The Astrophysical Journal, Vol. 157, September 1969 .157.13950 . (c) 1969 The University of Chicago All rights reserved Printed in U S A 9ApJ. 6 19 ON THE NATURE OF PULSARS. I. THEORY J. P. OSTRIKER AND J. E. GUNN Princeton University Observatory Received June 9, 1969 ABSTRACT We present in this paper the initial installment of a quantitative exploration of one particular pulsar model. We first make plausible and then assume that the seat of the pulsar phenomenon is a rotating neutron star having a dipolar magnetic field which is not parallel to the rotation axis. We then show that such stars may be expected to emit large amounts (1050-1052 ergs) of magnetic-dipole and gravitational- quadrupole radiation, that these energy losses are inevitably associated with losses of angular momentum and increases in the rotation periods, and that the emitted low-frequency magnetic-dipole radiation is extremely efficient at accelerating charged particles to relativistic energies. An explicit expression for the period as a function of time allows us to calculate the age of the Crab Nebula (with »20 percent accuracy) and to predict the so far unobserved second derivative of the period {d2P/dP). We also de- termine the luminosity of the nebula and the highest-energy electrons presently being injected into it— both numbers found to be in good agreement with independent observations. In extreme cases the ac- 2 2 l z 21 celeration mechanism can produce protons with energies up to mpc {e /Gm^) l or 10 eV, which is somewhat in excess of the most energetic cosmic rays yet observed. The theory predicts a relation be- tween period, P, and rate of change of period, dP/dt » 3 X 10“15/P(sec), which is well observed, in the mean, for pulsars. Finally, after determining the magnetic decay time to be about 4 million years, we predict that few pulsars should be found with periods in excess of 1.5 seconds, also in good accord with observations. We do not discuss the origin of the pulses themselves. I. INTRODUCTION Since the original announcement by Hewish et al. (1968) of astronomical objects emitting periodic pulsed radio radiation, a total of about 40 pulsars has been discovered, and their class characteristics are becoming apparent. We will discuss the observations in some detail in Paper III. For our present purpose of defining the model, two sets of characteristics are significant. First, the presently discovered pulsars have periods P and rates of change of period dP/dt satisfying the inequalities 0.033 sec < P < 3.7 sec and 0.0 < dP/dt < 4.3 X 10"13 -1 The individual pulses, which fill only a small fraction (~ 1/20) of the pulse period, are often strongly polarized. For the second set of characteristics we examine the distribu- tion of pulsars over the sky. Although obviously influenced by selection effects, the available data (see Paper HI) show a strong concentration in the galactic plane, no tendency toward clustering at the galactic center, and the presence of two pulsars (0833, 0532) in supernova remnants. The conclusions drawn from this data are, by now, well known (see Woltjer 1969; and Maran and Cameron 1969) ; we will review the arguments very briefly. The angular distribution of pulsars shows that they are galactic objects of disk or Population I type and enables us to restrict typical distances roughly to the range 102 pc < d < 104 pc; closer objects would be more isotropically distributed, and more distant ones would tend to lie in the longitude quadrant centered on (¿ = 0). With this estimate for distance {d = 103 pc) the radio luminosities are —10-4 Lo ; even with generous allowances for so far 1 Very recently Reichley and Downs (1969) and Radhakrishnan and Manchester (1969) appear to to have found a brief period in which the sign of dP/dt was negative in PSR 0833. 1395 © American Astronomical Society • Provided by the NASA Astrophysics Data System 1396 J. P. OSTRIKER AND J. E. GUNN Vol. 157 .157.13950 . unobserved optical flux, X-ray flux, etc., the pulsars are not intrinsically luminous ob- 9ApJ. jects in the usual electromagnetic spectrum. 