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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
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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":
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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
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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.
Representative terms from entire chapter:
field line
