Relativistic Gases. 1 Relativistic gases in astronomy. Optical. Radio. The inner part of M87. Astrophysical Gas Dynamics: Relativistic Gases 1/73

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1 Relativistic Gases 1 Relativistic gases in astronomy Optical The inner part of M87 Radio Astrophysical Gas Dynamics: Relativistic Gases 1/73

2 Evidence for relativistic motion Motion of the knots in the M87 jet indicates apparent velocities in excess of the speed of light Large scale component velocities are of the order of c Credits: Biretta and colleagues Astrophysical Gas Dynamics: Relativistic Gases 2/73

3 2 Some relevant aspects of relativity 2.1 Four vectors and tensors Special relativity is based on the geometry of space-time described by the metric The interval between two points is where diag (-1,1,1,1) ds 2 dx dx dx dx dx 2 2 +dx 3 2 (1) (2) x 0 ct (3) Astrophysical Gas Dynamics: Relativistic Gases 3/73

4 Indices Latin go from 1 to 3; Greek from 0 to 3 Timelike and spacelike Displacements are timelike if ds 2 0 and spacelike if ds 2 0. For timelike displacements, the proper time,, is given by Lorentz factor c 2 d 2 ds 2 The Lorentz factor is defined by v c (4) (5) Astrophysical Gas Dynamics: Relativistic Gases 4/73

5 Note that we use for the Lorentz factor corresponding to the bulk motion of the fluid. We use for the Lorentz factors of the particles making up the fluid. Since, for a timelike displacement (the displacement of a material particle) c 2 d 2 ds 2 dx 0 2 dx 1 2 dx 2 2 dx 3 2 dx dx dx 0 2 dx dx 0 v 2 c 2 dt c 2 dt 2 c 2 dx dx 0 (6) Astrophysical Gas Dynamics: Relativistic Gases 5/73

6 then Signature of metric cd 1 cdt Note the signature of the metric,+,+,+. Other expositions of special or general relativity adopt +,,,. The main 4 vector that we deal with here is the fluid 4-velocity, (7) u dx dx vi --- cd cdt c (8) Astrophysical Gas Dynamics: Relativistic Gases 6/73

7 where and x i v i dx i dt are the coordinates of a fluid element through space time. (9) 2.2 Raising and lowering indices Indices are raised and lowered by means of the Minkowski tensor. Some results to remember are as follows. The inverse of the Minkowski tensor, is defined by: 1111 (10) Astrophysical Gas Dynamics: Relativistic Gases 7/73

8 Indices are raised and lowered by the Minkowski tensor v v v v (11) 3 The stress-energy tensor 3.1 Definition The stress energy tensor is defined by the following components: T T 00 T 0i T i0 T ij (12) Astrophysical Gas Dynamics: Relativistic Gases 8/73

9 where T 00 ct 0i T i0 T ij We also have Energy density (incorporating rest mass energy etc.) Flux of energy in the i direction T 0i (13) Flux of the i component of momentum in the j direction 1 --T i0 c i component of momentum density c 2 Flux of energy in the i direction (14) Astrophysical Gas Dynamics: Relativistic Gases 9/73

10 The relationship of to both momentum and energy T 0i T 0i Why should we have these two different roles for the components? Remember that in special relativity, mass and energy are related by E mc 2 (15) Therefore a flux of energy is equivalent to times a flux of mass. But the mass flux is v i and this is just the momentum density. Hence c 2 Energy flux c 2 Momentum density (16) Astrophysical Gas Dynamics: Relativistic Gases 10/73

11 and Momentum density c 2 Energy flux (17) 3.2 Stress energy tensor for a perfect fluid The above characterisation of the stress-energy tensor is valid in general. For a perfect fluid in which the pressure is isotropic and normal to any surface we develop an expression for the stressenergy tensor as follows. Consider the rest-frame of the fluid. This is defined as that frame in which u 1000 (18) Astrophysical Gas Dynamics: Relativistic Gases 11/73

12 In this frame, Hence, Energy density e Energy flux 0 Momentum flux density p ij (19) e 000 T 0 p 00 00p 0 000p This can be covariantly expressed as e + p T e + pu u + p p p p p (20) (21) Astrophysical Gas Dynamics: Relativistic Gases 12/73 +

13 Relativistic enthalpy The quantity w e+ p (22) is called the relativistic enthalpy similar to the corresponding non-relativistic expression h + p (23) Note that the relativistic expression contains the rest-mass energy. Often we expression the relativistic enthalpy in the form: w c p (24) Astrophysical Gas Dynamics: Relativistic Gases 13/73

