IL NUOVO CIMENTO VOL. 106 B, N. 9 Settembre 1991

Can Young Pulsars Clarify the Odd Behaviour of Pulsars? (*).

M. AKBAR and A. QADIR

Mathematics Department, Quaid-i-Azam University - Islamabad, Pakistan

(ricevuto il 14 Gennaio 1990; approvato il 14 Gennaio 1991)

Summary. - - Though the basic model of the pulsar as a rotating neutron star is well established, its details may not be as clear as is often assumed. The pulsar drift, the value of the braking index and its relation to the age of the pulsar need to be explained. Suggestions for explaining these properties which are not in entire agreement with the standard model have been proposed. One such proposal includes a high initial pulsar frequency. With the millisecond pulsar expected in the supernova of 1987 it is possible that the proposal merits a more detailed investigation. Could young pulsars provide, more generally, tests for the theories of pulsar formation, evolution and radiation? We discuss here the possibility of testing it against the standard model.

PACS. 97.60.Gb - Pulsars.

1. - Introduction.

It is very difficult to really know what goes on in a star in general and in a pulsar in particular. Our ,,knowledge,) is actually only an elaborate conjecture based on a very meagre body of ~facts,, gleaned from observing electromagnetic radiation in a few frequency bands. As such the general problem of distinguishing between the theoretical construct, or model, and the observationally known facts is greatly compounded in Astrophysics. Nevertheless, for the pragmatic astrophysicist, it is clear that the ,,standard modeb, of pulsars is valid in its broad features. The standard model has many refinements, however, which, we would like to argue, are by no means so reliable. Since all refinements of the model were designed to explain the known facts, no real test between them could be provided by the older pulsars. However, with the birth of a pulsar in Supernova 1987A fresh opportunities for tests arise. It is important that the predictions of various refinements of the model be available before the data begin to arrive so that the ,,tests), are not reduced to mere explanations.

The ,,broad features, mentioned earlier are that the pulsar is a rotating neutron

(*) The authors of this paper have agreed to not receive the proofs for correction.

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1012 M. AKBAR and A. QADIR

star possessing a high magnetic field whose axis is inclined to the rotational axis. Rotational energy is converted into radiant energy through some ,,energy generation,, mechanism. This radiation is beamed along the line of intersection of the magnetic and rotational equatorial planes. The beam sweeps past the Ear th like a light-house beam so that the pulsar period is simply the rotational period of the neutron star.

Earlier it was believed that [1] the radiation was purely dipole in nature. In that case, ~o being the pulsar frequency and & its derivative,

(1) ~ = ~o2,

where a is a constant for a given pulsar, related to its moment of inertia. It was also noticed that extrapolating backwards the rate of slowing of the pulsar could provide an estimate for the age of the pulsar

T ~op

0

where T is the present age, Up is the present pulsar frequency and the initial pulsar frequency is taken to be much greater than the present value. The braking index

(3) n = ~ ~/~2,

should be 3 for purely dipole radiation. There are some problems with the early expectations. For the Crab pulsar n =

= 2.515 + 0.005 (as of 1969, when the age was 915 years). Also &p ~ 3.9.10 -l~ Hz/s and T ~ 2.9. 101~ Equations (1) and (2) taken together give T ~ 3.9. 101~ (about 1240 years). More generally, there was no explanation provided for the phenomenon of pulsar drift[2] (pulsars having a drift velocity ~ 10Zkm/s relative to their parent supernova renmants). It has been shown [3] that the low value of n cannot be accounted for by higher multiple terms in the radiation field.

The standard explanation[4] for the low value of n is that eq. (1) should be replaced by

(4) ~ = ao~ ~ ,

so that the correct value of n follows automatically. Also, putting in a finite initial pulsar frequency, oJi, eq. (2) can be used to determine wi instead of predicting T. This gives

(5) ~oi/~p = [1 + (n - 1) &p T/~p] 1/(1-n) .

For the Crab pulsar this ratio is about 1.715. This explanation is unsatisfactory, according to us, for the following reasons. Instead of predicting, it only ,,explains,,. It does not provide an explanation for the pulsar drift. Also, considering the initial temperature at the birth of the pulsar, the observed pulsars have an anomalously low temperature, given the energy loss rate implicit in eq. (4).

