Download 1– 14. SHOCK FRONTS AND IONIZATION FRONTS The interstellar

–1–
14.
SHOCK FRONTS AND IONIZATION FRONTS
The interstellar medium is intermittently disturbed by violent events—supernova explosions
being but one example—that cause large increases in the local pressure. As the result of the
pressure increase, the disturbed region will expand. If the pressure increase exceeds a minimum
value, a “shock front” will develop at the leading edge of this expanding disturbance, and the flow
in the neighborhood of this front is referred to as a “shock wave”. Besides supernova explosions,
interstellar shock waves may be driven by the pressure of photoionized gas, stellar winds, and
collisions between fast-moving clumps of interstellar gas.
Radiation can also drive fronts into the ISM, with EUV radiation impinging on neutral atomic
gas producing ionization fronts and FUV radiation impinging on molecular gas producing photodissociation fronts. These generally have small changes in pressure, but they can have large jumps in
the density. Another type of front can occur in the ISM when hot gas abuts cold gas, producing a
conduction front. In all these cases, the thickness of the front can be small compared to the overall
flow scale, so that the front is locally planar, and the time for the flow to cross the front can be
short compared to the overall flow time, so that the front is steady in a frame comoving with the
front.
14.1.
Jump Conditions
The Rankine-Hugoniot conditions, or “jump conditions,” relate the conditions ahead of the
front (labeled by “1”) to those behind the front (labeled “2”). We work in the frame of the front,
which moves at a velocity vs . We assume that the front is planar; if its thickness is ∆z, the front
will be planar if ∆z ¿ L and if the front is stable, so that it does not spontaneously generate
small scale structure. Second, we assume that the flow across the front is steady. In particular, the
acceleration of the front, if any, is small compared to that of the gas across the front, v̇ s ¿ vs2 /∆z.
Finally, we assume that gradients are negligible outside the shock front, so that the upstream and
downstream fluids obey ideal MHD. We make no such assumption inside the shock front, however,
and in fact viscosity, etc., are essential in effecting the transition from upstream to downstream
flow.
The jump conditions come from the conservation equations and Maxwell’s equations. Since
the flow across the front is steady, time derivatives vanish, and since the front is planar, the only
spatial derivative is parallel to the shock normal (the z-direction). Hence, Maxwell’s equations give
5 ·B =0
⇒
1 ∂B 5
+ ×E = 0
c ∂t
⇒
∂Bk
= 0,
∂z
∂
ẑ ×E = 0
∂z
(1)
(2)
We introduce the notation [f ] ≡ f (x2 ) − f (x1 ) to denote the jump in the function f across the
–2–
front. Integrating across the front, we obtain the first two jump conditions,
[Bk ] = 0,
(3)
c[ẑ ×E ] = −[ẑ ×(v ×B )] = [B⊥ vk − v⊥ Bk ] = 0,
(4)
where we have assumed that ideal MHD (E + v ×B /c = 0) obtains outside the shock front. The first
of these two equations is quite intuitive: a compression or rarefaction across the front cannot affect
the parallel component of the field. The other two Maxwell’s equations do not provide useful jump
conditions: In ideal MHD, the electric field parallel to B is zero, so we do not need 5 · E = 4πρ;
and we do not need to know the current, so we do not need Ampere’s Law.
The remaining jump conditions come from the conservation laws. Recall that the mass and
momentum conservation equations are of the form
∂
(density) + 5 · (flux) = 0.
∂t
(5)
The energy equation has this form on the LHS, but has two terms on the RHS, ∂φ/∂t and −n 2H L.
The first one vanishes because the flow is steady. We eliminate the second one by assuming that
cosmic ray heating is negligible in the front, so that the energy gains and losses are purely radiative;
in that case, the radiative transfer equation gives
5 · Frad = j − κJ = n2H L,
(6)
so that the energy flux is augmented by the flux of radiation, Frad . As remarked above, the partial
time derivatives vanish for a steady flow. As a result, the conservation equations reduce to
5 · flux = ∂ fluxk = 0,
∂z
(7)
Integrating across the front, we have
Z 2
∂
(fluxk ) dz = fluxk (2) − fluxk (1) ≡ [fluxk ] = 0.
