Magnetic Field Decay in Single Radio Pulsars: a Statistical Study

Home , Ajit Kembhavi

University of Texas Rio Grande Valley ScholarWorks @ UTRGV

Physics and Astronomy Faculty Publications and Presentations College of Sciences

1-1-1997

Magnetic field decay in single radio pulsars: A statistical study

Soma Mukherjee

Ajit Kembhavi

Follow this and additional works at: https://scholarworks.utrgv.edu/pa_fac

Part of the Astrophysics and Astronomy Commons

Recommended Citation Soma Mukherjee, et. al., (1997) Magnetic field decay in single radio pulsars: A statistical study.Astrophysical Journal489:2928. DOI: http://doi.org/10.1086/304798

This Article is brought to you for free and open access by the College of Sciences at ScholarWorks @ UTRGV. It has been accepted for inclusion in Physics and Astronomy Faculty Publications and Presentations by an authorized administrator of ScholarWorks @ UTRGV. For more information, please contact [email protected], [email protected]. THE ASTROPHYSICAL JOURNAL, 489:928È940, 1997 November 10 ( 1997. The American Astronomical Society. All rights reserved. Printed in U.S.A.

MAGNETIC FIELD DECAY IN SINGLE RADIO PULSARS: A STATISTICAL STUDY SOMA MUKHERJEE Indian Statistical Institute, 203, Barrackpore Trunk Road, Calcutta 700 035, India; soma=isical.ernet.in AND AJIT KEMBHAVI IUCAA, Post Bag 4, Ganeshkhind, Pune 411007, India; akk=iucaa.ernet.in Received 1996 December 3; accepted 1997 June 24 ABSTRACT We report on an extensive statistical study of single radio pulsar properties, which we undertook to estimate the decay timescale of pulsar magnetic Ðelds. We synthesized a population of pulsars using assumed theoretical distributions of age, initial magnetic Ðeld, period of rotation, position, luminosity, and dispersion measure, taking into account the new distance model for pulsars as well as new velocity estimates. We also took into account the selection e†ects that characterize pulsar observations. We compared the simulated population with the observed single pulsar population using one- and two- dimensional Kolmogorov-Smirnov tests to obtain a lower bound of D160 Myr on the decay timescale. Subject headings: pulsars: general È stars: evolution È stars: magnetic Ðelds È stars: rotation

