7 Shocks Reaing: Ryen chs 3 & 4, Shu chs 5 & 6 For the enthusiasts, Shu chs 3 & 4 A goo article for further reaing: Shull & Draine, The physics of interstellar shock waves, in Interstellar processes; Proceeings of the Symposium, Gran Teton National Park, WY, July -7, 986 A88-450 03-90 Dorrecht, D Reiel Publishing Co, 987, p 283-39 Consier the propagation of a finite amplitue soun wave The spee of propagation is higher at higher temperature, so the crest of the wave graually overtakes the trough T γ When faster moving gas overtakes slower moving gas, we get a iscontinuous change of ensity an velocity, a shock NB: In fact, this steepening happens even for γ because of non-linearity of the equations Shocks can also be prouce by any supersonic compressive isturbance This is the more common source of shocks in astrophysical situations What are places where astrophysical shocks occur? Clou-clou collisions HII regions expaning into neutral meium Stellar win encountering meium Supernova or GRB blast wave internal an external shocks Accretion onto compact objects: spherical or isk Accretion onto hyrostatic intracluster meium In general a shock wave is a pressure riven compressive isturbance propagating faster than the signal spee for compressive waves proucing irreversible change in flui state increase of entropy In most cases a shock involves a iscontinuous change of flui properties over a scale λ 7 Shock jump conitions Consier a propagating shock wave in the rest frame of the shock Unshocke gas approaches from the +x irection moving faster than its soun spee an passes through the shock Pre-shock conitions:, u, T Post-shock conitions: 2 >, u 2 < u, T 2 > T We woul like to erive the relations aka jump conitions between, u, T or, equivalently, P an 2, u 2, T 2 or P 2, for a steay-state, plane-parallel shock u perpenicular to shock front an flui properties epen only on istance to front 37
Within the shock front aka transition layer, viscous effects are important they cause the shock in the first place However, outsie this layer, viscous effects are small on scales larger than the mean free path We will erive conservation equations of the form Q, u, P 0 Q, u, P constant, an although the quantities Q involve viscous terms, we can ignore these outsie the shock zone an can therefore erive the jump conitions from equations that on t involve viscosity terms We start from the continuity equation an the momentum equation t + u 0, u t + u u g P + π, an from the thermal energy equation, which it is helpful to write in the form Ryen -43 t ɛ + ɛ u P u F + Ψ, analogous to the continuity equation, an to supplement with a separate equation Ryen -44 for conservation of kinetic energy t 2 u2 + 2 u2 u u g u P + u π We now assume steay-state, t 0, plane-parallel, y 0, z 0, an viscosity These equations then become Equation 4 immeiately gives x, an ignore gravity u 0 4 u u P u ɛu P 42 43 2 u2 u u P 44 u constant u 2 u 2 45 Using u2 2u u + u2 uu + u u + u u u + u u uu 38
allows the equation 42 to be written so u u + P u 2 + P 0 46 u 2 + P constant u 2 + P 2 u 2 2 + P 2 47 If we ha kept the viscosity terms in the erivation, equation 46 woul instea have been u 2 + P 4 3 µu 0 48 Within the transition zone, where µ an u are non-zero, u2 + P is not constant However, in the pre-shock an post-shock zones, µ an u are negligible, so equation 47 hols We coul use equation 48 together with our other equations an the constitutive relation for viscosity to follow what happens within the transition zone However, the flui approximation itself breaks own within this region Aing together equations 43 an 44 gives Since so 0 u 2 u2 + ɛ u 2 u2 + ɛ + P 2 u2 + ɛ + P + P u u + u u 0 an u 0, this equation implies that 2 u2 + ɛ + P 2 u2 + ɛ + P 0 2 u2 + ɛ + P constant, 2 u2 + ɛ + P 2 u2 2 + ɛ 2 + P 2 2 If we ha been more complete, the conserve quantity woul also inclue viscosity an heat conuction terms, but once again, these are unimportant outsie of the transition zone In summary, we have the Rankine-Hugoniot jump conitions for a plane-parallel shock: u 2 u 2 49 u 2 + P 2 u 2 2 + P 2 50 2 u2 + ɛ + P 2 u2 2 + ɛ 2 + P 2 5 2 Even though the physics of the shock region may be complicate an varie, these conitions follow from conservation of mass, momentum, an energy alone More precisely, the first follows 39
