RADIO PULSAR STATISTICS

D. R. LorimerUniversity of ManchesterJodrell Bank Observatory

Duncan.Lorimer@manchester.ac.uk

Abstract Radio pulsars are a major probe of the Galactic neutron star population and itsevolution. Our attempts to derive the properties of the underlying, as opposed tothe observed, population of radio pulsars in our Galaxy have gradually improvedover the years thanks to large-scale surveys and more detailed simulations of thepopulation. This review attempts to summarize our current state of knowledge inthis area. We discuss the currently observed pulsar population, selection effectsand correction techniques used to deduce the Galactic distribution and birth rate.Two outstanding problems in pulsar statistics are then reviewed: the period evo-lution of normal pulsars and the number of isolated ‘recycled’ pulsars. Finally,an exciting new search project with the Arecibo telescope is described alongwith the ultimate future pulsar survey with the Square Kilometre Array.

Keywords: stars — neutron; methods — statistical

1. Basic properties and evolutionary ideas

Thirty-seven years after the discovery of radio pulsars by Jocelyn Bell andAntony Hewish at Cambridge in 1967 (Hewish et al. 1968), the observed pop-ulation exceeds 1600 objects with spin periods in the range 1.5 ms to 8.5 s.Pulsar astronomy is currently enjoying a golden era, with over half of thesediscoveries in the past five years due largely to the phenomenal success of theParkes multibeam survey (Manchester et al. 2001). From the sky distribution inGalactic coordinates shown in Figure 1, it is immediately apparent that pulsarsare concentrated strongly along the Galactic plane. This is particularly strik-ing for the youngest pulsars known to be associated with supernova remnants.Also shown in Figure 1 are the millisecond pulsars which have spin periodsin the range 1.5–30 ms. The more isotropic sky distribution of the millisec-ond pulsars does not imply that they have a different spatial distribution; thedifference simply reflects the observational bias against detecting short-periodpulsars with increasing distance from the Sun. This is one of many selectioneffects that pervades the observed sample.

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From such a violent birth in supernovae, it is perhaps not surprising to learnthat pulsars are high-velocity objects. The right-hand panel of Figure 1 showspulsar proper motions on the plane of the sky taken from a recent study byHobbs et al. (in preparation). The mean transverse speed of the current sampleof 233 pulsars is

�����������km s � . From a sample of proper motions for

pulsars younger than 3 Myr, Hobbs et al. find the mean 3-D velocity of pulsarsto be �� � � �� km s � . This implies that pulsars receive an impulsive ‘kick’ ofseveral hundred km s � at birth. The origin of these kicks probably lies in smallasymmetries in the supernova explosions (Fryer, this volume). Millisecondpulsars have significantly lower space velocities; their mean transverse speedis only

���������km s � , while a study by Lyne et al. (1998) showed the mean 3-

D speed to be��� ��� km s � . Despite these differences, population syntheses

indicate that the two populations are consistent with the idea that all neutronstars share the same velocity distribution (Tauris & Bailes 1996).

Figure 1. Left: the distribution of pulsars in Galactic coordinates. Pulsar–supernova remnantassociations and millisecond pulsars are shown by the filled and open circles respectively. Right:pulsar proper motions in Galactic coordinates (provided by George Hobbs). The solid linesshow the proper motion (neglecting the unknown radial velocity) over the last million years.

The observed emission from radio pulsars takes place at the expense of therotational kinetic energy of the neutron star. As a result, in addition to observ-ing the pulsar’s spin period, � , we also observe the corresponding rate of spin-down,

�� . Such measurements give us unique insights into the spin evolution

of neutron stars and are summarized on the � –�� diagram shown in Figure 2.

The diagram contrasts the normal pulsars ( ���� ���� s and���� � ���� s s �

which populate the ‘island’ of points) and the millisecond pulsars ( � � � msand��!� � #"%$ s s � which occupy the lower left part of the diagram).

