Binary and Millisecond Pulsars at the New Millennium
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We review the properties and applications of binary and millisecond pulsars. Our knowledge of these exciting objects has greatly increased in recent years, mainly due to successful surveys which have brought the known pulsar population to over 1300. There are now 56 binary and millisecond pulsars in the Galactic disk and a further 47 in globular clusters. This review is concerned primarily with the results and spin-offs from these surveys which are of particular interest to the relativity community.
KeywordsNeutron Star Gravitational Wave White Dwarf Globular Cluster Radio Pulsar
In the 34 years that have elapsed since the discovery  of pulsars, rapidly rotating highly magnetised neutron stars, the study of these fascinating objects has resulted in many applications in physics and astronomy. Striking examples include the confirmation of the existence of gravitational radiation  as predicted by general relativity [239, 240] and the first detection of an extra-solar planetary system [273, 186].
1.1 What’s new in this review?
Since the first version of this article was written back in 1997/8  a number of pulsar surveys using the Parkes radio telescope  have discovered almost 700 pulsars. As a result, the sample size is now double what it was in 1997. Many of the exciting new discoveries from these searches are discussed in this review. Up-to-date tables of parameters of binary and millisecond pulsars are included as an appendix. Several new sections/figures have been added and existing sections reworked and modularized to make the review more self-contained and (hopefully!) easier to read in an html setting. We begin in § 2 with an overview of the pulsar phenomenon, the key observed population properties, the origin and evolution of pulsars and an introduction to pulsar search techniques. In § 3, we review present understanding in pulsar demography, discussing selection effects and the techniques used to correct for them in the observed sample. This leads to robust estimates of the total number of normal and millisecond pulsars (§ 3.3) and relativistic binaries (§ 3.4) in the Galaxy and has implications for the detection of gravitational radiation from these systems. We discuss pulsar timing in § 4. One application of these exceptional clocks, a sensitive detector of long-period gravitational waves, is discussed in § 5. We conclude with a brief outlook to the future in § 6.
2 An Introduction to Pulsar Astronomy
Many of the basic observational facts about radio pulsars were established shortly after their discovery  by Bell and Hewish in 1967. In the intervening years, theoretical and observational progress has flourished. Although there are many remaining questions, particularly about the emission mechanism, the basic model has long been established beyond all reasonable doubt, viz.: Pulsars are rapidly rotating, highly magnetised neutron stars formed during the supernova explosions of massive (∼ 5–10 M⊙) stars. In the following, we discuss the basic observational properties most relevant to this review.
2.1 The lighthouse model
As the neutron star spins, charged particles are accelerated out along magnetic field lines in the magnetosphere (depicted by the light blue cones). This acceleration causes the particles to emit electromagnetic radiation, most readily detected at radio frequencies as a sequence of observed pulses produced as the magnetic axis (and hence the radiation beam) crosses the observer’s line of sight each rotation. The repetition period of the pulses is therefore simply the rotation period of the neutron star. The moving “tracker ball” on the pulse profile in the animation shows the relationship between observed intensity and rotational phase of the neutron star.
Neutron stars are extremely stable rotators. They are essentially large celestial flywheels with moments of inertia ∼ 1045 g cm2. The rotating neutron star model, independently developed by Pacini and Gold in 1968 [184, 85], predicts a gradual increase in the pulse period as the outgoing radiation carries away rotational kinetic energy. This model became universally accepted when a period increase of 36.5 ns per day was measured for the pulsar in the Crab nebula , enabling Gold  to show that a rotating neutron star with a large magnetic field must be the dominant energy supply for the nebula.
2.2 Pulse profiles
Pulsars are weak radio sources. Measured flux densities, usually quoted in the literature for a radio frequency of 400 MHz, vary between 0.1 and 5000 mJy (1 Jy≡10-26 W m-2 Hz-1). This means that, even with a large radio telescope, the coherent addition of many thousands of pulses is required in order to produce an integrated profile. Remarkably, although the individual pulses vary dramatically from pulse to pulse, at any particular observing frequency the integrated profile is very stable. The pulse profile can thus be thought of as a fingerprint of the emission beam.
2.3 The pulsar distance scale
The electron density model is calibrated from the pulsars with independent distance estimates and measurements of scattering for lines of sight towards various Galactic and extragalactic sources. Independent distance estimates now exist for over 100 pulsars based on three basic techniques: neutral hydrogen absorption, trigonometric parallax (measured either with an interferometer or through pulse time-of-arrival techniques) and from associations with objects of known distance (i.e. supernova remnants, globular clusters and the Magellanic Clouds). Based on these data, Taylor & Cordes  have developed an electron density model which is free from large systematic trends and can be used to provide distance estimates with an uncertainty of ∼30%. However, use of this model to estimate distances to individual pulsars may result in uncertainties by as much as a factor of two. This model is currently being refined following recent pulsar discoveries and independent distance and scattering measurements (J. Cordes, private communication).
2.4 Normal and millisecond pulsars
2.4.1 Spin parameters
The present public-domain catalogue, available on-line at Princeton , contains up-to-date parameters for 706 pulsars. Parameters for many of the new Parkes multibeam pulsar surveys are also available on-line [28, 197]. Most of these are “normal” in the sense that their pulse periods P are of order one second and are observed to increase secularly at rates Ṗ of typically 10-15 s/s. A growing fraction of the observed sample are the so-called “millisecond pulsars”, which have spin periods primarily in the range 1.5 and 30 ms and rates of slowdown ≲10-19 s/s. The first millisecond pulsar discovered, B1937+21 , with P=1.5578 ms, remains the most rapidly rotating neutron star presently known.
2.4.2 Binary companions
As can be inferred from Fig. 1, just under 4% of all known pulsars in the Galactic disk are members of binary systems. Timing measurements (§ 4) place useful constraints on the masses of the companions which, supplemented by observations at other wavelengths, tell us a great deal about their nature. The present sample of orbiting companions are either white dwarfs, main sequence stars, or other neutron stars. Two notable hybrid systems are the “planet pulsars” PSR B1257+12 and B1620–26. B1257+12 is a 6.2-ms pulsar accompanied by at least three Earth-mass bodies [273, 186, 272] while B1620—26, an 11-ms pulsar in the globular cluster M4, is part of a triple system with a white dwarf and a high-mass planet [247, 15, 245].
