The Interstellar Medium

IK-LECTURE

– Fall 2007 –

Dieter Breitschwerdt

Institut für Astronomie, Wien

http://homepage.univie.ac.at/dieter.breitschwerdt

Email: breitschwerdt@astro.univie.ac.at

Prologue...it would be better for the true physics if there were no

mathematicians on earth. Daniel Bernoulli

For those who want some proof that physicists are human,

the proof is in the idiocy of all the different units which they

use for measuring energy. Richard P. Feynman

A THEORY that agrees with all the data at any given time

is 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

Overview

LECTURE 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 1

Non-Thermal ISM Components

• 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, diagmagnetic behaviour, Faraday rotation

I.1 Basic Plasma Physics

• 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 forces

2. Criteria & Parameters for Plasmas:

i. Macroscopic neutrality: Net Charge Q=0

L >> ( ... Debye length) for collective behaviour

ii. Many particles within Debye sphere

iii. Many plasma oscillations between ion-neutral

damping collisions

D D13 DeD nN

1 en

Quasi-neutral Plasma

• Physical volume: L

• Test volume: l

• Mean free path:

• Requirement:

Net Charge: Q = 0

L

l

Ll

• To violate charge neutrality within radius r

requires electric potential:

K 107or V 106.03

of voltagea need we

01.0 ,cm 10 cm, 1 and

s g cm 1 esu 1 esu, 10803.4

For

3

4

3

4)(

3

4

73

311

1-1/23/210

2

33

T

nnnnr

e

nnerr

q

nnerenenrq

iei

ei

eiei

Debye Shielding

• Violation of Q=0 only within Debye sphere

e e e

eee

+ +q x

yz

r

Plasma

q ... positive test charge

Equilibrium is disturbed

Is there a new equil. state?

Look for steady state!

)(

exp )( , )(

exp )( 00

Tk

enn

Tk

enn

B

i

B

e

rr

rr

)(rE

• Electrostatics:

• Equilibrium: Boltzmann distribution

• Total charge density:

)( )(

exp )(

exp

)())()(()(

0 rrr

rrrr

QTk

e

Tk

een

Qnne

BB

ie

)(4 rE

)( 4 )(

exp )(

exp 4)( 0

2r

rrr

Q

Tk

e

Tk

ene

BB

• Relation between charge distribution and

charge density (Maxwell):

• Disturbed potential thus given by diff.eq.:

• Test charge with small electrostatic potential energy:

Tke B )( r

)( 4 )(

2 4)( 0

2r

rr

Q

Tkene

B

0

24 ne

TkBD

)( 4)(2

)( 2

2 rrr

QD

• Thus the transcendental equation can be approximated

• Defining the Debye length:

TkTke

BB

)r(1

)r(exp

• Electrostatic forces are central forces:

In spherical coordinates:

0)( :

)()( :0

:conditionsboundary with

0)(2

)(1

2

2

2

rr

r

Qrrr

rrdr

dr

dr

d

r

C

D

r

r

Qr

D

D 2

exp )(

Debye-Hückel Potential

)()( r r

)(rD

)(rC

Radius r

Coulomb and Debye Potential

D~2.

1

• Result:

• Is total charge neutrality fulfilled: qtot = 0 ?

rrqrrr

qrrq

VVD

Vtot

33

D

2

3 d )( d2

exp 1

2d )(

q

00D

rrqr rrr

r q

d )( 4 d 2

exp 1

42

2

D

2

2

qr rq

rr q

0D

D

d 2

exp 2

42

2

exp 2

4 2

D

D

2

0D

D

2

q.e.d. 20 2

2

exp2

42

00 2

2

2

0D

2

22

0qqqq

qrqq

D

D

D

DD

Partial

Integration

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, D=0.2 cm

ISM: T=104 K, ne~1 cm-3, D=6.9 m; LISM~3 1016 m

Discharge: T=104 K, ne=1010 cm-3, D=6.9 10-3 cm

Dr

0r )(rDTke B

0nni D

r

Dr

Qr

e )(

cm cm][

[K]9.6

3-

e

Dn

T

• 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 << D for each

particle: only here violation of Q=0 possible

• Debye shielding is due to collective behaviour

• g<<1: mfp << D

3 3

4DeD nN

11 gND

Plasma parameter: 13 -

Deng

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 4

Oscillations damped by collisions between

electrons and neutral particles

timecollision mean ... /1~ enen

1 enen

Exercise:

Check the validity of plasma criteria: ISM – DIG:

Te ~ 8000 K, ne ~ 1 cm-3

To restore charge neutrality we need:

i. L >>

ii.

iii.

D13 DeD nN

1 en

Plasma Criteria

• 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 is high, thus large scale

electric fields are negligible and

I.2 Magnetic Fields

j

B

B

B

//B

Bc

j

4

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 ,,

1. Low frequency limit:

– Consider large scale ( big, small) gas motions

– Consider volume V of extension L:

– If then as assumed

– Note that also << g (for el. + ions) holds

– Ampére‘s law:

simplifies to since

Basic MHD assumptions

c

1

~~1

~

KnL

v

L

v

mfp

mfp

thc

th

(hydrodynamic limit)

ei ie PP

t

E

cj

cB

14

Bc

j

4

1c

1

~

1

2

c

vL

LB

E

B

t

E

c

using (see 2.) Bc

vE ~

2. Non-relativistic limit : v/c << 1

– Electric + magnetic field in plasma rest frame are then:

0 and speed),(drift with

eieiee

eeeeii

enZenvvu

uenvenvZenj

Evc

BB

Bvc

EE

1

1

jjBBvvcBBBvc

E

E

/1 , 1

00

2

– For v<<c, e- due to high mobility prevent large scale

E-fields, i.e.

– Current density

Negligible drift speed between electrons and ions:

Consider ISM with B-field: B~ 3 G, 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:

Len

Bc

en

ju

ee

e4

~~

for , 1

1 have we, Since

Bvc

E

Bvc

EEjjBB

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

11

Magnetic pressure and tension

extmag FFPvvt

v

dt

vd

BBBjc

Bvc

Fmag

4

111

BBB 04

1

Potential field!

• Magnetic pressure and tension:

– Aside: killing vector cross products use -tensor

and write with summation convention

and the identity:

Thus:

ijkijkjibaba

ijk klm il jm im jl

21

2

i j k jkl ilm i j k jkl ilm

i j k jkl lim i j k ji km jm ki

i i m i m i

B B B B

B B B B

B B B B

B B B B

B B B

• 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)

844

1 2BBBBBFmag

Magnetic tension Magnetic pressure

0 BB

Ideal MHD Equations

0

4

1

0

P

dt

d

Bvt

B

FBBPvvt

v

vt

ext

• Here we used the simple adiabatic energy equation

0B

• Note that is included in Faraday‘s law as

initial condition!

• For finite conductivity field lines can diffuse away

• Analyze induction equation:

Magnetic Viscosity and Reynolds number

BBv

BBc

Bv

Bvjc

t

B

Bvc

jE

Bvc

EEjj

Ect

B

m

2

22

4

!!! with term thekeep ... 1

1

4

2cm

... magnetic viscosity

• First term is the kinematic MHD term, second is

diffusion term

• Comparing both terms:

2

2 ~

~

L

BB

L

vBBv

mm

mm

m

vL

LB

LvB

R

2

• Rm >> 1: advection term dominates

Rm << 1: diffusion term dominates

cf. analogy to laminar and turbulent motions!

• Magnetic Reynolds number:

• For Rm << 1 we get:

Magnetic field diffusion

Bt

Bm

2

2

22

2

4~~

c

LL

L

BB

m

Dm

D

• Field diffusion decreases magnetic energy: field

generating currents are dissipated due to finite

conductivity -> Joule heating of plasma

Note: Form identical to heat conduction and particle

diffusion

• Deriving a magnetic diffusion time scale:

• Block of copper: L=10 cm, =1018 s-1

• Sun: R

~ L=7 1010 cm, =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 Rm

high!