According to the standard view, pulsars cannot have an initial frequency greater than about 100 Hz as they would break up due to centrifugal forces. This is not withstanding the observed ,,millisecond pulsar,, with frequencies of 108Hz. These

CAN YOUNG PULSARS CLARIFY THE ODD BEHAVIOUR OF PULSARS ? 1013

pulsars have been seen to have o, ~ 0 and are explained as neutron stars which are accreting matter. Thus they ,,spin up,~ due to accretion as much as they <~spin down, due to radiation. However, using a formalism similar to the pseudo-Newtonian formalism [5], Abramowicz has shown [6] that the , , r e l a t i v i s t i c centrifugal force,~ need not always to be repulsive. In fact in the regime relevant for the pulsar it will be a t t r a c t i v e and will hold the pulsar together instead of causing it to break up.

According to the standard explanation each pulsar has its own initial frequency which could be very different from that of any other pulsar. The value of n, however, is bet ter fixed. It is argued that the low value of n is due to slippage between the crust and superfluid core of the pulsar. It is, therefore, a slightly modified dipole radiation and n should be about 2.5 for all pulsars. If the rate of slowing down and its rate of change are fLxed, it is clear that the ratio of the frequency at any time to the initial frequency should be more or less the same. In 1000 years the ratio should be 1/1.7 or so. For the Crab pulsar the average rate of change of the pulsar frequency would then be

(6) = (oJp - o J i ) / T ~ - 0.715 O~p/T ~- - 7.5 x 10-1~ H z / s .

Using eqs. (4) and (5) the initial rate of change of the pulsar period is

(7) r ---- ( ~ r176 ~p ~ - - 1 . 5 X 10-9 Hz/s.

2. - A n o t h e r m o d e l .

An alternative suggestion [7] was that the low value n was due to ohmic radiation caused by plasma heating in the rotating magnetic field. In effect the pulsar acts like a dynamo generator with the plasma playing the role of a resistance. Here the energy generation mechanism is provided by the plasma. The radiation field could then be expressed in terms of a multipole expansion with an ,<ohmic>, term added. A natural assumption for the ohmic term gives

(8) & = ao~ + bo~ ~ + coJ ~ + ....

The coefficients in this expansion would be slowly varying functions of time (compared with the variation of o~). Over a short duration they could be regarded as constants. Taking a relatively high initial pulsar frequency these constants could be estimated by fitting with the age estimation and the observed value of n. The high initial pulsar frequency was predicted[8], using angular-momentum considerations, to be about 2-103Hz. This value of oJ i opened up[9] an intriguing relativistic explanation of the pulsar drift. For the current values of the Crab pulsar four terms in the expansion were required.

A major difference between the two suggestions is that the latter envisages all pulsars as essentially different evolutionary stages of more or less the same object while the former allowed for much greater variation between pulsars at birth. We would regard this as an advantage for the latter suggestion as it is more predictive. Since there is a higher rate of change of the pulsar frequency, at the initial stages, in the latter model there will be a greater rate of energy loss and hence greater rate of cooling than is allowed by the standard model. This, again, is a point in favour of the alternative suggestion. What is required

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is a decisive test between the two views. can provide the required test.

M, AKBAR and A. QADIR

It may be that the 1987A pulsar

3. - T e s t i n g t h e m o d e l s .

In one sense the brief (7 h) observations of the 1987A pulsar already favour the alternative model over the standard model. The pulsar period was seen to be about 0.5ms, or a pulsar frequency of about 2-103Hz, more or less exactly as predicted! However, it is possible that this confirmation is just a freak coincidence, or that only part of the model is verified but not the rest of it. t t is necessary to provide additional predictions to distinguish between the two models observationaily.

Our test must provide predictions on the basis of both models so that a comparison can be made. A problem in this programme is that the standard model does not give an a priori slowing rate for the pulsar. It did not even, as it stood, allow for such a short pulsar period. Estimates based on the standard model will not really be (,predictions,, but ,,expectations,, based on an extrapolation of the explanation for one of the older pulsars, say the Crab, in view of the quantity and quality of the data available for it.