2.4.3 Evolutionary scenarios
Starting with a binary star system, a neutron star is formed during the supernova explosion of the initially more massive star which has an inherently shorter main sequence lifetime. From the virial theorem it follows that the binary system gets disrupted if more than half the total pre-supernova mass is ejected from the system during the explosion [98, 31]. In addition, the fraction of surviving binaries is affected by the magnitude and direction of any impulsive “kick” velocity the neutron star receives at birth [98, 18]. Those binary systems that disrupt produce a high-velocity isolated neutron star and an OB runaway star . The high binary disruption probability during the explosion explains, qualitatively at least, why so few normal pulsars have companions. Over the next 107 yr or so after the explosion, the neutron star may be observable as a normal radio pulsar spinning down to a period ≳ several seconds. After this time, the energy output of the star diminuishes to a point where it no longer produces significant radio emission.
For those few binaries that remain bound, and in which the companion is sufficiently massive to evolve into a giant and overflow its Roche lobe, the old spun-down neutron star can gain a new lease of life as a pulsar by accreting matter and therefore angular momentum at the expense of the orbital angular momentum of the binary system . The term “recycled pulsar” is often used to describe such objects. During this accretion phase, the X-rays produced by the liberation of gravitational energy of the infalling matter onto the neutron star mean that such a system is expected to be visible as an X-ray binary system. Two classes of X-ray binaries relevant to binary and millisecond pulsars exist, viz. neutron stars with high-mass or low-mass companions. For a detailed review of the X-ray binary population, including systems likely to contain black holes rather than neutron stars, the interested reader is referred to .
The high-mass companions are massive enough to explode as a supernova, producing a second neutron star. If the binary system is lucky enough to survive this explosion, it ends up as a double neutron star binary. The classic example is PSR B1913+16 , a 59-ms radio pulsar with a characteristic age of ∼108 yr which orbits its companion every 7.75 hr [239, 240]. In this formation scenario, PSR B1913+16 is an example of the older, first-born, neutron star that has subsequently accreted matter from its companion. So far there are no clear examples of systems where the second-born neutron star is observed as a radio pulsar. In the case of PSR B1820-11 , which may be an example, the mass of the companion is not well determined, so either a main-sequence  or a white dwarf companion  are plausible alternatives. This lack of observation of second-born neutron stars as radio pulsars is probably reasonable when one realises that the observable lifetimes of recycled pulsars are much larger than those of normal pulsars. As discussed in § 3.4.1, double neutron star binary systems are very rare in the Galaxy — another indication that the majority of binary systems get disrupted when one of the components explodes as a supernova. Systems disrupted after the supernova of the secondary form a mildly-recycled isolated pulsar and a young pulsar formed during the explosion of the secondary.
Although no system has so far been found in which both neutron stars are visible as radio pulsars, timing measurements of three systems show that the companion masses are 1.4M⊙ — as expected for neutron stars . In addition, no optical counterparts are seen. Thus, we conclude that these unseen companions are neutron stars that are either too weak to be detected, no longer active as radio pulsars, or their emission beams do not intersect our line of sight. The two known young radio pulsars with main sequence companions massive enough to explode as a supernova probably represent the intermediate phase between high-mass X-ray binaries and double neutron star systems [110, 115].
The companions in the low-mass X-ray binaries evolve and transfer matter onto the neutron star on a much longer time-scale, spinning it up to periods as short as a few ms . This model has gained strong support in recent years from the discoveries of quasi-periodic kHz oscillations in a number of low-mass X-ray binaries , as well as Doppler-shifted 2.49-ms X-ray pulsations from the transient X-ray burster SAX J1808.4-3658 [267, 53]. At the end of the spin-up phase, the secondary sheds its outer layers to become a white dwarf in orbit around a rapidly spinning millisecond pulsar. Presently ∼10 of these systems have compelling optical identifications of the white dwarf companion [25, 27, 139, 138]. Perhaps the best example is the white dwarf companion to the 5.25-ms pulsar J1012+5307 [177, 133]. This 19th magnitude white dwarf is bright enough to allow measurements of its surface gravity and orbital velocity .
The range of white dwarf masses observed is becoming broader. Since this article originally appeared in 1998, the number of “intermediate-mass binary pulsars”  has grown significantly . These systems are distinct to the “classical” millisecond pulsar-white dwarf binaries like PSR J1012+5307 in several ways: (1) the spin period of the radio pulsar is generally longer (9–200 ms); (2) the mass of the white dwarf is larger (typically close to 1M⊙); (3) the orbital eccentricity, while still essentially circular, is often significantly larger (∼10-3). It is not presently clear whether these systems originated from either low- or high-mass X-ray binaries. It was suggested by van den Heuvel  that they have more in common with high-mass systems, the difference being that the secondary star was not sufficiently massive to explode as a supernova. Instead it formed a white dwarf. Detailed studies of this sub-population of binary pulsars are required for further understanding in this area.
Another relatively poorly understood area is the existence of solitary millisecond pulsars in the Galactic disk (which comprise just under 20% of all Galactic millisecond pulsars). Although it has been proposed that the millisecond pulsars have got rid of their companion by ablation, as appears to be happening in the PSR B1957+20 system , it is not clear whether the time-scales for this process are feasible. There is some observational evidence that suggests that solitary millisecond pulsars are less luminous than binary millisecond pulsars [20, 122]. If confirmed by future discoveries, this would need to be explained by any viable evolutionary model.
2.4.4 Space velocities
Pulsars have long been known to have space velocities at least an order of magnitude larger than those of their main sequence progenitors, which have typical values between 10 and 50 km s-1. The first direct evidence for large velocities came from optical observations of the Crab pulsar in 1968 , showing that the neutron star has a velocity in excess of 100 km s-1. Proper motions for about 100 pulsars have subsequently been measured largely by radio interferometric techniques [142, 21, 78, 92]. These data imply a broad velocity spectrum ranging from 0 to over 1000 km s-1 .
Based on the proper motion data, recent studies have demonstrated that the mean birth velocity of normal pulsars is ∼450 km s-1 ([148, 132, 58, 84]; see, however, also [93, 91]). This is significantly larger than the velocities of millisecond and binary pulsars. Recent studies suggest that their mean birth velocity is likely to be in the range ∼80–140 km s-1 [129, 57, 152]. The main reason for this difference surely lies in the fact that about 80% of the millisecond pulsars are members of binary systems (§ 2.4) which could not have survived had the neutron star received a substantial kick velocity.