1. INTRODUCTION that the characteristic age of the low-velocity pulsars is not Ever since the discovery of pulsars and the development representative of the time since they acquired the bulk of of early theoretical models, there has been much interest their velocities, as at that time they possessed spin periods and conjecture about the decay of their magnetic Ðelds. of several hundred milliseconds. Thus the evidence for Ðeld Strong magnetic Ðelds are essential in understanding the decay on the basis of comparison of kinetic and spin-down way pulsars radiate and spin down. Assuming that the ages is discredited. Our ignorance about how far from the observed spin-down is a result of the energy loss due to Galactic plane a pulsar was born, as well as the inability to magnetic dipole radiation, it follows straightforwardly that measure the radial component of the velocity, add to the the surface dipolar Ðelds have a strength ofBS D 1012 G in uncertainty of the results. young pulsars(Taylor & Stinebring 1986). However, there On the theoretical side, it has been suggested(Ostriker & is no consensus yet on the origin and stability of the mag- Gunn1969) that the magnetic Ðeld should decay from the netic Ðelds. The lack of a fundamental understanding of the crust on a timescale of a few million years because of ohmic pulsar emission mechanism and of the origin of the Ðelds dissipation of electrical currents in the neutron star. Calcu- have led to considerable debate regarding the timescale of lations based on the ohmic decay show that the Ðeld cannot decay of the magnetic Ðeld(Sang & Chanmugam 1990). decay signiÐcantly on the scale of the Hubble time if there is In their pioneering statistical study of pulsar properties, any penetration of the magnetic Ðeld lines into the interior Gunn& Ostriker (1970) concluded that pulsar magnetic of the star(Sang & Chanmugam 1990; Sang & Chanmu- dipole Ðelds decay on a timescale of D106 yr. However, this gam1987). With realistic thermal evolution and conductivi- conclusion was based on only about a dozen pulsars on ties, isolated neutron stars will maintain large magnetic which data were then available. A similar conclusion Ðelds for more than 1010 yr (Romani 1990). regarding the temporal behavior of the magnetic Ðelds can Purely observational evidences have also been signiÐcant be reached by comparing their kinetic age with their spin- in contributing to the controversy. Discovery of very old down age. The kinetic age is given byqK D z/vz, where z is neutron stars with substantial Ðeld strength in binary their current distance from the Galactic plane, andvz is pulsars(Bhattacharya et al. 1992, hereafter B92; their velocity transverse to it.qK corresponds to the time Bhattacharya& Srinivasan 1991; Bhattacharya & van den since the formation of the pulsar if it is assumed that pulsars Heuvel1991) and the presence of radio pulsars in globular are all formed in the Galactic plane and are ejected with a clusters indicate that the magnetic Ðeld of these neutron high velocity relative to it. The other indicator of the age of stars cannot continue to decay signiÐcantly on the short a pulsar is the spin-down timeP/2P0 , where P andP0 are the timescales discussed in the context of young radio pulsars. observed period and the period derivative, respectively. Given the uncertain observational and theoretical situ- This corresponds to the time since formation if it is assumed ation, it is useful to approach the problem of magnetic Ðeld that the initial spin periodPi > P and that the magnetic decay from a statistical consideration of the large pulsar Ðeld is constant. It was noticed that as pulsars aged, the database that is now available. The idea here is to simulate spin-down age lengthened relative toqK, which can be inter- a population of pulsars from assumed theoretical distribu- preted as a decay of the dipole magnetic Ðeld(Helfand & tions of various quantities and to subject this set of pulsars Tademaru1977; Lyne, Anderson, & Salter 1982; Lyne, to the same observational selection e†ects as were in oper- Manchester, & Taylor1985). Though a plot, using recent ation when real pulsars were discovered in di†erent surveys. data, of the kinetic age against the spin-down age does show This Ðltered set of pulsars can then be compared with the this trend, there is a great deal of scatter, and one can say real population to see whether the simulated and real popu- only that a constant magnetic Ðeld deÐnitely seems incon- lations have similar distributions of various parameters sistent with the distribution of points in the kinetic age/ such as period, magnetic Ðeld, position around the Galaxy, spin-down age plane.Bailes (1989) has subsequently shown and so on. Parameter values that best reproduce observed 928 MAGNETIC FIELD DECAY 929 distributions are most likely to be those that occur in the surface magnetic Ðeld, and the period and the initial posi- real population. One of the parameters that can be Ðxed in tion coordinates from appropriate distributions. All other this manner is the decay timescale for the magnetic Ðeld. essential properties, such as the magnetic Ðeld, luminosity, A statistical approach of this kind was taken by Narayan and Ñux, can then be calculated from these initial conditions & Ostriker(1990), who found that the magnetic Ðeld decays using evolution equations. In the present study, we synthe- on a timescale of D1 Myr. However, in a detailed investiga- size a population of single radio pulsars from the assumed tion using the same approach,Bhattacharya et al. (1992) theoretical distributions of the properties using Monte came to the di†erent conclusion that the decay timescale Carlo techniques. This is followed by application of selec- must be at least as long as D100 Myr. RecentlyHartman et tion procedures similar to those that were in e†ect in the al.(1997) have obtained a similar result from the simulation surveys in which pulsars were discovered. The population of of the birth and evolution of radio pulsars throughout the these selected pulsars is then compared with the observed Galaxy. While it has been usual to consider a simple expo- population of single radio pulsars. nential decay model in these calculations, several authors 2.1. Assumptions have also performed pulsar statistics with a nonexponential decay model of the magnetic Ðeld(Narayan & Ostriker Age.ÈThe age of a pulsar can be chosen from a uniform 1990; Wakatsukiet al. 1992). However, such models did not distribution between zero and the Hubble time, under the prove to be a marked improvement over the exponential assumption that pulsar birthrate is constant (B92). decay model in obtaining a good Ðt between the observed However, a maximum of 5 times the magnetic Ðeld decay and simulated data. constant q suffices, as any pulsar older than this would In the present paper we closely follow the approach of evolve beyond the death line. Thus, Bhattacharyaet al. (1992) in order to reexamine the issue of t D U(0, 5q) , (1) the Ðeld decay timescale. The new aspects of our study are i.e., the age is chosen from a uniform distribution between 0 as follows. and 5q. However, the assumption that the pulsar birthrate is 1. We incorporate the most recent catalog of pulsars constant is not a foolproof one. This will be discussed in ° 6. (Tayloret al. 1995) from the Princeton database, which Magnetic Ðeld.ÈThe initial magnetic Ðeld strengthBi is contains data on 706 pulsars. First, a large database is chosen from a normal distribution in logBi, i.e., always helpful in drawing stronger inferences in statistical 1 studies of this nature. More importantly, many of the p(log Bi)d(log Bi) \ missingP0 values in the older pulsar catalog (as used in B92) J2np B can be obtained in the new catalog. log 0 (log B log B ) 2. We use a new distance model(Taylor & Cordes 1993) C[ i [ 2D ] exp 0 d(log Bi) . (2) that provides estimates accurate to D25% or better. The 2p2 log B0 Princeton catalog incorporates the Taylor & Cordes (1993) The mean value and the dispersion that deÐne the distribu- distance model to calculate the distances of the pulsars. tion are to be determined from the comparision of the simu- 3. We use the most recent estimate of the mean pulsar lated and observed populations as described below. birth velocity from direct pulsar proper motion measure- Following general practice, we assume that the magnetic ments(Lyne & Lorimer 1994). Ðeld has an exponential time dependence: 4. We adopt the Galactic potential discussed in Carlberg & Innanen(1987). This is a three-component axisymmetric A tB B(t) \ B exp [ . (3) model representing the disk/halo, nucleus, and bulge mass i q distribution in the galaxy (Lorimer 1995). Period.ÈThe period of rotation is calculated from its 5. We use two-dimensional Kolmogorov-Smirnov (K-S) relation to the magnetic Ðeld using the standard dipole tests to compare the distributions of pulsars in the magnetic rotator model(Gunn & Ostriker 1970): ÐeldÈperiod plane and the periodÈdispersion measure sin b plane. This has some advantages in sensitivity over the pre- 8n2R6 PP0 \ B2 , (4) viously used one-dimensional K-S tests, which we also 3Ic3 introduce for comparision with previous work. 6. Our simulations span a greater range of values; for where R \ 106 cm is the radius and I \ 1045 gcm2is the example, we vary the decay timescale from 10 Myr to a few moment of inertia of the neutron star. The initial period is hundred million years so that the parameter space is thor- taken to be Ðxed at a valuePi \ 0.1 s because, as explained oughly tested. inBhattacharya et al. (1992), pulsars with an initial period 7. We carry out a large number of independent simula- less than this value evolve to it on a timescale negligible tions to test the level of statistical noise and the repro- compared to the age of the pulsars as considered in this ducibility of results. simulation. Beaming fraction.È For a pulsar to be observable, the In the following sections we discuss the simulation pro- pulsar beam should intersect the line of sight to an observer cedure, the statistical tests involved, and, Ðnally, the results on Earth. The fraction of the sky covered by a pulsar beam obtained from the analysis. depends on its period of rotation. We have used the 2. ASSUMPTIONS AND SELECTION CRITERIA USED IN beaming functionfb as modeled byNarayan & Vivekanand SIMULATIONS (1983)and Rankin (1990). It is given by The simulation process for evolving a pulsar from the fb D P~1@2 (5) time of its birth to the present time is based on choosing for P º 0.1 s. We have independently done the calculation initial conditions such as the age of the pulsar, the initial relatingfb and P using nonlinear regression methods to 930 MUKHERJEE & KEMBHAVI Vol. 489 re-establish the P~1@2 dependence and determine the con- where M is the disk/halo mass, a and h are the scale length stant of proportionality in the process. The expression and scale height of the disk, respectively, and b is the core obtained by us is given by radius of the halo. The summation refers to old disk, dark matter, and young disk contributions. Thebi are pro- fb \ 0.36075 ] P~0.45928 (6) portionality factors. For the nucleus and the bulge contributions, the scale height terms are set equal to zero so that for P º 0.1 s. For pulsar periods that led tofb \ 0.2, we set the form of the potential for these contributions becomes fb \ 0.2.Using other laws (Lyne & Manchester 1988) to relatef and P does not signiÐcantly a†ect our results. b GM Position.ÈThe radial positionRi of a pulsar in the ' \ ' \[ . (12) Galactic plane, in Galactocentric coordinates, is chosen nuc bulge Jb2]R2 from a normal distribution(Narayan 1987) given by The values of a, b, h, and thebi are obtained from Table A.1 inLorimer (1995). The Galactocentric radius, R, is taken to 1 A[Ri2B p(Ri)dRi \ exp dRi . (7) be equal to (x2]y2)1@2 and is generated from a normal J2nR2 2R2 0 0 distribution as described above. The circular rotation speed on the Galactic plane,v , is obtained from the centrifugal The Galactocentric radius of the Sun is assumed to be 8.5 rot kpc, and the disk scale lengthR is 4.8 kpc. Only those force per unit mass and the gravitational force per unit mass pulsars were retained in our simulation0 for which the helio- experienced by a test particle in a gravitational Ðeld. It is centric distance was found to be less than 3 kpc. This is given by done in order to keep the selection e†ects in our simulation C L D1@2 similar to those ofB92 for comparison of results. Increasing v (R) \ R ' (R,0) . (13) this distance adds just a few observed pulsars to the sample, rot LR gal when all other selection criteria are applied (see below), The circular rotation speed,v , is vectorially added to the while considerably increasing the required computing time. components of velocity generatedrot from the normal prob- The initial height of the pulsar above (or below) the ability distribution described above. For the present study, Galactic plane is chosen from an exponential distribution only the vertical motion of the pulsar is followed keeping x with scale height h of the pulsar progenitors(Mihalas & and y constant, as will be explained in more detail in ° 5. Binney 1981): L uminosity at 400 MHz.ÈIt is possible to determine the 1 C o z oD luminosity of a pulsar from its period and period derivative. p(zi)dzi \ exp [ dzi . (8) A model to determine the 400 MHz luminosity has been 2h h provided byProszynski & Przybycien (1984) and Narayan FollowingB92, we have chosen h \ 0.06 kpc. (1987) and is given by The initial velocity,vzi, of the pulsars in the vertical direc- 1 P0 tion was chosen from a Gaussian distribution with mean log SL T \ 6.64 ] log , (14) 400 0 3 P3 Svzi \ 0Tpand dispersionv\200 km s~1 (Lyne & Lorimer 1994, 1995): where the units of luminosity are mJy kpc2. In an alternative model byStollman (1986), the expression relating the 1 1 v C 2ziD variables is p(vzi)dvzi \ exp [ dvzi . (9) J2np 2 pv2 v B New determinations of proper motion(Harrison, Lyne, & log SL T \ ([10.05 ^ 0.84) ] (0.98 ^ 0.03) log , 400 0 P2 Anderson1993) and the new electron density model of Taylor& Cordes (1993) have led to a re-estimation of the B log ¹ 13 (15) velocities of pulsars. The present estimate ofpv \ 200 km P2 s~1 exceeds the value of 110 km s~1 used byB92 because the old distance model(Lyne et al. 1985) used in their paper and underestimated the distance to nearby pulsars and also B because of the discovery of a number of young and higher log SL T \ 2.71 ^ 0.60 , log [ 13 . (16) velocity pulsars in recent astrometric surveys(Harrison et 400 0 P2 al.1993; Fomalont et al. 1992). These relations are based on the distribution of observed The simulated pulsars were allowed to move under the luminosity and provide the mean luminosity of a popu- Galactic potential given byCarlberg & Innanen (1987) (see lation. The observations, and therefore the mean, are, Lorimer 1995). This is a three-component, axisymmetric however, biased toward the higher luminosities. The dis- model representing the disk/halo(' ), nucleus (' ), and DH nuc tribution of the intrinsic luminosity around the mean has bulge mass(' ) distribution in the Galaxy. The total been modeled byNarayan & Ostriker (1990) and is given by Galactic potentialbul is the sum of these three contributions: o(j) \ 0.5j2e~j , (17) ' \ ' ] ' ] ' . (10) gal DH nuc bul where The disk/halo potential is given by A L B GM j \ a log 400 ] b , (18) ' (R, z) \[ , SL T DH J[a ] ;3 b (z2]h2)1@2]2]b2]R2 400 0 i/1 i i with a \ 3.6 and b \ 1.8 for Proszynski & PrzybycienÏs law (11) and a \ 3.0 and b \ 2.0 for the Stollman luminosity law. No. 2, 1997 MAGNETIC FIELD DECAY 931