∂z
1
(8)
This gives the jump conditions
£
¤
ρvk
·
¸
1 2
2
ρvk + p +
B
8π ⊥
·
¸
1
B B⊥
ρvk v⊥ −
4π k
· µ
¸
¶
¢
1 2
1 ¡ 2
vk
B⊥ vk − Bk B⊥ · v⊥ + Frad, k
ρv + p + ρe +
2
4π
= 0
(mass),
(9)
= 0,
k (momentum), (10)
= 0
⊥ (momentum), (11)
= 0
(energy).
(12)
In writing the momentum jump conditions, we have made use of the fact that [Bk ] = 0. As remarked
above, we have assumed that the viscous stress tensor π = 0 outside the front; we have made no
assumptions about its behavior in the front, other than that it is differentiable.
–3–
14.2.
Shock Fronts
Of the various types of fronts in the ISM, shocks are the most important. A shock wave is
an irreversible, pressure-driven fluid-dynamical disturbance. A shock travels faster than the sound
speed—it is a “hydrodynamic surprise.” The irreversible character is due to entropy generation as
ordered kinetic energy is dissipated into heat. A familiar example of a shock wave is the sonic boom
generated by a supersonic aircraft. Shocks are common in the ISM because cooling is efficient, so
that the sound speed is often less than the flow velocity. In neutral gas, the dissipation is due to
molecular viscosity in the shock transition (where large velocity gradients and viscous stresses are
present); the thickness is of order the collisional mean free path. In low-density, ionized plasmas,
the dissipation is collisionless, due to collective motions of the charged particles and the resulting
electromagnetic fields. The thickness of a collisionless shock is generallly . the ion gyroradius,
rL = mp vc/eB ' 109 v7 /B−6 cm, where v7 = v/(107 cm s−1 ) and B−6 = B/(10−6 G). In partiallyionized gases, the dissipation may sometimes be primarily due to “friction” associated with neutralion “slip”. Regardless of the dissipation mechanism, shock waves always are compressive.
Interstellar shocks can generally be divided into four regions. (1) First is the preshock gas,
which is irradiated by photons emitted from the hot, shocked gas behind the shock front. This
radiation may heat and ionize the preshock gas; this pre-heated and pre-ionized zone is referred to as
the “radiative precursor”. (2) Next is the “shock front,” or “shock transition,” where the bulk of the
energy dissipation (and entropy generation) occurs. As the gas flows through the shock transition
it is compressed by a factor ∼ 4, and its flow velocity (in the shock frame) is reduced by the same
factor. The decrease in ordered kinetic energy is accompanied by an increase in thermal energy.
(3) The hot postshock gas radiates its energy as it flows away from the shock transition; as it cools
it is further compressed, since the postshock region is at approximately constant pressure. The
region within which the bulk of the radiative losses occurs is designated the “radiative” zone. The
radiated energy that propagates toward the preshock gas is responsible for the radiative precursor.
(4) A fraction of the photons that propagate in the downstream direction are absorbed by the gas
(and perhaps dust) far downstream and contribute to the heating of this gas, which has attained
near-equilibrium between heating and cooling (the “thermalization zone”).
14.2.1.
Non-radiative, hydrodynamic shocks: Frad = B = 0
In order to determine the conditions behind the shock front, we apply the jump conditions.
They simplify considerably in the absence of a magnetic field and radiation:
ρ2 v22
ρ 2 v2 = ρ 1 v1 ,
(13)
ρ1 v12
+ p1 ,
(14)
1 2
p1
v + e1 + .
2 1
ρ1
(15)
+ p2 =
1 2
p2
v + e2 +
=
2 2
ρ2
–4–
Note that |v1 | = |vs |: the magnitude of the incoming flow velocity in the frame of the front is
identical to the magnitude of the front velocity in the lab frame.
