
r
C
Dλ
)()( rΦ=Φ rr
)(rDΦ
)(rCΦ
Radius r
Coulomb and Debye Potential
λD~2.1
• Result:
−=Φ rrQr
DD 2exp )(
λDebye-Hückel Potential
• Is total charge neutrality fulfilled: qtot = 0 ?rrqrr
rqrrq
VVD
Vtot ∫∫∫ +
−−== 33
D2
3 d )( d2exp 12
d )( rr δλπλ
ρ
44 344 21q
00D
rrqr rrr
r q d )( 4 d 2exp 142
2
D
22 ∫∫
∞∞
+
−−= δπ
λπ
πλ
qr rq
rr q
0D
D
d 2exp 2
42
2exp 2
4 2
D
D2
0D
D2
+
−
−+
+
−
−−=
∫∞
∞
λλπ
πλ
λλπ
πλ
[ ]
[ ] q.e.d. 20 2
2exp2
42
00 2
22
0D
2
22
0qqqq
qrqq
DD
D
DD
=−=+−=
+
−+−−=
∞
πλπλ
λλπ
πλπλ
PartialIntegration
Importance of Debye Shielding• Test charge q neutralized by neighbouring
plasma charges within „Debye sphere“• Charge neutrality guaranteed for • For : bec. breaks down• Factor „2“: due to non-equil. distr. of ions
If we neglect ion motions: : • Numbers:
Ionosphere: T=1000 K, ne=106 cm-3, λD=0.2 cmISM: T=104 K, ne~1 cm-3, λD=6.9 m; LISM~3 1016 mDischarge: T=104 K, ne=1010 cm-3, λD=6.9 10-3 cm
Dr λ>>0→r ∞→Φ )(rD Tke B<<Φ
0nni =D
r
D rQr λ
−=Φ e )(
cm cm][[K]9.6 3-
eD n
T=λ
• Number of particles in a Debye sphere:
• Charge neutrality can only be maintained for a sufficient number of particles in Debye sphere:
• Collective behaviour only for r << λD for each particle: only here violation of Q=0 possible
• Debye shielding is due to collective behaviour• g<<1: λmfp << λD
Plasma parameter: ( ) 13 -Deng λ=3
34
DeD nN λπ=
11 <<⇔>> gND
Plasma frequency• Violation of Q=0: strong electrostatic
restoring forces lead to Langmuir oscillations due to inertia of particles
• Electrons move, ions are immobile• Averaged over a period: Q=0• Longitudinal harmonic oscillations with
plasma frequency:e
ep m
n2e 4πω =
Oscillations damped by collisions betweenelectrons and neutral particles
timecollision mean ... /1~ enen ντTo restore charge neutrality we need:
1 >>⇔>> enen τωνω
Check the validity of plasma criteria: ISM – DIG: Te ~ 8000 K, ne ~ 1 cm-3
i. L >>ii.iii.
Dλ13 >>≈ DeD nN λ
1 >>enτω
Plasma Criteria
Exercise:
I.2 Magnetic Fields• Magnetic Fields (MFs) are ubiquitous in universe• Observational evidence in ISM:
– Polarization of star light –> dust –> gives – Zeeman effect –> HI –> gives– Synchrotron radiation –> relativistic e- –> gives – Faraday rotation –> thermal e- –> gives
• Sources of MFs are electric currents• In ISM conductivity σ is high, thus large scale
electric fields are negligible and
jr
⊥B//B
⊥B
//B
( )Bcjrrr
×∇=π4
Basic Magnetohydrodynamics (MHD)• Plasma is ensemble of charged (electrons + ions) and
neutral particles –> characterized by distribution function in phase space –> evol. by Boltzmann Eq.
• MHD is macroscopic theory marrying Maxwell‘s Eqs.with fluid dynamic eqs.: „magnetic fluid dynamics“
• Moving charges produce currents which interact with MF–> backreaction on fluid motion
• To see basic MHD effects, a single fluid MHD is treated (mass density in ions, high inertia compared to e-)
• For simplicity gas treated as perfect fluid (eq. of state)• Neglect dissipative processes: molecular viscosity,
thermal conductivity, resistivity
( )tpxfi ,, rr
Basic MHD assumptions
1. Low frequency limit: – Consider large scale (λ big, ω small) gas motions– Consider volume V of extension L:
– If then as assumed – Note that also ω<< ωg (for el. + ions) holds– Ampére‘s law:
simplifies to since
cνω <<
1
~~1~
<<≡⇔
<<
KnL
vLv
mfp
mfp
thc
th
λλ
ντ
ω
(hydrodynamic limit)
eiνω << ie PP ≈
tE
cj
cB
∂∂+=×∇r
rrr 14π
( )Bcjrrr
×∇=π4
1c1
~
1
2 <<=×∇
∂∂
cvL
LB
E
BtE
cτ
τrr
r
using (see 2.) BcvE ~
2. Non-relativistic limit : v/c << 1– Electric + magnetic field in plasma rest frame are then:
0 and speed),(drift with ≡−=−=−=−=
eieiee
eeeeii
enZenvvuuenvenvZenj
ρrrr
rrrr
[ ] ( )[ ] jjBBvvcBBBvc
E
Errrrrrrrrrr
rr
=′⇒≈××+=′×−=⇒
→′⇔→Φ∇
/1 , 100
2
– Current density
( )( )Ev
cBB
Bvc
EE
rrrr
rrrr
×−=′
×+=′
1
1
– For v<<c, e- due to high mobility prevent large scale E-fields, i.e.
Negligible drift speed between electrons and ions:
Consider ISM with B-field: B~ 3 µG, L~1 pc, ne~1 cm-3
ue ~ 4.8 10-11 km/s as compared to cs ~ 1 km/s motion of ions and electrons coupled via collisions;small drift speed for keeping up B-field extremely low!
3. High electric conductivity : ideal MHD– In principle ue is needed to calculate current density
however Ohm‘s law can be used instead:
LenBc
enju
eee π4
~~
∞→σ
( )
( ) ∞→×−=⇒
×+=′==′=′
σ
σσ
for , 1
1 have we, Since
BvE
Bvc
EEjjBB
rrr
rrrrrrrr
c
Thus Faraday‘s law is given by:
• The equation of motion (including Lorentz force term):
with
Note: if field is force free
[ ] [ ]( )Bvc
Ect
B rrrrrr
××∇−=×∇−=∂∂ 11
Magnetic pressure and tension
( ) extmag FFPvvtv
dtvd rrrrrr
rr
++∇−=∇+∂∂= ρρρ
[ ] [ ] [ ] BBBjc
Bvc
Fmag
rrrrrrrr××∇=×=×=
πρ
4111
[ ] ψπ
∇−=⇒=××∇rrrrr
BBB 041
Potential field!