The differences in � and�� imply fundamentally different ages and mag-

netic field strengths for the two populations. Considering the spin evolutionof the neutron star to be a due to magnetic dipole radiation, the inferred age&(' �*)

�� and magnetic field strength + '-, �

��/. 102" . Lines of constant + and& are drawn on Figure 2 from which we infer typical magnetic fields and ages

of� �" G and

� �3 yr for the normal pulsars, and� 54 G and

� �6 yr for the mil-

Radio Pulsar Statistics 3

Figure 2. The � –�� diagram showing radio pulsars, ‘radio-quiet’ pulsars, soft-gamma re-

peaters (SGRs) and anomalous X-ray pulsars (AXPs). Figure provided by Michael Kramer.

lisecond pulsars. The rate of loss of rotational kinetic energy�� ' ��/) ��� (also

known as the ‘spin-down luminosity’) is also indicated. As expected, these arehighest for the young and millisecond pulsars.

In addition to spin behaviour, a very important additional difference betweennormal and millisecond pulsars is binarity. Orbiting companions are observed

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around about 80% of all millisecond pulsars but less than 1% of all normal pul-sars. The companions are either white dwarfs, main sequence stars, or otherneutron stars. Pulsars with low-mass companions ( � ���� M � – predominantlywhite dwarfs) usually have millisecond spin periods and essentially circularorbits with orbital eccentricities in the range

� ##������� � �� . Measure-ments of white-dwarf ‘cooling ages’ (see van Kerkwijk 1996) agree generallywith millisecond pulsar characteristic ages and support the idea that these bi-nary systems have typical ages of a few Gyr. Binary pulsars with high-masscompanions ( � � M � – neutron stars or main sequence stars) have larger spinperiods ( � � ms) and are in more eccentric orbits: �� � ����� �� .

The existence of binary pulsars can be understood by a simple evolution-ary scenario which starts with two main-sequence stars (see Bhattacharya &van den Heuvel 1991). The initially more massive (primary) star evolves firstand eventually explodes in a supernova to form a neutron star. The high ve-locity imparted to the neutron star at birth and dramatic mass loss during thesupernova usually is sufficient to disrupt most (90% or more) binary systems(Radhakrishnan & Shukre 1985). Those neutron stars remaining bound to theircompanions spin down as normal pulsars for the next

� �� 3 yr. Later on, theremaining (secondary) star comes to the end of its main sequence lifetime andbegins a red giant phase. For favourable orbital parameters, the strong gravita-tional field of the neutron star attracts matter from the red giant and forms anaccretion disk. As a result, the system becomes visible as an X-ray binary.

The accretion of matter transfers orbital angular momentum to the neutronstar, spinning it up to short periods and dramatically reducing its magneticfield (Bisnovatyi-Kogan & Kronberg 1974; Shibazaki et al. 1989). A limitingspin period is reached due to equilibrium between the magnetic pressure of theaccreting neutron star and the ram pressure of the infalling matter (Ghosh &Lamb 1979; Arzoumanian et al. 1999; Lamb & Yu 2004). Such ‘spun-up’ neu-tron stars are often referred to in the literature as recycled pulsars. Unlike theyoung pulsars with high spin-down rates, the now weakly-magnetized recycledpulsars appear in the lower-left hand part of the � –

�� diagram and spin down

much more gradually and over a longer timescale.

2. The observed pulsar spatial distribution

Pulsar astronomers are extremely fortunate in that they have a reasonablyaccurate means of estimating distances to their objects from measurementsof pulse dispersion caused by free electrons in the interstellar medium (seeWeisberg 1996 and also the contribution by D’Amico in these proceedings).In Figure 3, the most recent Galactic electron density model (Cordes & Lasio2002) is used to project the current sample of pulsars in the ATNF catalogue( � � ������������������ ��!��"� #%$��'&���&����%�)(%$+*�#�,��-� ��$+*.�/�%�-�-� ) onto the Galactic plane.

