Binary and Millisecond Pulsars

2.1 The lighthouse model

Figure 2 shows an animation depicting the rotating neutron star, also known as the “lighthouse” model. As the neutron star spins, charged particles are accelerated along magnetic field lines forming a beam (depicted by the gold cones). The accelerating particles emit electromagnetic radiation, most readily detected at radio frequencies as a sequence of observed pulses. Each pulse is 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.

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Neutron stars are essentially large celestial flywheels with moments of inertia ∼10³⁸ kg m². The rotating neutron star model [290, 122] predicts a gradual slowdown and hence an increase in the pulse period as the outgoing radiation carries away rotational kinetic energy. This idea gained strong support when a period increase of 36.5 ns per day was measured for the pulsar B0531+21¹ in the Crab nebula [321], which implied that a rotating neutron star with a large magnetic field must be the dominant energy supply for the nebula [123].

2.2 Pulse periods and slowdown rates

The pulsar catalogue [248, 12] contains up-to-date parameters for 1775 pulsars. Most of these are “normal” with pulse periods P ∼ 0.5 s which increase secularly at rates Ṗ ∼ 10⁻¹⁵ s/s. A growing fraction are “millisecond pulsars”, with 1.4 ms ≲ P ≲ 30 ms and Ṗ ≲ 10⁻¹⁹ s/s. As shown in the “P−Ṗ diagram” in Figure 3, normal and millisecond pulsars are distinct populations.

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The differences in P and Ṗ imply fundamentally different magnetic field strengths and ages. Treating the pulsar as a rotating magnetic dipole, one may show [226] that the surface magnetic field strength B ∝ (PṖ)¹/² and the characteristic age τ_c = P/(2Ṗ). Lines of constant B and τ_c are drawn on Figure 3, from which we infer typical values of 10¹² G and 10⁷ yr for the normal pulsars and 10⁸ G and 10⁹ yr for the millisecond pulsars. For the rate of loss of kinetic energy, sometimes called the spin-down luminosity, we have Ė ∝ Ṗ/P³. The lines of constant Ė shown on Figure 3 indicate that the most energetic objects are the very young normal pulsars and the most rapidly spinning millisecond pulsars.

The most rapidly rotating neutron star currently known, J1748−2446ad,with a spin rate of 716 Hz, resides in the globular cluster Terzan 5 [140]. As discussed by Lattimer & Prakash [206], the limiting (non-rotating) radius of a 1.4 M_⊙ neutron star with this period is 14.3 km. If a precise measurement for the pulsar mass can be made through future timing measurements (Section 4), then this pulsar could be a very useful probe of the equation of state of super dense matter.

While the hunt for more rapidly rotating pulsars and even “sub-millisecond pulsars” continues, and most neutron star equations of state allow higher spin rates than 716 Hz, it has been suggested [38] that the dearth of pulsars with P < 1.5 ms is caused by gravitational wave emission from Rossby-mode instabilities [5]. The most rapidly rotating pulsars [140] are predominantly members of eclipsing binary systems which could hamper their detection in radio surveys. Independent constraints on the limiting spin frequencies of neutron stars come from studies of millisecond X-ray binaries [66] which are not thought to be selection-effect limited [67]. This analysis does not predict a significant population of neutron stars with spin rates in excess of 730 Hz.

2.3 Pulse profiles

Pulsars are weak radio sources. Measured intensities at 1.4 GHz vary between 5μJy and 1 Jy (1 Jy ≡ 10⁻²⁶ W m⁻² Hz⁻¹). As a result, even with a large radio telescope, the coherent addition of many hundreds or even thousands of pulses is usually required in order to produce a detectable “integrated profile”. Remarkably, although the individual pulses vary dramatically, the integrated profile at a given observing frequency is very stable and can be thought of as a fingerprint of the neutron star’s emission beam. Profile stability is of key importance in pulsar timing measurements discussed in Section 4.