• Magnetic pressure and tension:– Aside: killing vector cross products use ε-tensor
and write with summation conventionand the identity:
Thus:
ijkεijkjibaba ε=×
rr
jkiljlikklmijk δδδδεε −=
[ ] [ ]( ) ( )
( ) ( )[ ]( ) ( )
( ) ( )2
lim
21 BBB
BBBB
BBBB
BBBBBBBB
imimii
kijmkmjikjijklkji
ilmjklkjiilmjklkji
∇−∇=
∂−∂=
−∂=∂=
∂−=∂−=×∇×−=××∇
rrrr
rrrrrr
δδδδεεεεεε
• Thus
– If field lines are parallel i.e. no magnetic tension, but magnetic pressure will act on fluid
– If field lines are bent, magnetic tension straightens them
– Magnetic tension acts along the field lines(Example: tension keeps refrigerator door closed)
[ ] ( )πππ 844
1 2BBBBBFmag −∇=××∇=rrr
rrrr
Magnetic tension Magnetic pressure
( ) 0 =∇ BBrrr
Ideal MHD Equations0=∇ B
rr• Note that is included in Faraday‘s law as
initial condition!
( )
( ) ( )
( )
0
41
0
=
××∇=∂∂
+××∇+∇−=∇+∂∂
=∇+∂∂
γρ
πρρ
ρρ
Pdtd
BvtB
FBBPvvtv
vt
ext
rrrr
rrrrrrrrr
rr
• Here we used the simple adiabatic energy equation
Magnetic Viscosity and Reynolds number• For finite conductivity field lines can diffuse away • Analyze induction equation:
( )
( )
( )
( ) ( )
( ) ( )[ ]( ) BBv
BBcBv
BvjctB
Bvc
jE
Bvc
EEjj
EctB
m
rrrr
rrrrrrr
rrrrrr
rrr
r
rrrrrr
rrr
2
22
4
!!! with term thekeep ... 1
1
∇+××∇=
∇−∇∇−××∇=
××∇+×∇−=∂∂
⇒
×−=⇒
×+=′=′=
×∇−=∂∂
ηπσ
σ
σσ
σσ
πση
4
2cm =
... magnetic viscosity
• First term is the kinematic MHD term, second is diffusion term
• Comparing both terms:( )
22 ~
~
LBB
LvBBv
mm ηηr
rrr
∇
××∇
• Magnetic Reynolds number:
mmm
vL
LB
LvB
Rηη
=≡2
• Rm >> 1: advection term dominatesRm << 1: diffusion term dominatescf. analogy to laminar and turbulent motions!
Magnetic field diffusionB
tB
m
rr
2∇=∂∂ η• For Rm << 1 we get:
• Deriving a magnetic diffusion time scale:
2
22
2
4~~c
LLL
BB
mD
m
D
πση
τητ
=⇒
Note: Form identical to heat conduction and particlediffusion
• Field diffusion decreases magnetic energy: field generating currents are dissipated due to finite conductivity -> Joule heating of plasma
Examples:
• Block of copper: L=10 cm, σ=1018 s-1
• Sun: R ~ L=7 1010 cm, σ=1016 s-1
although conductivty is not so high, it is the large dimension L in the sun (as well as in ISM) that keep Rmhigh!
• Note that since turbulence decreases L and thus diffusion times
thus fields in ISM have to be regenerated (dynamo?)
s 2.1≈⇒ Dτ
yr! 102s106 1017 ×≈×≈⇒ Dτ
2LD στ ∝
Concept of Flux freezing
• Flux freezing arises directly from the MHD kinematic (Faraday) equation (Rm >> 1): magnetic field lines are advected along with fluid, magnetic flux through any surface advected with fluid remains constant
• Theorem: Magnetic flux through bounded advectedsurface remains constant with time
• Proof: Consider flux tube
dS
Flux tube
• Consider surface 2211 by )( and by bounded )( CttSSCtSS ∆+==rrrr
( ) ldtvSdrrr
×∆−=
ldr
tv∆r
1C
2CBr
( ) SdtrBSB
rrr ,∫=Φ
• Surface changes postion and shape with time• Magnetic flux through surface at time t:
• Rate of change of flux through open surface:
• Expand field in Taylor series
• So that
• Using Gauss law
and applying to the closed surface consisting of
( ) ( ) ( )K
rrrrrr +∆
∂∂+=∆+ t
ttrBtrBttrB ,,,
( )[ ] ( ) ( )
−∆+
∆= ∫∫∫ →∆
SdtrBSdttrBt
SdtrBdtd
SStS
rrrrrrrrr , ,1lim ,120
( )[ ] ( ) ( ) ( )
−
∆+
∂∂= ∫∫∫∫ →∆
SdtrBSdtrBt
Sdt
trBSdtrBdtd
SSS
tS
rrrrrrrrrrrr , ,1,lim ,12
20
0 3 =⋅∇=∫ ∫ rdBSdBV
rrrrr
tvSS ∆rrr
length of surface lcylindrica theand , 21
We obtain
• Noting that in the limit
and using the vector identityand Stokes‘ theoremone gets
( ) ( ) ( ) ( )[ ] 0, , ,121
=×∆−+−= ∫∫∫∫CSS
ldtvtrBSdtrBSdtrBSdBrrrrrrrrrrrr
)()()()( ,0 122 tStSttStStrrrr
=→∆+=→∆
( )[ ] ( ) ( ) [ ]ldvtrBSdt
trBSdtrBdtd
CSS
rrrrrrrrrr ×⋅+∂
∂= ∫∫∫ ,, ,
( ) ( ) ( )[ ] ldtrBvldvtrBrrrrrrrr , , ⋅×−=×⋅
( )[ ] ( )[ ]∫∫ ⋅××∇=⋅×SC
SdtrBvldtrBvrrrrrrrrr , ,
( )[ ] ( ) ( )[ ] SdtrBvt
trBSdtrBdtd
SS
rrrrrrrrrr∫∫ ⋅
××∇−∂
∂= , , ,
bottom top mantle
• For a highly conducting fluid ( ) and taking as fluid velocity, field lines are linked to fluid motion and according to ideal MHD „flux freezing“ holds
∞→σ vr
( )[ ] 0 , =∫ SdtrBdtd
S
rrr
• Note: motions parallel to field are not affected
vr1C
2C
1C
2C
• For finite conductivity field lines can diffuse out:
( )[ ] ( ) SdtrBSdtrBdtd
SmS
rrrrrrr , , 2∫∫ ∇=η
MHD waves• Perturbations are propagated with characteristic speeds• For simplicity consider linear time-dependent
perturbations in a static compressible ideal background fluid
• Ansatz:
1<<XXδ
jjj
EEE
BBB
vvvPPP
rrr
rrr
rrr
rrr
δδδ
δδδρρρ
+=
+=
+=
+=+=+=
0
0
0
0
0
0 where
00 =vr• Assume background medium at rest:
( )
( )
( )
00
0
00
0
10
1
4
1
0
ρδργδ
δδ
δ
δδ
δπδ
δδδρ
δρδρ
=
×−=
=∇∂
∂−=×∇
=×∇
×+∇−=∂
∂
=∇−=∂
∂
PP
Bvc
E
BtB
cE
jc
B
Bjc
Ptv
vt
rrr
rr
rrr
rrr
rrrr
rr Keep only first order terms!