Radio Pulsar Statistics 5

Two main features can be seen in this diagram: (a) pulsar positions trace thespiral-arm structure of our Galaxy (spiral arms are now incorporated into themodel); (b) rather than being distributed about the Galactic centre, the majorityof pulsars are clearly biased towards the bright/nearby objects.

Figure 3. Left: The currently known pulsar population projected onto the Galactic plane. TheGalactic centre is at the origin and the Sun is at (0.0,8.5) kpc. Right: Cummulative distributionas a function of projected distance from the Sun. The solid line is the observed sample while thedashed line is the expected distribution of a simulated population free from selection effects.

To get an idea of how biased the sample is, the right panel of Figure 3 showsthe cummulative distribution of pulsars as a function of distance from the Sunprojected onto the Galactic plane. Also shown is the expected distribution fora simulated population in which there are no selection effects. As can be seen,the two samples are closely matched only out to a kpc or so before the selectioneffects become significant. From these curves, we deduce that less than 10%of the potentially observable population in the Galaxy is currently detectable.

3. Selection effects in pulsar surveys

The inverse square law. Like all astronomical sources, observed pulsars ofa given luminosity � are strongly selected by their apparent flux density, � .For a pulsar at a distance � from the Earth which beams to a certain fraction�

of ��� sr, ����� � � ��������� . Since all pulsar surveys have some limiting fluxdensity, only those objects bright or close enough will be detectable. Note thatin the absence of prior knowledge about beaming, geometrical factors are usu-ally ignored and the resulting ‘pseudoluminosity’ is quoted at some standardobserving frequency; e.g., at 1400 MHz, ������������������������� .

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The radio sky background. A fundamental sensitivity limit is the systemnoise temperature, ������� . While every effort is made to minimize this at the tele-scope, synchroton radiating electrons in the Galactic magnetic field contributesignificantly with a ‘sky background’ component, � ����� . At observing frequen-cies � �� � GHz, � ����� dominates � ����� along the Galactic plane. Fortunately,� ����� ' #"� 4 so this effect is significantly reduced when � �� � GHz.

Emitted Pulse Detected Pulse

Pulsar Telescope

Figure 4. Left: Pulse scattering by irregularities in the ISM. Right: A simulation showing thefraction of pulsars undetectable due to scattering as a function of observing frequency.

Propagation effects in the interstellar medium (ISM). Dispersion and scatter-broadening of the pulses in the ISM hamper detection of short period and/ordistant objects. The effects of scattering are shown in Figure 4. Fortunately,like � ����� , the scatter-broadening time & �� ������ has a strong frequency dependence,scaling roughly as � � . Figure 4 shows that for survey frequencies below 1GHz, scattering ‘hides’ a large fraction of the population. Additionally, scin-tillation, the diffractive and refractive modulation of apparent flux densities byturbulences in the ISM (Rickett 1970) affects pulsar detection. For example,two northern sky surveys carried out 20 years apart with comparable sensi-tivity (Damashek et al. 1978; Sayer et al. 1997) detected a number of pulsarsabove and below the nominal search thresholds of one experiment but not theother. Surveying the sky multiple times minimizes the effects of scintillationand enhances the detection of faint pulsars through favourable scintillation.

Finite size of the emission beam. The fact that pulsars do not beam to ���sr means that we see only a fraction � of the total active population. For acircular beam, Gunn & Ostriker (1970) estimated � � � )�� . A consensuson the precise shape of the emission beam has yet to be reached. Narayan& Vivekanand (1983) argued that the beams are elongated in the meridionaldirection. Lyne & Manchester (1988), on the other hand, favour a circularbeam. Using the same database, Biggs (1990) presented evidence in favour ofmeridional compression! All these studies do agree that the beam size is period

Radio Pulsar Statistics 7

dependent, with shorter period pulsars having larger beaming fractions. Tauris& Manchester (1998) found that ��� �� - ������� , �*) .�� �� "�� �� � , where � isthe period. A complete model for � needs to account for other factors, such asevolution of the inclination angle between the spin and magnetic axes.

