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!!!

-- TheThe End End --