The procedure adopted in the simulations is to determine The cumulative distribution of * log q is represented very the mean luminosity given P andP0 and then to pick at well by the functionM1 ] exp [(x [ * log q)/a ]N~1, with random a luminosity from the distribution around this x \ 0.04 and a \ 0.342 (B92). 0 0 mean. Both of the luminosity laws were used for the 0The Ðnal width0 of the broadened pulse is given by goodness-of-Ðt tests that followed the simulations. (Narayan 1987) Determination of Ñux.ÈThe Ñux in mJy received from a W 2\W2]q2]q2]q2 , (23) pulsar at 400 MHz is given by e samp DM scatt whereq is the broadening caused by the Ðnite sampling L samp S \ 400 . (19) time of the data,q is due to dispersion smearing, andWe is 400 x2]y2]z2 the width of unbroadenedDM pulse. The variable q \ DM Calculation of the limiting Ñux.ÈCorresponding to the C DM, whereC is a constant. The values ofC and q DM for the pulsarDM surveys used in the present studyDM are observation in a given direction in a given survey, there is a samp limiting ÑuxS below which the detection of the pulsar is given inNarayan (1987). The minimum Ñux is then not guaranteed.min In order to ensure completeness of the obtained from survey, it is necessary to retain only those pulsars that have AT ] T BC PW D1@2 Ñux greater thanS . This limiting Ñux,S , is a function S \ S r sky , (24) min min min 0 T W (P[W ) of period, dispersion measure, and instrument-speciÐc 0 e parameters. We have used the detection criteria of one of whereS is the survey Ñux limit,Tr is the receiver excess the four major pulsar surveys conducted at 400 MHz: the noise temperature,0 T is the sky background temperature, Jodrell Bank survey(Davies, Lyne, & Seiradakis 1972), the andT is a normalizationsky constant. All the parameter University of MassachusettsÈNRAO survey (Damashek, values0 can be obtained fromHaslam et al. (1982), Dewey et Taylor, & Hulse1978), the University of MassachusettsÈ al.(1984), and Narayan (1987). Arecibo survey(Hulse & Taylor 1974), and Second Mol- onglo survey(Manchester et al. 1978). 2.2. Selection Criteria The dispersion measure can be calculated from the distance of the pulsar and the electron density at that speciÐed After a pulsar is generated with parameters chosen at location. We have adopted the new distance model (Taylor random from their known distributions, one has to make & Cordes1993) that incorporates the e†ects of Galactic sure that the pulsar would actually be observed in at least spiral arms, the shapes and locations of which are derived one of the surveys used in forming the observed ensemble of from the radio and optical observations of H II regions. The pulsars that is used in the comparision. For it to be model also includes the electron densities of the outer and detected, a pulsar must have the right combination of inner axisymmetric components and of spiral arms, with period and Ðeld strength so as to lie above the death line appropriate scale lengths and Ñuctuation parameters used and should be beamed toward the Earth. Also, the Ñux to relate the dispersion and scattering contributions of the measured should lie above a limiting Ñux as described features. Owing to its large angular size and proximity, the above. Our simulated pulsars were subjected to the selec- Gum Nebula is also a signiÐcant contributing factor to the tion criteria described below, so that the accepted set resem- dispersion measure. bles the set of observed pulsars as closely as possible. A The electron density of a speciÐed Galactic location is comparision between simulated and observed populations deÐned as the sum of contributions from each of the four then becomes meaningful, since both the populations su†er components: from the same selection e†ects. First selection criterion.ÈA pulsar is retained in the simu- 2 A z B A z B lation only if it lies above the death line(Ruderman & n (x, y, z) \ ; n g (r) sech2 ] n sech2 e i i h a h Sutherland1975; Rawley, Taylor, & Davis 1986), i.e., it i/1 i a satisÐes the condition 4 ] ; fj ga(r, sj) ] n g (u) , (20) B j Gum Gum [ 0.17 ] 1012 Gs~2 . (25) /1 P2 where r \ (x2]y2)1@2. The summation is taken over the Second selection criterion.ÈWe have seen in equation (6) four spiral arms. Thefj are scale factors connected to the how the fraction of the sky covered by a pulsar beam is spiral arms. The functionsg , g , ga, andg signify position dependences of the model.1 2 The valuesGum of the param- related to its spin period. We have to make sure that for a eters of the model can be found inTaylor & Cordes (1993). pulsar to be a part of the observed sample, the beam inter- The dispersion is obtained from the relation sects the line of sight. For a randomly oriented pulsar, the probability that this happens is proportional to the beam- P d width. Having obtainedf from P, we selected a random DM \ ne ds . (21) b number u (say), and if the conditionu ¹ fb was satisÐed, we 0 retained the pulsar in the simulation; otherwise, we rejected We have used the Fortran subroutine provided by Taylor it. This ensures that, of the pulsars that we begin with, we & Cordes(1993) to calculate the dispersion measure. retain only the fraction warranted by the beamwidth. The contribution of scatter broadening to the pulse width T hird selection criterion.ÈWe have used observed is given by pulsars that have been found in at least one of the four q \ 10~4.62B0.52`(1.14B0.53) log DM major surveys conducted at 400 MHz, viz. the Jodrell-Bank scatt survey, the University of MassachusettsÈArecibo survey, ] 10~9.22B0.62`(4.46B0.33) log DM . (22) the Second Molonglo survey, and the University of 932 MUKHERJEE & KEMBHAVI Vol. 489