The selection of integrated profiles in Figure 4 shows a rich diversity in morphology including two examples of “interpulses” — a secondary pulse separated by about 180 degrees from the main pulse. One interpretation for this phenomenon is that the two pulses originate from opposite magnetic poles of the neutron star (see however [249]). Since this is an unlikely viewing angle, we would expect interpulses to be a rare phenomenon. Indeed, the fraction of known pulsars in which interpulses are observed in their pulse profiles is only a few percent [194]. Recent work [392] on the statistics of interpulses among the known population show that the interpulse fraction depends inversely on pulse period. The origin of this effect could be due to alignment between the spin and magnetic axis of neutron stars on timescales of order 10⁷ yr [392]. If we take this result at face value, and assume that all millisecond pulsars descend from normal pulsars (Section 2.6), then the implication is that millisecond pulsars should be preferentially aligned rotators. However, their appears to be no strong evidence in favour of this expectation based on the pulse profile morphology of millisecond pulsars [203].

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Two contrasting phenomenological models have been proposed to explain the observed pulse shapes. The “core and cone” model [310] depicts the beam as a core surrounded by a series of nested cones. Alternatively, the “patchy beam” model [243, 128] has the beam populated by a series of randomly-distributed emitting regions. Recent work suggests that the observational data can be better explained by a hybrid empirical model depicted in Figure 5 which employs patchy beams in a core and cone structure [179].

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A key feature of this new model is that the emission height of young pulsars is radically different from that of the older population. Monte Carlo simulations [179] using this phenomenological model appear to be very successful at explaining the rich diversity of pulse shapes. Further work in this area is necessary to understand the origin of this model, and improve our understanding of the shape and evolution of pulsar beams and fraction of sky they cover. This is of key importance to the results of population studies reviewed in Section 3.2.

2.4 The pulsar distance scale

Quantitative estimates of the distance to each pulsar can be made from the measurement of pulse dispersion — the delay in pulse arrival times across a finite bandwidth. Dispersion occurs because the group velocity of the pulsed radiation through the ionised component of the interstellar medium is frequency dependent. As shown in Figure 6, pulses emitted at lower radio frequencies travel slower through the interstellar medium, arriving later than those emitted at higher frequencies.

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Because the electron density models are only as good as the scope of their input data allow, one should be mindful of systematic uncertainties. For example, studies of the Parkes multibeam pulsar distribution [199, 224] suggest that the NE2001 model underestimates the distances of pulsars close to the Galactic plane. This suspicion has been dramatically confirmed recently by an extensive analysis [120] of pulsar DMs and measurements of Galactic Hα emission. This work shows that the distribution of Galactic free electrons to be exponential in form with a scale height of \(1830_{- 250}^{+ 120}{\rm{pc}}\). This value is a factor of two higher than previously thought. A revised version of the NE2001 model which takes into account these and other developments is currently in preparation [79].

2.5 Pulsars in binary systems

As can be inferred from Figure 1, only a few percent of all known pulsars in the Galactic disk are members of binary systems. Timing measurements (see Section 4) place useful constraints on the masses of the companions which, often 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” B1257+12 and B1620−26. PSR B1257+12 is a 6.2-ms pulsar accompanied by at least three terrestrial-mass bodies [404, 293, 403] while B1620−26, an 11-ms pulsar in the globular cluster M4, is part of a triple system with a 1–2 M_Jupiter planet [370, 16, 369, 338] orbiting a neutron star—white dwarf stellar binary system. The current limits from pulsar timing favour a roughly 45-yr mildly-eccentric (e ∼ 0.16) orbit with semi-major axis ∼ 25 AU. Despite several tentative claims over the years, no other convincing cases for planetary companions to pulsars exist. Orbiting companions are much more common around millisecond pulsars (∼ 80% of the observed sample) than around the normal pulsars (≲ 1%). In general, binary systems with low-mass companions (≲ 0.7 M_⊙ — predominantly white dwarfs) have essentially circular orbits: 10⁻⁵ ≲ e ≲ 0.01. Binary pulsars with high-mass companions (≳ 1 M_⊙ — massive white dwarfs, other neutron stars or main sequence stars) tend to have more eccentric orbits, 0.15 ≲ e ≲ 0.9.