• Note that since turbulence decreases L and

thus diffusion times

thus fields in ISM have to be regenerated (dynamo?)

Examples:

s 2.1 D

yr! 102s106 1017 D

2LD

• 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 advected

surface remains constant with time

• Proof: Consider flux tube

Concept of Flux freezing

dS

Flux tube

• Consider surface 2211 by )( and by bounded )( CttSSCtSS

SdtrBS

B

,

ldtvSd

ld

tv

1C

2CB

• 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

t

t

trBtrBttrB

,,,

SdtrBSdttrBt

SdtrBdt

d

SStS

, ,

1lim ,

120

SdtrBSdtrBt

Sdt

trBSdtrB

dt

d

SSS

tS

, ,

1,lim ,

12

2

0

0 3 rdBSdBV

tvSS

length of surface lcylindrica theand , 21

We obtain

• Noting that in the limit

and using the vector identity

and Stokes‘ theorem

one gets

0, , ,

121

CSS

ldtvtrBSdtrBSdtrBSdB

)()()()( ,0 122 tStSttStSt

ldvtrBSd

t

trBSdtrB

dt

d

CSS

,

, ,

ldtrBvldvtrB

, ,

SC

SdtrBvldtrBv

, ,

SdtrBv

t

trBSdtrB

dt

d

SS

,

, ,

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

v

0 , SdtrBdt

d

S

v 1C

2C

1C

2C

SdtrBSdtrBdt

d

S

mS

, , 2

• Note: motions parallel to field are not affected

• For finite conductivity field lines can diffuse out:

• Perturbations are propagated with characteristic speeds

• For simplicity consider linear time-dependent

perturbations in a static compressible ideal background

fluid

• Ansatz:

MHD waves

jjj

EEE

BBB

vvv

PPP

0

0

0

0

0

0 1X

Xwhere

• Assume background medium at rest: 00 v

00

0

00

0

1

0

1

4

1

0

P

P

Bvc

E

B

t

B

cE

jc

B

Bjc

Pt

v

vt

Perturbed Equations

Keep only

first order terms!

• Combining equations and eliminate all variables in

favour of

• Defining sound speed:

yields single perturbation equation

• Defining Alfvén speed:

v

PPc

S

S

2

044 0

0

0

02

2

2

BBvvc

t

vs

0

0

4

BvA

yields finally

• We seek solutions for plane waves propagating

parallel and perpendicular to B-field

• Wave ansatz:

• Dispersion relation:

0 2

2

2

AAs vvvvc

t

v

txkiAtxv

exp ,

0222 AAAAAs vvkkvvvkvkvkvkvcv

• Case 1:

dispersion relation reads then

Phase velocity

• Case 2:

dispersion relation reads then

Case Study for different type of waves

Avk

kv

// Longitudinal magnetosonic wave

22

Asph vck

v

0// i.e. ,// Bkvk A

0 1 2

2

2222

AA

A

sA vvvk

v

cvvk

AAAA vvkkvvvkvkv

kvkvcv As

222 0

• 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 case

thus

the solution is a transverse Alfvén wave (pure MHD wave)

driven by magnetic tension forces

due to flux freezing gas (density 0 ) must be set in motion!

vvvk A

////

vvvv

vvvvvv A

AAAA

2

0v 222 kcs

0 vvvvvk AA

0v 222 kvA

Aph vvk

• Magnetosonic wave (longitudinal compression wave)

• Transverse Alfvén wave

Note 1: in both cases phase

velocity independent of and k:

dispersion free waves!