MassachusettsÈNRAO survey. Every simulated pulsar that Figure 1 shows the distribution of the age, vertical com- is included in the sample should therefore be detectable in ponent of dispersion measure, Ñux, magnetic Ðeld, period, at least one of these surveys and meet the selection criterion and period derivative of the observed set of 478 pulsars for completeness, i.e., (dotted lines) and of the selected pulsars (solid lines). S º S . (26) 400 min 4. STATISTICAL TESTS 3. SAMPLE SELECTION FROM REAL PULSARS The simulated population depends on a number of The data for real pulsars were obtained from the catalog assumptions and values for parameters. However, the most of 706 pulsars from the Princeton database(Taylor et al. important parameters in our simulation are just three: the 1995). We dropped binaries, supernova remnants, and decay timescale q, logB , and the variancep2 B . For a X-ray and c-ray pulsars and isolated a subset of 511 single given simulation, we assume0 a luminosity law andlog 0 generate radio pulsars. An additional 43 pulsars had to be dropped a sequence of populations by allowing logB , p2 B , and q from the population because of missingP0 values. The to vary in suitable steps. 0 log 0 remaining population of pulsars was then passed through For each choice of parameters, we have used the the same selection e†ects as those used for the simulated Kolmogorov-Smirnov test in its standard one-dimensional population, since the models and survey parameters used form and a generalization to two dimensions to compare for the calculation ofS in° 2 give only an approximate the simulated and observed populations. We run through minimum Ñux for whichmin the surveys were reasonably com- the range of parameters, generate a simulated and selected plete, rather than an absolute minimum Ñux(B92). This led to sample for each set of parameter values, and make the com- the retention of only 97 pulsars from the observed sample. parision. The aim of this procedure is to determine the

