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On Two-Dimensional Relativistic Stellar Winds M.E. GEDALIN, J.G. LOMINADZE, AND E.G. TSIKARISHVIL! Abastumani Astrophysical Observatory Stellar wind is of great interest, because many of the astrophysical systems, besides stars, possess wind-like structures. For a long time only nonrelatmstic winds have been investigated. However, recently (Kennel et al. 1983; Kennel and Coroniti 1984; Kennel et al. 1984) it was proposed that the relati~sitic pulsar Ad with the plasma, consisting of electrons and positrons, can be responsible for the observed features of the Crab nebula. The study in Kennel e! al. (1983) has revealed the inconsistency of the assumption of the wind zero temperature with the observational data. It has been shown that only for high relatn~sitic temperatures can high Mach numbers be reached, which allows the possibility of a shock formation. However, the analysis of Kennel et al. (1983) is rather incomplete, since it ignores the essentially three~imensional wind structure and deals with a one-dimensional object. It means in spherical coordinates only b/br = 0 and ~ = ~r/2 In this case, all connections between neighboring field lines are lost and only one is considered. The essentially three-dimensional structure requires a smooth transition from one field line to another, and thus a distortion of the "monopole solution" arises so that the last can be only an approximation at large distances. An attempt to investigate a two-dimensional wind structure has been made in Okamoto (19783. However, temperature is taken at zero and parameter variations across field lines are not considered. In the present paper we extend the analysis of Kemel, e! al. (1983) into the two-dimensional case 0~¢ = 0, d/~8 ~ 0, bIbr ~ 0. As it was proposed in Okamoto (1978) and Kennel e! al. (1983) and confirmed in Kennel et al. (1988), we use conventional MHD equations for a relativistic plasma with an isotropic relativistic temperature. The state equation is assumed pol~rtropic. One can readily see that plasma is always cold at r 108
HIGH-ENERGY ASTROPHYSICS 109 ~ oo, so that one should rather say that the temperature is high at the Ejection pout. ~ be mathematically formal we introduce curvilinear orthogonal co- ordinates ((, ,7, C) with the metric dl2 = h:2~2 + h2d2 ~ h2 d2 where ~ is the coordinate along the poloidal magnetic field line and ?? numbers field lines. When ~ ~ oo: ~ (, ~ ~ 8. With the new coordinates one can derive the following set of constants of motion along the field line: Bh,'h<~, = fat Arty nuh,7h~ = f2~?7) he (u~B-ABE) = ~ f3 (~) f27p-f3hyB~/4~ = few) f2puy ho-fit ho B~/4~ = fig (~) . (1) The equation for the transverse variations of parameters looks as follows: nh2h2 0~ = (npu2h2 /2) a~ 02 _ (npu2 h2 /2) ~ he = (~/~e (h~h~f3/hg)2-(h2V/8~) o_b2he2-(hV/8~) ~ B2h2 (2) ' C777 ~r ad It does not give a constant of motion, but a constraint on the proceeding set. Physically the constraint arises because of the distortion of the field lines due to the interaction with the neighboring ones. It is essential and describes the global features of the wind structure, since it cannot be one-dimensional. One could assume that (2) could be ignored when the equatorial plane is considered (cf. Kennel et al. 1983~. We show below, that even if this is the case, (23 defines a new "global" parameter. The set (1) can be used in the same manner, as it has been used in 1 to derive the 'mind algebraic equation":
110 AMERICAN AND SOVIET PERSPECTIVES N2 = K2(_B2 ~ [S2(EY-H)2 _ (qY-HS2)]2)/S2Y2(Y ~ 52 _ 1)2) (3) Y = KXoo(l+ or N,-1)/N where Y = M2 (where M is the Alfven~c Mach number), S = Qhp (where Q is angular velocity), and Q. K, E, H. Q. XCO, or are constants along the field line (cf. [1~. The analysis is straightforward and the results of Kennel et al. (1983) can be easily rederived. The system posesses critical points of three ~es: slow, intermediate, and fast. The only was to have MF > 1 when ~ ~ x is to pass through the fast critical point of to have initially ME > 1 at the injection point. We will not discuss the topological features of the solutions of (3~. The reader is referred lo Kennel et al. (1983) where he should ignore all the points where B ~ r~2 is assumed. Here we bneDy discuss the most interesting asymptotics s2 ~ oo (open field lines). One can easily see that there are no solutions with y/s2 ~ 0. It can be also shown that if y/s2 ~ x, then BS2 - ~ 0. However, there must be B ~ 1/S2 on the open field lines. Therefore, the only possibility is YIS2 ~ C2 = COnSL One can obtain ~ , Ec2/(c2 ~ 1) ,u*~*/m, it* - initial specific enthalpy, SU<p ~ (qc2 - H)/(c2 ~ 1) , Us' ~ 1/S, SBs, , -E/(c2 + 1) ~ B<p ~1/S, U ~ U<' = const Ail. Since we are mostly interested in the wind structure at large distances, we may assume ~ ~ r + 0~1), ,7 ~ ~ + 0~1/4r). One can approximately solve (3) for this conditions to have Bohr , 1~), 12/72 = coot. (4) This ratio does not depend on 8. It is easy lo obtain the important relation QZ/Xoo =p,z = c2 (5) where now p is constant not only on a particular field line, but for the whole two-dimensional structure. It is a "global" parameter. Thus, one should replace one of the parameters in (1) with the global constant of motion p. The asymptotic velocity is then determined by p: U00 = p~/Q, ~ = BS2, S26 ~
HIGH-ENERGY ASTROPHYSICS 111 Thus, the asymptotic energy is completely determined by the magnetic flux, angular velocity, and p. One can see mat for a given p cob one solution exists for r ~ oo. It means that the two solutions of Kennel et al. (1983) with MF > 1 and MF < 1, cannot coexist: only one family is consistent with the given p. As a conclusion we state that the two-dimensional relativistic winds with MF > 1 at r ~ oo can exist, when the plasma is relativistically hot at the injection point In this case the structure is defined by the set (1) of constants of motion along the field line. Additional constraint arises from the transverse continuity requirement, which gives life to a global parameter, that is constant for the whole structure. The number of asymptotic solutions is reduced to one. REFERENCES Kennel, C.F., and F.V. Coroniti. 1984. Ap. J. 283: 694. Kennel, OF., and F.V. Coroniti. 1984. Ap. J. 283: 710. Kennel, C.F., F.S. Fujimura, and I. Okamoto. 1983. Fluid Dynamics. Geophys. Astrophys. 26: 147. Kennel, CF., M.E. Gedalin, and J.G. Lominadze. 1988. Prod Joint Varenna-Abastumani Int. School and Workshop Plasma Astrophys p 137. Okamoto, I. 1978. MNRAS. 185: 69.