2.5 Searching for pulsars
Pulsar searching is, conceptually at least, a rather simple process — the detection of a dispersed, periodic signal hidden in a noisy time series collected using a large radio telescope. The search strategy is to form a large number of time series for different trial DM values. These data are then analysed for periodic signals. We give here a brief description of the basic search techniques. More detailed discussions can be found elsewhere [141, 178, 130].
After the data have been processed for a suitable range of DM, a list of pulsar candidates is compiled and the raw time series data are folded modulo each candidate period. In practise the analysis is often hampered by the presence of periodic interference sources which can often look very “pulsar-like”. Although interference excision schemes (usually based on coincidence analyses of data taken from different points on the sky) work fairly well, interference is an ever-increasing problem in radio astronomy and considerable efforts are required to carry out sensitive searches.
2.6 Where to look for binary and millisecond pulsars
Following the above discussions on demography and evolution, it is instructive to briefly summarize the rationale behind the major searches being carried out at the present time.
2.6.1 All-sky searches
The oldest radio pulsars form a virialised population of stars oscillating in the Galactic gravitational potential. The scale height for such a population is at least 500 pc, about 10 times that of the massive stars which populate the Galactic plane. Since the typical ages of millisecond pulsars are several Gyr or more, we expect, from our vantage point in the Galaxy, to be in the middle of an essentially isotropic population of nearby sources. All-sky searches for millisecond pulsars at high Galactic latitudes have been very effective in probing this population. Much of the initial interest and excitement in this area was started at Arecibo when Wolszczan discovered two classic recycled pulsars at high latitudes: the neutron star binary B1534+12  and the planets pulsar B1257+12 . Surveys carried out at Arecibo, Parkes, Jodrell Bank and Green Bank by others in the 1990s found many other millisecond and recycled pulsars in this way. Camilo has written several excellent reviews of these surveys [41, 44, 45]. See also Tables 2, 3 and 4 in the appendix.
2.6.2 Searches close to the plane of our Galaxy
Young pulsars are most likely to be found near to their place of birth, close to the Galactic plane. This is the target region of one of the Parkes multibeam surveys and has already resulted in the discovery of around 600 new pulsars [48, 159], almost half the number currently known! Such a large haul inevitably results in a number of interesting individual objects such as: PSR J1141-6545, a young pulsar in a relativistic 4-hr orbit around a white dwarf ; PSR J1740-3052, a young pulsar orbiting an ∼11M⊙ star (probably a giant ); several intermediate-mass binary pulsars  and a likely double neutron star system, PSR J1811-1736 .
2.6.3 Searches at intermediate Galactic latitudes
In order to probe more deeply into the population of millisecond and recycled pulsars than possible at high Galactic latitudes, Edwards et al. have recently completed a survey of intermediate latitudes with the Parkes multibeam system [74, 75]. The results of this survey are extremely exciting — 58 new pulsars including 8 relatively distant recycled objects. Two of the new recycled pulsars from this survey recently announced by Edwards & Bailes  are mildly relativistic neutron star-white dwarf binaries. An analysis of the full results from this survey should significantly improve our knowledge on the Galaxy-wide population and birth-rate of millisecond pulsars.
2.6.4 Targeted searches of globular clusters
Globular clusters have long been known to be breeding grounds for millisecond and binary pulsars. The main reason for this is the high stellar density in globular clusters relative to most of the rest of the Galaxy. As a result, lowmass X-ray binaries are almost 10 times more abundant in clusters than in the Galactic disk. In addition, exchange interactions between binary and multiple systems in the cluster can result in the formation of exotic binary systems. Since a single globular cluster usually fits well within a single telescope beam, deep targeted searches can be made. Once the DM of a pulsar is known in a globular cluster, the DM parameter space for subsequent searches is essentially fixed. This allows computation power to be invested in so-called acceleration searches for short-period binary systems (see § 3.1.3). To date, searches have revealed 47 pulsars in globular clusters (see Table 5 in the appendix for a list and the review by Kulkarni & Anderson ). Highlights include the double neutron star binary in M15  and a low-mass binary system with a 95-min orbital period in 47 Tucanae , one of 20 millisecond pulsars currently known in this cluster alone . On-going surveys of clusters continue to yield new discoveries [199, 62].
2.7 Going further
Two excellent graduate-level monographs are available: the classic 1970s text Pulsars by Manchester & Taylor  and the more up-to-date Pulsar Astronomy by Lyne & Smith . Those wishing to approach the subject from a more theoretical viewpoint are advised to read Michel’s The Theory of Neutron Star Magnetospheres  and The Physics of the Pulsar Magnetosphere by Beskin, Gurevich & Istomin . Our summary of evolutionary aspects serves merely as a primer to the vast body of literature available. The reader is referred to the excellent review by Bhattacharya & van den Heuvel  for further insights. For an excellent overview of pulsar distance measurements and their implications, see the review by Weisberg .
Pulsar resources available on the Internet are continually becoming more extensive and useful. Good starting points for pulsar-surfers are the pages maintained at Arecibo , Berkeley , Bonn , Jodrell Bank , Princeton , Swinburne  and Sydney .
3 The Galactic Pulsar Population
The extent to which the pulsar sample is incomplete is shown in the right panel of Fig. 12 where the cumulative number of pulsars is plotted as a function of the projected distance from the Sun. The observed distribution is compared to the expected distribution for a simple model population in which there are errors in the distance scale, but no selection effects. We see that the observed sample becomes strongly deficient in terms of the number of sources for distances beyond a few kpc. We now discuss the main selection effects that hamper the detection of pulsars in some detail.
3.1 Selection effects in pulsar searches
3.1.1 The inverse square law and survey thresholds
3.1.2 Pulse dispersion and scattering
It follows from Equation (3) that the sensitivity decreases as W/(P−W) increases. Also note that if W≳P, the pulsed signal is smeared into the background emission and is no longer detectable, regardless of how luminous the source may be. The detected pulse width W will be broader than the intrinsic value largely as a result of pulse dispersion and scattering by free electrons in the interstellar medium. As discussed above, the dispersive smearing scales as Δν/ν3, where ν is the observing frequency. This can largely be removed by dividing the pass-band into a number of channels and applying successively longer time delays to higher frequency channels before summing over all channels to produce a sharp profile. This process is known as incoherent dedispersion.