FIG. 1.ÈDistribution of various pulsar properties before and after selection. The dotted lines show the distribution for all observed single radio pulsars for which we have all the required data, and the solid lines show the distribution after selection criteria are applied to the observed population, as explained in the text. No. 2, 1997 MAGNETIC FIELD DECAY 933 parameter values for which we get the best match between (P,P0 ) and (P,DMsinobo), and the distributions in these the simulated and the real populations. The value of q for planes of the simulated and real pulsars can be considered. which we get this match gives an estimate of the magnetic We adoptFasano & FranceschiniÏs (1987) version of the Ðeld decay timescale for the population of single radio two-dimensional K-S test. In this test, the absolute di†er- pulsars. ence between the observed and the predicted (normalized) two-dimensional cumulative distributions is computed 4.1. T he One-dimensional Kolmogorov-Smirnov Statistic separately in the four quadrants of the plane deÐned by the The degree of similarity between the simulated and variables of interest. Thus, for any data point with coordi- observed populations can be ascertained by comparing the nates(Xi, Yi), the observed data and predictions are distribution of various attributes of individual pulsars, such summed up in the four quadrants of the plane deÐned by as the period and magnetic Ðeld for the two populations. (x \ Xi, y \ Yi), (x\Xi,y[Yi), (x[Xi,y\Yi), The one-dimensional K-S test provides a simple measure of the overall di†erence between two distributions. The K-S (x[Xi,y[Yi), i\1,..., n . (28) statistic is deÐned as (seePress et al. 1992 for a brief The statisticD v is taken to be the largest of the di†er- discussion) ences of the integratedK S2 probabilities when all data points are considered. The probability that a di†erence exceeding D \ max o S (x) [ S (x) o , (27) N1 N2 D is obtained under the null hypothesis is approx- KvS2 whereSN (x) andSN (x) are two cumulative probability dis- imately given byPress et al. (1992). tribution1 functions of2 the parameter x.N andN are the We evaluate the two-dimensional K-S probability for dis- number of objects in the two samples,1 respectively.2 The tributions in the (P, B) and (P,DMsinobo) planes and, as one-dimensional test does not depend in any way on the in the one-dimensional case, obtain the product probability. shape of the parent distribution and does not require any We will see that this appears to provide a better discrimi- binning of the data. nant than the one-dimensional product probability. We The K-S probability can be obtained for each of the inde- shall compare the relative merits of the one-dimensional pendent attributes that deÐne the population. The three and two-dimensional tests in detail in a future publication. parameters we use are the period, P, the magnetic Ðeld, B, and the vertical component of the dispersion measure, 4.3. Parameter Ranges DM sin o b o, as these can be independently speciÐed for each As explained at the beginning of this section, the variable pulsar.Battacharya et al. (1992) also use luminosity as an parameters in our simulations are (1) the magnetic Ðeld independent attribute, but we Ðnd that it is not necessary to decay constant, q; (2) the mean of the logarithm of the initial include it in the tests. The luminosity of each pulsar can be magnetic Ðeld, logB ; and (3) its variance,p B . In order obtained as described above, given a law that determines to determine the optimum0 step size of these parameterslog 0 and mean luminosity, as inequation (15), for instance, and the the number of objects to be used in the simulation, we Ðrst distribution of individual luminosity about the mean, which generated a ““ real population ÏÏ using some set of input has the Ðxed form given in equations(17) and (18). It is parameters and tried to recover these through the process of therefore not an independent parameter. We have veriÐed simulation and comparision using the K-S test. that the conclusions we reach about the population are We began by varying q from 10 to 370 Myr in steps of 10 independent of whether or not the luminosity is used as an Myr, logB from 12.0 to 13.0, andp B from 0.2 to 0.4, extra attribute in the comparision, and we do not consider both in steps0 of 0.01. We found thatlog the0 K-S probability it any further. values were insensitive to such small step variations of log From the individual probabilities for the three attributes B andp B and showed Ñuctuations. By increasing the P,P0 , and DM sin o b o, we obtain an overall probability, step0 sizelog and0 repeating the simulations several times, we which is the product of the three probabilities, and use it as found that step sizes of 0.1 for logB andp B and 30 Myr the indicator of the degree of similarity in the distributions. for q led to stable probabilities, and0 peak probabilitieslog 0 were We vary the three parameters logB , p , and q in the 0 log B0 always obtained for the values of parameters that were manner described above and Ðnd the combination for equal to the input values. which the maximum product probability is obtained. If too few simulated pulsars are used in the comparision with the observed population, the results obtained will 4.2. T he Two-dimensional Kolmogorov-Smirnov Statistic change owing to ““ noise ÏÏ from one set of simulated pulsars In the one-dimensional K-S test, the distribution of each to another. On the other hand, if the simulated set is very attribute in the simulated and real populations is compared large, the requirement on computer time will be prohibitive. independently from the distributions of the other attributes. In order to decide on the size of the simulated sample, we However, it would be most natural to consider the joint again had trial runs with samples of 200 and 2000 objects distribution of the triplet (P,P0 , DM sin o b o ) for the two each and tried to recover known input values that went into populations. While this would be difficult to do computa- deÐning a ““ real population.ÏÏ We found that with 2000 tionally, schemes have been developed for comparing two- simulated pulsars, the input parameter values could always dimensional distributions(Peacock 1983; Fasano & be recovered, independent of the seed value used in random Franceschini1987; see Press et al. 1992 for a brief descrip- number generation for simulating a population. However, tion and numerical algorithms of the Fasano & Frances- when the same tests were carried out with 200 simulated chini version), which are analogous to the pulsars, we observed unacceptable Ñuctuations in the prob- Kolmogorov-Smirnov test applicable in one dimension and ability values for test runs with di†erent seeds. Therefore, in are loosely termed as two-dimensional K-S tests. In our all the simulations that followed, we kept the sample size of case, the three attributes can be divided into the two planes the simulated pulsars at 2000. Nevertheless, test runs with 934 MUKHERJEE & KEMBHAVI Vol. 489 the smaller number provided valuable insights during the paper. We shall present details of these simulations in a early stages of the project. later work. As explained above, the simulation procedure involves We have performed the one- and two-dimensional K-S choosing pulsars with parameters that are picked at tests between the simulated and observed samples by random from certain distributions and then passing them varyingSlog B Tp from 12.0 to 12.3 in steps of 0.1, B through selection Ðlters to ensure that the simulated objects from 0.3 to 0.50 in steps of 0.1, and q from 10 to 370 Myrlog in0 could actually be observed on Earth under the conditions of steps of 30 Myr. We have used the luminosity laws of the discovery surveys. When survey conditions were not Stollman (1986)as well as Proszynski & Przybycien (1984) met, the pulsar had to be dropped from consideration. The in order to arrive at the best-Ðt model. Our samples of rejection level of simulated pulsars because of the imposi- simulated and selected pulsars always have 2000 objects. tion of selection e†ects was quite high: in order to obtain When the mean and dispersion in the initial magnetic 2000 simulated pulsars that satisÐed the selection e†ects, we Ðeld are varied over their range, we Ðnd that the best Ðt, i.e., found it was necessary to make D106 trials, i.e., only one of the maximum product probability, is always obtained for every D500 simulated pulsars could meet the selection cri- logB \ 12.4 andp B \ 0.4. We have presented in Table teria of at least one of the four surveys from which our 1 the0 results of one-log and0 two-dimensional K-S tests when observed sample was derived. The requirement that the cal- the decay constant was varied over its range, with log B 0 culated Ñux should be above a minimum limiting Ñux led to andp B held Ðxed at 12.4 and 0.4, respectively. Three sets maximum rejection. of valueslog 0 of the one- and two-dimensional K-S probabilities We found that in all trial cases of comparision between corresponding to three sets of simulations with di†erent simulated and observed pulsars, maximum probability seed values are listed. The table shows results for the lumi- values were obtained for logB in the range 12.3È12.5 and nosity laws ofStollman (1986) as well as those of Proszynski 0 p B in the range 0.3È0.5. In order to reduce the computer & Przybycien (1984). timelog 0 needed, we restricted the variation in these parameters It is seen fromTable 1 that for low values of q, the simula- to these short ranges. We ensured from time to time that tions poorly reproduce the observed distributions. Prob- probability maxima were not situated outside these ranges abilities are very low for values of q up to 100 Myr. by having runs over wider ranges. However, beyond q \ 160 Myr, there is a distinct rise in the D RESULTS OF SIMULATIONS probability values (by a factor of 2È4), which tend to 5. reach saturation for q[220 Myr. It is important to note In the simulations reported here we have considered that in the case of the one-dimensional K-S test, probability pulsar motion only in the direction normal to the Galactic values for q[160 Myr can occasionally drop to values plane, after obtaining their initial positions in the plane comparable to those corresponding to ¹100 Myr. However using the appropriate probability distribution. We assume this is never so with the two-dimensional K-S probability. that the velocity dispersion for the pulsars is 200 km s~1. Even though there is statistical Ñuctuation in the probabil- Evolution of pulsar positions by considering their motion in ity values for q º 220 Myr, the values at any stage beyond three dimensions slows down the simulation process con- q º 220 Myr never approach values corresponding to siderably. We have made some preliminary simulation q ¹ 100 Myr. studies with three-dimensional motion taken into account, We have shown inTable 2 the peak K-S probabilities and and the results support the conclusions reported in this the corresponding q values obtained for many runs, each