2.6 Evolution of normal and millisecond pulsars

A simplified version of the presently favoured model [39, 110, 339, 2] to explain the formation of the various types of systems observed is shown in Figure 7. Starting with a binary star system, a neutron star is formed during the supernova explosion of the initially more massive star. From the virial theorem, in the absence of any other factors, the binary will be disrupted if more than half the total pre-supernova mass is ejected from the system during the (assumed symmetric) explosion [142, 36]. In practice, the fraction of surviving binaries is also affected by the magnitude and direction of any impulsive “kick” velocity the neutron star receives at birth from a slightly asymmetric explosion [142, 19]. Binaries that disrupt produce a high-velocity isolated neutron star and an OB runaway star [40]. The high probability of disruption explains qualitatively why so few normal pulsars have companions. Those that survive will likely have high orbital eccentricities due to the violent conditions in the supernova explosion. There are currently four known normal radio pulsars with massive main sequence companions in eccentric orbits which are examples of binary systems which survived the supernova explosion [169, 182, 348, 349, 236]. Over the next 10^7–8 yr 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 diminishes to a point where it no longer produces significant amounts of radio emission.

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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 angular momentum at the expense of the orbital angular momentum of the binary system [2]. The term “recycled pulsar” is often used to describe such objects. During this accretion phase, the X-rays produced by the frictional heating of the infalling matter onto the neutron star mean that such a system is expected to be visible as an X-ray binary. Two classes of X-ray binaries relevant to binary and millisecond pulsars exist: neutron stars with high-mass or low-mass companions. Detailed reviews of the X-ray binary population, including systems likely to contain black holes, can be found elsewhere [36, 320].

2.6.2 Low-mass systems

The companion in a low-mass X-ray binary evolves and transfers matter onto the neutron star on a much longer timescale, spinning it up to periods as short as a few ms [2]. Tidal forces during the accretion process serve to circularize the orbit. 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. 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 [396], as well as Doppler-shifted 2.49-ms X-ray pulsations from the transient X-ray burster SAX J1808.4-3658 [398, 65]. Seven other “X-ray millisecond pulsars” are now known with spin rates and orbital periods ranging between 185–600 Hz and 40 min −4.3 hr respectively [397, 263, 173, 204].

Numerous examples of these systems in their post X-ray phase are now seen as the millisecond pulsar—white dwarf binary systems. Presently, 20 of these systems have compelling optical identifications of the white dwarf companion, and upper limits or tentative detections have been found in about 30 others [384]. Comparisons between the cooling ages of the white dwarfs and the millisecond pulsars confirm the age of these systems and suggest that the accretion rate during the spin-up phase was well below the Eddington limit [132].

Further support for the above evolutionary scenarios comes from two correlations in the observed sample of low-mass binary pulsars. Firstly, as seen in Figure 8, there is a strong correlation between orbital period and eccentricity. The data are in very good agreement with a theoretical relationship which predicts a relic orbital eccentricity due to convective eddy currents in the accretion process [298]. Secondly, as shown in Panel b of Figure 9, where companion masses have been measured accurately, through radio timing (see Section 4.4) and/or through optical observations [384], they are in good agreement with a relation between companion mass and orbital period predicted by binary evolution theory [361]. A word of caution is required in using these models to make predictions, however. When confronted with a larger ensemble of binary pulsars using statistical arguments to constrain the companion masses (see Panel a of Figure 9), current models have problems in explaining the full range of orbital periods on this diagram [347].