14 where c 2 is the rest-mass energy density and e c 2 is the internal energy. Therefore, relativistic enthalpy rest mass energy + internal energy + pressure. Astrophysical Gas Dynamics: Relativistic Gases 14/73

15 3.3 Expressions for the components of in an arbitrary frame The various components of the stress-energy tensor can then be expressed as: T T 00 e + pu 0 u 0 p 2 e + p p T 0i e + p 2 v --- i c T ij e + p 2 v i v j + p ij c 2 (25) Astrophysical Gas Dynamics: Relativistic Gases 15/73

16 4 The relativistic fluid equations 4.1 Energy and momentum equations The relativistic fluid equations are T, T x (26) To see why these are the equations, let us examine the different equations corresponding to 0 and i. Astrophysical Gas Dynamics: Relativistic Gases 16/73

17 Time component 0 T 00,0 + T 0j,j 0 1 c t T00 T 0j x j (27) t x j T 00 ct 0j In words: Energy density divergenceenergy flux 0 t (28) Astrophysical Gas Dynamics: Relativistic Gases 17/73

18 Spatial component i T i0,0 + T ij,j 0 1 c t Ti0 T ij x j (29) c 1 T i t T ij x j In words: Momentum density divergence Momentum flux density 0 t These are the appropriate conservation laws for energy and momentum. Astrophysical Gas Dynamics: Relativistic Gases 18/73

19 4.2 The conservation of number density Assuming that our fluid consists of particles that do not transform into other particles, then their number is conserved. Let n Number of particles per unit proper volume (30) i.e. n is the number density of particles in the rest frame of the fluid. We often use the baryon number density for n. The conservation law for number is nu x c n t n t nv i c x i nv i x i (31) Astrophysical Gas Dynamics: Relativistic Gases 19/73

20 This makes sense since the number density in an arbitrary frame is n. Rest mass density Let m be the average rest mass of particles in the fluid. The restmass energy density in the rest frame is nmc 2. The rest-mass energy density in an arbitrary frame is 5 Non-relativistic limit c 2 nmc 2 (32) The non-relativistic limit is instructive since it Confirms that we have the right equations Clarifies the relationship between energy flux and momentum Astrophysical Gas Dynamics: Relativistic Gases 20/73

21 density 5.1 Nonrelativistic limits of the components of the energy-momentum tensor In proceeding to the non-relativistic limit, we express the energy density in the form: where e nmc 2 + is the internal energy density. (33) The component T 00 T 00 2 e + p p 2 nmc p p nmc p p (34) Astrophysical Gas Dynamics: Relativistic Gases 21/73

22 Now v / 2 1 c 2 v c v c 2 (35) So T v c p p c v c 2 (36) So we can see that the energy density in an arbitrary frame, breaks up into the rest mass energy density plus the kinetic energy density plus the internal energy density. Astrophysical Gas Dynamics: Relativistic Gases 22/73

23 The components T 0i T 0i e + p 2 v --- i nmc c p c 2 v i c p 1 2 c 2 v --- i c -- v c 2 vi --- c 2 v --- i c c 2v i --- c 1 v p 1 --v 2 2 v i + + p c c vi c v i --- c (37) Astrophysical Gas Dynamics: Relativistic Gases 23/73

24 Momentum density The components 1 --T c 0i v i 1 -- v 2 c - 2 v i + p v i Energy flux density ct 0i c 2 v i v 2 2 v i + + pv i T ij T ij e+ p 2 v i v j + p ij nmc p 2 v i v j + p ij c 2 c 2 v i v j 2 + p vi v j + p ij c 2 c 2 v i v j + p ij + p vi v j + Higher order in v (38) (39) Astrophysical Gas Dynamics: Relativistic Gases 24/73 c 2 c 2 c 2 c 2

25 The term nmv i This originates from the continuity equation for the number of particles. nmv i v i (40) 5.2 Non-relativistic equations Continuity equation The conservation equation for rest-mass nm t nmv i x i (41) Astrophysical Gas Dynamics: Relativistic Gases 25/73

26 is, without approximation: Momentum equation The relativistic form t v i x i 0 c 1 T i t T ij x j (42) (43) Astrophysical Gas Dynamics: Relativistic Gases 26/73