Pulse nulling. The abrupt cessation of the pulsed emission for many pulseperiods, was first identified by Backer (1970). Ritchings (1976) presented evi-dence that the incidence of nulling became more frequent in older long-periodpulsars, suggesting that it signified the onset of the final stages of the neutronstar’s life as an active radio pulsar. Since most pulsar surveys have short ( �few min) integration times, there is an obvious selection effect against nullingobjects. Means of overcoming this effect are to look for individual pulses insearch data (Nice 1999), survey the sky many times, or use longer integrations.

4. Techniques to correct for observational selection

From an observationally-biased sample, we seek to characterise the under-lying population accounting for the aforementioned selection effects. For agiven survey of integration time, & , and bandwidth, � , the quantity

������� � � ��� ��� �) �� & (1)

is the limiting sensitivity to pulsars of a certain period, � , and pulse width,�, given an antenna with gain,

�, and system temperature, � ��� � . For further

details, see the review by D’Amico in this volume. For a given�������

, then, thereis a maximum detection volume � � ��� � , � ) ������� . � 02" to pulsars of luminosity,� . This idea is used to correct the sample in two ways described below.

Population inversion techniques

The first method, originally developed by Large (1971), is of particular in-terest to determine the spatial distribution of the parent population. Given theobserved distribution ! , �#"%$&"%'(" � . in terms of period, � , distance from theGalactic plane, $ , Galactocentric radius, ' , and luminosity, � , we may write) ! , �#"%$&"%'(" � .*�+� , �#"%$&"%'(" � .-, , �#"%$&"%'(" � . ) � ) $ ) ' ) � " (2)

where � is the volume of the Galaxy effectively searched and , is the underly-ing (true) distribution of the population. Since we know ! and can estimate �on the basis of pulsar survey sensitivities, we can invert equation (2) to solvefor , . The only simplification required to do this is to assume that � , $ , ' and� are independent quantities. Fortunately, apart from a very weak couplingbetween � and $ , there are no significant relationships between any of theseparameters. The problem then reduces to four equations which can be solvedfor the underlying distributions: ,/. , ��. , ,10 , $ . , ,12 , ' . and ,13 , � . .

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Galactocentric radius (kpc) Galactocentric radius (kpc)

Figure 5. Left: the observed radial distribution and corrected Galactic radial density function������� as derived by Lyne, Manchester & Taylor (1985). Right: corrected radial density functionsproposed by Narayan (1987), Lorimer (2004) and Yusifov & Kucuk (2004).

Of particular interest is ,&2 , the underlying radial pulsar density. For manyyears, the standard reference for ,/2 was Lyne, Manchester & Taylor (1985)As can be seen from the results of this work shown in Figure 5, the form of,12 at small ' is poorly constrained. These results were approximated in mostsubsequent work using a Gaussian distribution for , 2 (e.g. Narayan 1987). Asthe left panel of Figure 5 clearly shows, there is no reason to prefer a Gaus-sian over a function which tends to zero at small ' . Using the results of theParkes multibeam survey, which has discovered many more pulsars in the innerGalaxy, I revisited this method recently (Lorimer 2004) and found strong evi-dence in favour of a non-Gaussian radial distribution. This result substantiatesearlier work by Johnston (1994) and more recently Yusifov & Kucuk (2004).Whether the deficit in the inner Galaxy is a real effect is not yet clear.

Scale factors and pulsar current

If the form of the Galactic distribution is known, a related approach, pio-neered by Phinney & Blandford (1981) and Vivekanand & Narayan (1981),can be used to estimate the pulsar birth rate. The method involves binning thepopulation in period and computing the flow or ‘current’ of pulsars

� , ��.*� �� �

��� � �� ���������� (3)

Here, ��� ��� is the number of pulsars in a period bin of width � � ,

���is the

‘scale factor’ and ��

is the beaming fraction of the � ��� pulsar. As discussedearlier, �

�is based on some beaming model. For a given pulsar, its scale fac-

tor

� �represents the number of pulsars with similar parameters in the Galaxy

and is computed using a Monte Carlo simulation of ! pulsars with identical

Radio Pulsar Statistics 9

periods and luminosities. Using accurate models for the various pulsar sur-veys, it is relatively straightforward to calculate the number of pulsars � thatare detectable from that population. As a result,

��� ! ) � .