Note 2: in ISM density is low,

Therefore Alfvén velocity high

~ km/s

k

B

kv

//

B

k

v

kv

I.3 Cosmic Rays

Cosmic Radiation

Includes -

• Particles (2% electrons, 98% protons and

atomic nuclei)

• Photons

• Large energies ( )

-ray photons produced in collisions of high

energy particles

eVEeV 209 1010

• Increase of

ionizing

radiation with

altitude

• 1912 Victor

Hess‘ balloon

flight up to

17500 ft.

(without

oxygen mask!)

• Used gold leaf

electroscope

Extraterrestrial Origin

• Diffuse -

emission

maps the

Galactic HI

distribution

• Discrete

sources at

high lat.:

AGN (e.g.

3C279)

Inelastic collision of CRs with ISM

2 0

0

pppp For E > 100 MeV dominant process

HI

• 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

• Probability that CR proton undergoes inelastic collision

with ISM nucleus

• 1/3 of pions are 0 decaying with <E > ~ 180 MeV

• If Galactic disk is uniformly filled with gas + CRs

the total diffuse -ray luminosity is

• Galaxy with half thickness H=200 pc, nH~1 cm-3, CR~1

eV/cm3 Vgal ~ 2 1066 cm3

• in agreement with obs.!

• Thus -rays are tracer of Galactic CR proton distribution

-ray luminosity of Galaxy

226cm105.2 , ppHppcoll cnP

galCRcollCRHpp VPEEncnL 3

1)(

3

1

erg/s 1039L

Chemical composition

Groups of nuclei Z CR Universe

Protons (H) 1 700 3000

(He) 2 50 300

Light (Li, Be, B) 3-5 1 0.00001*

Medium (C,N,O,F) 6-9 3 3

Heavy (Ne->Ca) 10-19 0.7 1

V. Heavy >20 0.3 0.06

Note: Overabundance of light elements spallation!

• 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

Origin of light elements

Differential Energy Spectrum • differential

energy spectrum

is power law for

109 < E < 1015 eV

1016 < E < 1015 eV

• for E > 1019 eV CRs are

extragalactic (rg ~ 3 kpc

for protons)

~ E-2.7

~E-3.1

„knee“

„knee“

„ankle“

• 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 = 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, ... pitch angle

Primary CR energy spectrum

dEKEdEENEEIN

)(or 70.2 with )(

BcR

BcZe

pc

ZeB

vmr L

g

sinsinsin0

Example: CR with E=1 GeV

has rg ~ 1012 cm! For B ~ 1G

@ 1015 eV, rg ~ 0.3 pc

• 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 H~ 1.6 Myr -> -decay into 10B

Important CR facts:

4106 I

I

– From amount of 10Be relative to other Be isotopes and 10B

and H the mean CR residence time can be estimated to be

esc~ 2 107 yr for a 1 GeV nucleon

CRs 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 of rigidity R:

for 1 GeV particle, x ~ 9 g/cm2

– Mean measured CR energy density:

0.6 ,g/cm GV 20

9.6)( 2

RRx

3eV/cm 1~~~ turbthmagCR

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 radiation

assuming energy equipartion between B-field and CRs

radio continuum observations of starburst galaxy M82

give

CR production rate proportional to star formation rate

no constant high background level!

CR interact strongly with B-field and thermal gas

)(100~)82( GalaxyM CRCR

• 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

CR propagation:

esc

4 yr 103~ c

Llc

BBB reg

B

cLvr

diff

3-7 10~km/s 490yr102

kpc 10~

ISMesc

h cx

4

1~g/cm 105~ 325

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~ 228esc

CRLh

pc 1~/3~ cCR

• CR electrons (~ 1% of CR paeticle density) must be of

Galactic origin due to strong synchrotron losses in

Galactic magnetic field and inverse Compton losses

Note: radio continuum observations of edge-on

galaxies show strong halo field

CR origin:

• Estimate of total Galactic CR energy flux:

Note: only ~ 1% radiated away in -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~ 41esc

confCRCR

VF

erg/s 10erg10 yr 100

3~ 4251 SNRSNSNR EF

eV)10~(for kpc 100 20ERr galg

• 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 exist

thus strong induced E-fields from strongly time varying large scale B-fields could help -> no strong evidence!