Perturbed Equations
• Combining equations and eliminate all variables in favour of
• Defining sound speed:
yields single perturbation equation
• Defining Alfvén speed:
vrδ
ργ
ρPPc
SS =
∂∂=2
( ) 044 0
0
0
022
2
=×
××∇×∇−∇∇−
∂∂
πρπρδδδ BBvvc
tv
s
rrrrrrrrr
0
0
4πρBvA
rr =
yields finally• We seek solutions for plane waves propagating
parallel and perpendicular to B-field• Wave ansatz:
• Dispersion relation:
( ) ( )[ ]{ } 0 22
2
=××∇×∇+∇∇−∂
∂AAs vvvvc
tv rrrrrrrrr
δδδ
( ) ( )[ ]txkiAtxv ωδ −= rrrr exp ,
( )( ) ( ) ( ) ( )[ ] 0222 =−−+++− AAAAAs vvkkvvvkvkvkvkvcv rrrrrrrrrrrrrrr δδδδδω
• Case 1:dispersion relation reads then
Phase velocity
• Case 2:dispersion relation reads then
Avk rr⊥
kvrr //δ Longitudinal magnetosonic wave
22Asph vc
kv +== ω
0// i.e. ,// Bkvk A
rrrr
( ) ( ) 0 1 22
2222 =⋅
−+− AA
A
sA vvvk
vcvvk rrrr δδω
( ) ( ) ( )[ ]AAAA vvkkvvvkvkv rrrrrrrrrrr δδδ −−+( )( )kvkvcv As
rrrr δδω 222 ++− 0=
Case Study for different type of waves
• Two different types of waves satisfy this DR:– Case A:
thus and
the solution is just an ordinary (longitudinal) sound wave– Case B:
the DR reads in this casethus
the solution is a transverse Alfvén wave (pure MHD wave)driven by magnetic tension forcesdue to flux freezing gas (density ρ0 ) must be set in motion!
vvvk Arrrr
δδ //// ⇒( ) vvv
vvvvvvv A
AAAA
rrrrr δδδ
δδ 2 ==⋅
( ) 0v 222 =− rδωkcs
0=⋅⇔⊥⇒⊥ vvvvvk AArrrrrr
δδδ( ) 0v 222 =− rδωkvA
Aph vvk
==ω
• Magnetosonic wave (longitudinal compression wave)
• Transverse Alfvén wave Note 1: in both cases phase velocity independent of ω and k: dispersion free waves!
Note 2: in ISM density is low, Therefore Alfvén velocity high ~ km/s
kr
Br
kvrr //δ
Br
kr
vrδ
kvrr ⊥δ
I.3 Cosmic Rays
Cosmic RadiationIncludes -• Particles (2% electrons, 98% protons and
atomic nuclei)• Photons
• Large energies ( )γ-ray photons produced in collisions of high energy particles
eVEeV 209 1010 ≤≤
Extraterrestrial Origin• Increase of
ionizing radiation with altitude
• 1912 Victor Hess‘ balloon flight up to 17500 ft. (without oxygen mask!)
• Used gold leaf electroscope
Inelastic collision of CRs with ISM
• Diffuse γ-emission maps the Galactic HIdistribution
• Discrete sources at high lat.: AGN (e.g. 3C279)
γππ
2 0
0
→++→+ pppp
HI
For E > 100 MeV dominant process
• Top: CO survey of Galaxy mapping molecular gas (Dame et al. 1997)
• Bottom: Galactic HI survey (Dickey
& Lockman 1990)
HI
Columbia 1.2m radio telescope
γ-ray luminosity of Galaxy• Probability that CR proton undergoes inelastic collision
with ISM nucleus• 1/3 of pions are π0 decaying with <Eγ > ~ 180 MeV• If Galactic disk is uniformly filled with gas + CRs
the total diffuse γ-ray luminosity is
• Galaxy with half thickness H=200 pc, nH~1 cm-3, εCR~1 eV/cm3 Vgal ~ 2 1066 cm3
• in agreement with obs.!• Thus γ-rays are tracer of Galactic CR proton distribution
226 cm105.2 , −×== ppHppcoll cnP σσ
galCRcollCRHpp VPEEncnL εσγ 31)(
31 == ∑
erg/s 1039≈γL
Chemical composition
Groups of nuclei Z CR UniverseProtons (H) 1 700 3000α (He) 2 50 300Light (Li, Be, B) 3-5 1 0.00001* Medium (C,N,O,F) 6-9 3 3Heavy (Ne->Ca) 10-19 0.7 1V. Heavy >20 0.3 0.06
Note: Overabundance of light elements spallation!