The beauty of the pulsar-current analysis is that it makes just two funda-mental assumptions about pulsars: (i) they are a steady-state population; (ii)they are spinning down steadily from short to long periods. The first of theseis justified since the ages of pulsars, while not well known (

� 3 4 yr), are cer-tainly less than the age of the Galaxy,

� �$ yr. The second is, of course, wellin accord with timing observations. The birth-rate can be computed from thisanalysis by simply plotting

�as a function of � . In the standard model where

pulsars are born spinning rapidly, there should be a peak in the current at shortperiods followed by a decline in the current as pulsars end their life with longerperiods. The birth rate is then just the height of this peak.

Somewhat controversially, the first such analysis by Vivekanand & Narayan(1981) found a step function at � � ���� s in their distribution of

�. This

was claimed as evidence for an ‘injection’ of pulsars into the population with� � ���� s The most recent analysis of this kind was carried out by Vranesevicet al. (2005) using a sample of 815 normal pulsars from the Parkes multibeamsurvey. The total birthrate of the population was found to lie between 1–2pulsars per century for 1400-MHz luminosities above 1 mJy kpc " . Dividingthe population into groups according to magnetic field strength, Vranesevicet al. found that over half of the total birthrate is contributed by pulsars withfields � � ��� � � �" G. This is in spite of the fact that such pulsars make up lessthan 30% of the observed sample and, based on their scale factors, only about5–10% of the total population. While no evidence was found for a significantpopulation of pulsars injected into the population with intermediate spin peri-ods, the observed distribution of pulsar current is consistent with up to 40% ofall pulsars being born with periods in the range 0.1–0.5 s.

5. Some of the many outstanding problems

In view of the difficulties in correcting for these selection effects, and the in-herent problem of small-number statistics, many controversies have pervadedpulsar statistics over the years. I review here only two topics: the period evo-lution of normal pulsars and the apparent paucity of isolated recycled pulsars.

Period evolution and field decay of isolated pulsars

The classic model for spin-down of an isolated pulsar is to write the brakingtorque as a generalized power law. For an angular velocity

� � � � ) � , theequation of motion is given by

�� ��� � � , where � is proportional to thebraking torque and � is the so-called braking index. For a constant value of �

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and pure magnetic dipole braking � � � , the equation of motion on the � –��

diagram is such that pulsars follow a slope of –1 in a log-log plot like Figure 2.The dipolar braking hypothesis can be tested for a handful of young pulsars,

where timing measurements provide � . So far, all 6 measured values of � areconsistent with a flat distribution in the range 1.4–2.9. In other words, all of thepulsars with measured values of � are moving along lines with slopes greaterthan –1 on the � –

�� diagram. When these vectors are plotted (see, for example

Lyne 2004) one sees that the directions these young pulsars are moving wouldplace them above the pulsar island! So the conundrum is, either the pulsars inthe island have a different set of progenitors than the young objects, or there issome evolution in the braking index as a function of time.

The evolution in braking index can either be provided by integrating theequation of motion assuming that � is genuinely a function of time, or that� decays with time. In all simulations of the � –

�� plane that I am aware

of to date, the shape of the diagram is reproduced by modeling the evolutionof � with time. Excellent fits to the observed diagrams (see, for example,Figure 8 in Gonthier et al. 2004) can be obtained by decay laws of the form� ,�� . '������ , � � ) �� . for decay times �� of a few million years. This is usuallyinterpreted as exponential decay of the magnetic moment of the neutron staron a timescale of a few million years. While earlier versions of these simula-tions were criticised by van Leeuwen (2004) as not taking into account perioddependent beaming, the work of Gonthier et al. (2004) does, I believe, accountfor this effect and still prefers a short magnetic field decay time.