Dispersion and scattering are most severe for distant pulsars in the inner Galaxy where the number of free electrons along the line of sight becomes large. The strong frequency dependence of both effects means that they are considerably less of a problem for surveys at observing frequencies ≳1400 MHz [56, 109] compared to the usual 400 MHz search frequency. An added bonus for such observations is the reduction in Tsky, since the spectral index of the non-thermal Galactic emission is about -2.8 . Pulsars themselves have steep radio spectra. Typical spectral indices are -1.6 , so that flux densities are roughly an order of magnitude lower at 1400 MHz compared to 400 MHz. Fortunately, this can usually be compensated for by the use of larger receiver bandwidths at higher radio frequencies. For example, the 1370-MHz system at Parkes has a bandwidth of 288 MHz  compared to the 430-MHz system, where 32 MHz is available .
3.1.3 Orbital acceleration
Standard pulsar searches use Fourier techniques to search for a-priori unknown periodic signals and usually assume that the apparent pulse period remains constant throughout the observation. For searches with integration times much greater than a few minutes this assumption is only valid for solitary pulsars, or those in binary systems where the orbital periods are longer than about a day. For shorter-period binary systems, as noted by Johnston & Kulkarni , the Doppler-shifting of the period results in a spreading of the signal power over a number of frequency bins in the Fourier domain, leading to a reduction in signal-to-noise ratio. An observer will perceive the frequency of a pulsar to shift by an amount aT/(Pc), where a is the (assumed constant) line-of-sight acceleration during the observation of length T, P is the (constant) pulse period in its rest frame and c is the speed of light. Given that the width of a frequency bin is 1/T, we see that the signal will drift into more than one spectral bin if aT2/(Pc)>1. Survey sensitivities to rapidly-spinning pulsars in tight orbits are therefore significantly compromised when the integration times are large.
It is clearly desirable to employ a technique to recover the loss in sensitivity due to Doppler smearing. One such technique, the so-called “acceleration search” , assumes the pulsar has a constant acceleration during the integration. Each time series can then be re-sampled to refer it to the frame of an inertial observer using the Doppler formula to relate a time interval τ in the pulsar frame to that in the observed frame at time t, as τ(t)∝(1+at/c). Searching over a range of accelerations is desirable to find the time series for which the trial acceleration most closely matches the true value. In the ideal case, a time series is produced with a signal of constant period for which full sensitivity is recovered (see right panel of Fig. 14). Anderson et al.  used this technique to find PSR B2127+11C, a double neutron star binary in M15 which has parameters similar to B1913+16. Camilo et al.  have recently applied the same technique to 47 Tucanae to discover 9 binary pulsars, including one in a 96-min orbit around a low-mass (0.15M⊙) companion. This is currently the shortest binary period for any known radio pulsar.
For the shortest orbital periods, the assumption of a constant acceleration during the observation clearly breaks down. Ransom et al.  have developed a particularly efficient algorithm for finding binaries whose orbits are so short that many orbits can take place during an integration. This phase modulation technique exploits the fact that the pulses are modulated by the orbit to create a family of periodic sidebands around the nominal spin period of the pulsar. This technique has already been used to discover a 1.7-hr binary pulsar in NGC 6544 . The existence of these short-period radio pulsar binaries, as well as the 11-min X-ray binary X1820–303 in NGC 6624 , implies that there must be many more short-period binaries containing radio or X-ray pulsars in globular clusters that are waiting to be discovered by more sensitive searches.
3.2 Correcting the observed pulsar sample
3.2.1 Scale factor determination
3.2.2 The small-number bias
For small samples of observationally-selected objects, the detected sources are likely to be those with larger-than-average luminosities. The sum of the scale factors (Equation (5)), therefore, will tend to underestimate the true size of the population. This “small-number bias” was first pointed out by Kalogera et al. [112, 113] for the sample of double neutron star binaries where we know of only three clear-cut examples (§ 3.4.1). Only when the number of sources in the sample gets past 10 or so does the sum of the scale factors become a good indicator of the true population size.
3.2.3 The beaming correction
When most of these beaming models were originally proposed, the sample of millisecond pulsars was ≲5 and hence their predictions about the beaming fractions of short-period pulsars relied largely on extrapolations from the normal pulsars. A recent analysis of a large sample of millisecond pulsar profiles by Kramer et al.  suggests that the beaming fraction of millisecond pulsars lies between 50 and 100%.
3.3 The population of normal and millisecond pulsars
3.3.1 Luminosity distributions and local number estimates
The most recent use of the scale factor approach to derive the characteristics of the true normal and millisecond pulsar populations is based on the sample of pulsars within 1.5 kpc of the Sun . The rationale for this cut-off is that, within this region, the selection effects are well understood and easier to quantify by comparison with the rest of the Galaxy. These calculations should give reliable estimates for the local pulsar population.
3.3.2 Galactic population and birth-rates
Integrating the local surface densities of pulsars over the whole Galaxy requires a knowledge of the presently rather uncertain Galactocentric radial distribution [153, 107]. One approach is to assume that pulsars have a radial distribution similar to that of other stellar populations and scale the local number density with this distribution to estimate the total Galactic population. The corresponding local-to-Galactic scaling is 1000±250 kpc2 . With this approach we estimate there to be ∼160,000 active normal pulsars and ∼40,000 millisecond pulsars in the Galaxy. Based on these estimates, we are in a position to deduce the corresponding rate of formation or birth-rate. From the P−Ṗ diagram in Fig. 7, we infer a typical lifetime for normal pulsars of ∼107 yr, corresponding to a Galactic birth rate of ∼1 per 60 yr — consistent with the rate of supernovae . As noted in § 2.4, the millisecond pulsars are much older, with ages close to that of the Universe τu (we assume here τu=10 Gyr ). Taking the maximum age of the millisecond pulsars to be τu, we infer a mean birth rate of at least 4×10-6 yr-1. This is consistent, within the uncertainties, with the birth-rate of low-mass X-ray binaries .
3.3.3 Implications for gravitational wave detectors
The estimates of the local surface density of active pulsars allow us to deduce the likely distance of the nearest neutron star to Earth. For the combined millisecond and normal pulsar populations, with a surface density of 193±35 pulsars kpc-2, the nearest neutron star is thus likely to be ≲40 pc. This number is of interest to those building gravitational wave detectors, since it determines the likely amplitude of gravitational waves emitted from nearby rotating neutron stars . According to Thorne , currently planned detectors will be able to detect neutron stars with ellipticities greater than 7.5×10-11(Pd)2, where P is the rotation period in ms and d is the distance in kpc. The recent probable detection of free precession in the radio pulsar B1828-11  does indicate that ellipticities exist in neutron stars so that nearby objects may be continuous sources of gravitational radiation.