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2.7 Intermediate mass binary pulsars

2.8 Isolated recycled pulsars

The scenarios outlined qualitatively above represent a reasonable understanding of binary evolution. There are, however, a number of pulsars with spin properties that suggest a phase of recycling took place but have no orbiting companions. While the existence of such systems in globular clusters are more readily explained by the high probability of stellar interactions compared to the disk [337], it is somewhat surprising to find them in the Galactic disk. Out of a total of 72 Galactic millisecond pulsars, 16 are isolated (see Table 2). Although it has been proposed that these millisecond pulsars have ablated their companion via their strong relativistic winds [192] as may be happening in the PSR B1957+20 system [119], it is not clear whether the energetics or timescales for this process are feasible [208].

There are four further “anomalous” isolated pulsars with periods in the range 28–60 ms [58, 229]. When placed on the P—Ṗ diagram, these objects populate the region occupied by the double neutron star binaries. The most natural explanation for their existence, therefore, is that they are “failed double neutron star binaries” which disrupted during the supernova explosion of the secondary [58]. A simple calculation [229], suggested that for every double neutron star we should see of order ten such isolated objects. Recent work [29] has investigated why so few are observed. Using the most recent population synthesis models to follow the evolution of binary systems [28], it appears that the discrepancy may not be as significant as previously supposed. In particular, the space velocity distribution of surviving binary systems is narrower than for the isolated objects that were during the second supernova explosion. The isolated systems occupy a larger volume of the Galaxy than the surviving binaries and are harder to detect. When this selection effect is accounted for [29], the relative sample sizes appear to be consistent with the disruption hypothesis.

2.9 A new class of millisecond pulsars?

One of the most remarkable recent discoveries is the binary pulsar J1903+0327 [70]. Found in an on-going multibeam survey with the Arecibo telescope [275, 82], this 2.15-ms pulsar is distinct from all other millisecond pulsars in that its 95-day orbit has an eccentricity of 0.43! In addition, timing measurements of the relativistic periastron advance and Shapiro delay in this system (see Section 4.4), show the mass of the pulsar to be 1.74 ± 0.04M_⊙ and the companion star to be 1.051 ± 0.015M_⊙. Optical observations show a possible counterpart which is consistent with a 1 M_⊙ star. While similar systems have been observed in globular clusters (e.g. PSR J0514−4002A in NGC 1851[117]), presumably a result of exchange interactions, the standard recycling hypothesis outlined in Section 2.6 cannot account for pulsars like J1903+0327 in the Galactic disk.

How could such an eccentric binary millisecond pulsar system form? One possibility is that the binary system was produced in an exchange interaction in a globular cluster and subsequently ejected, or the cluster has since disrupted. Statistical estimates [70] of the likelihood of both these channels are in the range 1–10%, implying that a globular cluster origin cannot be ruled out.

Another possibility is that the pulsar is a member of an hierarchical triple system with a one solar mass white dwarf in the 95-day orbit, and a main sequence star in a much wider and highly inclined orbit which has so far not been revealed by timing. The origin of the high eccentricity is through perturbations from the outer star, the so-called Kozai mechanism [197]. Formation estimates based on observational data on stellar multiplicity [304] find that around 4% of all binary millisecond pulsars are expected to be triple systems [70]. The existence of a single triple system among the current sample of millisecond pulsars appears to be consistent with this hypothesis.

If future observations of the proposed optical counterpart confirm it as the binary companion through spectral line measurements of orbital Doppler shifts, the above triple-system scenario will be ruled out. Such an observation would favour a hybrid scenario suggested by van den Heuvel [383] in which the white dwarf and pulsar merge due to gravitational radiation losses. Tidal disruption of the white dwarf in the inspiral would produce an accretion disk and induce an eccentricity in the orbit of the outer star leaving behind an eccentric binary system. This idea could naturally account for the high pulsar mass observed in this system which could arise from accretion of a white-dwarf debris disk following coalescence. Alternatively, as suggested by Champion et al. [70], the millisecond pulsar might have ablated the white dwarf companion in a triple system leaving only the unevolved companion in an elliptical orbit.