Despite the good agreement on the � –�� plane, there are a number of vexing

issues: (a) spontaneous decay of the magnetic field on such short timescalesis inconsistent with the observations of millisecond pulsars which have Gyrages and yet field strengths at the level of

� 4 G; (b) the exponential model isinconsistent with all braking index measurements, since it always predicts aneffective ��� � ; (c) in principle, the same behaviour could be reproduced bymodeling the evolution of � rather than field decay; (d) what is the ultimatefate of low-braking-index pulsars? For example, the Vela pulsar has � � � � �(Lyne et al. 1996) and is moving towards the magnetars on the � � �� diagram,rather than the pulsar island. Lyne (2004) proposed that such objects might bethe progenitors of the magnetars. This idea requires further investigation.

Where are all the isolated ‘recycled’ pulsars?

The discovery of new pulsars often sheds light on previously unseen areas ofthe neutron star ‘zoo’ which likely represent quite rare evolutionary processes.One example is the discovery of two isolated pulsars J2235+1506 (Camilo etal. 1993) and J0609+2130 (Lorimer et al. 2004) with spin properties similarto the double neutron star binaries. Camilo et al. suggested that J2235+1506

Radio Pulsar Statistics 11

might be the remains of a high-mass binary system that disrupted during thesecond supernova explosion.

Is this hypothesis consistent with the observations? One way to test this is toconsider the fraction, � , of binary systems that remain bound after the secondsupernova explosion. Numerous authors have followed the orbital evolutionof a wide variety of binary systems containing neutron stars using detailedMonte Carlo simulations. For example, Portegies Zwart & Yungelson (1998)find � ����� . We therefore expect for each double neutron star system weobserve to find of order 20 systems which disrupted. Currently we know of 8double neutron star binaries. Why, then, do we not see of order 160 pulsars likeJ0609+2130 or J2235+1506? This currently outstanding problem may indicatea different evolutionary scenario for these objects and warrants further study.

6. Current and future pulsar search projects

Pulsar astronomy is currently enjoying the most productive phase of its his-tory, with applications providing a wealth of new information about compact-object astrophysics, general relativity, the Galactic magnetic field, the interstel-lar medium, binary evolution, planetary physics and even cosmology. Our un-derstanding of the Galactic pulsar population has improved dramatically thankslargely to the success of the Parkes multibeam survey. Like our colleagues inother parts of the electromagnetic spectrum, radio astronomers are highly ac-tive in a number of areas which will bring new advances in sensitivity.

The first of these is the Arecibo L-band feed array project (ALFA), a seven-beam system which is currently in active use in large-scale surveys for pul-sars and neutral hydrogen. The excellent sensitivity of Arecibo means thatALFA will be able to probe much deeper into the Galaxy than was possiblein the Parkes surveys. A simulation of a survey with ALFA that is currentlyin progress along a narrow strip of the Galactic plane ( � ���.� ��� ) shows that itshould discover over 200 pulsars. Over the next 5–10 years, as surveys withALFA extend to higher latitudes, we can reasonably expect to detect of order1000 pulsars. This sample the population will provide a further quantum leapto pulsar statistical analyses and sample to the edge of the Galaxy.

The ALFA system, however, is only a precursor for what might be possiblewith the Square Kilometre Array (SKA), an ambitious world-wide collabora-tion currently planned for the year 2020 (see �����-��&�,�&�����!�* & �"!-� ). Simula-tions for pulsar surveys with this instrument demonstrate that the increase insensitivity of the SKA (around two orders of magnitude over current radio tele-scopes!) would mean that essentially every Galactic pulsar beaming towardsus (of order 30,000 objects!) could be detectable. Perhaps by the year 2030,the sample of radio pulsars will be finally free of selection effects.

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Acknowledgments

I wish to thank the Royal Society and NATO for supporting my attendance atthis meeting, and to the organizers for putting together an excellent program.

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