from mass conservation, the secon from mass an momentum conservation, an the thir from mass an energy conservation If, u, an P are known, we have three equations for the three unknowns 2, u 2, an P 2 Using ɛ i γ i P i i, the last of the jump conitions can be written 2 u2 + γ P γ 2 u2 2 + γ 2 P 2, γ 2 2 for a gas that has a polytropic equation of state Note that γ 2 may be ifferent from γ if, for example, the shock issociates molecules, or raises the temperature so that previously inaccessible egrees of freeom become accessible 72 The Mach Number The imensionless number that characterizes the strength of a shock is the Mach number, the ratio of the shock spee to the upstream soun spee: M u u 2 /2 52 a γp The factor in can be viewe as a ratio of ram pressure to thermal pressure in the pre-shock gas, or as a ratio of kinetic energy ensity to thermal energy ensity In terms of the Mach number, the shock jump conitions are Ryen eqs 3-5 Together these conitions imply A strong shock is one with M, yieling 2 u u 2 γ + M 2 γ M 2 + 2 P 2 2kT 2 /m P kt /m 2γM 2 γ γ + T 2 [γ M 2 + 2][2γM 2 γ ] T γ + 2 M 2 2 u γ + u 2 γ 4 53 2γ P 2 γ + M 2 P 2 γ + u 2 3 4 u 2 54 T 2 2γγ γ + 2 T M 2 where the last equalities are for γ 5/3 2γ γ + 2 m k u2 3 6 In the rest frame of a strong shock, with γ 5/3, the post-shock kinetic energy is 2 u2 2 32 u2, 40 m k u2, 55
an the post-shock thermal energy is 3 kt 2 2 m 9 32 u2 So roughly half of the pre-shock kinetic energy is converte to thermal energy The total energy of the post-shock gas is lower in the shock rest frame because of the work one on the gas by viscosity an pressure in the shock A weak shock has M ɛ In this limit where the last equalities are again for γ 5/3 2 + 4 γ + ɛ + 3 2 ɛ 56 P 2 + 4γ P γ + ɛ + 5 2 ɛ 57 T 2 4γ + ɛ + ɛ, T γ + 58 The post-shock Mach number is In the weak shock limit, M 2 ɛ In the strong shock limit M 2 u 2 a 2 u a u 2 u a a 2 M u 2 u [ γ γ + 2 M 2 M γ + 2γγ M 2 /2 T T 2 ] /2 γ /2 045 2γ A shock converts supersonic gas into enser, slower moving, higher pressure, subsonic gas increases the specific entropy of the gas by an amount s 2 s c V ln P2 γ 2 c V ln P γ c V ln P2 P c V γ ln 2 It One can also write this in the form s 2 s c P ln T2 which is Ryen s eq 3-55, using c P c V + k m γc V T k m ln P2, P In another terminology, a shock shifts gas to a higher aiabat An aiabat is a locus of constant entropy T γ in the ensity-temperature plane Gas can move aiabatically along an aiabat, while changes in entropy move it from one aiabat to another 4
73 Numerical hyroynamics an artificial viscosity The physical scale of a shock transition layer is λ In most systems of astrophysical interest, the scale of the system is L λ In numerical hyroynamics, it is usually impractical to make an iniviual gri cell comparable to or smaller than λ especially in more than one imension, so the physical scale of a shock is generally much smaller than one gri cell Fortunately, we usually on t care about the region right aroun the shock, just about the influence of the shock on the physical state of the gas The relation between the physical properties of the upstream an ownstream gas are etermine by the shock jump conitions, an they follow from conservation laws, inepenent of the etails of the shock Shock capturing coes work by searching for places where shocks are likely to occur eg, where the velocity fiel is eveloping a iscontinuity an imposing shock jump conitions over the scale of one or two gri cells An alternative metho, introuce by von Neumann an Richtmeyer in the 950s, is to use an artificial viscosity that is much larger than the true viscosity one woul calculate base on the microscopic properties of the gas High viscosity makes shocks broaer, so that they can be resolve over several gri cells Since the jump conitions follow from conservation laws, the pre- an post-shock gas shoul still have the correct relations even if the viscosity is not the true viscosity There is a fair amount of experimentation