Assuming that the standard model would lead to the expectation that the pulsar frequency drops by a factor of less than 2 in 100 years, taking ~i ~ 2-108 Hz gives O~p ~> 108 Hz. Then eq. (6) gives - ~ <~ 3.2.10 -8 Hz /s . Taking n -~ 2.5 and using eq. (7) we have -&i ~< 5.4 .10-SHz/s . On the other hand, the alternative model predicts that in 1000 y e a r s t h e new pulsar will have a frequency comparable to that of the Crab. This gives - & ~ 6.25-10-SHz/s. We are unable to predict &i on the basis of eq. (8) as the coefficients a, b, c, ... are not genuinely constants. As such the best fit for these coefficients for the Crab pulsar cannot be taken as they stand for the new pulsar. Though they are slowly varying after 100 years they would not have been varying so slowly in the early stages of the evolution of the pulsar. Further, the changes would be cumulative. Though we can not give a definite value of ~ it is clear that -&i >> - ~ and we can reasonably put -&i > 7 .10-SHz/s (about 2 Hz/year). Thus, when steady observations of the new pulsar can be recommenced, ff its slowing rate can be measured to an accuracy of even 10% of the mean value we should be able to distinguish between the models.

4. - R e f i n e m e n t s o f the a l t e r n a t i v e m o d e l .

I f the alternative, nonstandard model is verified by observation it will be necessary to consider how to extract and interpret the data obtained. In this model it has only a minor dependence on the internal structure of the pulsar (through the magnetic field) and mainly depends on the plasma surrounding the pulsar. As such the information obtained will essentially be of the plasma. This information would have to be decoded from fitting the data with eq. (8).

Given any data a fit could always be obtained by taking enough terms in the expansion in eq. (8), even with the requirement that each term must be separately negative. It is conceivable that a finite expansion would continue to fit the data forever, but this seems unlikely. Clearly, just increasing the number of terms in the expansion to maintain a fit with the data would be an exercise in futility. More relevant would be to limit the number of terms and see how the coefficients in the

CAN YOUNG PULSARS CLARIFY THE ODD B E H A V I O U R OF P U L S A R S ? 1015

expansion vary with time and how good the <<slow variation,> assumption is. To compare with the Crab pulsar it is necessary to take a four-term expansion and see ff extrapolating the time variation of the coefficients over a period of 915 years gives a result similar to that for the Crab pulsar. For this purpose it is necessary to work out the coefficients for a four-term expansion for the best fit to the data in a reasonable amount of observational time.

To be able to work out the four coefficients in eq. (8) we need to observe the pulsar frequency for at least five distinct times. For simplicity we take the intervals between two consecutive times to be the same, say St. Writing the pulsar frequency observed at the j 4 h time as coj, the coefficients are given by

(9)

where

f a = sJktmWjA~ A~ A ~ / A , b -- ~Jk'~WjAkA~ A ~ / A ,

c = dk~'~WjAkn~A~/zi,

d = dk~mWjAkn3 A~ / z l ,

(lo)

A = (oJj+l + ~ j ) /2 ,

W --- (0)3"+ 1 - oJj)/~t, ,~ = ~#tmAjA~ A~ A ~ .

Since the initial rate of decrease of the pulsar frequency of the new pulsar is expected to be about 2 Hz/year, a period of 5 years would be required to obtain the coefficients if the accuracy of observation of the pulsar frequency is 10 -8. If the accuracy can be improved by an order of magnitude the observational period could be reduced to half a year. By repeating the procedure after some time we could t ry to measure the time variation of the coefficients.

5. - S u m m a r y a n d c o n c l u s i o n .