Origin of light elements
• Over-abundance of light elements caused by fragmentation of ISM particles in inelastic collision with CR primaries
• Use fragmentation probabilites and calculate transfer equations by taking into account all possible channels
Differential Energy Spectrum • differential
energy spectrumis power law for109 < E < 1015 eV
1016 < E < 1015 eV
~ E-2.7
~E-3.1
„knee“
„knee“
„ankle“• for E > 1019 eV CRs are
extragalactic (rg ~ 3 kpcfor protons)
Primary CR energy spectrum
• Power law spectrum for 109 eV < E < 1015 eV:
[IN] = cm-2 s-1 sr-1 (GeV/nucleon)-1
• Steepening for E > 1015 eV with γ = 3.08 („knee“)and becoming shallower for E > 1018 eV („ankle“)
• Below E ~ 109 eV CR intensity drops due to solar modulation (magnetic field inhibits particle streaming)gyroradius:
R ... rigidity, θ ... pitch angle
dEKEdEENEEI Nγγ γ −− =≈∝ )(or 70.2 with )(
BcR
BcZepc
ZeBvmr L
gϑϑϑγ sinsinsin0 =
==
Example: CR with E=1 GeVhas rg ~ 1012 cm! For B ~ 1µG@ 1015 eV, rg ~ 0.3 pc
Important CR facts:• CR Isotropy:
– Energies 1011 eV < E < 1015 eV: (anisotropy) consistent with CRs streaming away from Galaxy
– Energies 1015 eV < E < 1019 eV: anisotropy increases -> particles escape more easily (energy dependent escape)
Note: @ 1019 eV, rg ~ 3 kpc– Energies E > 1019 eV: CRs from Local Supercluster?
particles cannot be confined to Galactic disk
• CR clocks: – CR secondaries produced in spallation (from O and C) such
as 10Be have half life time τH~ 3.6 Myr -> β-decay into 10B
4106 −×≈IIδ
– From amount of 10Be relative to other Be isotopes and 10B and τH the mean CR residence time can be estimated to be τesc~ 2 107 yr for a 1 GeV nucleonCRs have to be constantly replenished!What are the sources?
– Detailed quantitative analysis of amount of primaries and secondary spallation products yields a mean Galactic mass traversed („grammage“ x) as a function rigidity R:
for 1 GeV particle, x ~ 9 g/cm2
– Mean measured CR energy density:
( ) 0.6 ,g/cm GV 209.6)( 2 ==−
ξξRRx
3eV/cm 1~~~ ≅turbthmagCR εεεε
If all CRs were extragalactic, an extremely high energy production rate would be necessary (more than AGN and radio galaxies could produce) to sustain high CR background radiationassuming energy equipartion between B-field and CRs radio continuum observations of starburst galaxy M82 giveCR production rate proportional to star formation rateno constant high background level! CR interact strongly with B-field and thermal gas
)(100~)82( GalaxyM CRCR εε
CR propagation:• High energy nucleons are ultrarelativistic -> light
travel time from sources• CRs as charged particles strongly coupled to B-field• B-field: with strong fluctuation
component -> MHD (Alfvén) waves • Cross field propagation by pitch angle scattering
random walk of particles! CRs DIFFUSE through Galaxy with mean speed
• Mean gas density traversed by particles
esc4 yr 103~ ττ <<×≈c
Llc
BBB reg
rrrδ+=
Br
δ
cLvr
diff3-
7 10~km/s 490yr102kpc 10~ =×≈τ
ISMh cx ρτρ 1~g/cm 105~ 325−×≈
esc 4
particles spend most time outside the Galactic disk in the Galactic halo! „confinement“ volume– CR „height“ ~ 4 times hg (=250 pc) ~ 1 kpc– CR diffusion coefficient:
– Mean free path for CR propagation:strong scattering off magnetic irregularities!
• Analysis of radioactice isotopes in meteorites: CR flux roughly constant over last 109 years
s/cm105~ 228×≈×esc
CRLh τκ
pc 1~/3~ cCR κλ
CR origin:• CR electrons (~ 1% of CR paeticle density) must be of
Galactic origin due to strong synchrotron losses in Galactic magnetic field and inverse Compton lossesNote: radio continuum observations of edge-on galaxies show strong halo field
• Estimate of total Galactic CR energy flux:
Note: only ~ 1% radiated away in γ-rays!• Enormous energy requirements leave as most
realistic Galactic CR source supernova remnants (SNRs)– Available hydrodynamic energy:
about 10% of total SNR energy has to be converted to CRs – Promising mechanism: diffusive shock acceleration
• Ultrahigh energy CRs must be extragalactic
erg/s 10~ 41≈esc
confCRCR
VF τε
erg/s 10erg10 yr 1003~ 4251 ≈⋅≈SNRSNSNR EF ν
eV)10~(for kpc 100 20ERr galg >≥
Diffusive shock acceleration:
• Problem: can mechanism explain power law spectrum?• Diffusive shock acceleration (DSA) can also explain
near cosmic abundances due to acceleration of ISMnuclei
• Acceleration in electric fields?
due to high conductivity in ISM, static fields of sufficient magnitude do not existthus strong induced E-fields from strongly time varying large scale B-fields could help -> no strong evidence!
( ) [ ]
×+= Bv
cEevm
dtd
L
rrrr 1γ
Fermi mechanism:
• Fermi (1949): randomly moving clouds reflect particles in converging frozen-in B-fields („magnetic mirrors“)processes is 2nd order, because particles gain energy by head-on collisions and lose energy by following collisions (2nd order Fermi process)particles gain energy stochastically by collisions
• 1st order Fermi process (more efficient):– Shock wave is a converging fluid– Particles are scattered (elastically) strongly by field
irregularities (MHD waves) back and forth
2nd order Fermi Shock Front
1st order Fermi
Cloud
• Energy gain per crossing:
Fermi)order (1 34 st
cV
EE S≈∆
Fermi)order (2 34 nd
2
≈∆
cV
EE
• Shock is convergingfluid
– Energy gain per collision: ∆E/E ~ (∆v/c)– Escape probability downstream increases with energy
– Essence of statistical process:let be average energy of particle after one collisionand P be probability that particles remains in acceleration process -> after k collisions:
0 EE β=
( )( )
β
β
β
lnln
00
0
0
00
lnln
lnln
energies with particles )(
P
kk
EE
NN
PEENN
EEPNEN
=⇒
=⇒
==>
speed) particle ... ( vv
VP Sesc ≈ 11
vVPP S
esc −≈−=
cV
EE
EE S+≈∆+=≡ 11
0
β
)(for ,11
1
1
1ln
1ln
lnln cvV
cv
vV
cv
cV
vVc
V
P SS
S
S
S
≤<<−≈−≈
−−
+≈
−
+
=β
– Note: that N=N(>E), since a fraction of particles is accelerated to higher energiestherefore differential spectrum given by
– Note: statistical process leads naturally to power law spectrum!– Detailed calculation yields:– Hence the spectrum is:
– The measured spectrum E < 1015 eV gives spectral index -2.7However: CR propagation (diffusion) is energy dependent ∝ E0.6
source spectrum ∝ E-2.1 : excellent agreement! PROBLEM: Injection of particles into acceleration mechanism
Maxwell tail not sufficient „suprathemal“ particles
( ) dEEconstdEEN P 1lnln.)( −×= β
1lnln −≈
βP e.g. Bell (1978)
dEEconstdEEN 2.)( −×=
- remains unsolved! -
LECTURE 2Dynamical ISM Processes
• ISM is a compressible magnetized plasma•• Pressure disturbances due to energy + momentum
injection: SNe, SWs, SBs, HII regions, jets• Speed of sound:
ISM motions are supersonic: • Shocks (collisionless) propagate through ISM
(λmfp >> ∆)
II.1 Gas Dynamics & Applications
Lmfp <<λ
, mTkc B
s µ= km/s 1203.0 ≤≤ sc
1>>=sc
uM
II.2Shocks• Assumption: Perfect gas, B=0• Shock thickness time
independent across shock discontinuity: steady shock
• Conservation laws: Rankine-Hugoniot conditions
mfpλ≤∆ Pu , ,ρ
(energy) 12
1 12
1(momentum)
(mass)
2
2222
1
1211
2222
2111
2211
ργγρ
ργγρ
ρρρρ
PuPu
uPuP
uu
−+=
−+
+=+
=
1 2
Shock Frame
111 ,, Puρ 222 ,, Puρ
Analoga1. Traffic JamTraffic Jam::
„speed of sound“ cs= vehicle distance/reaction time=d/τ• If car density is high and/or people are „sleeping“: cs decreases• If vcar ≥ cs then a shock wave propagates backwards due to
„supersonic“ driving• Culprits for jams are people who drive too fast or too slow because
they are creating constantly flow disturbances
• For each traffic density there is a maximum current density jmax and hence an optimum car speed to make djmax/dρ =0!