When radiative cooling and heating are negligible, the composition is constant, so that e I =
const. For a gas with a constant ratio of specific heats, γ, we have p = (γ − 1)ρe; the Mach number
of the shock is
vs
vs
.
(16)
=
M=
cs1
(γp1 /ρ1 )1/2
The solution of the jump conditions then gives
ρ2
ρ1
p2
p1
=
=
γ+1
γ+1
→
→ 4,
2
γ − 1 + 2/M
γ−1
2γM2 − (γ − 1)
2γM2
5
→
→ M2 ,
γ+1
γ+1
4
(17)
(18)
where the cases of strong shocks (M À 1) and γ = 53 are given explicitly. The jump in the flow
velocity is v2 /v1 = ρ1 /ρ2 ; in the lab frame, the velocity of the postshock gas is
vps = v1 − v2 =
2(1 − 1/M2 )
2
3
vs →
vs → vs .
γ+1
γ+1
4
(19)
The postshock temperature can be inferred by taking the ratio of these equations, but it can
be inferred more directly for strong shocks by noting that in the frame of the shocked gas, the
energy per unit mass is simply the incident kinetic energy per unit mass,
1
e2 = (v1 − v2 )2
2
(strong shock).
(20)
Note that this implies that there is equipartition of kinetic and internal energies behind a strong
shock. Since e2 = kT2 /[(γ − 1)µ], the density jump implies
kT2 =
3
2(γ − 1)
µvs2 → .
2
(γ + 1)
16
Numerically, the postshock temperature behind a strong shock is

5 2

(fully ionized),
 1.38 × 10 vs7 K
5
2
T2 =
2.90 × 10 vs7 K
(atomic),

 5.30 × 103 v 2 K
(molecular),
s6
(21)
(22)
where vs7 = vs /(107 cm s−1 ), etc.
For strong, non-radiative shocks, our neglect of the magnetic field is self-consistent: Because
of flux freezing, we have
B⊥2
ρ2
γ+1
=
≤
→ 4.
(23)
B⊥1
ρ1
γ−1
Since Bk is constant across the shock, the magnetic pressure increases by at most a factor 4 2 = 16
2 /8π ¿ ρ v 2 in a strong shock.
for γ = 5/3, so that B⊥2
1 s
–5–
14.2.2.
Radiative shocks
When the column density of shocked gas becomes large enough, the shocked gas cools and
compresses. The cooling behind a shock is generally isobaric (at constant pressure), and we shall
assume that here. Because the gas can compress by a factor much larger than 4, the magnetic
pressure behind the shock can become important. We assume that the shock is strong—i.e., that
the shock velocity vs À cs1 , vA1 , or equivalently that M, MA À 1, where the Alfven Mach number
is MA = vs /vA1 . The mass and momentum jump conditions then give
ρ 2 v2 = ρ 1 v1 ,
B2
ρ2 v22 + p2 + ⊥2 ' ρ1 v12 .
8π
(24)
(25)
Now the first term in the last equation is
ρ2 v22 =
ρ1
ρ1 v12 ¿ ρ1 v12
ρ2
(26)
for large compressions. The momentum jump condition then reduces to
µ ¶ 2
ρ2 B⊥1
p2 +
' ρ1 v12 = ρ1 vs2 .
ρ1
8π
(27)
2 /8π À p ). In this case, equation (27) gives
Case 1: Magnetically supported (B⊥2
2
ρ2 √
= 2 MA⊥ ,
ρ1
(28)
where MA⊥ = vs /vA⊥1 is defined with respect to the perpendicular component of the upstream magnetic field. Numerically, we have
Ã
!
B⊥, µ
B⊥
= 1.84
vA⊥ =
km s−1 ,
(29)
1/2
(4πρ)1/2
n
H
and
nH2 = 77
Ã
3/2
nH1 vs7
B⊥1, µ
!
cm−3 ,
(30)
where Bµ = B/(1 µG).
