ISM Lecture 9

Shock waves;

Supernova remnant evolution

9.1 Shock waves

A shock wave is a pressure-driven compressive disturbance propagating faster than “signal speed”

Shock waves produce an irreversible change in the state of the fluid

Literature: Landau & Lifshitz, Fluid mechanics, Ch. IV Dyson & Williams 1997, Ch. 6 + 7 Tielens, Ch. 11

Sound speed and Mach number

Sound speed: P = Kργ with γ = 5/3 for adiabatic flow,

γ = 1 for isothermal flow

C 1/3 for adiabatic flow sound speed larger in denser gas

for isothermal gas in ISM Mach number: M ratio of velocity w.r.t. sound

speed, M v/C

CdP

d2

CkT

m 1 km s 1

Steepening of non-linear acoustic wave

speed of trough

speed of crest

Speed is larger in denser gas

Speed of crest is larger than speed of trough

Crests “catch up” with troughs

Steepening of waveform

Steepening halted by viscous forces

Development of shock

“Signal speed” in ISM

In the absence of magnetic fields, information travels with sound speed M < 1 subsonic M > 1 supersonic shocks

If magnetic field is present, disturbances will travel along B at Alfvén speed vA:

Interstellar magnetic field (empirical):

vA B2 2 4 /

B n 1G H for 10 < nH<106 cm-3

Examples of shock waves in the ISM

Cloud-cloud collisions Expansion of H II regions Fast stellar winds (“interstellar bubbles”) Supernova blast waves Accretion and outflows during star formation Spiral shocks in Galactic disk

Cloud-cloud collisions

v0 v0

shocked gas

shock front

Δv = 2 v0

v0 5-10 km/s vs v0

Expansion of H II region

vs 10 km/s

ambient neutral gas(cold, medium density)

compressed neutral gas(cold, high density)

H II region

(hot, lowdensity)

Fast stellar wind (“Interstellar Bubble”)

inner shockouter shock

contact discontinuity(wind meets ISM)

unshocked cloud material

shocked cloud material

shocked wind material

unshocked wind material

Terminal velocity of wind from hot stars

Supernova blast wave

Early: Mswept < Mejecta like stellar wind

Late: Mswept >> Mejecta ignore Mejecta

hot interiorunshocked ISM

shocked ISM (cool shell)

shocked ISM (did not cool radiatively)

shock front

Star formation: infall and outflow

shock front

hot, shocked accreted matter

infalling ambientcold material (lowpressure, free fall)

Spiral shocks in Galactic disk

shock frontflow

compressed material formsspiral arm material expands and

leaves spiral arm

9.2 Shock jump conditions

Adopt frame in which shock is stationary Consider plane-parallel shock: fluid properties

depend only on distance x from shock;

Neglect viscosity, except in shock transition zone:large v-gradient viscous dissipation transform bulk kinetic energy into heat irreversible change (entropy increases)

v v shockx

Shock profile (shock frame)

Shock jump conditions (cont’d)

Thickness of shock transition mean free path of particles This is always << thickness of radiative zone (need

collisions to get radiative cooling)

Regard shock thickness as 0 discontinuity Find physical conditions at 2 (immediately

behind shock) or 3 (in post-radiative zone) given those at 1 (pre-shock) and shock velocity vs

Schematic depiction of a radiative shock

1. Undisturbed pre-shock material2. Immediate post-shock conditions3. Cold post-radiative conditions

radiativezone

radiativezone

Mass conservation across shock

Steady plane-parallel shock:

Integrate across shock:

t

( )v 0

t y z x x0 0 0 0, , ( )v

( ) ( ) v vx 1 2 x

Momentum conservation across shock

Note:

no change in gradient of gravitational potential =>

Dv

DtP

BB B ( ) ( )

2

8

1

4

D

Dv

t t

vv

v

xx x x

xx

x xP

B B B

x x

xP

B B

x

( )

(( )

)

2

22 2

8 4

8

v vxy z

xy zP

B BP

B B22 2

22 2

8 1 8 2

FHG

IKJ

FHG

IKJ

Energy conservation across shock

Same procedure as before

D

DtU

B

xP

B B B

x

T

ii i

k i ki

i

1

2 8

8 4

22

2

v

v vv

v

LNM

OQP

LNM

OQP

e j a f

Heat conduction Heating Cooling

1

2 8 4 42

2 2

1

2

1

2

v v v v vv v

v d

x x xy z

xx y y x z z

x

U PB B B B B B dT

dx

x

LNM

OQP

z b g ~0 if 1 and 2 are close

Magnetic field equation

The electric field vanishes in the fluid frame Under this condition the Maxwell Equations

can be simplified to give

The last term describes diffusion of the magnetic field and vanishes if the conductivity (ideal MHD)

B

tB

cB( )v

2 2

4

Special (often used) case: simplifying assumptions

Planar shock: No heat conduction: No net heating / cooling:

No gravitational force:

Internal energy density: with=cP/cV

B-field flow velocity: :

v vz y 0

T 0

z )dx1

2

0

v dx x1

2

0z

U P1

1

B Bx y 0 0may choose

0

/

/

x B B

x B B

x y y x

x x z

v v

v vz

c hb g

Rankine-Hugoniot equations for shock jump conditions

Follow general practice and write vx=u1(=vS)

Mass conservation: Momentum conservation:

Energy conservation:

Flux conservation:

2211 uu

8222281

211

22

21 BB pupu

8221

3222

18111

3112

1222

211 BuBu puupuu

2211 BuBu

Solution of shock jump equations

These are the jump conditions: 4 eqs. in 4 unknowns: 2, u2, P2, B2

Define Now we have two equations for P2 and x (cubics in x):

Define Mach number M by M2=u12/vms

2 , where vms is the

magnetosonic speed given by

xu

u

B

B 1

2

2

1

2

1

1 1

21

12

1 12

222

2

8 8u P

B u

xP

Bx

1

2 1 8

1

2 1 81 13

1 11 1

2

113

2 21 1 1

2

u u Pu B u

xP

u

x

u Bx

vms

P B2 1

1

12

14

Solution of shock jump equations (cont’d)

The trivial solution 1 = 2 always exists A second solution exists for M > 1 Strong shock: M >> 1

xu

u

B

B

1

2

2

1

2

1

1

1

5

3

( )4 for

Tu

k

u

k2 212

12

21

1

3

16

5

3

FHG

IKJ

( )

for

Tm

u

H2

1

2

2300 FHG

IKJ

10 km / s

K

Some properties of strong shocks

Compression ratio x = 4

If 16B12/1u1

2 and P1<< 1u12 , then momentum

conservation gives

P23/4 1u12=3/4 . ram pressure

B2 = 4 B1 magnetic pressure increases by factor 16

Post-shock flow is at P constant, and subsonic:

22

2

1

2

2

4 1

4

3

5

16

3

1

50 45

u u

Pmsv ,

/. .

Adiabatic vs. radiative shocks

So far considered shocks without radiative cooling: “adiabatic shocks” This is a bad terminology, since shocks are abrupt and

irreversible The term ‘adiabatic’ refers to the fact that no heat is

removed during shock If post-shock gas radiates lines cools down

“radiative shock” If temperature in region 3 is the same as in region 1

“isothermal shock” This is also a bad terminology since T always rises at shock

front

Isothermal shock structure

Solve jump conditions with T3 = T1 => B = 0:

compression factor can become much larger than 4!

B 0:

where

typical compression factor 100!

3

1

2 M

3

1

1 12

12

1

1

182 77

1

u

B

u u

bAv 100 km / s,

B n b1 1610 G

J vs. C shocks: MHD effects

So far considered only single-fluid shocks (“J-shocks”)

Normally interstellar gas consists of 3 fluids: (i) neutral particles, (ii) ions, (iii) electrons Fluids can develop different flow velocities and

temperatures For B 0 information travels by MHD waves Perpendicular to B the propagation speed is the

magnetosonic speed v vms s AC 2 2

Magnetic Precursors: C-shocks

Alfvén speed for decoupled ion-electron plasma can be much larger than Alfvén speed for coupled neutral-ion-electron fluid

In many cases: CS<vA,n < vs < vA,ie

Ion-electron plasma sends information ahead of disturbance to “inform” pre-shock plasma that compression is coming: magnetic precursor

Compression is sub-sonic and transition is smooth and continuous “C-shock”

Ions couple by collisions to neutrals

vA 1 2/

v v

50 km / s) H H

A A ie i

i

i i

B

n n m

n

FH IKFHG

IKJ

,/

/ /

/ ( )

(/

/

4

10

20

1 2

4

1 2 1 2

Comparison of J-shocks and C-shocks

J (“jump”)-shocks: vS50 km/s Shock abrupt Neutrals and ions tied into single fluid T high: T40 (vS/km s-1)2

Most of radiation in ultraviolet

C (“continuous”)-shocks: vS50 km/s Gas variables (T, ρ, v) change continuously Ions ahead of neutrals; drag modifies neutral flow Ti Tn; both much lower than in J-shocks Most of radiation in infrared

Schematic structure of J- and C-shocks

C-shock structure

Draine & Katz 1986

Post-shock temperature structure of fast J-shock

Three regions: Hot, UV

production Recombination,

Lyα absorption Molecule

formation Grains are weakly

coupled to gas, so that Tgr << Tgas

Computed fine structure line intensities for J-shock

Computed H2 near-IR line intensities for J-shock

9.3 Single point explosion in uniform cold medium

Idealized treatment of SN explosion due to Sedov (1959) and Taylor See also Chevalier (1974)

Equally applicable to atomic bombs Assumptions:

E conserved no radiative cooling no charateristic tcool

Fluid treatment adequate (mean free path << R) Neglect viscosity, heat conduction Ambient pressure small

Then there is only 1 characteristic Length: R=size SNR Time: t=age Velocity: R/t

Self-similar solution of blast-wave problem

Three characteristic quantities: Explosion energy E Ambient density ρ0

Time since explosion t Self-similar solution: the space-time evolution of all

relevant quantities (ρ, v, T) can be described by universal functions that depend only on one dimensionless parameter v(r,t) = R/t fv(r/R) ρ(r,t) = ρ0 fρ (r/R) T(r,t)=m/k (R/t)2 fT(r/R)