2. HydraulicHydraulic JumpJump::„speed of sound“ cs= („shallow water“ theory)gh
Kitchen sink experiment • total pressure: Ptot =Pram+Phyd
ghvP 2tot ρρ +=
• At bottom: v > cs • Therefore: abrupt jump• Behind jump:
h2 > h1
V2 < V1
• V1 > c1 „ supersonic“ V2 < c2 „ subsonic“
II.3 HII Regions• Rosette Nebula
(NGC2237):– Exciting star cluster
NGC 2244, formed ~ 4 Myr ago
– Hole in the centre: Stellar Windscreating expanding bubble
Trifid Nebula:Stellar photons heat molecular cloud gasGasdynamical expansion (e.g. jets)
Facts• Lyc photon output S* of O- and B stars ionizes
ambient medium to T ~ 8000 K– In ionization equilibrium: – Energy input Q per photon:
– Energy loss per recombination: Equil. Temperature:
• Q depends on stellar radiation field and frequency dependence of ioniz. cross section
recI NN && =QNQN recI
&& =
receB NTk & 23
=
Be k
QT32=
• Assumption: stellar rad. field is blackbodyaverage kinetic energy per photo-electron:
For a blackbody at temperature T*:
For
T* = 47000 K (for O5 star) Te ~ 31300 K
LHH hEdSdSEhQLL
ννννν
ννν
=−= ∫∫∞∞
, )( **
( )[ ] 1*2** 1exp2)( −−=∝ Tkh
chTBSh Bννν νν
1* >>Tkh Bν
*32 TT
TkQ
e
eB
≈⇒
≈
Bad agreement with observation!Reason: forbidden line cooling of heavy elements, like [OII], [OIII], [NII] is missing!
• For H the total recombination coefficient to all excited states is:
• Thus • If [OII] is the dominant ionic state: • Collisional excitation rate (all ions are approx.
in the ground state):
-134/310)2( s cm 102)( −−×= ee TTβ
*)2( TknnQN BHerec β=&
eOII nn 4106 −×≈
( ) ]exp[ )(
)(2/1
eBijeijeij
eijIeij
TkETATC
TCnnN
−=
=
• Radiative energy loss by [OII] for
taking and demanding
For T*=40000 K, we obtain: Te~8500 K,in excellent agreement with observations!• HII regions are thermostats!
( ) -1-342/1232 scm erg ]1089.3exp[ 101.1 eeOIIOII TKTnyL ×−×≈ −
1≈OIIy
QNQNL recIOII&& ==
*644/1 105.2]1089.3exp[ TTKT ee
−×=×−
levels and 2/32
2/52 DD
Dynamics of HII Regions• Spherical symmetric Model:
– Ionization within radius r:
– Recombination within r:
– Ion. fract.– Balance of ion. + recomb.
Lyc photons
SR
Case A: Static HII Region
JrS 2* 4π=
eHs nnR (2)3 34 βπ
nnnx He =≈⇒≈ 1
3/1
2)2(*
4 3
=
nSRS πβ
... „Stroemgren“ radius
Example:
pc 3 type)(O6.5 s 10 ,cm 10n
K) 8000(T scm1021-49
*3-2
H
-1313)2(
≈⇒==
≈×= −
SRS
β
Case B: Evolving HII Region• Photon flux at IF:
(conservation of photons)• Thickness of IF ~ photon mfp:
planar geometry; no rec. in IF• Ambient medium at rest• IF velocity: • Define dimensionless quant.
RnR
SJ 20
)2(2
*
31
4β
π−=
JdtdRn =0
( ) 1)2(0rec
recrec
, ,t , −==== βτττ
τη nRVRR S
RS
( )0 0, :BC
1 3/1
→→−= −
ητη τeCSolution:
IFRHII
Lyc photons
IFR
SR
Equation of motion:
( ) 23 31 ηηη −=&
Evolution of Stroemgren sphere
• RS is reached only within 1% after
• IF velocity >> cII until R~ RStherefore then gasdynamical expansion
• IF velocity slows down rapidly
• However: NOT possible
η
τ
τ
η&
=
dtdRR IF
rec
S
τη&
recττ 4≥
0nnII ≈
IIIF c
dtdR <<
Case C: Gasdynamical Expansion of HII Region
• When sound waves can reach IF
• : HII gas acts as pistonshock driven into HI gas
• Gas pressed into thin shell:
• Pressure uniform in shell & HII region:
• Ionization balance in HII reg.BUT: HII grows in size + mass
IIIF cR ≤&
III PP >>
RRR shIF =≈ :
dynsc ττ <<
01 34
2)2(*3 >−=
=
dtdn
nSmmnR
dtd
dtdM II
IIIIS
II
βπ
Lyc photons
shR
HII
HI
Shock
IFR
Simple Model:•• For strong shock:• Ion. Balance:
• Ambient medium at rest:• Thus equation of motion: • Dimensionless quantities:
• Solution:
22 IIIIIIBIIIIsh cmnTknPP ==≈
( )1for 1
2 20
20 ==
+= γρρ
γ shshsh VVP3)2(2
03)2(2
* 34
34
SII RnRnS βπβπ ==
RVsh&=
22/3
2
0
2 IIS
IIII c
RRc
nnR
=
=&
2/3
0
=
RR
nn SII
0 1, :BC
1/ ,/ , 4/3
→→
=⇒===
N
cRRtcNRR
IISIIS
η
ηηηη &&&
7/37/4
471 ,
471
−
+=
+= NN ηη &
• At• Note: • Is pressure equilibrium
reached: ?