Diffusive shock acceleration:

Bv

cEevm

dt

dL

1

• 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

Fermi mechanism:

Shock Front

Cloud

2nd order Fermi 1st order Fermi

• Energy gain per

crossing:

Fermi)order (1 3

4 st

c

V

E

E S

Fermi)order (2 3

4 nd

2

c

V

E

E

• Shock is converging

fluid

– 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 collision

and P be probability that particles remains in acceleration

process -> after k collisions:

0 EE

lnln

00

0

0

00

ln

ln

ln

ln

energies with particles )(

P

kk

E

E

N

N

P

EE

NN

EEPNEN

speed) particle ... ( vv

VP S

esc 11v

VPP S

esc

c

V

E

E

E

E S

110

)(for ,11

1

1

1ln

1ln

ln

lncvV

cv

v

V

c

vc

V

v

V

c

V

PS

S

S

S

S

– Note: that N=N(>E), since a fraction of particles is accelerated to

higher energies

therefore 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 -2.7

However: 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.)(

1ln

ln

Pe.g. Bell (1978)

dEEconstdEEN 2.)(

- remains unsolved! -

LECTURE 2

Dynamical 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

(mfp >> )

II.1 Gas Dynamics & Applications

Lmfp

, m

Tkc B

s

km/s 1203.0 sc

1sc

uM

II.2Shocks

• Assumption: Perfect gas, B=0

• Shock thickness time

independent across shock discontinuity: steady

shock

• Conservation laws: Rankine-Hugoniot conditions

mfp Pu , ,

(energy) 12

1

12

1

(momentum)

(mass)

2

22

22

1

12

11

2

222

2

111

2211

Pu

Pu

uPuP

uu

1 2

Shock Frame

111 ,, Pu 222 ,, Pu

Analoga1. Traffic Jam:

„speed of sound“ cs= vehicle distance/reaction time=d/

• 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/d =0!

2. Hydraulic Jump:

„speed of sound“ cs= („shallow water“ theory)

• total pressure: Ptot =Pram+Phyd

ghvP 2

tot

• At bottom: v > cs

• Therefore: abrupt jump

• Behind jump:

h2 > h1

V2 < V1

• V1 > c1 „ supersonic“

V2 < c2 „ subsonic“

Kitchen sink experiment

gh

II.3 HII Regions

• Rosette Nebula

(NGC2237):

– Exciting star cluster

NGC 2244, formed

~ 4 Myr ago

– Hole in the centre:

Stellar Winds

creating expanding

bubble

Trifid Nebula:

Stellar photons heat molecular cloud gas

Gasdynamical 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 2

3

=

B

ek

QT

3

23

2rec B e recN Q k T N

• Assumption: stellar rad. field is blackbody

average kinetic energy per photo-electron:

For a blackbody at temperature T*:

For

T* = 47000 K (for O5 star) Te ~ 31300 K

LHH hEdSdSEhQ

LL

, )( **

1

*2** 1exp2

)(

Tkhc

hTBSh B

1* Tkh B

*3

2TT

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 TT

*

)2( TknnQN BHerec

eOII nn 4106

]exp[ )(

)(

2/1

eBijeijeij

eijIeij

TkETATC

TCnnN

• Radiative energy loss by [OII] for

taking (i.e. all O is in O+)

and demanding

For T*=40000 K, we obtain: Te~8500 K,

in excellent agreement with observations!

• HII regions are thermostats!