an black magic in artificial viscosity tuning things to get the esire physical behavior in as many situations as possible 74 Raiative shocks Now suppose that the gas is able to raiate energy in a raiative relaxation layer rrl after the shock Immeiately after the shock front the flui properties are still enote 2, u 2, P 2, T 2, but they change in the raiative relaxation layer, settling to new values 3, u 3, P 3, T 3 further ownstream The rrl is always large compare to the shock front itself because many collisions are require to cool the gas so that the size of the rrl is λ Within the rrl, the temperature rops, an the gas is squeeze to higher ensity Since u constant, the velocity rops We can emonstrate this by returning to the steay-state equations 4-43 but incluing the cooling term Γ Λ L, T > 0, [L] erg g s The equations are u 0 42
Combining the last equation with an since u implies an substituting P an thus u u P ɛ u ɛu u P L ɛ P γ u u implies ɛ [ P γ P ] 2 u u [ u P γ + P ] u + P u u L, u2 u γ + γ γ P u L a 2 u 2 γ u L with a 2 γp/ Since the post-shock flow is sub-sonic u < a an L > 0, we conclue that u < 0, an mass conservation therefore implies that > 0 The mass conservation an momentum conservation jump conitions apply as before, since their erivation i not involve ɛ which is the quantity affecte by raiative cooling Therefore 3 u 3 2 u 2 u 3 u 2 3 + P 3 2 u 2 2 + P 2 u 2 + P As u rops in the rrl, u 2 uu rops an the pressure must rise For a strong shock with γ 5/3, M 2 u 2 5 3 P /2 2 045, 2 so P 2 3 2 u 2 2 an only a small rise in pressure is require to satisfy the secon jump conition even if u 3 rops to zero The important point is that the pressure oes not go own, an therefore a rop in temperature must be accompanie by a proportional rise in ensity to maintain the pressure If there is a lot of post-shock cooling, then the ensity ratio 3 / can be very high 43
This is probably the most important ifference between a non-raiative shock an a raiative shock: a non-raiative shock can only increase the ensity by a factor of 4, but a raiative shock can increase the ensity by a very large factor A specific interesting case is that of an isothermal shock, where T 3 T This becomes the thir jump conition This can arise if, for instance, T T 3 is a temperature where the cooling time becomes very long, or a temperature where heating an cooling processes balance Combine with the equation of state P m kt a2 T, where a T kt/m /2 is the isothermal soun spee, the solution to the shock jump conitions is, Since 3 u 2 3 + P 3 u 2 3 2 u at a T u u 3 T 3 T u 4 + a 2 T u this solution satisfies the momentum jump conition a T u 2 M 2 T a T 2 a 2 T + u 2 P + u 2, The fact that 3 / MT 2 arbitrarily high means that the compression factor in an isothermal shock can be 75 Oblique shocks If the flui enters the shock at an angle other than 90, the shock jump conitions apply only to the perpenicular component u of u The parallel component of u is unchange, so the flui is refracte by the shock, changing its irection so that it is closer to parallel to the front 76 Shocks with magnetic fiels: a few comments Hyromagnetic shocks are often important in the ISM For a shock with B parallel to the shock front: B u B 2 u 2, ie, the magnetic fiel lines are compresse by the same factor as the ensity The momentum jump conition becomes u 2 + P + B2 8π 2u 2 2 + P 2 + B2 2 8π, ie, there is an aitional magnetic pressure term B 2 /8π The units of B 2 are erg cm 3 yne cm 2 44
The energy jump conition becomes 2 u2 + γ P + B2 γ 4π 2 u2 2 + γ P 2 + B2 2 γ 2 4π 2 Coupling of ions to magnetic fiels can allow a shock front to be much narrower than the mean free path λ for particle-particle interactions The situation is more complicate when B is not parallel to the shock front, an it can be much more complicate in multi-flui shocks when electrons, ions, an neutrals are couple but not perfectly couple One important effect is that magnetosonic waves aka Alfven waves can warn upstream ions an electrons about an approaching ensity iscontinuity There are many aitional classes of solutions, some of them iscusse in the Shull & Draine article 45