Thus, in order to detect small ellipticities, nearby sources with short spin periods are required. One of the best candidates is the nearby 5.75-ms pulsar J0437-4715 . At a distance of 178±26 pc  this is currently the closest known millisecond pulsar to the Earth. The closest known neutron star is RX J185635-3754 discovered in the ROSAT all-sky survey . Multi-epoch HST observations show that this isolated neutron star is located at a distance of 61±9 pc . In keeping with other radio-quiet isolated neutron stars, the period of this pulsar is likely to be several seconds .
3.4 The population of relativistic binaries
Of particular interest to the astronomical community at large are the numbers of relativistic binary systems in the Galaxy. Systems involving neutron stars are: double neutron star binaries, neutron star-white dwarf binaries and neutron star-black hole binaries. Interest in these systems is two-fold: (1) to test the predictions of general relativity against alternative theories of strong-field gravity using the radio pulsar as a highly stable clock moving in the strong gravitational field; (2) to detect strong gravitational wave emission from coalescing binaries with upcoming gravitational-wave observatories like GEO600 and LIGO, VIRGO and TAMA [244, 212]. For a Living Reviews article on this new generation of interferometric gravitational wave detectors see .
3.4.1 Double neutron star binaries
Known DNS binaries and candidates. Listed are the pulse period P, the orbital period Pb, the orbital eccentricity e, the pulsar characteristic age τc, and the expected binary coalescence time-scale τg due to gravitational wave emission calculated from Equation (6). Cases for which τg is a factor of 100 or more greater than the age of the Universe are listed as ≫ τu. To distinguish between definite and candidate DNS systems, we also list whether the masses of both components have been determined.
τc [108 y]
τg [108 y]
τc [108 y]
τg [108 y]
Despite the uncertainties in identifying DNS binaries, for the purposes of determining the Galactic merger rate, the systems for which τg is less than τu (i.e. PSRs B1534+12, B1913+16 and B2127+11C) are primarily of interest. Of these PSR B2127+11C is in the process of being ejected from the globular cluster M15 [195, 192] and is thought to make only a negligible contribution to the merger rate . The general approach with the remaining two systems is to derive scale factors for each object (as outlined in § 3.2.1) and then divide these by a reasonable estimate for the lifetime. In what follows we summarize the main studies of this kind. The most comprehensive investigation of the DNS binary population to date is the recent study by Kalogera et al. (hereafter KNST; ).
As discussed in § 3.2.1, scale factors are dependent on the assumed pulsar distribution. The key parameter here is the scale height of the population with respect to the Galactic plane which itself is a function of the velocity distribution of the population. KNST examined this dependence in detail and found scale heights in the range 0.8–1.7 kpc. Based on this range, KNST revised earlier scale factor estimates  to 145–200 for B1534+12 and 45–60 for PSR B1913+16. As mentioned in § 3.2.2 scale factors calculated from a small sample of objects are subject to a significant bias. KNST find the bias in their sample to be anywhere between 2 and 200. This boosts the scale factors to the range 190–40000 for B1534+12 and 90–12000 for B1913+16.
The above scale factors also require a beaming correction. As noted in § 3.2.3, current radio pulsar beaming models vary considerably. Fortunately, for the two pulsars under consideration, detailed studies of the beam sizes [9, 119, 262] lead KNST to conclude that both pulsars beam to only about a sixth of the entire sky. The beaming-corrected numbers suggest a total of between 1680 and 312,000 active DNS binaries in our Galaxy. Many of these systems will be extremely faint objects. These estimates are dominated by the small-number bias factor. KNST’s study highlights the importance of this effect.
Some debate exists about what is the most reasonable estimate of the lifetime. Phinney  defines this as the sum of the pulsar’s spin-down age plus τg defined above. A few years later, van den Heuvel and myself argued  that a more likely estimate can be obtained by appealing to steady-state arguments where we expect sources to be created at the same rate at which they are merging. The mean lifetime was then found to be about three times the current spin-down age. This argument does, however, depend on the luminosity evolution of radio pulsars which is currently only poorly understood. Arzoumanian, Cordes & Wasserman  used kinematic data to constrain the most likely ages of the DNS binaries. They note that the remaining detectable lifetime should also take account of the reduced detectability at later epochs due to acceleration smearing as the DNS binary becomes more compact due to gravitational wave emission. KNST concluded that the lifetimes are dominated by the latter time-scale which, following Arzoumanian et al., they took to be the time for the orbital period to halve. The resulting lifetimes are 2.5×109 yr for B1534+12 and 2.5×108 yr for B1913+16.
Taking these number and lifetime estimates, KNST find the Galactic merger rate of DNS binaries to range between 3×10-6 and 4×10-4 yr-1. Extrapolating this number out to include DNS binaries detectable by LIGO in other galaxies á la Phinney , KNST find the expected event rate to be <0.25 yr-1 for LIGO-I and 2–1300 yr-1 for LIGO-II. Thus, despite the uncertainties, it seems that the prospects for detecting gravitational-wave emission from DNS inspirals in the near future are most promising.
3.4.2 White dwarf-neutron star binaries
The population of white dwarf-neutron star (WDNS) sources containing a young radio pulsar has only recently been confirmed by observers following the identification of a white dwarf companion to the binary pulsar B2303+46 by van Kerkwijk et al. . Previously, this eccentric binary pulsar was thought to be an example of a DNS binary in which the visible pulsar is the second-born neutron star [229, 143]. The optical identification rules this out and now strongly suggests a scenario in which the white dwarf was formed first. In this case, material was transfered onto the secondary during the giant phase of the primary so that the secondary became massive enough to form a neutron star [256, 232].
PSR B2303+46 has a long orbital period and does not contribute significantly to the overall merger rate of WDNS binaries. The new discovery of PSR J1141-6545 , which will merge due to gravitational-wave emission within 1.3 Gyr, is suggestive of a large population of similar binaries. This is particularly compelling when one considers that the radio lifetime of the visible pulsar is only a fraction of total lifetime of the binary before coalescence due to gravitational-wave emission. Edwards & Bailes  estimate there to be 850 WDNS binaries within 3 kpc of the Sun which will merge within τu.
Population syntheses by Tauris & Sennels  suggest that the formation rate of WDNS binaries is between 10–20 times that of DNS binaries. Based on the merging rate estimates for DNS binaries discussed in the previous section, this translates to a merging rate of WDNS binaries of between 3×10-5 yr-1 and 8×10-3 yr-1. In summary, although statistics are necessarily poor at this stage, coalescing WDNS binaries look to be very promising sources for gravitational wave detectors.