• major cooling by forbidden lines

-1-342/1232 scm erg ]1089.3exp[ 101.1 eeOIIOII TKTnyL

1OIIy

OII heat I recL G N Q N Q

*

644/1 105.2]1089.3exp[ TTKT ee

levels and 2/3

2

2/5

2 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

* 4

eHs nnR (2)3 3

4

nnnx He 1

3/1

2)2(

*

4

3

n

SRS

... „Stroemgren“ radius

Example:

pc 3

type)(O6.5 s 10 ,cm 10n

K) 8000(T scm102

1-49

*

3-2

H

-1313)2(

SR

S

• 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.

IFR

Case B: Evolving HII Region

RnR

SJ 2

0

)2(

2

*

3

1

4

HII

Jdt

dRn 0

1)2(

0rec

recrec

, ,t

,

nR

VR

R SR

S

23 31

Equation of motion:

0 0, :BC

13/1

eCSolution:

Lyc photons

IFR

SR

• RS is reached only within 1% after

• IF velocity >> cII until R~ RS

therefore

then gasdynamical expansion

• IF velocity slows down rapidly

• However:

NOT possible

Evolution of Stroemgren sphere

dt

dRR IF

rec

S

rec 4

0nnII

IIIF c

dt

dR

• When sound waves

can reach IF

• : HII gas acts as piston

shock 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

Case C: Gasdynamical Expansion of HII Region

IIIF cR

III PP

RRR shIF :

Lyc photons

shR

HII

HI

IFR

dynsc

01

3

42)2(

*3

dt

dn

n

SmmnR

dt

d

dt

dM II

II

IISII

•

• For strong shock:

• Ion. Balance:

• Ambient medium at rest:

• Thus equation of motion:

• Dimensionless quantities:

• Solution:

Simple Model:22 IIIIIIBIIIIsh cmnTknPP

1for 1

2 2

0

2

0

shshsh VVP

3)2(2

0

3)2(2

* 3

4

3

4SII RnRnS

RVsh

2

2/3

2

0

2 IIS

IIII c

R

Rc

n

nR

2/3

0

R

R

n

n SII

0 1, :BC

1/ ,/ , 4/3

N

cRRtcNR

RIISII

S

7/37/4

4

71 ,

4

71

NN

• At

• Note:

• Is pressure equilibrium

reached: ?

• Ionization balance must

hold:

• Result:

• Final mass:

Gasdynamical Expansion:

t [sec]

t [sec]

IIcRN ,0

rect exp

III PP

IBIIIBfIII TknTknPP 2

3)2(2

* 3

4ff RnS

SIIIf

IIIIf

RTTR

nTTn

3/2/2

2/

I

II

S

ff

S

f

T

T

Rn

Rn

M

M 23

0

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, 100

3

0

-3

0

SfSff

III

MMRRnn

nTT

yr 108.7

yrn 107.1/ 273~ 34For

7

-2/3

0

13

IISf cRt

-33

0 cm103n

• Massice star

BD+602522

blows bubble into

ambient 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

•O- and B-stars:

–Burn Hot

–Live Fast

–Die young

• Strong UV radiation field

• Momentum transfer to gas

• Resonance lines show P Cygni

profiles (e.g. CIV, OVI)

610~

km/s 30002000~

W

W

M

V

M

/yr

Total outward force:

2 4 r

LFW

In reality:

)(

• Assume a stationary wind flow (ignore Pth)

• Stars with L*>Lc are radiatively unstable

Line Driven Stellar Wind:

22

*

4 r

L

r

mGM

dr

dvmv

)luminosity (Eddington 4

4

,1

*

*

*

*

2

*

mcGML

L

L

mcGM

L

r

GM

dr

dvv

c

c

radiation pressure

• Integration:

• If v(R*)=v0=0:

This is CAK velocity profile (L* > Lc, i.e.>1 needed)

• If L >> Lc

rR

GMvvdrr

GMvdv

r

R

v

v

111

2

1 1

*

*

2

0

2

2

*

*0

112

1)(

*

2/1

*

*

2/1

*

c

escL

Lv

R

GMv

r

Rvrv

mcR

Lv

*

*

2

• 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. dep., multiple scattering

• Note:

Example:

km/s 2566

km/s 10

2/1

*

0

2/1

0

*

2/12/1

v

R

R

L

L

m

mv

p

T

vMc

LW* Momentum rate

6103.6 WM M

/yr

erg/s 103 erg/s 103.12

1 39

*

372 LVML WWW

Most of energy lost as radiation!