• Ionization balance must hold:
• Result:
• Final mass:
Gasdynamical Expansion:η
t [sec]
t [sec]η&
IIcRN == & ,0
rect τ>>exp
III PP ≈
IBIIIBfIII TknTknPP =⇒≈ 2
3)2(2*
34
ff RnS βπ=( )
( ) SIIIf
IIIIf
RTTR
nTTn3/2/2
2/
=⇒
=
I
II
S
ff
S
f
TT
RnRn
MM 2
30
3
==
• Example:
• Initial mass in Stroemgren sphere is a SMALL fraction of final mass
• Equilibrium never reached, because star leaves main sequence before, unless density is high!
100/ ,34/ ,105/
cm 100 K, 8000 K, 1003
0
-30
≈≈×=⇒
===−
SfSff
III
MMRRnn
nTT
yr 108.7
yrn 107.1/ 273~ 34For 7
-2/30
13
×≈
×≈⇒= IISf cRtη
-330 cm103×≥n
• Massice star BD+602522 blows bubble intoambient medium
• Ionizing photons produce bright nebula NGC7635
• Diameter is about 2 pc
• Part of bubble network S162 due to more OB stars
II.4 Stellar Winds
Some Facts
610~
km/s 30002000~−
−
W
W
M
V& M /yr
Total outward force:
2 4 rLFW π
σ=
In reality: )(λσσ =
•Strong UV radiation field•Momentum transfer to gas•Resonance lines show P Cygni
profiles (e.g. CIV, OVI)
•O- and B-stars: –Burn Hot–Live Fast–Die young
Line Driven Stellar Wind:• Assume a stationary wind flow (ignore Pth)
• Stars with L*>Lc are radiatively unstable
22*
4 rL
rmGM
drdvmv
πσ+−=
( )
)luminosity (Eddington 44
,1
*
*
*
*
2*
σπ
πσ
mcGML
LL
mcGML
rGM
drdvv
c
c
=
==Γ
−Γ−=
radiation pressure
• Integration:
• If v(R*)=v0=0:
This is CAK velocity profile (L* > Lc, i.e.Γ>1 needed)
• If L >> Lc
( ) ( ) ( )
−−Γ=−⇔−Γ= ∫∫ rR
GMvvdrr
GMvdvr
R
v
v
11121 1
**
20
22
*
*0
( ) 112
1)(
*
2/1
*
*
2/1*
−=
−Γ=
−=
∞
∞
cesc L
LvR
GMv
rRvrv
mcRLv*
*
2πσ=∞
Example:
• For an O5 star: L*=7.9 105 L , R*=12 R
• Mass loss rate (assume one interaction per photon):
Thus we get: • Shortcomings: cross sect. λ dep., multiple scattering• Note:
km/s 2566
km/s 102/1
*
0
2/1
0
*
2/12/1
≈⇒
=
∞
∞
vRR
LL
mm
v p
Tσσ
∞= vMcL
W&* Momentum rate
6103.6 −×≈WM& M /yr
erg/s 103 erg/s 103.121 39
*372 ×=<<×≈= LVML WWW
&
Most of energy lost as radiation!Less 1 % converted into mechanical luminosity
Effects of stellar winds (SWs) on ISM:
• Observationally is better determined than• We use here for estimates:
• Note: kinetic wind energy >> thermal energy• Star also has Lyc output: • Hypersonic wind flows into HII region:• SW acts as a piston shock wave formed (once
wind feels counter pressure) facing towards star• Hot bubble pushed ISM outwards facing shock• Contact surface separating shocked wind and ISM
WV WM&
WM&
610−=WM& M /yr
km/s, 2000=WV erg/s 1021 362 == WWW VML &
-149* s 10=S
200 km/s 10
km/s 2000M =≈=II
W
cV
Flow Pattern:
• Free expanding windshocked at S-
• Energy driven shocked wind bubble bounded by contact discontinuity CDNo mass flux across C
• Shell of compressed HII region, bounded by IF
• Shell of shocked HI region(ambient gas) bounded by S+
• ambient HI gas
S-
CDIF
S+
5
2
43
1
1
2
3
4
5
Qualitative Discussion:• Region : free expanding wind has mainly ram
pressure,
• Region : shocked wind; the post-shock temperature is given by
– Note: S- moves slowly with respect to wind– Since cb ~ 600 km/s and CD slows down ,
pressure is uniform and energy is mostly thermal– nb low, therefore region is adiabatic– Density jump by factor 4 region extended
1
2
W
WWWW Vr
MVP 22
4 ,
πρρ
&=≈
K 104323
2163
43
72
22
×≈=⇒
===
WB
b
bBbWWWbb
VkmT
TknVVP ρρ
bbdyncool RR &/=>>ττ
)( bb Rc &>>
• Region : expansion of high pressure region drives outer shock S+; compresses HII gas into thin shell– Density high (at least 4 times n0)– Post-shock temperature since – HII region trapped in dense outer shell, once
– Since wind sweeps up ambient medium into shell:
– Pressure uniform since
• Region : between IF and S+, shock isothermal• Region : ambient gas at rest at
*22(2) 4 SnRRN shrec ≥= δβπ&
mnRRRM shshsh 434 23
0 δππρ ==
3
cooling highbII TT << Wb VR <<&
dynshsc cR τδτ <<= /
4III TT <<
5 IT
Simple Model for stellar wind expansion:• Extension of Regions + is
• Extension of Region >> Region therefore it is assumed that occupies all space
• Equations:
3 4
bbsh
b
RRRRRR
≈+=⇒<<
δδ
2 1
2
on)conservati(energy PR R 4LE
on)conservati (momentum P R 4)R M(
energy) (thermal )(1
1E
on)conservati (mass )(M
bb2bW
th
b2bbsh
3th
3sh
&
&
π
π
γ
ρ
−=
=
′′−
=
′′=
∫
∫
dtddtd
rdrp
rdr
r
r
Similarity Solutions:
• No specific length and time scales involved, i.e. r and t do not enter the equations separately
PDE ODE, with variable • Thermal energy given by • Combining the conservation equations yields
• Substituting the similarity variable into equation
αAtRb =33 2
23
34
bbbbth RPPRE ππ ==
0
3234
231512
ρπW
bbbbbbbLRRRRRRR =++ &&&&&&&
( )( ) ( )[ ]0
35325
23 1511221πρ
αααααα α WLtA =+−+−− −
• RHS is time-independent
• Thus the solution reads:
• Note: we have assumed
5/1
0
5/1
154125
53035
=
=⇒=−
ρπ
αα
WLA
5/2
5/35/1
0
5/15/3
53
53
154125
−===
==
tAt
RVR
tLtAR
bshb
Wb
&
ρπ
20 shshI VPP ρ=<<
Numerical Simulation
• NGC 3079: edge-on spiral, D~17 Mpc
• Starburst galaxy with active nucleus
• Huge nuclear bubble generated by massive stars in concert rising to z~700 pc above disk
• Note: substantial fraction of energy blown into halo!