2.10 Pulsar 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. Proper motions for over 250 pulsars have now been measured largely by radio timing and interferometric techniques [237, 24, 111, 133, 144, 406]. These data imply a broad velocity spectrum ranging from 0 to over 1000 km s⁻¹ [242], with the current record holder being PSR B1508+55 [72], with a proper motion and parallax measurement implying a transverse velocity of \(1083_{- 90}^{+ 103}{\rm{km}}\,{{\rm{s}}^{- 1}}\). As Figure 10 illustrates, high-velocity pulsars born close to the Galactic plane quickly migrate to higher Galactic latitudes. Given such a broad velocity spectrum, as many as half of all pulsars will eventually escape the Galactic gravitational potential [242, 81].

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Such large velocities are perhaps not surprising, given the violent conditions under which neutron stars are formed. If the explosion is only slightly asymmetric, an impulsive “kick” velocity of up to 1000 km s⁻¹ can be imparted to the neutron star [336]. In addition, if the neutron star progenitor was a member of a binary system prior to the explosion, the pre-supernova orbital velocity will also contribute to the resulting speed of the newly-formed pulsar. The relative contributions of these two factors to the overall pulsar birth velocity distribution is currently not well understood.

The distribution of pulsar velocities has a high velocity component due to the normal pulsars [242, 144, 106], and a lower velocity component from binary and millisecond pulsars [217, 80, 244, 144]. One reason for this dichotomy appears to be that, in order to survive and subsequently form recycled pulsars through the accretion process outlined above, the binary systems contain only those neutron stars with lower birth velocities. In addition, the surviving neutron star has to pull the companion along with it, thus slowing the system down.

Further insights into pulsar kicks from analyses of proper motion and polarization data [165, 166, 311] find strong evidence for an alignment between the spin axis and the velocity vector at birth. These data have recently been combined with modeling of pulsar-wind nebulae [281], where strong evidence is found for a model in which the natal impulse is provided by an anisotropic flux of neutrinos from the proto-neutron star on timescales of a few seconds.

2.11 Pulsar searches

The radio sky is being repeatedly searched for new pulsars in a variety of ways. In the following, we outline the major search strategies that are optimized for binary and millisecond pulsars.

2.12 Going further

Two books, Pulsar Astronomy [241] and Handbook of Pulsar Astronomy [226], cover the literature and techniques and provide excellent further reading. The morphological properties of pulsars have recently been comprehensively discussed in recent reviews [340, 333]. Those seeking a more theoretical viewpoint are advised to read The Theory of Neutron Star Magnetospheres [261] and The Physics of the Pulsar Magnetosphere [34]. Our summary of evolutionary aspects serves merely as a primer to the vast body of literature available. Further insights can be found from more detailed reviews [36, 300, 345].

Pulsar resources available on the Internet are continually becoming more extensive and useful. A good starting point for pulsar-surfers is the Handbook of Pulsar Astronomy website [227], as well as the Pulsar Astronomy wiki [307] and the Cool Pulsars site [375].

3.1 Selection effects in pulsar searches

3.1.2 Interstellar pulse dispersion and multipath scattering

It follows from Equation (3) that the minimum flux density increases as W/(P − W) and hence 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 may be broader than the intrinsic value largely as a result of pulse dispersion and multipath scattering by free electrons in the interstellar medium. 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.

The smearing across the individual frequency channels, however, still remains and becomes significant at high dispersions when searching for short-period pulsars. Multipath scattering from electron density irregularities results in a one-sided broadening of the pulse profile due to the delay in arrival times. A simple scattering model is shown in Figure 12 in which the scattering electrons are assumed to lie in a thin screen between the pulsar and the observer [331]. The timescale of this effect varies roughly as ν⁻⁴, which can not currently be removed by instrumental means.