TABLE 1 ONE- AND TWO-DIMENSIONAL K-S PRODUCT PROBABILITIES

SIMULATION 1SIMULATION 2SIMULATION 3

LUMINOSITY LAW q (Myr) P1D P2D P1D P2D P1D P2D product product product product product product Stollman ...... 10 8.3E[8 1.5E[9 3.2E[8 1.18E[10 8.5E[9 8.6E[11 40 4.9E[4 5.2E[5 6.8E[3 1.4E[4 4.0E[4 4.9E[5 70 0.001 0.001 9.5E[4 1.8E[4 0.002 8.0E[4 100 0.008 0.005 0.013 0.010 0.007 0.009 160 0.015 0.022 0.007 0.020 0.021 0.032 220 0.019 0.024 0.014 0.013 0.006 0.017 250 0.019 0.025 0.020 0.020 0.011 0.026 280 0.017 0.034 0.013 0.022 0.009 0.014 310 0.030 0.035 0.022 0.015 0.018 0.032 370 0.023 0.035 0.069 0.050 0.041 0.072 Proszynski & Przybycien ...... 10 4.7E[7 9.9E[9 1.1E[7 9.8E[10 4.7E[7 9.9E[9 40 6.4E[4 2.6E[4 1.3E[4 1.3E[4 6.4E[4 2.6E[4 70 6.0E[4 2.6E[4 6.2E[4 0.001 5.8E[4 2.0E[4 190 0.002 1.4E[4 7.3E[4 6.9E[5 0.002 1.4E[4 310 0.006 7.7E[4 0.005 2.2E[4 0.006 7.7E[4 340 0.007 2.9E[4 0.004 3.3E[4 0.007 2.9E[4 370 0.019 0.002 0.014 0.001 0.019 0.002

NOTE.ÈFor di†erent values of decay timescales for logB \ 12.4 andp B \ 0.4. The pulsar motion is assumed to be normal to the Galactic plane, with a velocity dispersion of 2000 km s~1. Thelog K-S0 probabilities for three sets of simulations corresponding to di†erent seed values are shown. The number of selected pulsars in each run is 2000. No. 2, 1997 MAGNETIC FIELD DECAY 935

TABLE 2 s~1. The Ðgure shows the sensitivity of the probability to ONE- AND TWO-DIMENSIONAL K-S BEST-FIT VALUES CORRESPONDING TO variations of di†erent parameters. The vertically elongated PEAK PRODUCT PROBABILITIES nature of the contours indicates that the probabilities are more sensitively dependent on the initial magnetic Ðeld ONE DIMENSION TWO DIMENSIONS value than the decay timescale. The dots in each panel show LUMINOSITY LAW q P1D q P2D the maximum probability, and it is clear that the maximum product product in every case occurs at logB \ 12.4 andp \ 0.4. While Stollman ...... 340 0.029 250 0.044 0 log B0 280 0.033 190 0.040 the position of highest probability on the q-axis depends on 220 0.045 220 0.060 the other parameters, it is always high, which indicates that 310 0.035 310 0.056 there is almost no Ðeld decay during the lifetime of the 370 0.049 370 0.057 pulsar. 310 0.052 310 0.090 Another point of interest inTable 2 is that the maximum 250 0.043 160 0.066 220 0.034 310 0.054 two-dimensional product probabilities are always greater 250 0.026 310 0.060 than the corresponding one-dimensional product probabil- 370 0.041 370 0.072 ities, which indicates that the two-dimensional test provides 310 0.029 370 0.036 a better way to discriminate between the di†erent sets of 250 0.048 340 0.066 Proszynski & Przybycien ...... 160 0.013 160 0.010 input parameter values so as to obtain the best-Ðt model. 310 0.011 100 0.010 This also shows that a multiparameter comparison of the 340 0.009 220 0.010 simulated and observed pulsars is better brought out by 370 0.017 130 0.019 testing the distributions in the parameter space by using a 280 0.018 100 0.018 higher dimensional K-S test than by using the one- 250 0.016 340 0.011 dimensional K-S test to compare the properties individ- NOTE.È Each entry for q is the decay timescale for which the maximum ually. product probability is obtained, in a run with q varying over the range Figure 3 shows the distributions of the observed pulsars 10È370 Myr, logB varying between 12.3 and 12.5, andp varying B0 against the simulated pulsars corresponding to log B \ between 0.3 and 0.5.0 In every run the maximum was obtainedlog for log 0 B 12.4 andp 0.4, so all entries shown are for these values. Other 12.4,p \ 0.4, and q \ 280 Myr. The distributions \ B \ log B0 assumptions0 arelog as in0 Table 1. shown are for period, period derivative, the magnetic Ðeld, DM sin b, and the luminosity at 400 MHz. The close coin- cidence of the distributions of the simulated and the initiated with a di†erent seed value. For each seed value, the observed pulsars show that our simulations can reproduce parameters logB , p B , and q were varied over their the actual observed distribution of the respective properties respective ranges,0 whichlog 0 were mentioned above. Since in of the pulsars well.Figures 4 and 5 show the cumulative every run the maximum probability was obtained for log distribution of the pulsars for the period, the period deriv- B \ 12.4 andp B \ 0.4, these values are not separately ative, the magnetic Ðeld, and the luminosity for two values indicated0 in the table.log 0 It is seen from the table that in every of q, viz. 280 and 10 Myr. It is quite clear from the Ðgures run the best Ðt was obtained for q º 160 Myr, even though q that the simulated pulsars follow the observed pulsar dis- was varied over a wide range. tribution closely for q \ 280 Myr, but show a rather poor Ðt It is seen from both Tables12 and that the probabilities for q \ 10 Myr.Figure 6 shows the distribution of the obtained in the simulations carried out with StollmanÏs pulsars in the B-P plane. luminosity law are always greater than those corresponding to Proszynski & PrzybycienÏs luminosity law. StollmanÏs 6. DISCUSSION luminosity law therefore provides a better overall Ðt between simulated and observed pulsars. This is true for Though we began with an observed sample of 467 both one- and two-dimensional K-S tests. pulsars, for each of which we had all the information neces- The values of q corresponding to the best Ðt varied from sary to do our tests, the selection e†ects lead to the retention 160 to 370 Myr in the di†erent sets of simulations using of only 97 pulsars in our sample. We have found that most di†erent initial seed values. It is important to note that even of the rejection occurs because the observed Ñux falls below though we have repeated the entire procedure several times the minimum Ñux calculated on the basis of the model using di†erent seed values, in none of the cases do we obtain based on survey parameters. However, rejection of so many the best-Ðt model for values of q\160 Myr, though the observed pulsars for this reason implies that all assumptions starting value in our simulation loop for q is 10 Myr. This in the model for the calculation ofS should be thor- consistently sets a lower bound to the value of q, which oughly checked, and a better model, whichmin can give a more shows that the existing model favors a longer value of the realistic minimum Ñux estimate, should be developed such magnetic Ðeld decay constant. This is equally true for one- that fewer observed pulsars are deleted from the sample. dimensional as well as two-dimensional tests. The selected pulsars tend to have their luminosity toward Figure 2 illustrates the above results explicitly through the higher values of the luminosity distribution of the full equal probability contours. The contours in each panel sample. This happens because of the minimum Ñux selec- show the locus of equal probability points as logB and the tion criterion. However, the distributions of all other magnetic Ðeld decay timescale, q, are varied. In each0 panel parameters are similar for the selected subsample and the contours forp B \ 0.3 and 0.4 are shown. The four panels full observed sample. correspond tolog two0 di†erent seed values for the random The main goal that we have addressed in this paper is to number generators used in the simulations and one- and Ðnd a good estimate of the magnetic Ðeld decay timescale. two-dimensional K-S tests. Results shown are for Stoll- Our simulations have consistently shown that the best-Ðt manÏs luminosity law and a velocity dispersionpv \ 200 km models correspond to q º 160 Myr. In fact it can be seen 936 MUKHERJEE & KEMBHAVI Vol. 489

