Characteristics of plerions powered by pulsars

Adv. Space Res. Vol. 9, No. 12. pp. (12)87—(12)90. 1989 Printed in Great Britain. Alt rights reserved.

0273—1177/89 $0.00 + .50 Copyrtght © 1989 COSPAR

CHARACTERISTICS OF PLERIONS POWERED BY PULSARS R. Cowsik Tata Institute of Fundamental Research, Bombay 400005, India and McDonnell Center for the Space Sciences, Physics Department, Washington University, St. Louis, MO 63130, U.S.A.

ABSTRACT

It is generally believed that energy flows from a pulsar initially as magnetic dipole radiation and then largely as a wind of relativistic particles and magnetic field which fill Out an expanding spherical volume around the pulsar. We calculate here the nature and evolution of the non—thermal radiation from such a plerion, giving adequate attention to the betatron effect on the electrons induced by the non—adiabatic behaviour of the magnetic field. These calculations are of interest in the modelling of supernova remnants (SNR) like the Crab—nebula, in discussing pulsar—SNR associations and in predicting the observational features expected if a pulsar is embedded inside the remnant of SN1987A when the debris becomes transparent. INTRODUCTION The observations of pulsars and the related astrophysics has been reviewed extensively by Manchester and Taylor /1/ and by Radhakrishnan /2/. It was irmediately clear after the discovery that a rotating neutron star with intense magnetic fields will accelerate particles very efficiently /3—6/ and would thereby control the nature of electromagnetic radiations from the nebula associated with it /7/. It was soon realised by Pacini and Salvati /8/ and by Rees and Gunn /9/ that besides the relativistic particles, the pulsar could also be the source of the magnetic fields of the remnant, the field lines having been stretched out during the rotation of the neutron star. It is observed that the birth rate of pulsars and supernova rate in our galaxy are comparable, a few per century and this fact lends support to the idea that neutron stars are born in supernova explosions /1,2,10—12/. Therefore it is quite surprising that even though some 500 pulsars and 150 SNRs are known there are only four known cases where the pulsars reside inside the remnant. Beaming and other selection effects have been suggested as the possible cause of the low rate of association /13/ or that the neutron stars are rotating rather slowly and/or with very weak magnetic fields so that they fail to function as pulsars /14,15/. The reduction in the associations due to beaming could be overcome by searching for the effects of injection of relativistic particles and magnetic fields into the nebula, as these would manifest themselves as enhanced synchrotron emission from the remnant. By such studies one could get information regarding the magnetic fields and spin frequencies of neutron stars at their birth ,14,16/ and it has been suggested that the absence of plerions inside most supernova remnants is because the pulsars are born rotating very slowly. To follow up on these suggestions, it is necessary to model carefully the evolution of the plerion generated by a pulsar and calculate the radiations emitted from the plerion due to the synchrotron and synchro—compton processes. Such a study would also be useful in estimating the non—thermal emissions, especially in the X—ray and the gaimna—ray bands from the SN1987A whose debris would become transparent to these radiations as the debris expands in the next few years. In this paper, we present a calculation of the evolution of the spectrum of plerion created by a pulsar, we include the effects of the betatron effects electrons, induced by the non—adiabatic behaviour of the magnetic field and the effects of breaks in the spectrum of electrons injected by the pulsar. ment of the synchrotron and the synchro—compton spectra and their relevance studies will be presented elsewhere.

(12)87

electrons in the on the energy of also explicitly Detailed assessto observational

(12)88

R. Cowsik

THE MODEL

The model presented here would be applicable between 1 year and 100 years after the supernova explosion during which time one can reasonably assume that the expansion velocity, v, of the debris and the luminosity L of the pulsar remain sensibly constant. As discussed earlier by Pacini and Salvati /8/ the magnetic field, B, inside the remnant would have been built up rapidly, within a small fraction of a year, being fed by a fraction, h 0.1, of the total luminosity of the pulsar. The strength of the magnetic field for times, t > 1 yr is given by 3Lh 1/2 B(t) = —i— (1) rv where r is the radius of the remnant at time, t, viz. r=r+vt

(2)

with obvious notation. In our discussions, here, we neglect any contributions the pulsar might have made before t 1 yr 3.1 x iü~s, corresponding to a radius r 1 vt 1016 cm. For a more complete discussion this contribution has to be included. 2 expected the are flux to were to be conserved; the flux that Two aspects of the magnetic iffield be noted; first, thesecondly, field behaves as Btubes — are stretched from the rotating neutron star surface elongate and the same time instead of as out r ‘random—walk’ to fill the nebular volume. These two aspects of the field have important effects on the energy of the electrons in the remnant. The two adiabatic irtvariants of charged particle motion in magnetic fields yield dp 11 i’ll dp~ 2 and Setting E

p~ 2/3 E2 1+

dlnr dt

~.dlnB

4

dt2 dt () 2 we and get averaging the above equations over pitch angles to get ’~ +E dE_E(ldlnrldlnB) dt 3 dt 3 dt

(5)

Note that in the standard case with flux conservation equation (5) reduces to the oft used expression dE dt

Ed~r dt

(6)

However, with pulsar feeding in the fields at a constant rate we have dE 2Ed~r dt ~3 dt Now including the radiative energy losses due to synchrotron emission we can write

(7)

dE

dlnr 22 dt —bEB (8) The notation in the above equation is standard, other than that we have written x in place of 2/3 so that the familiar expressions presented in earlier literature can be obtained by setting x 1. Eqn. (8) can be solved in a straightforward way to get E(r 1) B E~ E (r/ri)x ~

3LhbrxE {r~~ v (1+x)

-

r_(1~~

.