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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 ≳ 1.4 GHz [76, 168] compared to the 400-MHz search frequency used in early surveys. An added bonus for such observations is the reduction in T_sky which scales with frequency as approximately ν^−2.8 [207]. Pulsar intensities also have an inverse frequency dependence, with the average scaling being ν^−1.6 [234], so that flux densities are roughly an order of magnitude lower at 1.4 GHz compared to 400 MHz. Fortunately, this can be at least partially compensated for by the use of larger receiver bandwidths at higher radio frequencies. For example, the 1.4-GHz system at Parkes has a bandwidth of 288 MHz [240] compared to the 430-MHz system, where nominally 32 MHz is available [251].

3.1.3 Orbital acceleration

Standard pulsar searches use Fourier techniques [226] 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 binary systems with orbital periods longer than about a day. For shorter-period binary systems, 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 S/N [162]. 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) pulsar period in its rest frame and c is the speed of light. Given that the width of a frequency bin in the Fourier domain 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.

As an example of this effect, as seen in the time domain, Figure 13 shows a 22.5-min search mode observation of the binary pulsar B1913+16 [150, 366, 367]. Although this observation covers only about 5% of the orbit (7.75 hr), the severe effects of the Doppler smearing on the pulse signal are very apparent. While the standard search code nominally detects the pulsar with S/N = 9.5 for this observation, it is clear that this value is significantly reduced due to the Doppler shifting of the pulse period seen in the individual sub-integrations.

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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” [262], assumes the pulsar has a constant acceleration during the observation. 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 Figure 13). This technique was first used to find PSR B2127+11C [4], a double neutron star binary in M15 which has parameters similar to B1913+16. Its application to 47 Tucanae [56] resulted in the discovery of nine binary millisecond pulsars, including one in a 96-min orbit around a low-mass (0.15 M_⊙) companion. This is currently the shortest binary period for any known radio pulsar. The majority of binary millisecond pulsars with orbital periods less than a day found in recent globular cluster searches would not have been discovered without the use of acceleration searches.

For intermediate orbital periods, in the range 30 min — several hours, another promising technique is the dynamic power spectrum search shown in Figure 14. Here the time series is split into a number of smaller contiguous segments which are Fourier-transformed separately. The individual spectra are displayed as a two-dimensional (frequency versus time) image. Orbitally modulated pulsar signals appear as sinusoidal signals in this plane as shown in Figure 14.

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This technique has been used by various groups where spectra are inspected visually [245]. Much of the human intervention can be removed using a hierarchical scheme for selecting significant events [71]. This approach was recently applied to a search of the globular cluster M62 resulting in the discovery of three new pulsars. One of the new discoveries — M62F, a faint 2.3-ms pulsar in a 4.8-hr orbit — was detectable only using the dynamic power spectrum technique.

For the shortest orbital periods, the assumption of a constant acceleration during the observation clearly breaks down. In this case, a particularly efficient algorithm has been developed [78, 312, 172, 315] which is optimised to finding binaries with periods so short that many orbits can take place during an observation. This “phase modulation” technique exploits the fact that the Fourier components are modulated by the orbit to create a family of periodic sidebands around the nominal spin frequency of the pulsar. While this technique has so far not resulted in any new discoveries, the existence of short period binaries in 47 Tucanae [56], Terzan 5 [317] and the 11-min X-ray binary X1820−303 in NGC 6624 [354], suggests that there are more ultra-compact radio binary pulsars that await discovery.

3.2 Correcting the observed pulsar sample

In the following, we review common techniques to account for many of the aforementioned selection effects and form a less biased picture of the true pulsar population.

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 (5), therefore, will tend to underestimate the true size of the population. This “small-number bias” was first pointed out [174, 178] for the sample of double neutron star binaries where we know of only five systems relevant for calculations of the merging rate (see Section 3.4.1). Only when N_obs ≳ 10 does the sum of the scale factors become a good indicator of the true population size.

