last class: adiabatic and isothermal shocksastrom/ay126/pdf/ay126_notes10.pdflast class: adiabatic...
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Last class: Adiabatic and Isothermal shocks
adiabatic and cooling regions in discontinuity
1
In both cases R-H equations conserving mass and momentum apply to material on either side of shock
R-H energy conservation equation applies to adiabatic case For isothermal, condition becomes T1 = T2
Adiabatic: Isothermal: maximum compression factor ≤ 4 ρ2/ρ1 = u1
2/cs2 = M1
2
compression can be large cs isothermal sound speed
dx infinitely thin → “jump”
Shock wave theory applied: Supernovae Clarke & Carswell Chapter 8; Draine Chapter 39
Supernovae explosions from gravitational collapse of stellar core + envelope ejection
Type II SN – end point of massive stars M > 8M.
“instantly” ~ 1 – 5 M of ejecta velocities ~ 104 km/s
energy released ~ 1051 ergs
Gas sweeps up, accelerates, compresses, heats ISM → blast wave
Typical ISM: nHI ~ 1 cm-3, ρcloud ~ 1.6 x 10-24 g/cm3, Tcloud ~ 104 K
2
Phases of a Supernova Explosion (interaction between ejecta & ambient medium) 1. Mejecta >> Mswept up ISM
Free expansion of expelled gas at constant vE
2. Mswept up ISM > Mejecta Expansion slows somewhat, shocked gas has no time to cool
→ adiabatic shock 3. Expansion slows and radiative losses become important
→ isothermal shock 4. SN remnant slows to point where pressure balance with
ambient medium is achieved → merges
Idealized case: homogeneous gas explosion energy E instantaneously from point surrounded
by region of uniform density ρ 3
Phase I –free expansion Free expansion, vE const; radius increases with time;
assume uniform density ρ in front of ejecta Phase I terminates when mass swept up = mass ejected
,energy emitted ε = 4 x 1050 ergs, vE
2 = 2ε/ME and vE ~ 104 km/s
∴ ME = (8x1050) /1018x 2x 1033) = 0.4 M when mass swept up = mass ejected, 4/3πρR3 ~ 0.4 M
R3 = 0.4 x 2x 1033/4 x 1.6 x 10-24 ~ 1057/8 ∴R ~ 0.5 x 1019 cm ~ 1.7 pc
∴duration = R/vE = 1.7 x 3 x 1018/109 x3x107 ~ 170 yrs in practice, phase I lasts about 100 years
“bubble” continues to expand at high velocity → shock
energy conserved → shock adiabatic
4
Phase II Blast Wave [Adiabatic or Sedov Phase] Explosion propagates outwards → shock Assume all swept up medium in narrow shell ∴ for small D, Adiabatic shock:
for γ = 5/3, D/R = 1/12 << 1 i.e. OK to assume shell thin
diatomic gas, γ = 7/5, ρ0/ρ1 =6 – even thinner ∴ assume all gas in thin shell moves at same
average velocity at any instant
In frame of shock, diagram as seen earlier
Words to fill an cover
1
0
11
ρ γρ γ
+=
−
3 20 1
4 43
R R Dπ ρ πρ=
1 13 1
D Rγγ
−∴ = +
5
R-H (1) from continuity equation: ρ0u0 = ρ1u1, so that velocity of shocked to unshocked gas = U As shell grows, its radial momentum increases as (momentum due to pressure on inside of shell, Pin outside ρ, T are 0)
Let pressure within shell. P1 scale as pressure Pin Pin = αP1 For strong shock And rate of change of radial momentum = force over area ∴
01 0 0
1
11
u u uρ γρ γ
−= = +
00 1
21
uu uγ
= − = +
3 00
243 1
ud Rdt
π ργ
+
20 0
21isP uρ
γ
= +
3 2 200 0 0
24 243 1 1
ud R R udt
π ρ πα ργ γ
= + +
6
1
From Obtain i.e. shock advances on undisturbed gas at u0 = dR/dt,
d/dt[R3Ŕ] = 3αR2Ŕ2
Substituting R ∝ tb, here → equn w. solution b = 1/(4-3α) And power law scalings for R and u0 that describe
how radius of shell increases with time in terms of α
[Clarke & Carswell p 93]
How to determine α?