27 becomes (44) v i 1 -- v 2 c - 2 v i + p c 2 v i t v i v j + p ij + p vi v j c x j 0 (45) Astrophysical Gas Dynamics: Relativistic Gases 27/73

28 The leading terms are the usual non-relativistic terms, the others are of order v c 2 by comparison. Hence, the momentum equations become: v i t The energy equation The relativistic form is: T t v i v j + p ij x j x j ct 0j (46) (47) Astrophysical Gas Dynamics: Relativistic Gases 28/73

29 With the non-relativistic expressions: c v t c 2 v j v 2 2 v j + + pv j x j 0 (48) In this case we need to go beyond the zeroth order expression since this is c t c 2 v j x j 0 i.e. the continuity equation for rest mass energy density. (49) Astrophysical Gas Dynamics: Relativistic Gases 29/73

30 The next order in v c 2 gives: 1 --v t 1 --v 2 2 v j + + pv j x j 0 (50) which is the non-relativistic form of the energy equation. Note that both the momentum equation and the energy equation T 0i have involved the same term. It is the different contributions from terms of different orders in v c 2 which have given rise to the different contributions to both the energy and momentum equations. Astrophysical Gas Dynamics: Relativistic Gases 30/73

31 6 Equations of state 6.1 Ultrarelativistic particles and the distribution function Often when we have relativistic motion, the internal composition of the fluid is relativistic, as in the case of the M87 jet. Most extragalactic jets are made up of ultrarelativistic particles with Lorentz factors much greater than unity. In this case the energy of a particle is the speed of light times the momentum, i.e. as it is for photons. E cp (51) Astrophysical Gas Dynamics: Relativistic Gases 31/73

32 To derive the equation of state, we use a distribution function fp so that Number density of particles in volume d 3 p of momentum space (52) For the so-called perfect fluids we are dealing with here, the distribution function is isotropic, i.e. fp fpd 3 p fp and depends only only on the magnitude of the momentum. (53) Astrophysical Gas Dynamics: Relativistic Gases 32/73

33 6.2 The energy density and the pressure The energy density is e Ef p d3p 4 Ef p p 2 dp p 4c 0 0 fpp 3 dp (54) The pressure tensor (equivalently, the stress tensor in the rest frame) is T ij p i v j fpd 3 p p space 4 0 p i p j fp p m 2 dp (55) Astrophysical Gas Dynamics: Relativistic Gases 33/73

34 Here we take m m 0 to be the mass of the particle, to be the Lorentz factor and to be the rest-mass energy. The isotropy of the distribution function guarantees the isotropy of the stress tensor. Hence and the value of Now p m 0 T ij P ij can be estimated from the trace, i.e. 3P T i i 4 m 0 E ---- c 2 p -----f 2 pp m 2 dp (56) (57) (58) Astrophysical Gas Dynamics: Relativistic Gases 34/73 p -- c

35 Hence 0 3P 4c pfpp 2 dp P This is of course is also true for a photon gas where exact. (59) We have used the symbol P above, to distinguish he pressure from the momentum. Henceforth, we use p for pressure and the 1 ultra-relativistic equation of state is p --e e 3 e E cp is Astrophysical Gas Dynamics: Relativistic Gases 35/73

36 7 The relativistic fluid equations 7.1 The fluid equations in terms of the 4-velocity Let us now consider the fully relativistic equations derived from the divergence of the stress-energy tensor, i.e. T, 0 where T wu u + p (60) The divergence is: T, u wu, + wu u, + p, (61) (Note that p, p, ) Astrophysical Gas Dynamics: Relativistic Gases 36/73

37 This has a form similar to that of the non-relativistic equations. However, the term wu, is a bit anomalous. We deal with this as follows. Take the scalar product of this equation with Since u u u wu, + wu u u +u, p,. Then, (62) u u 1 (63) then u, u + u u, 2u u, 0 (64) Astrophysical Gas Dynamics: Relativistic Gases 37/73

38 Hence, i.e. wu, + u p, 0 wu, u p, (65) (66) The equations of motion of the fluid then become: wu u, + p, + u u p, 0 (67) Astrophysical Gas Dynamics: Relativistic Gases 38/73

39 u v Projection operator The terms involving the pressure can be written: q v p, + u u p, + u u p, (68) The tensor q + u u (69) is the projection operator. Every vector contracted with u othorgonal to. q is Astrophysical Gas Dynamics: Relativistic Gases 39/73