FIG. 2.ÈEqual probability contours in the logB -qp plane. Probability contours are shown for two values ofB (0.3, 0.4), for two seed values, and for one- and two-dimensional K-S tests. StollmanÏs luminosity0 law and a velocity dispersion of 200 km s~1 were used. log 0

from the tables that the q º 200 Myr is the most frequently 1997). We intend to include this e†ect in a subsequent study. reproduced value that gives the best-Ðt solution. It is clear We have used theTaylor & Cordes (1993) distance model from Figures12 and that properties such as the period, the to calculate the dispersion measure. However, we Ðnd that period derivative, the magnetic Ðeld, and the luminosity are the DM sin b factor is not well reproduced in any of the simultaneously well reproduced for the value of q[200 simulations. This is a bit surprising, since we have seen that Myr. However, none of the above mentioned properties this distance model takes into account all relevant factors of could be matched with the observed sample for q D 10 Myr. the structure of the Galaxy. In fact, the reason that It is difficult (and perhaps meaningless) to pinpoint an exact Bhattacharyaet al. (1992) obtained 130 pulsars in the selec- value of q when the model is that of a truncated joint dis- ted observed sample (with 3 kpc cut o† in the projected tribution having so many parameters. However, it is of con- distance on the Galactic plane) while only 97 pulsars were siderable signiÐcance to note that in all the sets of included in the present study is that the distance estimates simulations that we have carried out, we have obtained a obtained from the old distance model(Lyne et al. 1985), as consistent lower bound for q. In a former study,Wakatsuki et used by Bhattacharya and coworkers, were considerably al.(1992) proposed a constant Ðeld model as the better lower than the most recent estimates given in theTaylor & alternative to a decaying model (exponential or power law). Cordes(1993) model. This is also demonstrated in Figure 7, However, exponential decay with timescale º160 Myr sig- which shows that most of the pulsar distances were under- niÐes practically no Ðeld decay at all during the active life- estimated in the earlier model(Lyne et al. 1985). time and hence cannot be distinguished from a truly As mentioned earlier, we are in the process of carrying constant magnetic Ðeld with the data presently available. out similar simulations that take into account the motion of We would, however, like to add that experimenting with the pulsars in three dimensions. Preliminary studies with 200 parameters (a and b) in the calculation ofL will lead to a and 500 simulated pulsars show the same trends as much wider range of acceptable values of q400(Hartman et al. discussed in this paper. No. 2, 1997 MAGNETIC FIELD DECAY 937

FIG. 3.ÈDistribution of the various properties of the simulated and observed pulsars after selection, for parameter values that give a good match. The solid lines show simulated and selected pulsars, while the dotted lines show observed and selected pulsars. Parameter values use in this simulation were log B \ 12.4, p \ 0.4, and q \ 280 Myr. StollmanÏs luminosity law was used, and the number of simulated and selected pulsars was 2000. 0 log B0

The assumption regarding the uniform distribution of the this as ““ insensitive to small step variations ÏÏ and decided to age of the pulsars should be critically examined. This is use larger steps (\0.1) that led to stable peak probability because the existence of a correlation between the present values and also reduced the computation load to a large pulsar distribution and the estimated locations of the spiral extent. We shall show in a subsequent paper (which is about arms at earlier epochs(Ramachandran & Deshpande 1994) to be communicated) that if we retain parameter variation might be a crucial factor in inÑuencing birthrates. This has over such Ðne grid, a much wider range of acceptable value not been modeled in the present simulation but would be of q can be obtained. However, we feel that caution must be important in any future study. taken in the interpretation of this result, since what might Last, we would like to include a detailed discussion appear to give more acceptable values of q might actually be regarding the justiÐcation of the choice of step size in the caused by statistical noise Ñuctuations. As we can see, the variation of logB and p in the simulation. The choice of whole model is a complex one with the inclusion of many the step size of the0 variables has an important e†ect on the parameters and variables. The quantity q is not a mono- inference that we draw on the Ðeld decay constant, q, and tonic function of these variables, nor is the dependence of q thus a thorough discussion on the various aspects of it is on these parameters a predictable one. This aspect tends to relevant. As is explained in° 4.3, we started by varying log add to the statistical noise associated with complex numeri- B and p over a very Ðne grid of parameter space, as in cal simulations, such as those undertaken in the present Bhattacharya0 et al. (1992) and Hartman et al. (1997). study. However, it is also important to note that noise is However, we found that the best-Ðt values obtained were present in any real data set, and one should allow enough strongly correlated for such Ðne variation of the parameters. degrees of freedom so as not to suppress it as far as possible. The Ðt was indeed equally good everywhere. We described InFigure 2, we have plotted the probability contours as 938 MUKHERJEE & KEMBHAVI

FIG. 4.ÈCumulative distribution of various properties of simulated and observed pulsars after selection. Details are as in Fig. 3.