Dimensional analysis

Use dimensional analysis to determine dependence of R(t) on E, ρ0, t:

=>

R t At E( ) 0

[ ]:

[ ]:

[ ]:

, ,

M

L

t

0 0

1 0 2 32

5

1

5

1

50 2 0

}

R t AEt

t( )/

/FHGIKJ

2

0

1 5

2 5

Calculation of A

Expect A~1. To calculate, put in physics Assume ‘adiabatic’:

E=Ekin+Ethermal=const=E0

Assume ‘self-similar’: Ekin/Ethermal=const

Behind shock:

E

Ek

k

S

s

kin

thermal

v

v

FH IKFHG

IKJ

12

34

32

316

1

2

2

More detailed calculation of A

Assume most of mass concentrated just behind shock (O.K. from detailed solution)

Ekin~Ethermal for entire SNR Ekin~1/2MvS

2, M= 4/3R30

Then E0=2Ekin=2.1/2. 4/3R30.vS2

R=cst.t=>vS=.cst.t-1

=> E0=4/30(cst)52t5-2=const. => =2/5

=> (cst)5=3/(4(5/2)2(E0/0)

=> (cst)=1.083 (E0/0)1/5 => A=1.083 Exact: A=1.15169 for =5/3

Sedov solution

50% of mass in outer 6% of radius T P/ρ drops from center towards edge

9.4 Evolution of SNR in homogeneous ISM1. Early phase

Early: Mswept < Mejecta (‘Free expansion’)

RS=vSt

vSv0=ejecta velocity=(2E0/Mejecta)1/2 =10000 km/s (E0/1051 erg)1/2 (Msun/Mejecta)1/2

Phase ends at t=tM when Mswept=Mejecta

For density 0, Mejecta=Mswept=04/3(v0tM)3 =>

tM

M E nM FHG

IKJFHG

IKJFHG

IKJ

190 yr 10

5 6 51

0

1 2 1 3

ejecta

Sun

3

H

erg cm/ / /

2. Sedov phase

Mejecta<Mswept and t<trad

RE t R

tSFHGIKJ 115167

2

50

2

0

1 5

.

/

, v

RE n t

E n tS

FHG

IKJFH IK

FHG

IKJ

FHG

IKJFH IK

FHG

IKJ

32 pc erg cm yr

v / s erg cm yr

H

H

051

1 5

3

1 5

5

2 5

051

1 5

3

1 5

5

3 5

10 10

120 km10 10

/ / /

/ / /

Cooling at end of Sedov phase

cool dense shell

hot, low density, pressure pi

radiative cooling negligible

shock front

Estimate of trad

Estimate t=trad when cooling becomes important and Sedov phase ends

Most cooling just behind shock where highest and T lowest

Define trad when 50% of Ethermal radiated

tE n

tE n

R tE n

S

S

rad4 H

radH

radH

4.4 10 yr erg cm

v km / s erg cm

pc erg cm

FHG

IKJFH IK

FHG

IKJFH IK

FHG

IKJFH IK

10

20010

2310

51

2 9

3

5 9

51

1 15

3

2 15

51

13 45

3

19 45

/ /

/ /

/ /

( )

( )

almost constant!

3. Post-Sedov or ‘radiative’ phase

Cold dense shell slows down due to sweeping of ISM; momentum is conserved

Naïve snowplow (neglect Pi):

d

dtM M M

R R R

( )

/

v v v

v v v

cool cool

cool3

cool

0

13 3

v R t/ Rt1/4

Radiative phase (cont’d)

Pressure-modified snowplow (Oort snowplow):

Pi drops due to adiabatic expansion: d

dtM P R

P PR

R

i

i

( )

/

v

with coolcool

FH IK

4

5 3

2

3

FH IKv vv

coolcool4

4

342

03

0

52 R R

d

dtP

R

RR

R Rt

t

t

tS SFHGIKJ

FHGIKJ

coolcool

coolcool

v v2 7 5 7/

,

/

,

4. Merging phase

SNR fades away when expansion velocity = velocity dispersion ambient gas (~10 km/s)

Use Oort snowplow:

t

tfade

cool

S,coolv

km / s

FHGIKJ

5 7

10

200

1020

/

tE n

RE n

fade6

51H

3

fade 51H

3

2.9 10 yr 10 erg cm

pc 10 erg cm

FHG

IKJFH IK

FHG

IKJFH IK

0 32 0 27

0 32 0 16

76

. .

. .

Summary phases of supernova shell expansion

1. Early phase (mswept < mejecta): Free expansion, Rs = vs t

2. Sedov phase (mswept > mejecta and t < trad): Energy conservation, R t2/5

3. Radiative “snowplow” phase (t > trad): Momentum conservation, R t1/4 or R t2/7

4. Merging phase: vs drops below velocity dispersion of ambient ISM

Overview of SNR evolution

Overview of SNR evolution