40 Final form of fluid equations Using the projection operator, we have, wu u, q p, (70) The left hand side represents the relativistic generalisation of the non-relativistic differentiation following the motion of the three velocity, v t j v i x j vi (71) The right hand side is the negative of the pressure gradient perpendicular to the 4 velocity. Since the scalar product of these u equations with now leads to an identity, the above four equations are equivalent to three. Astrophysical Gas Dynamics: Relativistic Gases 40/73

41 7.2 The entropy equation in relativistic fluid dynamics The second law of thermodynamics Consider a comoving volume of gas (i..e an elementary volume with a fixed number of baryons) in which Entropy per baryon n baryon density e p Energy density (inc. rest mass) pressure (72) T temperature in energy units Astrophysical Gas Dynamics: Relativistic Gases 41/73

42 The equation of state of all fluids, including relativistic fluids has other consequences that can be derived by consideration of the second law of thermodynamics. Consider the second law of thermodynamics in the form TdS du + pdv (73) where, T is the temperature (in energy units), S is the entropy, U is the internal energy and V the volume of a comoving element of fluid which contains a fixed number, N of baryons. This equations applies in the rest frame of the fluid. Why concentrate on baryons? In most fluids, except under the most extreme of conditions, the baryon number is conserved. For example, even when nuclear reactions are being considered, the number of baryons is conserved. Astrophysical Gas Dynamics: Relativistic Gases 42/73

43 Also, the rest-mass energy of such a volume is constant so that we can replace the internal energy with the total energy, E. This appropriate for a relativistic approach. The entropy equation then becomes TdS de + pdv In terms of the variables in the above diagram (74) S N E N energy per baryon N e n -- V N --- n (75) Astrophysical Gas Dynamics: Relativistic Gases 43/73

44 Hence: TdN dn e n -- + pd N --- n Td d e n -- + pd 1 n -- (76) The equivalent equation in non-relativistic fluid dynamics is Tds d -- + pd 1 -- (77) In this case s is the entropy per unit mass. Astrophysical Gas Dynamics: Relativistic Gases 44/73

45 Entropy and enthalpy We have Td d e n -- pd 1 n -- 1 e + p + --de dn n (78) We can also express this equation in terms of the differential of the enthalpy, viz. n de n w -----dn n 2 Td d e p e dp dn n n d w --- n dp n n 2 p -----dn n 2 e + p dn n 2 (79) Astrophysical Gas Dynamics: Relativistic Gases 45/73

46 The constancy of entropy along a streamline Let us return to the equation that we derived from the scalar product of the fluid equations with We can write this as u wu, u p,, i.e. (80) wu, + u w, u p, Now use the conservation of baryon density nu, u n, + nu, 0 u, 1 --u n n, (81) (82) Astrophysical Gas Dynamics: Relativistic Gases 46/73

47 Therefore, wu, + u w, u 1 w, --u n n, nu w --- n The operator u w --- n 1 --u n p, 0, u x d ---- ds, u p (83) (84) Astrophysical Gas Dynamics: Relativistic Gases 47/73

48 i.e. differentiation along a world line. Hence the above equation for the relativistic enthalpy is d ---- w ds dp n n ds Consider the equation for the differential of the entropy (85) Td d w --- n T d ds dp n i.e. entropy is constant along a streamline. d ---- w ds dp n n ds (86) Astrophysical Gas Dynamics: Relativistic Gases 48/73

49 8 The speed of sound 8.1 Sound waves from perturbation of the relativistic fluid equations The speed of sound in a relativistic gas is of interest. This is best derived directly from perturbation of T x 0 We take as the unperturbed state, (87) v i 0 p p 0 e e 0 w w 0 e 0 + p 0 (88) Astrophysical Gas Dynamics: Relativistic Gases 49/73

50 and the perturbed variables are v i p p 0 + p 1 e e 0 + e 1 v 1 i w w 0 + e 1 + p 1 We expand to first order in these quantities. The perturbation to the Lorentz factor is given by: v / v O 2 2 c 2 c 2 (89) (90) (91) Astrophysical Gas Dynamics: Relativistic Gases 50/73

51 The perturbations to the stress-energy tensor are give by: T 00 2 e + p p e 0 + e 1 T 0i T ij w vi --- w c 0 + w 1 v 1 i vi w c c 2 w vi v j + p 0 + p 1 ij p 0 + p 1 ij c 2 (92) Astrophysical Gas Dynamics: Relativistic Gases 51/73