3.5 Going further
Studies of pulsar population statistics represent a large proportion of the pulsar literature. During this section we have tried to cite many of the key papers in this field. Good starting points for further reading can be found in other review articles [167, 31, 156]. Our coverage of compact object coalescence rates has concentrated on empirical methods and we have hopefully convinced the reader that these are fair and straightforward.
An alternative approach is to undertake a full-blown Monte Carlo simulation of the most likely evolutionary scenarios described in § 2.4.3. In this “scenario-machine” approach, a population of primordial binaries is synthesized with a number of underlying distribution functions: primary mass, binary mass ratio, orbital period distribution etc. The evolution of both stars is then followed to give a predicted sample of binary systems of all the various types. Since the full range of binary parameters is known, the merger rates of each type of binary are then automatically predicted by this model without the need to debate what the likely coalescence times will be. Selection effects are not normally taken into account in this approach. The final census is usually normalized to the star formation rate. Numerous examples of the scenario-machine approach (most often to populations of binaries where one or both members are NSs) can be found in the literature [69, 204, 252]. These include the widely-cited code, developed by Lipunov and collaborators to perform population syntheses of binary stars . Although extremely instructive, the uncertain assumptions about initial conditions, the physics of mass transfer and the kicks applied to the compact object at birth result in a wide range of predicted event rates which are currently broader than the empirical methods . Ultimately, the detection statistics from the gravitational wave detectors could provide far tighter constraints on the DNS merging rate than the pulsar surveys from which these predictions are made.
An excellent overview of gravitational-wave astronomy and the detection of gravitational waves from inspiraling binaries is presented by Thorne in his presentation at the centennial meeting of the American Physical Society which is available on-line .
4 Pulsar Timing
It became clear soon after their discovery that pulsars are excellent celestial clocks. In the original discovery paper , the period of the first pulsar to be discovered, PSR B1919+21, was found to be stable to one part in 107 over a time-scale of a few months. Following the discovery of the millisecond pulsar B1937+21 in 1982  it was demonstrated that its period could be measured to one part in 1013 or better . This unrivaled stability leads to a host of applications including time keeping, probes of relativistic gravity and natural gravitational wave detectors.
4.1 Observing basics
During the observation, the data regularly receive a time stamp, usually based on a caesium time standard or hydrogen maser at the observatory plus a signal from the Global Positioning System of satellites (GPS; ). The TOA of this mean pulse is then defined as the arrival time of some fiducal point on the profile. Since the mean profile has a stable form at any given observing frequency (§ 2.2), the TOA can be accurately determined by a simple cross-correlation of the observed profile with a high signal-to-noise “template” profile obtained from the addition of many observations of the pulse profile at the particular observing frequency.
One of the main problems of employing large bandwidths is pulse dispersion. As discussed in § 2.3, the velocity of the pulsed radiation through the ionised interstellar medium is frequency-dependent: pulses emitted at higher radio frequencies travel faster and arrive earlier than those emitted at lower frequencies. This process has the effect of “stretching” the pulse across a finite receiver bandwidth, reducing the apparent signal-to-noise ratio and therefore increasing ∊TOA. For most normal pulsars, this process can largely be compensated for by the incoherent de-dispersion process outlined in § 3.1.
To exploit the precision offered by millisecond pulsars, a more precise method of dispersion removal is required. Technical difficulties in building devices with very narrow channel bandwidths require another dispersion removal technique. In the process of coherent de-dispersion  the incoming signals are de-dispersed over the whole bandwidth using a filter which has the inverse transfer function to that of the interstellar medium. The signal processing can be done either on-line using finite impulse response filter devices  or off-line in software [219, 222]. The on-line approach allows for large bandwidths to be employed and real-time viewing of the data. Off-line reduction, while slow and computationally expensive, allows for more flexible data reduction schemes as well as periodicity searches to be carried out.
The maximum time resolution obtainable via coherent dedispersion is the inverse of the receiver bandwidth. For bandwidths of 10 MHz, this technique makes it possible to resolve features on time-scales as short as 100 ns. This corresponds to probing regions in the neutron star magnetosphere as small as 30 m!
4.2 The timing model
From Equation (9), an error in the assumed Ωo results in a linear slope with time. A parabolic trend results from an error in Ωo (Fig. 19b). Additional effects will arise if the assumed position of the pulsar (the unit vector ŝ̱ in equation (8)) used in the barycentric time calculation is incorrect. A position error of just one arcsecond results in an annual sinusoid (Fig. 19c) with a peak-to-peak amplitude of about 5 ms for a pulsar on the ecliptic; this is easily measurable for typical TOA uncertainties of order one milliperiod or better. A proper motion produces an annual sinusoid of linearly increasing magnitude (Fig. 19d).
After a number of iterations, and with the benefit of a modicum of experience, it is possible to identify and account for each of these various effects to produce a “timing solution” which is phase coherent over the whole data span. The resulting model parameters provide spin and astrometric information about the neutron star to a precision which improves as the length of the data span increases. Timing observations of the original millisecond pulsar B1937+21, spanning almost 9 years (exactly 165,711,423,279 rotations!), measure a period of 1.5578064688197945±0.0000000000000004 ms [117, 114] defined at midnight UT on December 5 1988! Astrometric measurements based on these data are no less impressive, with position errors of ∼20 µarcsec being presently possible.
4.3 Timing stability
Ideally, after correctly applying a timing model, we would expect a set of uncorrelated timing residuals scattered in a Gaussian fashion about a zero mean with an rms consistent with the measurement uncertainties. This is not always the case; the residuals of many pulsars exhibit a quasi-periodic wandering with time.
The relative dearth of timing noise for the older pulsars is a very important finding. It implies that, presently, the measurement precision depends primarily on the particular hardware constraints of the observing system. Consequently, a large effort in hardware development is presently being made to improve the precision of these observations using, in particular, coherent dedispersion outlined in § 4.1. Much of the pioneering work in this area has been made by Joseph Taylor and collaborators at Princeton University . From high quality observations made using the Arecibo radio telescope spanning almost a decade [206, 207, 117], the group has demonstrated that the timing stability of millisecond pulsars over such time-scales is comparable to terrestrial atomic clocks.