3 2 200 0 0
24 243 1 1
ud R R udt
π ρ πα ργ γ
= + +
3 2 20 03d R u R u
dtα =
14 3
3 33 34 3
o
R t
u t R
α
ααα
−
−−−
∝
∝ ∝
7
Energetics of explosion define α Pin = αP1 relates pressure in shocked gas to pressure on shell from explosion
Adiabatic case: energy (= KE + ε) is conserved Internal energy /unit volume = P/(γ-1)
Most energy is in material in cavity (little mass)
∴ Internal energy = 4πR3Pin/(γ-1) = (4/3) πR3αP1/(γ-1) and KE = ½ (4/3) πρ0 R3U2
E = KE + ε = ∴ Since energy must be conserved, 6α - 3 = 0, and α = ½
2 23 0 0 0
02 24 1
3 2 1 1 1u uaR ρπ ρ
γ γ γ
+ + − +
→ R ∝ t2/5, u0 ∝ t-3/5 P1 ∝ t-6/5
6 33 2 4 3
0E R u tα
α−
−∝ =
8
∝
with increasing R, u0 and P1 decreasing C& C §8.1, Kwok §16.2, Tielens §12.3.2
More rigorous derivation: Similarity/Sedov Solutions C&C §8.1.2
Let λ = scale parameter → size of blast wave at time t after explosion Assume λ variation with t depends on E, ρ0
∗
Dimensionally, E ≡ ML2T-2 and ρ ≡ ML-3
To keep E/ρ const, ML2T-2/ ML-3 ~ L5T-2 = const ∴(Et2/ρ)1/5 has dimensions of length → λ ≡ (Et2/ρ)1/5 Sedov-Taylor solution Replace radius of shell r by ξ = r/λ and assume solution self-similar; i.e. since no natural scale length all variables X(r,t) take form X = X(t)X(ξ); substitution → scaling relation for Rshock from ξ = r/λ = r/ (E/ρ0)1/5 t2/5
SIMILARITY SOLUTION - All physical properties scale wrt radius of shock front
Rshock = ξ0(E/ρ0)1/5t2/5 ∝ t2/5
(ξ0 ≡ value of similarity variable that labels location of shock front ~1) ∗ E and ρ are key parameters; ejecta expand without losing energy; density important since initial energy redistributed to larger and larger amount of gas.
X = X(t)X(ξ) has same shape But scaled by time-dependent X(t)
X
ξ ξ
9
Sedov: solved R-H equations analytically
(C and C §8.3 adapt, using ξ = r/λ = r (Et2/ρ0)1/5 and dimensionless variables ρ′(ξ), u′(ξ), P′(ξ),
so that ρ(r,t) = ρ1ρ′(ξ) = ρ0(γ+1)/γ-1) ρ′(ξ) etc. → set of equations independent of r and t
(integration “elementary but laborious”)
Exact solution traces u, ρ, P, T, behind shock (assumes γ = 7/5)
At r=o t= 0, instantaneous energy release and shock front values us, ρs, Ps, Ts
Plots show variation of u/us , ρ/ρs, P/ Ps, T/ Ts, with ξ
Similarity solutions more or less validated closer to shock, density , pressure high
inner regions : gas hotter , more tenuous, but pressure maintained velocity over-simplified?
1
0 ξ0
10
“Experiments” also validate self-similar solutions R ∝ t2/5 and R(t) = ξ0(Et2/ρ0)1/5
e.g. blast wave from atomic bomb Snapshots of fireball (C&C §8.2) confirm scaling of R5/2 with t (log r plotted v logt) Relation enables determination of explosion energy R(t) = ξ0(Et2/ρ0)1/5
For SN, expansion velocity U = dR/dt = 2/5ξ0(E/t3ρ0)1/5 = 2R/5t Energy E ~ (R/t)2
SN ejecta ~ 1 M expands with velocity ~ 104 km/s Energy ~ mv2 ~ 1033 x 1018 – 1051 ergs
Rewrite R(t) in terms of typical SN energy and density 1 gm/cm3
→ R(t) ~ 1013t2/5 and u = 4 x 1012t-3/5 . R ~ 0.3t2/5 pc, u ~ 105t-3/5 km/s
11
Alternatively can rewrite R(t) = ξ0(Et2/ρ0)1/5 as R(t) ~ 10-0,7(E50/ρ24)1/5t2/5 with t in yrs, R in pc
Again leads to R ~ 0.3t2/5 pc, and u ~ 105t-3/5 km/s
For Crab nebula, with R ~ 2 pc, E = 4 x 1050, derive age ~ 103 yrs
What next?