52 The perturbation equations are, 0 T x 0 T 0i x i 1e 0 + e 1 v w j c c t x j 0 (93) e 1 v w j t x j 0 Astrophysical Gas Dynamics: Relativistic Gases 52/73

53 and i T i x 0 + T ij x j 1 -- c v w i c t p 0 + p x j ij (94) w 0 c vi p t x i 0 Hence the two perturbation equations are: e 1 v w j t x j 0 w vi p 1 c t x i 0 (95) Astrophysical Gas Dynamics: Relativistic Gases 53/73

54 Take the divergence of the second equation: From the first equation Therefore, w v i c t x i w 0 t 2 v i x i 2 p 1 + x i x i 0 2 e 1 t 2 p 1 1 e 1 x i x i t 2 c 2 2 (96) (97) (98) Astrophysical Gas Dynamics: Relativistic Gases 54/73

55 Now relate the perturbations to the pressure and energy density via the equation of state, which we represent in the form: Hence, p pe (99) p 1 p e e 1 + p e 1 (100) For adiabatic perturbations: p 1 p e e 1 (101) Astrophysical Gas Dynamics: Relativistic Gases 55/73

56 and the wave equation for the pressure perturbations becomes: 2 e 1 1 e 1 x i x i c 2 0 p e t 2 This represents sound waves travelling with a speed: 2 (102) c2 s c 2 p e (103) 8.2 Sound speed for an ultrarelativistic equation of state If p 1 --e 3 (104) Astrophysical Gas Dynamics: Relativistic Gases 56/73

57 then p e 1 -- c2 3 s 1 --c 3 2 c s c (105) This speed is quite fast, c s 0.577c. 9 Relativistic shocks 9.1 The junction conditions The junction conditions for relativistic shocks follow from the same approach as for non-relativistic fluid dynamics, i.e. conservation of rest-mass or number density, conservation of momentum and conservation of energy across the shock. Astrophysical Gas Dynamics: Relativistic Gases 57/73

58 In the following and in much of relativistic fluid dynamics, we use i Oblique relativistic shock y 1 2 v i --- c 0 1 x (106) The relativistic Rankine-Hugoniot relations are derived from continuity of number, momentum and energy across the shock normal. Number v - c 1 n 1 1x 2 n 2 2x (107) Astrophysical Gas Dynamics: Relativistic Gases 58/73

59 Momentum (x-component) T 1 xx T 2 xx w x + p 1 w x 2 + p 2 (108) Momentum (y-component) T 1 yx w x 1y T 2 yx w x 2y (109) Energy w x w x (110) Astrophysical Gas Dynamics: Relativistic Gases 59/73

60 Summary of relativistic junction conditions 1 n 1 1x 2 n 2 2x w x + p 1 w x w x 1y 2 + p 2 w x 2y (111) w x w x 9.2 Solution of the relativistic junction conditions Dividing the third equation by the fourth: 1y 2y y (112) i.e. the component of velocity normal to the shock is unchanged, as in non-relativistic shocks. Astrophysical Gas Dynamics: Relativistic Gases 60/73

61 Velocity Equations (2) and (4) can be solved to give the velocity on both sides of the shock in terms of the hydrodynamic variables. Parametrisation of the velocity and Lorentz factor x y 2 (113) Put x 1 2 y 12 / tanh y 1 tanh y sech 2 (114) 1 2 y 12 / cosh Astrophysical Gas Dynamics: Relativistic Gases 61/73

62 Since, y is the same for both sides of the shock, then, we have 1x 1 2 y 12 / tanh y 12 / cosh 1 We make these substitutions into (115) 2x 1 2 y 12 / tanh y 12 / cosh 2 w x + p 1 w x w x w x 2 + p 2 (116) Astrophysical Gas Dynamics: Relativistic Gases 62/73

63 to obtain w 1 sinh p 1 w 2 sinh p 2 w 1 sinh 1 cosh 1 w 2 sinh 2 cosh 2 (117) Velocities The solutions for the velocities are algebraically long but straightforward and are derived in the appendix: 1x 1 2 y 12 / tanh y 12 / p 2 p 1 e 2 + p e 2 e 1 e 1 + p 2 (118) 2x 1 2 y 12 / tanh y 12 / p 2 p 1 e 1 + p e 2 e 1 e 2 + p 1 12 / 12 / Astrophysical Gas Dynamics: Relativistic Gases 63/73