4.4 Binary pulsars and Kepler’s laws
For binary pulsars, the simple timing model introduced in § 4.2 needs to be extended to incorporate the additional radial acceleration of the pulsar as it orbits the common centre-of-mass of the binary system. Treating the binary orbit using Kepler’s laws to refer the TOAs to the binary barycentre requires five additional model parameters: the orbital period (Pb), projected semi-major orbital axis (ap sin i, see below), orbital eccentricity (e), longitude of periastron (ω) and the epoch of periastron passage (T0). This description, using five “Keplerian parameters”, is identical to that used for spectroscopic binary stars.
4.5 Post-Keplerian parameters
Measurements of post-Keplerian parameters for PSR B1913+16 have been carried out by Taylor and a number of collaborators over the years with steadily improving precision. The first of these parameters to be measured was the advance of the longitude of periastron ω̇. This measurement is analogous to the perihelion advance of Mercury . For PSR B1913+16 this amounts to about 4.2 degrees per year , some 4.6 orders of magnitude larger than for Mercury. A measurement of ω̇ alone yields the total mass for this system, M=2.83M⊙, assuming this advance is due to general relativity.
Measurement of a second post-Keplerian parameter for B1913+16, γ (gravitational redshift and transverse Doppler shifts in the orbit), permits an unambiguous determination of mp, mc and i when combined with ω̇ and the five Keplerian parameters. The original measurements  have since been substantially refined [239, 240] and the mass of the pulsar and its unseen companion have been determined to be 1.442±0.003M⊙ and 1.386±0.003M⊙ respectively. Such phenomenal precision is a testament to the timing stability of radio pulsars as clocks, and the diligence of Taylor and collaborators in carrying out these long-term measurements. Similar mass measurements now exist for the two other double neutron star binary systems discussed in § 3.4.1: B1534+12  and B2127+11C . For a number of other systems, ω̇ measurements allow interesting constraints to be placed on the component masses [246, 179, 248].
An important general relativistic prediction for eccentric double neutron star systems is the orbital decay due to the emission of gravitational radiation (Ṗb in Equation (13)). Taylor et al. [236, 239, 240] were able to measure this for B1913+16 and found it to be in excellent agreement with the predicted value.
For those binary systems which are oriented nearly edge-on to the line-ofsight, a significant delay is expected for orbital phases around superior conjunction where the pulsar radiation is bent in the gravitational potential well of the companion star. The so-called “range” and “shape” of the Shapiro delay effect are parameterized by the last two post-Keplerian parameters r and s≡sin i that were introduced in Equations (14) and (15). This effect, analogous to the solar system Shapiro delay, has so far been measured for two neutron star-white dwarf binary systems, B1855+09 and J1713+0747 [206, 117, 46], and for the double neutron star binaries B1534+12  and B1913+16 .
4.6 Geodetic precession
In addition to the above results, there is now evidence for geodetic precession in the other classic neutron star binary, PSR B1534+12 [9, 223]. Although geodetic precession in binary pulsars is another successful test of general relativity (albeit at a lower precision than e.g. orbital decay measurements), what is perhaps more interesting are the various consequences it has. Geodetic precession only occurs when the spin and orbital axes are misaligned . This is most likely to occur if the neutron star received an impulsive “kick” velocity at birth (§ 2.4.4). Wex et al.  have investigated the B1913+16 observations and find that the kick magnitude was at least 250 km s-1 and was directed almost perpendicular to the spin axis of the neutron star progenitor. This places stringent constraints on any kick mechanism. Detailed monitoring of the pulse profile and polarization properties now underway [262, 120] will allow the first map of the emission beam of a neutron star to be made. This has important implications for the various beaming models described in § 3.2.3. There are already indications that the beam is circular .
The current results predict that B1913+16 will completely precess out of the line of sight by around 2025 and re-appear some 240 years later . Although we shall lose a most treasured pulsar, we can take comfort from the fact that other pulsars will precess into our field of view. Perhaps one example is the newly-discovered relativistic binary J1141-6545  discussed in § 2.6.2 and § 3.4.2. This relatively bright object was apparently missed by two previous searches during the early 1990s [109, 160, 152].
4.7 Going further
This chapter has outlined past and present progress in a number of areas related to pulsar timing. For further details on the technical details and prospects of pulsar timing, the interested reader is referred to a number of excellent review articles [269, 16, 233, 23, 24]. Two freely available software packages which are routinely used for time-of-arrival analyses by the pulsar community are available, viz. TEMPO [240, 196] and TIMAPR [72, 166]. These packages are based on more detailed versions of the timing model outlined in § 4.2. An up-to-date list summarising the various timing programmes is kept by Don Backer . An audio file and slides from a lecture on pulsar timing presented by Backer at the centennial meeting of the American Physical society is also available online . Another relevant lecture from that meeting is Will’s presentation  on tests of Einstein’s relativity which includes an excellent overview of Taylor and Weisberg’s measurements of PSR B1913+16. Kramer  has written a lucid review article discussing measurements of geodetic precession in binary pulsars and their implications.
The remarkable precision of these measurements, particularly for millisecond pulsars, allows the detection of radial accelerations on the pulsar induced by orbiting bodies smaller than the Earth. Alex Wolszczan detected one such “pulsar planetary system” in 1990 following the discovery of a 6.2-ms pulsar B1257+12. In this case, the pulsar is orbited by at least three Earth-mass bodies [273, 186, 272]. Subsequent measurements of B1257+12 were even able to measure resonance interactions between two of the planets , confirming the nature of the system beyond all doubt. Long-term timing measurements of the 11-ms pulsar B1620-26 in the globular cluster M4 indicate that it may also have a planetary companion [247, 15, 245]. For detailed reviews of these systems, and their implications for planetary formation scenarios, the interested reader is referred to .
5 Pulsars as Gravitational Wave Detectors
Many cosmological models predict that the Universe is presently filled with a stochastic gravitational wave background (GWB) produced during the big bang era . The idea to use pulsars as natural detectors of gravitational waves was first explored independently by both Sazhin and Detweiler in the late 1970s [209, 68]. The basic concept is to treat the solar system barycentre and a distant pulsar as opposite ends of an imaginary arm in space. The pulsar acts as the reference clock at one end of the arm sending out regular signals which are monitored by an observer on the Earth over some time-scale T. The effect of a passing gravitational wave would be to cause a change in the observed rotational frequency by an amount proportional to the amplitude of the wave. For regular monitoring observations of a pulsar with typical TOA uncertainties of ∊TOA, this “detector” would be sensitive to waves with dimensionless amplitudes ≳∊TOA/T and frequencies as low as ~1/T [29, 36]. This method, which already yields interesting upper limits on the GWB, is reviewed in § 5.1. The idea of a more sensitive detector based on an array of pulsar clocks distributed over the sky is discussed in § 5.2.