The Gold model of the pulsar as a fast-rotating neutron star with a high-magnetic field is entirely satisfactory as far as it goes. The finer details of the observations are explained by the standard model as being due to the internal structure of the neutron star. It also assumes a fairly gradual reduction of the pulsar frequency from its initial value. Problems of explaining the energy generation mechanism, the high cooling rate and pulsar drift persist here. An alternative to the standard model, within the framework of the Gold model, had been proposed and a typical pulsar frequency of 2130 Hz predicted on the basis of angular momentum considerations. (This value could be expected to vary by 10 to 20% from one pulsar to another). The pulsar formed by Supernova 1987A has provided one verification of one part of the alternative proposal.

The major difference between the two models is that the standard model allows wide variations of the initial pulsar frequency to account for the variations between the observed pulsar frequency. The alternative model takes a much more uniform initial pulsar frequency and requires that the variation in pulsar frequency arise from

1016 M. AKBAR and A. QADIR

the fact that they are at different evolutionary stages. As such the latter model is more predictive. Another significant difference between them is that the standard model puts the structure inside the neutron star while the alternative model puts it outside, in the plasma. (This plasma provides the energy generation mechanism, essentially like a dynamo.) It also provides a possible explanation for pulsar drift and is more consistant with the observed cooling rate. In this model the data regarding the intensity and spectrum of the radiation and the pulsar frequency and its derivatives gives information about the plasma, the magnetic field in which it is trapped and the rotational rate of the pulsar but not about other aspects of the neutron star. We would argue that this is a more realistic view.

Predictions of the rate of decrease of the pulsar frequency are significantly different in the two models. In fact, even in the early years an accuracy of 10% would be adequate to distinguish between them. Further , since the effects are cumulative, over a period of ten years or so the differences would be dramatic.

If the observations favour the alternative model it would be worthlike to see how much information regarding the details of the energy generation mechanism could be gleaned at a relatively early stage. For this purpose we would need to fit up to four terms (say) of the expansion in eq. (8) and see to what extent the assumption of (,slow variation, of the coefficients is good.

It is worth mentioning, here, that the (,glitches,) in the steady radiation of the pulsar do not seriously effect the analysis. The effect in the age estimation would be less than 1% and there would be no effect on n.

R E F E R E N C E S

[1] R. RUFFXNI and A. QADIR: Physics and Contemporary Needs, edited by RIAZUDDIN, Vol. 1 (Plenum Press, 1977).

[2] L. F. ZANO, A. QADIR and R. RUFFINI: Physics and Contemporary Needs, edited by RIAZUDDIN and A. QADIR, Vol. 5 (Plenum Press, 1983).

[3] A. QADIR, R. RUFFINI and L. VIOLINI: Lett. Nuovo Cimento, 27, 381 (1980). [4] R. N. MANCHESTER and J. H. TAYLOR: Pulsars (W. H. Freeman, 1977); J. M. IRVINE:

Neutron Stars (Oxford, 1978). [5] S. M. MAHAJAN, A. QADIR and P. M. VALANJU: Nuovo Cimento B, 65, 404 (1981); A. QADIR

and J. QUAMAR: Proceedings of the Third Marcel Grossmann Meeting, edited by Hu NING (World Scientific, 1983), p. 189; A. QADIR and J. QUAMAR: Europhys. Lett., 2, 423 (1986); J. QUAMAR: Ph.D. Thesis (Quaid-i-Azam University, 1984); A. QADIR: Proceedings of the First Chittagong Conference on Mathematical Physics 1986, edited by J. N. ISLAM (Chittagong University Press, 1987).

[6] M. A. ABRAMOWICZ: Centrifugal force: a few surprises, preprint from International Centre for Theoretical Physics.

[7] A. QADIR and M. RAFIQUE: Kexue Tongbao, 28, 1639 (1983); M. RAFIQUE: Math. Philos. Dissertation, Quaid-i-Azam University, 1981

[8] A. QADIR and M. RAFIQUE: Chinese Phys. Lett, 3, 189 (1985). [9] A. QADIR and M. RAFIqUE: Proceedings of the Fourth Marcel Grossmann Meeting, edited by

R. RUFFINI (Elsevier-North Holland, 1987), p. 1597; A. QADIR, M. RAFIQUE and A. W. SIDDIQUI: Chinese Phys. Lett., 4, 177 (1987); A. QADIR: Chinese Phys. Left., 4, 289 (1987).