The Interstellar Medium
IK-LECTURE
– Fall 2007 –
Dieter Breitschwerdt
Institut für Astronomie, Wien
http://homepage.univie.ac.at/dieter.breitschwerdt
Email: [email protected]
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!!!