6 19 What property of low-luminosity stars can provide such extraordinary stable clock- work with tick rate of about 1 per second? Stellar pulsations of any single class of star are not a likely explanation. The shorter periods are not in the range where any known, stable, astronomical object can oscillate in its fundamental mode—white dwarfs have periods P > 0.3 sec (Faulkner and Gribbin 1968; Ostriker and Tassoul 1968), and neu- tron stars have periods P < 0.05 sec (Meitzer and Thome 1966). Overtone oscillations of white dwarfs are theoretically possible, but the very wide range of observed period (over 100-fold) is difficult to understand if pulsars are, in fact, pulsating objects of any single class; in general (period) oc (density)“1/2, and an extremely wide range of density would be required. Finally, the periods are very stable, with characteristic times r = (dP/Pdt)*1 as great as 108 years, whereas the proposed pulsating stars would evolve on a very much shorter timescale. The clock cannot be provided by orbital motion in a binary system, since gravitational radiation of energy would lead to period changes much greater than observed—and of the wrong sign. There remained the possibility that pulsar time is kept by the same means as do- mestic time—the rotation of a massive object. White-dwarf stars were first suggested (Ostriker 1968), but even in their central regions, white dwarfs cannot rotate with a period less than about 1 sec, so pulsars, in general, cannot be rotating white dwarfs. The possible existence of stars more dense than white dwarfs had been guessed since the work of Baade and Zwicky (1934) and Oppenheimer and Volkoff (1938). Others, includ- ing Zwicky (1938) and Colgate and White (1966), had suggested that these neutron stars would be formed during supernova explosions, that they would initially be rapidly rotating (Hoyle, Narlikar, and Wheeler 1964; Tsuruta and Cameron 1966), and that the energy source of the Crab Nebula might be a rotating neutron star (Wheeler 1966). However, the first association between pulsars and such stars was made by Gold (1968, 1969), who argued convincingly that rotating neutron stars, with surface magnetic fields of « 1012 gauss, could account for many of the properties of the pulsars including pulse polarization and increase of period. In the rest of this paper we shall explore some of the many implications of Gold’s suggestion. We will not treat the origin of the pulses them- selves. Section II contains a discussion of the multipole radiation which is necessarily emitted by nonaxisymmetric rotating stars, and § III discusses the acceleration of relativistic particles by this radiation. Conclusions, a summary, and a brief discussion of applications are reserved for § IV. II. MULTIPOLE RADIATION a) Losses of Energy and Angular Momentum Any object, rotating in vacuo, having gravitational, electric, or magnetic fields which are not symmetric about the rotation axis must radiate energy and angular momentum. We would like to stress that the pulsars cannot be axisymmetric stars, since, from any aspect, the appearance of an axisymmetric star is independent of time, and the pulsars do emit pulses.2 To the lowest order in the angular velocity, fí, the energy losses from the time-varying gravitational and magnetic fields are (Landau and Lifshitz 1951) dE^/dt = -\2mPQl/â (1) and dEgJdt = -^GDPW/â . (2) 2 A more detailed argument, based on the time dependence of the polarization in PSR 0833 by Radha- krishnan et al, (1969), shows that an off-axis magnetic field is indicated for this object. © American Astronomical Society • Provided by the NASA Astrophysics Data System No. 3, 1969 PULSARS 1397 .157.13950 . Here and mi are the components of the mass-quadrupole and magnetic-dipole mo- 9ApJ. ments perpendicular to the rotation axes. There is no electric dipole radiation (by sym- 6 metry), and the electric quadrupole can be shown3 to be less than the gravitational 19 quadrupole by a factor of order (ßm^/e2) « 10-37. The probable emission of large quantities of multipole radiation from pulsars has already been pointed out by several authors, including Pacini (1968), Gunn and Ostriker (1969Ô), and Shklovskii (1969); see also Deutsch (1955). Once we have adopted the working hypotheses that the pulsar phenomenon is con- nected with a rotating neutron star, how do we estimate the magnetic-dipole-moment vector mi, the mass-quadrupole tensor Dab (from which mi and Di are derived), and the moment-of-inertia tensor I a? These quantities and the observed angular velocity Q will completely specify our model. If we use, for the sake of definiteness, the 1.4 Mo neutron-star model of Hartle and Thome (1968) constructed with the Vy equation of state, then the radius a and moment of inertia / (=fS/u) are M = 1.4Mo , a = 1.2 X 106 cm , and / = 1.4 X 1045 g cm2, (3) when rotational distortion is not significant (fí2 <<C 108 sec“2). Magnetic fields are observed on main-sequence stars, and there is considerable evi- dence (see, e.g., Wilson 1963) that young stars have larger fields than old ones; there is also some evidence (cf. Paper III) that pulsars are associated with a young, massive population, so that moderate-sized fields can be expected to occur in the progenitors of the neutron star. Fields of the order of 100 gauss seem not unreasonable, as an average, for stars slightly younger and more massive than the Sun, though of course much higher fields are known among the (relatively common) Ap stars.