Radiation losses increase when vs < 250 km/s ( T<106 K)
→ no longer adiabatic shock → isothermal at that time t3/5 = 1/250, t~ 4 x 104 yrs, R ~ 25 pc
i.e. Sedov phase ends after ~ a few x 104 yrs (Tielens)
Note: Approximations do not hold for t < 100 yrs (the free expansion stage with U = 104 km/s)
12
SN 1987A
In LMC, at 50kpc. Star (Sanduleak) SK-69, mass 20 M. Strong stellar wind 106 years ago → cavity in ISM
1987 explosion → blast wave (Sedov phase) . Shock moves at 4.5 x 103 km/s.
X-ray emission as it meets dense ring of gas deposited by wind 13
Why transition from adiabatic conditions at 106 K?
Here Λ = cooling rate (ergs cm-1 s-1) Below 106 K Λ increases as T decreases Time scale for Sedov phase: Λ= φne
2T-1, where φ= 8x1017 (cgs) For sphere radius R, cooling ~ 4/3(πR3) φne
2T-1, i.e. fastest cooling for larger R, low T
Since kT ~ ½ m(3/4)v2, T =3/32(mv2/k)
At T = 106 K, v2 ~ 32/3 x10-16/10-24x 106 ~ 1015, V ~ 3 x 107 ~ 300 km/s
From Sedov tables, v =300 km/s and t = 104 yrs Adiabatic phase lasts ~ 104 yrs
14
More formally: recall R-H conditions P0 negligible, u1 much decreased behind shock ∴
Isothermal shock: ρ1/ρ0 ~ M0
2 = u02/cs
2, Suppose blast wave at 100 km/s moves into gas with cs = 1 km/s,
M02 = 104 = ρ1/ρ0
→ density increase of 104 ?
→ In practice, radiation inside SNR ionizes and heats gas ahead cs ~ 10 km/s more typical, and ρ1/ρ0 ~ 100
Note: since u1 is small, material remains near shock → thin, highly compressed shell
1 1 0 02 2
1 1 1 0 0 22
1 0 0~
u uu P u P
P u
ρ ρ
ρ ρ
ρ
=
+ = +
15
SN Evolution (Sedov-like tables for γ = 5/3) t(yrs) 1 10 100 1000 10,000 100,000 R(pc) 0.315 0.791 0.791 4.91 12.5 31.5 V(km/s) 124,000 81,000 7820 1970 4.94 1.24
As shock speed decreases, Tshock decreases - after about 104 yrs, T ~ 4 x 106 K and still decreasing Recombination of C, N, O ions → subsequent line cooling i.e. radiative losses and shock becomes isothermal temperature determined by conservation of momentum
- snowplow phase mass swept up slows front to sub-sonic
Sedov onset radiative cooling
16
Phase III Snowplow or Momentum-Conserving Phase (Kwok 16.2.1)
Begins at t= 0, when momentum of shell p0 = Msvs = const ∴equn of motion Integrating → with Solutions:
4 310 0 04c R R Rt
•
= −
( )( )
340 03
3403
4 014
0
34
o
o
p R R
p R R
pR t c
π ρ
π ρ
πρ
•
•
=
=
= +
( ) ( )
1 34 4
0 00 000
0 0
4 41 1
R t t R t tR R and R R
R R
−
• •• •
− −
= + = +
R ∝ t1/4 dR/dt ∝ t-3/4
Adiabatic phase: R ∝ t2/5, dR/dt ∝ t-3/5
Snowplow phase: expansion slowing down much more
17
Supernova remnant evolution – schematic, realistic?
Remaining questions: - effect of magnetic field on SNR growth? -interaction of shock front with ISM inhomogeneities?
-how does the real ISM (irregular clouds - e.g. SN 1987A) affect Sedov phase?
-role of dynamical instabilities in structure of SNR - Rayleigh-Taylor instabilities in Crab nebula filaments
-energy input into ISM - Kwok 16.2.1
Draine & McKee 1993 AnnRevAstrAstrophys 31, 373 18
Cygnus Loop - 7.5 kpc - 2 x 105 yrs - radiative phase
19
20
Expansion of HII Regions - dynamics
So far: expansion of static HII regions → Strömgren sphere rate of input of ionizing photons ≡ recombinations
Assumed density in HII region same as before gas ionized But: gas much hotter inside HII region 10,000K versus 100K
∴pressure inside ~ 100 x pressure outside → expansion and drop in density
recombination rate ∝ ne2 - more ionized gas than in static
Stages of expansion: 1) Initial Strömgren sphere Produced by ionizing photons from star, very little motion α = recombn coefft = 2.6 x 10-11/T1/2.
For O5 star, Rinit ~ 5.6 pc
13
*2
34 ( )init
H
SRn Tπ α
=
21
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