II.5 Superbubbles
HST
More superbubbles:
• Ring nebula Henize 70 (N70) in the LMC
• Superbubble with ~100 pc in diameter, excited by SWs and SNe of many massive stars
• Image by 8.2m VLT (+FORS)
Some Facts• Massive OB stars are born in associations (N*~102-103)• If this happens approx. coeval, SWs and SN explosions
are correlated in space and time!• About >50% of Galactic SNe occur in clusters• MS time is short:
stars occupy small volume during SB formationSB evolution can be described by energy injectionfrom centre of association
• Initially energy input from SWs + photon output rate of O stars dominates for O7 star
( ) yrM10/103 6.10
7MS
−×= mτ
erg/s 10621 352 ×≈= WWW VML &
Simple expansion model:• In early wind phase SB expansion is given by:
• After last O star leaves main sequence and energy input is dominated by succesive SN explosions
• Thus for until last SN occurs (M ~ 7 M ) subsequent ejecta input at energy E~1051 E51 erg mimic a stellar wind!
• If the number of OB stars is N* energy input is given by
yr 105 6×≈MSτ
yr 105tyr 105 76 ×≤≤×
yr10 ,erg/s10
pc 269
773838
5/37
5/1
0
38
ttLL
tnLR
W
SB
==
=
McCray & Kafatos, 1987
cf. Cygnus supershell: RS~225 pc(Cash et al. 1980)
5151
*10 ENLSB = ( )minMMSτ
• Here we have assumed:– A common bubble is formed– Bubble is energy driven (like SW case)– Energy input by SNe is constant with time (therefore
taking MS life time of least massive SN)– Ambient density is constant
• The expansion is given by:
• Most of SB size is due to SN explosions!• SB velocity larger than stellar drift
explosions occur always INSIDE bubble
5/27
5/1
0
51*
5/37
5/1
0
51*
km/s 7.5
pc 97
−
=
=
tnENV
tnENR
SB
SB
Example: Local (Super-)Bubble
• Solar system shielded from ISM by heliosphere
• Nearest ISM is diffuse warm HI cloud (LIC)
• LIC embedded in low density soft X-ray emitting region: LocalBubble (RLB ~ 100 pc)
• Origin of LB: multi-SN?
Breitschwerdt et al. 2001
NaI absorption line studies
• 3 sections through Local Bubble
• LB inclined ~ 20°wrt NGP
• LB open towardsNGP?
NGP
GC
NGP
Gal.Plane
Sfeir et al. (1999) LB ⊥ Gould‘s Belt
Local Chimney
RASSCrabVela
LMCNorth Polar SpurCygnus Loop
Anticorrelation: SXRB – N(HI)• Anticorrel. on large
angular scales forsoft emission
• Increase of SXR flux: disk/pole ~3
• Absorption effect: ~50% local em.
RASS
HI
Local Stellar Population• Local moving groups
(e.g. Pleiades, subgr. B1)• 1924 B-F-MS stars (kin.):
Hipparcos + photometric ages (Asiain et al. 1999)
• Youngest SG B1: 27 B,
• Use evol. track (Schaller 1992):det. stellar masses
• IMF (Massey et al 1995):
pc 120D Myr, 1020 0 ≈±≈τ
1.1 ,N N(m) 1
0 −=Γ
=⇒−Γ
mm
Berghöfer and Breitschwerdt 2002
0
Young Stellar Content and Motion
• Adjusting IMF (B1):
• 2 B stars still active• SNe explode in LB
1
0
0
Mm 551.6 N(m)
7)8MN(m−Γ
=⇒
==
∫ ≈=⇒
≅⇒≤max
min
21dm N(m)N
M 20m1)N(m
SN
0maxmaxm
m
Berghöfer and Breitschwerdt 2002
Formation and Evolution of the Local Bubble
• Energy input by sequential SNe
• Assumption: coeval star formationStar deficiency:First explosion: 15 Myr ago
)m(m1.6 yr,M 10
1030
7MS τατ
α
=⇒=
×=
−m
3.0)1/( ,tLNEL 0SN
SNSB −≈+Γ−=== αδδ
dtd
yr102.5M 10 7cl0 ×≤⇒≥ τm
)M 20(m 0max =
Superbubble Evolution
on)conservati(energy PR R 4)(LE
on)conservati (momentum P R 4)R M(
energy) (thermal )(1
1E
on)conservati (mass )(M
bb2bSB
th
b2bbsh
3th
3sh
&
&
π
π
γ
ρ
−=
=
′′−
=
′′=
∫
∫
tdt
ddtd
rdrp
rdr
r
r
AnalyticModel:
ββ
µ ρρβ
δµ −−
=
=
−+== rK
RrAt 0
0b
~ ,5
3 ,R
54.01.1 ,6.1 ,0 :IF ≈⇒−=Γ== µαβSimilarity solution:
• Note that similarity exponent µ is between SNR (Sedov phase: µ=0.4) AND SW/SB (µ=0.6)
• The mass of the shell is given by:
• After some tedious calculations we find:
ββ
βππ −−
−== ∫ 3
000
2
344 b
R
sh RKdrrKrMb
( ) ( )( )( )( )
β
αββααββααπββ −
+−−+−−+−−=
51
0
03
7241173235
KLA
Local Bubble (Analytic results)• Local Bubble at present:
• Present LB mass:
• Average SN rate in LB: • Cooling time:
• Energy input by recent SNe:
km/s 9.5V pc, 146RMyr 13 ,cm 30n shbexp-3
0 ≈≈⇒== τ
0ej0LB M 200M ,M 600M ≤≥
Mass loading! (decreases radius)yr) 105.6/(1 5
SN ×≅fK)10T ,cm105(nMyr 13 6
b33
b ≈×≈≈ −−cτ
X-ray emission due to last SN(e)!
