Shock waves; Supernova remnant evolution 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 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

Steepening of non-linear acoustic wave 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 speed of crest speed of trough

“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): 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 shock front v0  Δv = 2 v0 shocked gas v0  5-10 km/s vs  v0

Expansion of H II region compressed neutral gas (cold, high density) H II region ambient neutral gas (cold, medium density) (hot, low density) vs  10 km/s

Fast stellar wind (“Interstellar Bubble”) inner shock outer shock contact discontinuity (wind meets ISM) unshocked wind material shocked wind material shocked cloud material unshocked cloud material

Terminal velocity of wind from hot stars

Supernova blast wave Early: Mswept < Mejecta  like stellar wind Late: Mswept >> Mejecta  ignore Mejecta shock front unshocked ISM hot interior shocked ISM (cool shell) shocked ISM (did not cool radiatively)

Star formation: infall and outflow shock front hot, shocked accreted matter infalling ambient cold material (low pressure, free fall)

Spiral shocks in Galactic disk flow shock front compressed material forms spiral 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)

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 zone radiative zone Undisturbed pre-shock material Immediate post-shock conditions Cold post-radiative conditions

Mass conservation across shock Steady plane-parallel shock: Integrate across shock:

Momentum conservation across shock Note: no change in gradient of gravitational potential F =>

Energy conservation across shock Same procedure as before Heat conduction Heating Cooling ~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)

Special (often used) case: simplifying assumptions Planar shock: No heat conduction: No net heating / cooling: No gravitational force: Internal energy density: with g=cP/cV B-field  flow velocity:   :

Rankine-Hugoniot equations for shock jump conditions Follow general practice and write vx=u1(=vS) Mass conservation: Momentum conservation: Energy conservation: Flux conservation:

Solution of shock jump equations These are the jump conditions: 4 eqs. in 4 unknowns: r2, u2, P2, B2 Define Now we have two equations for P2 and x (cubics in x): Define Mach number M by M2=u12/vms2 , where vms is the magnetosonic speed given by

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 

Some properties of strong shocks Compression ratio x = 4 If 16B12/8p<<r1u12 and P1<< r1u12 , then momentum conservation gives P23/4 r1u12=3/4 . ram pressure B2 = 4 B1  magnetic pressure increases by factor 16 Post-shock flow is at P  constant, and subsonic:

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 => r3>>r1 B = 0: compression factor can become much larger than 4! B  0: where  typical compression factor  100!

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

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

Comparison of J-shocks and C-shocks J (“jump”)-shocks: vS50 km/s Shock abrupt Neutrals and ions tied into single fluid T high: T40 (vS/km s-1)2 Most of radiation in ultraviolet C (“continuous”)-shocks: vS50 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: =>

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:

More detailed calculation of A Assume most of mass concentrated just behind shock (O.K. from detailed solution) Ekin~Ethermal for entire SNR Ekin~1/2MvS2, M= 4/3pR3r0 Then E0=2Ekin=2.1/2. 4/3pR3r0.vS2 R=cst.ta =>vS=a.cst.ta-1 => E0=4/3pr0(cst)5a2t5a-2=const. => a=2/5 => (cst)5=3/(4p)(5/2)2(E0/r0) => (cst)=1.083 (E0/r0)1/5 => A=1.083 Exact: A=1.15169 for g=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 ISM 1. Early phase Early: Mswept < Mejecta (‘Free expansion’) RS=vSt vSv0=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 r0, Mejecta=Mswept=r04/3p(v0tM)3 =>

2. Sedov phase Mejecta<Mswept and t<trad

Cooling at end of Sedov phase shock front cool dense shell hot, low density, pressure pi radiative cooling negligible

Estimate of trad Estimate t=trad when cooling becomes important and Sedov phase ends Most cooling just behind shock where r highest and T lowest Define trad when 50% of Ethermal radiated 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):  Rt1/4

Radiative phase (cont’d) Pressure-modified snowplow (Oort snowplow): Pi drops due to adiabatic expansion:

4. Merging phase SNR fades away when expansion velocity = velocity dispersion ambient gas (~10 km/s) Use Oort snowplow:

Summary phases of supernova shell expansion Early phase (mswept < mejecta): Free expansion, Rs = vs t Sedov phase (mswept > mejecta and t < trad): Energy conservation, R  t2/5 Radiative “snowplow” phase (t > trad): Momentum conservation, R  t1/4 or R  t2/7 Merging phase: vs drops below velocity dispersion of ambient ISM

Overview of SNR evolution

Overview of SNR evolution