Less 1 % converted into mechanical luminosity

• 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

Effects of stellar winds (SWs) on ISM:

WV WM

WM

610WM M

/yr

km/s, 2000WV erg/s 102

1 362 WWW VML

-149

* s 10S

200 km/s 10

km/s 2000M

II

W

c

V

• Free expanding wind

shocked at S-

• Energy driven shocked

wind bubble bounded by

contact discontinuity CD

No 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

Flow Pattern:

S-

CDIF

S+

5

2

43

1

1

2

3

4

5

• 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

Qualitative Discussion:

1

2

W

WWWW

Vr

MVP

2

2

4 ,

K 10432

3

216

3

4

3

72

22

W

B

b

bBbWWWbb

Vk

mT

TknVVP

bbdyncool RR /

)( 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

mnRRRM shshsh 43

4 23

0

3

cooling highbII TT

Wb VR

dynshsc cR /

4III TT

5 IT

• Extension of Regions + is

• Extension of Region >> Region

therefore it is assumed that occupies all space

• Equations:

Simple Model for stellar wind expansion:

3 4

bbsh

b

RRRR

RR

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

bb

2

bWth

b

2

bbsh

3

th

3

sh

dt

d

dt

d

rdrp

rdr

r

r

• 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

Similarity Solutions:

AtRb 33 2

2

3

3

4bbbbth RPPRE

0

3234

2

31512

W

bbbbbbb

LRRRRRRR

0

35325

2

3 1511221

WL

tA

• RHS is time-independent

• Thus the solution reads:

• Note: we have assumed

5/1

0

5/1

154

125

5

3035

WLA

5/2

5/3

5/1

0

5/1

5/3

5

3

5

3

154

125

tAt

RVR

tL

tAR

bshb

Wb

2

0 shshI VPP

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

• 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)

More superbubbles:

• 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 formation

SB evolution can be described by energy injection

from centre of association

• Initially energy input from SWs + photon output rate of

O stars dominates for O7 star

Some Facts

yrM10/1036.1

0

7

MS

m

erg/s 1062

1 352 WWW VML

• 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

Simple expansion model:

yr 105 6MS

yr 105tyr 105 76

yr10 ,

erg/s10

pc 269

773838

5/3

7

5/1

0

38

ttL

L

tn

LR

W

SB

McCray & Kafatos, 1987

cf. Cygnus supershell: RS~225 pc(Cash et al. 1980)

min

51

51

*10

M

ENL

MS

SB

• 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

5/2

7

5/1

0

51*

5/3

7

5/1

0

51*

km/s 7.5

pc 97

tn

ENV

tn

ENR

SB

SB

• 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: Local

Bubble (RLB ~ 100 pc)

• Origin of LB: multi-

SN?

Example: Local (Super-)Bubble

Breitschwerdt et al. 2001

NaI absorption line studies

• 3 sections through

Local Bubble

• LB inclined ~ 20

wrt NGP

• LB open towards

NGP?

NGP

GC

NGP

Gal.

Plane

Sfeir et al. (1999)LB Gould‘s

Belt

Local Chimney

RASSCrab

VelaLMCNorth Polar Spur

Cygnus Loop

Anticorrelation: SXRB – N(HI)

• Anticorrel. on large

angular scales for

soft 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

1.1 ,N N(m)

1

0

0

m

m

Berghöfer and Breitschwerdt 2002

Young Stellar Content and Motion

• Adjusting IMF (B1):

• 2 B stars still active

• SNe explode in LB

1

0

0

M

m 551.6 N(m)