Despite a limited sample size, it has been demonstrated [190] that rigorous confidence intervals of NG can be derived using Bayesian techniques. Monte Carlo simulations verify that the simulated number of detected objects N_detected closely follows a Poisson distribution and that N_detected = αN_G, where α is a constant. By varying the value of N_G in the simulations, the mean of this Poisson distribution can be measured. The Bayesian analysis [190] finds, for a single object, the probability density function of the total population is

$$P({N_{\rm{G}}}) = {\alpha ^2}{N_{\rm{G}}}\exp (- \alpha {N_{\rm{G}}}).$$

(6)

Adopting the necessary assumptions required in the Monte Carlo population about the underlying pulsar distribution, this technique can be used to place interesting constraints on the size and, as we shall see later, birth rate of the underlying population.

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3.2.3 The beaming correction

The “beaming fraction” f in Equation (5) is the fraction of 4π steradians swept out by a pulsar’s radio beam during one rotation. Thus f is the probability that the beam cuts the line-of-sight of an arbitrarily positioned observer. A naïve estimate for f of roughly 20% assumes a circular beam of width ∼ 10° and a randomly distributed inclination angle between the spin and magnetic axes [365]. Observational evidence summarised in Figure 16 suggests that shorter period pulsars have wider beams and therefore larger beaming fractions than their long-period counterparts [269, 243, 37, 360].

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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. An analysis of a large sample of millisecond pulsar profiles [203] suggests that their beaming fraction lies between 50 and 100%. Independent constraints on f for millisecond pulsars come from deep Chandra observations of the globular cluster 47 Tucanae [126] and radio pulsar surveys [56] which suggest that f > 0.4 and likely close to unity [135]. The large beaming fraction and narrow pulses often observed strongly suggests a fan beam model for millisecond pulsars [261].

3.3 The population of normal and millisecond pulsars

3.3.1 Luminosity distributions and local number estimates

Based on a number of all-sky surveys carried out in the 1990s, the scale factor approach has been used to derive the characteristics of the true normal and millisecond pulsar populations and is based on the sample of pulsars within 1.5 kpc of the Sun [244]. Within this region, the selection effects are well understood and easier to quantify than in the rest of the Galaxy. These calculations should therefore give a reliable local pulsar population estimate.

The 430-MHz luminosity distributions obtained from this analysis are shown in Figure 17. For the normal pulsars, integrating the corrected distribution above 1 mJy kpc² and dividing by π × (1.5)² kpc² yields a local surface density, assuming a beaming model [37], of 156±31pulsars kpc⁻² for 430-MHz luminosities above 1 mJy kpc². The same analysis for the millisecond pulsars, assuming a mean beaming fraction of 75% [203], leads to a local surface density of 38 ± 16 pulsars kpc⁻² also for 430-MHz luminosities above 1 mJy kpc².

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3.4 The population of relativistic binaries

Although no radio pulsar has so far been observed in orbit around a black hole companion, we now know of several double neutron star and neutron star-white dwarf binaries which will merge due to gravitational wave emission within a reasonable timescale. The current sample of objects is shown as a function of orbital period and eccentricity in Figure 18. Isochrones showing various coalescence times τ_g are calculated using the expression

$${\tau _{\rm{g}}} \simeq 9.83 \times {10^6}{\rm{yr}}{\left({{{{P_{\rm{b}}}} \over {{\rm{hr}}}}} \right)^{8/3}}{\left({{{{m_1} + {m_2}} \over {{M_ \odot}}}} \right)^{- 2/3}}{\left({{\mu \over {{M_ \odot}}}} \right)^{- 1}}{(1 - {e^2})^{7/2}},$$

(7)

where m₁ and m₂ are the masses of the two stars, μ = m₁m₂/(m₁+ m₂) is the “reduced mass”, P_b is the binary period and e is the eccentricity. This formula is a good analytic approximation (within a few percent) to the numerical solution of the exact equations for τ_g [294, 295].