obtained from the results of our repeated simulations using more sensitive indicator for the comparision of distribu- di†erent seed values using the above criterion of the choice tions than the one-dimensional tests used so far. The of grids. In the earlier study byBhattacharya et al. (1992), a present study conÐrms the result ofBhattacharya et al. contour level of 1/e2 below the maximum on the grid was (1992) that there is no signiÐcant Ðeld decay in single radio used, and the authors found that its location was fairly pulsars during their active lifetime. stable to the variations of the seed. This is a clear alternative way of deÐning the area of best Ðt. To conclude, we stress that inclusion of the most recent The authors thank E. P. J. van den Heuvel, Frank distance and velocity estimates make our simulations more Verbunt, Yashwant Gupta, M. Vivekanand, and, particu- realistic than the previous studies. We also avoid selecting larly, Dipankar Bhattacharya for many helpful discussions the x- and y-position coordinates from a uniform distribu- and Ralph Wijers for valuable comments. The authors tion around the Sun (which was a doubtful assumption as acknowledge Anup Dewanji and Probal Chaudhuri for pointed out inB92), generate the coordinates from appro- useful comments and suggestions and for constructive criti- priate distributions as explained in° 2, and evolve them cism regarding the statistical analysis. The authors acknow- under the Galactic potential. We have used a more detailed ledge Professors Anil Gore and S. Kunte for helpful form of this potential(Lorimer 1995) than used in any of the discussions regarding statistics. S. M. would like to express previous simulations of a similar nature. We have also thanks to Partha Pramanik of Computer and Statistical taken into account Galactic rotation in solving the equa- Service Centre, Indian Statistical Institute, Calcutta for the tions of motion in order to model the pulsar position with cooperation extended during the initial stages of computa- as much accuracy as possible. The use of a two-dimensional tion and to the Inter-University Centre for Astronomy and Kolmogorov-Smirnov test to examine the distributions of Astrophysics for funding several visits during which this the pulsars in the B-P and PÈDM sin b planes provides a paper was completed. FIG. 5.ÈThe cumulative distribution of various properties as inFig. 4 but for q \ 10 Myr. The mismatch between observation and simulation for this low value of the magnetic Ðeld decay timescale is clearly seen.

FIG. 6.ÈDistribution of pulsars on the B-P plane. The dots represent FIG. 7.ÈComparision of pulsar distances obtained using the new dis- the simulated and selected pulsars, while the squares represent observed tance model and as inBhattacharya et al. (1992). The distances are in and selected pulsars. kiloparsecs. 940 MUKHERJEE & KEMBHAVI

REFERENCES Bailes,M. 1989, ApJ, 342, 917 Lyne,A. G., Manchester, R. N., & Taylor, J. H. 1985, MNRAS, 213, 613 Bhattacharya, D., & Srinivasan, G. 1991, in Neutron Stars: Theory and Manchester, R. N., Lyne, A. G., Taylor, J. H., Durdin, J. M., Large, M. I., & Observations, ed. J. Ventura & D. Pines (Dordrecht: Kluwer), 219 Little, A. G. 1978, MNRAS, 185, 409 Bhattacharya,D., & van den Heuvel, E. P. J. 1991, Phys. Rep., 203, 1 Mihalas, D., & Binney, J. 1981, Galactic Astronomy (San Francisco: Bhattacharya, D., Wijers, R. A. M. J., Hartman, J. W., & Verbunt, F. 1992, Freeman) A&A, 254,198 Narayan,R. 1987, ApJ, 319, 162 Carlberg,R. G., & Innanen, K. A. 1987, AJ, 94, 666 Narayan,R., & Ostriker, J. P. 1990, ApJ, 352, 222 Damashek,M., Taylor, J. H., & Hulse, R. A. 1978, ApJ, 225, L31 Narayan,R., & Vivekanand, M. 1983, A&A, 122, 45 Davies,J. G., Lyne, A. G., & Seiradakis, J. H. 1972, Nature, 240, 229 Ostriker,J. P., & Gunn, J. E. 1969, ApJ, 157, 1395 Dewey, R. J., Stokes, G. H., Segelstein, D. J., Taylor, J. H., & Weisberg, Peacock,J. A. 1983, MNRAS, 202, 615 J. M. 1984, Millisecond Pulsars, ed. S. P. Reynolds & D. R. Stinebring Press, X., et al. 1992, Numerical Recipes (2d ed; Cambridge: Cambridge (Greenbank: NRAO), 234 Univ. Press) Fasano,G., & Franceschini, A. 1987, MNRAS, 225, 155 Proszynski, M., & Przybycien, D. 1984, Millisecond Pulsars, ed. S. P. Fomalont, E. B., Goss, W. M., Lyne, A. G., Manchester, R. N., & Justta- Reynolds & D. R. Stinebring (Greenbank: NRAO), 234 nont, K. 1992, MNRAS, 258, 497 Ramachandran,R., & Deshpande, A. A. 1994, J. Astro. Astron., 15, 69 Gunn,J. E., & Ostriker, J. P. 1970, ApJ, 160, 979 Rankin,J. M. 1990, ApJ, 352, 247 Harrison,P. A., Lyne, A. G., & Anderson, B. 1993, MNRAS, 261, 113 Rawley,L. A., Taylor, J. H., & Davis, M. M. 1986, Nature, 319, 383 Hartman, J. W., Bhattacharya, D., Wijers, R. & Verbunt, F. 1997, A&A, in Romani,R. W. 1990, Nature, 347, 741 press Ruderman,M. A., & Sutherland, P. G. 1975, ApJ, 196, 51 Haslam, C. G. T., Salter, C. J., Sto†el, H., & Wilson, W. E. 1982, A&AS, 47, Sang,Y., & Chanmugam, G. 1987, ApJ, 323, L61 1 ÈÈÈ.1990, ApJ, 363, 597 Helfand,D., & Tademaru, E. 1977, ApJ, 216, 842 Stollman,G. M. 1986, A&A, 171, 152 Hulse,R. A., & Taylor, J. H. 1974, ApJ, 191, L59 Taylor,J. H., & Cordes, J. M. 1993, ApJ, 411, 674 Lorimer,D. 1995, Ph.D. thesis, Univ. of Manchester Taylor, J. H., Manchester, R. N., Lyne, A. G., & Camilo, F. 1995, Catalog Lyne,A. G., Anderson, B., & Salter, M. J. 1982, MNRAS, 201, 503 of 706 Pulsars (Princeton: Princeton Univ. Press) Lyne,A. G., & Lorimer, D. 1994, Nature, 369, 127 Taylor,J. H., & Stinebring, D. R. 1986, ARA&A, 24, 285 ÈÈÈ.1995, J. Astrophys. Astron., 16, 97 Wakatsuki,S., Hikita, A., Sato, N., & Itoh, N. 1992, ApJ, 392,628 Lyne,A. G., & Manchester, R. N. 1988, MNRAS, 234, 477