dtd N EE SNSN =&
erg/s105.2)1(103.4E 3631.07
36SN ×≈+×≈ t&
Disturbed background ambient medium
z
x
y
Local Bubble Evolution: Realistic Ambient MediumDensityCut through Galactic planeLB originates at (x,y) = (200 pc, 400 pc)SNe Ia,b,c & II at Galactic rate60% of SNe in OB associations40% are randomGrid resolution:
1.25 and 0.625 pc
DensityCut through galactic planeLB originates at (x,y) = (200 pc, 400 pc)Loop I at (x,y) = (500 pc, 400 pc)
LB Loop I Results Bubbles collided
~ 3 Myr agoInteraction shell
fragments in ~3Myrs
Bubbles dissolve in ~ 10 Myrs
Numerical Modeling (II)
II.6 Instabilities• Equilibrium configurations (especially in plasma
physics) are often subject instabilities• Boundary surfaces, separating two fluids are
susceptible to become unstable (e.g. contact discontinuity, stratified fluid)
• Simple test: linear perturbation analysis– Assume a static background medium at rest – Subject the system to finite amplitude disturbances– Test for exponential growth– However: no guarantee, system can be 1st order stable and
2nd or higher order unstable
Kelvin-Helmholtz instability
• Why does the surface on a lake have ripples?
Cloud – wind interface:
• Shear flow generates ripples and kinks in the cloud surface due to Kelvin-Helmholtz instability
• „Cat‘s eye“ pattern typical
Examples:
Vincent van Gogh knew it!2D Simulation of Shear Flow
2U
1U
Normal modes analysis:
• Assume incompressible inviscid fluid at rest ( )• Two stratified fluids of different densities move
relative to each other in horizontal direction x at velocity U
• Let disturbed density at (x,y,z) be corresponding change in pressure isthe velocity components of perturbed state are:
in x-, y- and z-directionthus the perturbed equations read:
0=∇ vrr
δρρ +
Pδ
wvuU and ,+
U
xz1ρ
2ρ gr Stratified fluidgr can stands for any
acceleratiion!
( )
0
=∂∂+
∂∂+
∂∂
=∂
∂+∂
∂
−=∂
∂+∂
∂
−∂∂−=
∂∂+
∂∂
∂∂−=
∂∂+
∂∂
∂∂−=+
∂∂+
∂∂
zw
yv
xu
zwxzU
tz
dzdw
xU
t
gPzx
wUtw
Pyx
vUtv
Pxdz
dUwxuU
tu
ss
ss δδ
ρδρδρρ
δρδρρ
δρρ
δρρρ where ( )ss zUU =is the surface at which changes discontinously
szρ
Disturbances vary as[ ]tykxki yx exp σ++
( ) 0Im <σInstability if
• Inserting the normal modes yields a DR
• At the interface U is discontinous, but the perturbation velocity w must be unique
• Integrate over a small „box“ , with
where• Since we have a discontinuity in U and ρ the DR becomes
[ ] [ ] [ ] dzd
UkwgkwUkkw
dzdUk
dzdwUk
dzd
xxxx
ρσ
σρρσρ+
=+−
−+
22
( )εε +− ss zz , 0→ε
[ ] ( )sx
sxxs Ukwgkw
dzdUk
dzdwUk
+
∆=
−+∆
σρρσρ 2
( ) 00 −=+= −=∆ss zzzzs fff
BC
022
2
=
− wk
dzd 0 since ==
dzd
dzdU ρ
• Since must be continous at and w must notincrease exponentially on either side, we must have
• Applying the BC to solutions:
• Expanding and rearranging yields the growth rate
where and
Ukw
x+σ sz
( ) [ ] ( )( ) [ ] ( )0 ,exp
0 ,exp
22
11
>−+=<++=
zkzUkAwzkzUkAw
x
x
σσ
( ) ( ) ( )212
112
22 ρρσρσρ −=+++ gkUkUk xx
( )( )
21
221
221212
21
21
21
2211
+−−
+−±
++−=
ρρρρ
ρρρρ
ρρρρσ UUkgkUUk xx
ϑcoskUUk =rr
ϑcoskkx =
• Two solutions are possible– If
the growth rate is simply
– For every other directions of wave vector instability occurs if:
– Even for stable stratification (against R-T)There is ALWAYS instability no matter how SMALL
is!– For large velocity difference large wavelength instability
0=xk
21
21
ρρρρσ
+−±= gk
Perturbations transverse to streaming are unaffected by it
>k
( )( ) ϑρρ
ρρ22
2121
22
21
cosUUgk
−−>
21 ρρ >
21 UU −
Kelvin-Helmholtz instability
R-T instability
• Qualitative picture:
• CRs are coupled to magnetic field which is frozen into gas• Magnetic field is held down to disk by gas!• CRs exert buoyancy forces on field -> gas slides down
along lines to minimize potential energy -> increases buoyancy -> generates magnetic („Parker“) loops!
• Note: CRs and magnetic field have tendency to expand if unrestrained
Parker instability
Gas + CRs Gas
CRsB
B
Do not despair if you did not understand everything right away!
BUT
Literature:• Textbooks:
– Spitzer, L.jr.: „Physical Processes in the Interstellar medium“, 1978, John Wiley & Sons
– Dyson, J.E.: Williams, D.A.: „Physics of the Interstellar Medium“, 1980, Manchester University Press
– Longair, M.S.: „High Energy Astrophysics“, vol. 1 & 2, 1981, 1994, 1997 (eds.), Cambridge University Press
– Chandrasekhar, S.: “Hydrodynamic and hydomagnetic stability“, 1961, Oxford University Press
– Landau, L.D., Lifshitz, E.M.: „Fluid Mechanics“, 1959, Pergamon Press
– Shu, F.H.: „The Physics of Astrophysics“, vol. 2, 1992, University Science Books
– Choudhuri, A.R.: „The Physics of Fluids and Plasmas“, 1998, Cambridge University Press
• Original Papers:– Dyson, J.E., de Vries, 1972, A&A 20, 223– Weaver, R. et al., 1977, ApJ 218, 377– McCray, R., Kafatos, M., 1987, ApJ 317, 190– Bell, A.R., 1978, MNRAS 182, 147– Blandford, R.D., Ostriker, J.P., ApJ 221, L29– Drury, L.O‘C., 1983, Rep. Prog. Phys 46, 973– Berghöfer, T.W., Breitschwerdt, D., 2002, A&A 390, 299
Bubbles everywhere!!!
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