7)8MN(m

max

min

21dm N(m)N

M 20m1)N(m

SN

0maxmax

m

m

Berghöfer and Breitschwerdt 2002

Formation and Evolution of the Local Bubble

• Energy input by sequential SNe

• Assumption: coeval star formation

Star deficiency:

First explosion: 15 Myr ago

)m(m1.6 yr,M 10

1030

7

MS

m

3.0)1/( ,tLN

EL 0SN

SNSB

dt

d

yr102.5M 10 7

cl0 m

)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

bb

2

bSBth

b

2

bbsh

3

th

3

sh

tdt

d

dt

d

rdrp

rdr

r

r

Analytic

Model:

• Note that similarity exponent is between SNR

(Sedov phase: =0.4) AND SW/SB (=0.6)

• The mass of the shell is given by:

• After some tedious calculations we find:

Similarity solution:

rK

R

rAt 0

0

b~ ,

5

3 ,R

54.01.1 ,6.1 ,0 :IF

3

00

0

2

3

44 b

R

sh RKdrrKrMb

5

1

0

0

3

72411732

35

K

LA

• Local Bubble at present:

• Present LB mass:

• Average SN rate in LB:

• Cooling time:

• Energy input by recent SNe:

Local Bubble (Analytic results)

km/s 9.5V pc, 146RMyr 13 ,cm 30n shbexp

-3

0

0ej0LB M 200M ,M 600M

Mass loading! (decreases radius)

yr) 105.6/(1 5

SN f

K)10T ,cm105(nMyr 13 6

b

33

b

c

X-ray emission due to last SN(e)!

dt

d N EE SNSN

erg/s105.2)1(103.4E 3631.0

7

36

SN t

Disturbed background ambient medium

z

x

y

Local Bubble Evolution: Realistic Ambient Medium

Density

Cut through

Galactic plane

LB originates at

(x,y) = (200 pc,

400 pc)

SNe Ia,b,c & II

at Galactic rate

60% of SNe in

OB associations

40% are random

Grid resolution:

1.25 and 0.625 pc

Numerical Modeling (II) Density

Cut through

galactic plane

LB 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 ago

Interaction shell

fragments in

~3Myrs

Bubbles dissolve in

~ 10 Myrs

• 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

II.6 Instabilities

• Why does the

surface on a lake

have ripples?

Kelvin-Helmholtz instability

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

• 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 is

the velocity components of perturbed state are:

in x-, y- and z-direction

thus the perturbed equations read:

Normal modes analysis:

0v

P

wvuU and ,

Stratified fluid

0

z

w

y

v

x

u

zwx

zU

t

z

dz

dw

xU

t

gPzx

wU

t

w

Pyx

vU

t

v

Pxdz

dUw

x

uU

t

u

ss

ss

where ss zUU

is the surface at which

changes discontinouslysz

Disturbances vary as

tykxki yx exp

Instability if 0Im

U

xz

1

2g

g

can stands for any

acceleratiion!

• 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 the DR becomes

dz

d

Uk

wgkwUkkw

dz

dUk

dz

dwUk

dz

d

x

xxx

22

ss zz , 0

sx

sxxsUk

wgkw

dz

dUk

dz

dwUk

2

00 ss zzzzs fff

BC

02

2

2

wk

dz

d0 since

dz

d

dz

dU

• Since must be continous at and w must not

increase exponentially on either side, we must have

• Applying the BC to solutions:

• Expanding and rearranging yields the growth rate

where and

Uk

w

x sz

0 ,exp

0 ,exp

22

11

zkzUkAw

zkzUkAw

x

x

21

2

11

2

22 gkUkUk xx

21

2

21

2

21212

21

21

21

2211

UUkgk

UUk xx

coskUUk

coskkx

• 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

0xk

21

21

gk

Perturbations transverse to streaming are unaffected by it

k

22

2121

2

2

2

1

cosUU

gk

21

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

• 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

Literature:

• 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

- The End -

Bubbles everywhere!!!