(9)

—

Now, the spectrum of electrons injected by the pulsar into the nebula needs to be specified. Taking the cue from the radio spectra of plerions like the Crab—nebula /17/, one expects that the low energy part of the spectrum is rather flat; ste assume the injection to be S(E,t)

=

s

E~

for t < and E < E < E , with 8 5 2. During the first is roughly constLt, inde~endentof time and the focus is with such a flat spectral index, it is necessary that the beyond some energy Ex~ to keep the luminosity finite. We steepens as S(E,t) = S0 ~ 2~’ for E

>

E~. Again with crab—nebula as a reference E~ —

(10) 100 years or so the injection rate on this phase in this paper. Also injection spectrum should steepen assume that beyond E~ the spectrum (11) GeV.

Characteristics of Plerions Powered by Pulsars

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CALCULATION OF THE SPECTRA

The model has been specified adequately to calculate all the spectra explicitly. The spectrum of electrons resident in the nebula at a time when the radius is r is given by F(E,r)

f

=

dEi Here

S(E)

—~

(12)

.

i mm is the Jacobian given by

-~-

=

dE

rL~

—2

f

3LhbrXE —(1+x) v2 (1+x) ~ri

—

—(1+x)

—

r

(13)

and r~ in is the larger of r

1 and rmin obtained by inverting equation (9) in which E. is 1

set toLco

1

+ 3Lhb ~2 ~v (1+x)Y(1+x) (14) mm ~E r v2(1+x)j3LbhrX_I r Remembering that S(E) has a simple power law behav~.ourbelow and above Ex the integral in eq. 13 can be written explicitly as For E < E < Ex F = (1-8)~ 3LhbrXE {r(1+x) - r1~~ (82)

H

~

xE

[

v(1+x)

j

v (y—2)

+

r~ rmin SE~8 E~

r ri

L

(1—y)x ~

3LhbrxE v2(l+x) {r_(1+x) ~

—

]

r_(1~~l

drV (15 a)

Here r

is the larger of r 1 and r~obtained from inverting eq. 9 with E.

E

>

E

x

F

=

mm

S E y8 (l—y)x 1 ri 0 X ~r] E

rL

1

v2(1+x) ~ 3Lhbr E {r_(1~

—

r_(1~~}j

~

=

E.

v

For

(15 b)

Thus the magnetic field and the spectra of the electrons are completely specified as a function of the radius of the nebula and this allows the calculation of the synchrotron and synchro—compton spectra in a straight forward manner. RESULTS AND CONCLUSIONS Detailed numerical work for a broad range of the pulsar and nebular parameters would be presented elsewhere. We note here that the integrals given in eq. 15 become simple when one chooses 8 = 2 and y = 3, not unlikely for a typical case. For example, at very high energies with E >> (v2(1+x)r}/3Lhb the spectrum of electrons is given by S F

=

v2 E 3Lhb X

2(1—x) ~y+1

(16)

1. The synchrotron luminosity increases rapidly for r < r reaches a broad maximum and decreases for r >> r 05). ~emembering that x = 1 when the betatron effect is neglected x = 2/3 with it, the synchrotron luminosity falls more 1 as Lsynand r(x+0.5)(1_~)r(~ gently in the latter case. This slower decrease of the luminosity is more pronounced for the synchro—compton radiation, i.e. synchrotron radiation rescattered by the electrons. 2. Other than at the lowest energies all the spectra show curvature, because of the effect ofthe lower limit in the integrals given in eq. 15; simple power law approximations to the spectra, especially at high energies are inadequate for their description, even though asymptotically one does obtain a power law. 3. One expects an intense X—ray emission due to synchro—compton emission when the debris in SN1987A becomes transparent by - 1990; the subsequent decrease in the intensity would be as — r3~8 in case there is an active pulsar inside the remnant /18/. REFE3.ENCES 1.

R.N. Manchester and J.H. Taylor, Pulsars, Freeman, San Francisco, (1977).

2.

V. Radhakrishnan, Contemp. Phys. 23, 207 (1982).

JASS 9:12-C

(12)90

R. Cowsik

3.

J.E. Gunnand J.P. Ostriker, Ap. J., 157, 1395 (1969).

4.

P. Goldreich and W.H. Julian,

5.

T. Gold, Nature, 218, 215 (1968).

6.

L. Woltjer, A.R.A.Ap., 10, 129 (1972).

7.

R. Cowsik, Y. Pal, T.N. Rengarajan, Ap. & Sp. Sci., 6, 390 (1970); also Proc. ICRC, Budapest, (1969).

8.

F. Pacini and M. Salvati, Ap. J., 186, 249 (1973).

9.

M.J. Rees and J.E. Gunn, M.N.R.A.S.,

Ap. J.,

157, 869 (1969).

167, 1 (1974).

10.

D.H. Clark and F.R. Stephenson, M.N.R.A.S., 179, 47 (1977).

11.

G.A. Tamann, Supernovae, p. 371, Ed. Rees & Stonehani (1982).

12.

A. Blauw, Birth & Evolution of Massive Stars & Stellar Groups, p. 211, Eds. Boland et al., Reidel Dordrecht (1985).

13.

A.G. Lyne, R.N. Manchester and J.H. Taylor, M.N.R.A.S., 213, 613 (1985).

14.

V. Radhakrishnan and G. Srinivasan, J. Ap. Astr., 1, 25 (1980).

15.

R. Narayan and K.J. Schaudi, Ap. J., 325, L47 (1988).

16.

D. Bhattacharyya, Ph.D. Thesis, Indian Inst. of Sci., Bangalore, (1987).

17.

K. Davidson and R.A. Feson, Ann. Rev. Astron. Astrophysics., 23, 119 (1985).

18.

F.D. Seward and Z. Wang, (C. of. A. preprint No. 2668, 1988) (To appear in Ap. J., September 1, 1988).