Volume SlA,number
2
PHYSICS LETTERS
ELECI’ROHYDRODYNAMICS
10 February 197s
OF THE PULSAR MAGNETOSPHERE
H. HEINTZMANN Institut f?lr theoretische Physik der Um’versitci’tzu Kiiln, D5 Koln 41, West Germany
and W. KUNDT and J.P. L4SOTA* I. Znstitutfiir theoretische Physik der Universittit Humburg, 02 Hamburg 36, West Germuny Received 16 January 1975 The dispersion relations for weak waves in a cold, chargeseparated plasma (due to a strongrotating magneticf=ld) showthat radio waves, and even low frequencywavescan propagatethrougha (one+omponent)pulsar magnetosphere.
It is very likely [l] that within the velocity of light cylinder KZX rl = c (and probably even way beyond), a pulsar plasma shows nearly complete charge separation. Such a plasma differs from ordinary (neutral) plasmas in that it cannot be globally at rest, i.e. there does not exist an inertial rest frame, and electric fields play an important role. The wellknown MHD approach to electromagnetic wave propagation has therefore to be replaced by an electrohydrodynamic (EHD) treatment (with the proper conduction law). ‘Ihis letter reports on results obtained under the following simplifications: The plasma is assumed completely charge separated, cold, forcefree and effectively homogeneous. Forcefreeness (in the local rest system) implies 6 = (a X r/c) X B where B is the (strong but otherwise arbitrary) magnetic field, and 52 is the rotation frequency of the magnet. The magnetosphere is characterized by a charge density [2] p w n*B%nc, a plasma frequency op = (4nep/m)1/2 a 3 X lOlo B{~Ql(me/m)1~2 setl, and a Jxmor frequency 52, = eB/mc as2 X 10lg B&r,lm) set l where typical pulsar parameters were inserted. In order to derive the dispersion relation, we use Maxwell’s equations (in an inertial system) together with the conduction law j= PC
(1)
and the equations of motion m,(&+
VWYV =e(E+fXB),
(2)
7: = (1  r?/C+“/2,
linearize around the unperturbed state, and Fouriertransform. The main difference between MHD of a neutral plasma and EHD of a one component plasma lies in the relation between current and electric field: in EHD this relation is only indirect through the equation of motion, and results in a nonhermitean, wave vector dependent dielectric tensor. After Lorentztransforming to the rest system, or via straightforward calculation we fmd the following dispersion relation f&r2
 l)[r$l
 9>  u;  ;;2(?22  1)3&l
[email protected]
= V2[ 1 + z? {2(n2  1)  r&2} + ;;4(?22  1) {?r2 1  2912)  z&r2
 l)V_12]
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* On leavefrom Institute of Astronomy of the Polish Academy of Sciences. 105
PHYSICS LETTERS
Volume 51 A, number 2
10 February 1975
is;:=
or: = W&9 Gj: = w/or, wlwp, VI: = 1 n*fiI, p: = v/c, v: = 1 n$, and with the indices II,1 referring to the&direction. Letting n + 00, 0 respectively we find no resonances, and three cutoffs at the approximate angular frequencies aL, wp , or (for w,/SZL 4 1). Remarkable is the low cutoff at wr = S? which in a way replaces the plasma frequency in the dispersion relation for a neutral plasma. More specifically, eq. (3) simplifies for not too high frequency, small wave number waves G2 In2  1i 4 1) to
with n: = ck/w,
ij$?
 I)[$(1

p,“> I$
= V2
(4)
where v, vI, and (1  @) tend to 1 for nonrelativistic plasma velocities (p + 0). In this limit the dispersion relation can be solved exactly: w2 = f [#(
1 + cos%) + 0; f (c4k4 sin%3t2cVw;(
1tcos%)tw~)1’2]
(5)
with 0: = angle between R and B. The corresponding group velocity va = Vkw, and the fact that electric and mag netic amplitudes are comparable in magnitude show that within the velocityoflight cylinder of an arbitrarily inclined magnetic rotator, electromagnetic waves of frequency 2 wr can propagate in all directions. This result strongly modifies existing pulsar theories if charge separation is a valid assumption, and strengthens the earlier proposal by Gunn and Ostriker [3] that electromagnetic waves at the rotation frequency would be emitted by an oblique rotator which could boost the relativistic electrons needed to illmte the Crab nebula [4]. Of course, a cogent treatment would have to allow for both the inhomogeneity of the magnetosphere on the scale of their wavelength [S] , and their probably large amplitude. The two polarization states of radio waves propagating along magnetic field lines can interfere to show Faraday rotation. Their differential rotation [6] of polarization is found from eq. (3) to be of order
(6) for 1n # I 4 1, “w < 1, The differential group (time) delay [6] for such waves is l
w& [@‘3(g) ‘p:]. Both expressions are much smaller than the corresponding interstellar contributions.
[l] [2]
[3] [4] [5] [6]
106
L.G. KuoPetravic,M. Petravic, K.V. Roberta, Phys. Rev. Letters 32 (1974) 1019; L., Meatel,Astrophys. Space Sci. 24 (1973) 289. P. Goldreich, W.H. Julien, Ap.J. 157 (1969) 869; L. Mestel,Nature Phys. Sci. 233 (1971) 149; F.C. Michel,Astrophys. J. (1974) to appear. J.E. Gunn, J.P. Gstriher, Ap.J. 157 (1969) 1395. R.M. Kulsrud, J.P. Gstriher, J.E. Gunn, Phys. Rev. Lett. 28 (1972) 636. K.G. Budden, Radio wavesin the ionospherech. 21 (CambridgeUniv. Press, Cambridge1966). D. ter Haar, Physics Letters 3C (1972) 57, p. 71 eq. (5) and p. 112, eq. (66).
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