2 /8π ¿ p ). In order for this case to occur, the cooling
Case 2: Thermally supported (B⊥2
2
must stop at a high enough temperature that the thermal pressure dominates the magnetic
pressure. In this case, the compression is determined by
ρ2 C22 = ρ1 vs2 ⇒
v2
ρ2
= s2 .
ρ1
C2
(31)
–6–
14.3.
Ionization fronts
The response of the ISM to the ionizing radiation from a star can be treated by the same
formalism used for shock fronts. Consider the response of the ambient ISM, assumed neutral, when
a newly formed star first emits ionizing radiation. Initially, recombination is negligible and each
ionizing photon ionizes one atom crossing the ionization front,
S = 4πr 2 n0 v1 ,
so that
cnπ
S
=
= cU :
2
4πr n0
n0
v1 =
(32)
(33)
the ratio of the velocity of the ionization front to the speed of light is just the ionization parameter
U . Since H II regions have U ∼ 10−2.5 , it follows that v1 À C2 , the sound speed behind the
ionization front. This is termed an R-type ionization front. (Actually, if the ionizing radiation were
to turn on suddenly, the ionization parameter close to the star could exceed unity, and the velocity
of the front would be limited to the speed of light.)
Eventually, recombinations slow the ionization front and v1 drops below 2C2 (the origin of this
criterion will become apparent below). The high pressure of the ionized gas now drives a shock
into the H I, and the ionization front must advance into dense, shocked gas; this is termed a D-type
ionization front.
To describe ionization fronts, we solve the mass and momentum jump conditions in terms
of the isothermal sound speed behind the front, C2 ≡ (p2 /ρ2 )1/2 ; since the temperature of the
photoionized gas is about 104 K, it follows that C2 ' 10 km s−1 . Inserting the mass condition into
the momentum condition, we obtain
ρ2 C22 +
which has the solution
ρ21 v12
= ρ1 C12 + ρ1 v12 ,
ρ2
£
¤1/2
C12 + v12 ± (C12 + v12 )2 − 4v12 C22
ρ2
=
.
ρ1
2C22
(34)
(35)
We require that the root be real, so that
v12 + C12 ≥ 2v1 C2 .
(36)
This is always true for C1 > C2 . However, we are interested in the opposite case (C2 > C1 ) for
ionization fronts, since the front heats the gas from O(102 K) to O(104 K). The critical velocity at
which the root vanishes is
vcr = C2 ± (C22 − C12 )1/2 ,
(37)
so that either vcr > C2 (+) or vcr < C2 (-). These two cases correspond to:
–7–
R-front: For vcr ≥ C2 , we require v1 ≥ vcr in order for the root to be real. It follows that
v1 ≥ vcr > C2 > C1 : the front is supersonic. One can also show that ρ2 > ρ1 : the front is
compressive. For C2 À C1 (which is often the case in practice), the minimum velocity of an
R-front is
vcr & 2C2 .
(38)
As an ionization front advances away from a star, its velocity drops from a high value down
to 2C2 , below which it can no longer propagate as an R-front. Note that in the terminology
of ionization fronts, shocks are R-type fronts.
D-front: For vcr ≤ C2 , we require v1 ≤ vcr in order for the root to be real. We then have
vcr = C2 − (C22 − C12 )1/2 =
C12
< C1 ,
C2 + (C22 − C12 )1/2
(39)
so that the front is subsonic. One can also show that ρ2 < ρ1 , so that the front is a rarefaction.
For C2 À C1 ,
C2
vcr ' 1
(40)
C2
is very subsonic. Keep in mind, however, that a D-front often follows a shock front, so that
the combination of the D-front and shock front moves supersonically into the unperturbed
gas. Since mass builds up between the shock front and the ionization front, however, one
cannot apply the jump conditions across the combination of the two fronts.
It is often the case that D-type fronts are not driven by pressure, so that the gas flows freely
behind the front. In that case, v2 = C2 , and one can show that this corresponds to v1 = vcr ;
this is termed a D-critical front. For C2 À C1 , one can readily show that such fronts have
p2 = 21 p1 .