5.1 Limits from individual pulsars
Romani & Taylor  applied this technique to a set of TOAs for PSR B1237+12 obtained from regular observations over a period of 11 years as part of the JPL pulsar timing programme . This pulsar was chosen on the basis of its relatively low level of timing activity by comparison with the youngest pulsars, whose residuals are ultimately plagued by timing noise (§ 4.3). By ascribing the rms scatter in the residuals (σ=240 ms) to the GWB, Romani & Taylor placed a limit of ρg/ρc≳4×10-3h-2 for a centre frequency f=7×10-9 Hz.
For those pulsars in binary systems, an additional clock for measuring the effects of gravitational waves is the orbital period. In this case, the range of frequencies is not limited by the time span of the observations, allowing the detection of waves with periods as large as the light travel time to the binary system . The most stringent results presently available are based on the B1855+09 limit Ωgh2<2:7×10-4 in the frequency range 10-11<f<4.4×10-9 Hz. Kopeikin  has recently presented this limit and discusses the methods in detail.
5.2 A pulsar timing array
5.3 Going further
6 Summary and Future Prospects
The main aim of this article was to review some of the many astrophysical applications provided by the present sample of binary and millisecond radio pulsars. The topics covered here, along with the bibliography and associated tables of observational parameters, should be useful to those wishing to delve deeper into the vast body of literature that exists. We now briefly recap on the main issues.
Through an understanding of the Galactic population of radio pulsars summarised in § 3 it is possible to predict the detection statistics of terrestrial gravitational wave detectors to nearby rapidly spinning neutron stars (§ 3.3), as well as coalescing relativistic binaries at cosmic distances (§ 3.4). Continued improvements in gravitational wave detector sensitivities should result in a number of interesting developments and contributions in this area. These developments and contributions might include the detection of presently known radio pulsars, as well as a population of coalescing binary systems which have not yet been detected as radio pulsars. The phenomenal timing stability of radio pulsars leads naturally to a large number of applications, including their use as laboratories for relativistic gravity (§ 4.5) and as natural detectors of gravitational radiation (§ 5). Long-term timing experiments of the present sample of millisecond and binary pulsars currently underway appear to have tremendous potential in these areas and perhaps detect the gravitational wave background (if it exists) within the next decade.
A dual-line binary pulsar, i.e. a double neutron star system in which both components are observable as radio pulsars. The additional clock in such a binary system would be most valuable in further tests of strong-field gravity.
A radio pulsar with a black-hole companion would undoubtably also be a fantastic laboratory for studying gravity in the strong-field regime.
A sub-millisecond pulsar. The original millisecond pulsar, B1937+21, rotating at 642 Hz is still the most rapidly rotating neutron star known. Do kHz neutron stars exist? Searches now have sensitivity to such objects  and a discovery of even one would constrain the equation of state of matter at high densities.
A binary system in which the neutron star is in the process of transforming from an X-ray-emitting neutron star to a millisecond radio pulsar.
Many thanks to Maura McLaughlin, Jiannis Seiradakis and Michael Kramer who read and commented on earlier incantations of this revised review, as well as a number of other colleagues who gave me useful feedback on the original article. Jiannis Seiradakis urged me to include the tables of parameters given in the appendix after I promised to put them in the original article, but didn’t. The tables and references should be useful to both observers and theorists. Thanks also to Fernando Camilo for allowing me to include details on a number of pulsars in these tables prior to publication.
7 Appendix: Tables of Binary and Millisecond Pulsars
Parameters for the 9 isolated millisecond pulsars currently known in the Galactic disk. Listed are the spin period P, the base-10 logarithms of the characteric age τc and surface magnetic field strength B (§ 2.4.1), the distance d derived from the Taylor & Cordes electron density model  or independently (when available), and the transverse speed vt inferred from d and a proper motion measurement (when available). Key publications for each pulsar are referenced to the bibliography.
Parameters for the 9 high-eccentricity (e>0.15) binary pulsars currently known in the Galactic disk. Listed are the spin period P, the base-10 logarithms of the characteric age τc and surface magnetic field strength B (§ 2.4.1), the distance d derived from the Taylor & Cordes electron density model  or independently (when available), the transverse speed vt inferred from d and a proper motion measurement (when available), the binary period Pb, the projected semi-major axis of the orbit x in units of light seconds, the orbital eccentricity e, and the companion mass m2 evaluated from the mass function assuming a pulsar mass of 1.4M⊙ and an inclination angle of 60 degrees (§ 4.4) or (when known) from independent measurements. Key publications for each pulsar are referenced to the bibliography.
Parameters for 38 low-eccentricity binary pulsars currently known in the Galactic disk. Listed are the spin period P, the base-10 logarithms of the characteric age τc and surface magnetic field strength B (§ 2.4.1), the distance d derived from the Taylor & Cordes electron density model  or independently (when available), the transverse speed vt inferred from d and a proper motion measurement (when available), the binary period Pb, the projected semi-major axis of the orbit x in units of light seconds, the orbital eccentricity e (when measured), and the companion mass m2 evaluated from the mass function assuming a pulsar mass of 1.4M⊙ and an inclination angle of 60 degrees (§ 4.4) or (when known) from independent measurements. Key publications for each pulsar are referenced to the bibliography.
Parameters for the 47 pulsars currently known in globular clusters. Listed are the spin period P, cluster name and the distance d to the cluster. For binary pulsars we also list the binary period Pb, the projected semi-major axis of the orbit x in units of light seconds, the orbital eccentricity e, and the companion mass m2 evaluated from the mass function assuming a pulsar mass of 1.4M⊙ and an inclination angle of 60 degrees (§ 4.4) or (when known) from independent measurements. Key publications for each pulsar are referenced to the bibliography.
currently no orbital solution
1 Pulsars are named with a PSR prefix followed by a “B” or a “J” and their celestial coordinates. Those pulsars discovered prior to 1990 are usually referred to by their “B” names (Besselian 1950 system). More recent discoveries are only referred to by their “J” names (Julian 2000 system).
2Pulsar astronomers usually define the luminosity L=Sd2, where S is the mean flux density at 400 MHz (a standard observing frequency) and d is the distance derived from the DM (§ 2.3).
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