64 9.3 Ultrarelativistic equation of state For an ultrarelativistic equation of state, the above equations become: p 1 --e 3 1x 1 2 y 12 / 1 3p 2 + p p 1 + p 2 2x 1 2 y 12 / 1 3p + p p 2 + p 1 12 / 12 / (119) (120) Astrophysical Gas Dynamics: Relativistic Gases 64/73

65 9.4 Normal shocks Mainly for convenience, let us now concentrate on normal shocks in which y 0. There are a couple of interesting results here Product of velocities Weak shock In a weak shock p 1 p 2 1 1x 2x This implies that: (121) 1 1 1x x (122) Astrophysical Gas Dynamics: Relativistic Gases 65/73

66 i.e. the velocities, before and after the shock are equal to the sound speed. Strong shock The densities on either side of the shock are related by: Since ñ p 2 p 1 for a strong shock x 1 2x -- 3 n is the lab frame density then ñ 1 1x 1 n 1 1x 2 n 2 2x ñ 2 2x ñ ñ 3 1 (123) (124) (125) Astrophysical Gas Dynamics: Relativistic Gases 66/73

67 Important points The points to note here, are In a relativistic fluid, a weak shock can travel quite fast, essentially because the benchmark speed, the sound speed, is fast. The strong shock limit has quite different velocity and density solutions than that in a non-relativistic fluid. 10 Appendix 10.1 Solution of shock equations w 1 sinh p 1 w 2 sinh p 2 w 1 sinh 1 cosh 1 w 2 sinh 2 cosh 2 (126) Astrophysical Gas Dynamics: Relativistic Gases 67/73

68 Taking the parameters, and as given, we now have to w 12 p 12 solve for the two unknowns 1 and 2. Let x 1 cosh 2 1 x 2 cosh 2 2 of the above equations: w 1 x 1 1+ p 1 w 2 x 2 1+ p 2 w 2 1 x 1 1x 1 w 2 2 x 2 1x 2 and square the second (127) Astrophysical Gas Dynamics: Relativistic Gases 68/73

69 In the first of the above equations equations are: p w e w 1 x 1 e 1 w 2 x 2 e 2 w 2 1 x 1 1x 1 w 2 2 x 2 1x 2 so that these two (128) Astrophysical Gas Dynamics: Relativistic Gases 69/73

70 The aim of the following is use these two equations to obtain linear equations for x 1 and x 2. To do so we square the first and subtract from the second. w 2 1 x2 1 2e 1 w 1 x 1 + e2 1 w 2 2 x2 2 2e 2 w 2 x 2 + e2 2 w 2 1 x2 1 w 2 1 x 1 w 2 2 x2 2 w 2 2 x 2 Subtract w 1 2e 1 w 1 x 1 e2 1 w 2 2e 2 w 2x2 e2 2 (129) Astrophysical Gas Dynamics: Relativistic Gases 70/73

71 Use then w 1 2e 1 p 1 e 1 In matrix form: and the other previous linear equation, w 1 x 1 e 1 w 2 x 2 e 2 p 1 e 1 w 1 x 1 e2 1 p 2 e 2 w 2x2 e2 2 (130) w 1 w 2 e 1 p 1 w 1 e 2 p 2 w 2 x e 1 2 e 1 x 2 e2 2 e2 1 (131) Astrophysical Gas Dynamics: Relativistic Gases 71/73

72 The solutions are (using Maple/Mathematica) x 1 cosh 2 1 e 2 e 1 e 1 + p e 1 + p 1 p 1 + e 2 p 2 e 1 x 2 cosh 2 2 e 2 e 1 e 2 + p e 2 + p 2 p 1 + e 2 p 2 e 1 (132) Since tanh 2 1 sech x then tanh 2 p 2 p 1 e 2 + p e 2 e 1 e 1 + p 2 tanh 2 p 2 p 1 e 1 + p e 2 e 1 e 2 + p 1 (133) Astrophysical Gas Dynamics: Relativistic Gases 72/73

73 Velocities We now have the solutions for the velocities: 1x 1 2 y 12 / tanh y 12 / p 2 p 1 e 2 + p e 2 e 1 e 1 + p 2 (134) 2x 1 2 y 12 / tanh y 12 / p 2 p 1 e 1 + p e 2 e 1 e 2 + p 1 12 / 12 / Astrophysical Gas Dynamics: Relativistic Gases 73/73