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In addition to tests of strong-field gravity through observations of relativistic binary systems (see Section 4.4), estimates of their Galactic population and merger rate are of great interest as one of the prime sources for current gravitational wave detectors such as GEO600 [265], LIGO [51], VIRGO [153] and TAMA [273]. In the following, we review empirical determinations of the population sizes and merging rates of binaries where at least one component is visible as a radio pulsar.

3.4.1 Double neutron star binaries

As discussed in Section 2.2, double neutron star (DNS) binaries are expected to be rare. This is certainly the case; as summarized in Table 1, only around ten DNS binaries are currently known. Although we only see both neutron stars as pulsars in J0737−3039 [239], we are “certain” of the identification in five other systems from precise component mass measurements from pulsar timing observations (see Section 4.4). The other systems listed in Table 1 have eccentric orbits, mass functions and periastron advance measurements that are consistent with a DNS identification, but for which there is presently not sufficient component mass information to confirm their nature. One further DNS candidate, the 95-ms pulsar J1753−2243 (see Table 3), has recently been discovered [187]. Although the mass function for this pulsar is lower than the DNS systems listed in Table 1, a neutron star companion cannot be ruled out in this case. Further observations should soon clarify the nature of this system. We note, however, that the 13.6-day orbital period of this system means that it will not contribute to gravitational wave inspiral rate calculations discussed below.

Table 1

Known and likely DNS binaries. Listed are the pulse period P, orbital period Pb, orbital eccentricity e, characteristic age τ_c, and expected binary coalescence timescale τ _g due to gravitational wave emission calculated from Equation (7). To distinguish between definite and candidate DNS systems, we also list whether the masses of both components have been determined via the measurement of two or more post-Keplerian parameters as described in Section 4.4.

	J0737−3039	J1518+4904	B1534+12	J1756−2251	J1811−1736
P [ms]	22.7/2770	40.9	37.9	28.5	104.2
Pb[d]	0.102	8.6	0.4	0.32	18.8
e	0.088	0.25	0.27	0.18	0.83
log10(τc/[yr])	8.3/7.7	10.3	8.4	8.6	9.0
log10(τg/[yr])	7.9	12.4	9.4	10.2	13.0
Masses measured?	Yes	No	Yes	Yes	Yes
	B1820−11	J1829+2456	J1906+0746	B1913+16	B2127+11C
P [ms]	279.8	41.0	144.1	59.0	30.5
Pb[d]	357.8	1.18	0.17	0.3	0.3
e	0.79	0.14	0.085	0.62	0.68
log10(τc/[yr])	6.5	10.1	5.1	8.0	8.0
log10(τg/[yr])	15.8	10.8	8.5	8.5	8.3
Masses measured?	No	No	Yes	Yes	Yes

Despite the uncertainties in identifying DNS binaries, for the purposes of determining the Galactic merger rate, the systems for which τ_g is less than τ_u (i.e. PSRs J0737−3039, B1534+12, J1756−2251, J1906+0746, 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 [305, 301] and is thought to make only a negligible contribution to the merger rate [297]. The general approach with the remaining systems is to derive scale factors for each object, construct the probability density function of their total population (as outlined in Section 3.2.1) and then divide these by a reasonable estimate for the lifetime. Getting such estimates is, however, difficult. It has been proposed [178] that the observable lifetimes for these systems are determined by the timescale on which the current orbital period is reduced by a factor of two [7]. Below this point, the orbital smearing selection effect discussed in Section 3.1.3 will render the binary undetectable by current surveys. More recent work [73] has suggested that a significant population of highly eccentric binary systems could easily evade detection due to their short lifetimes before gravitational wave inspiral. If this selection effect is significant, then the merger rate estimates quoted below could easily be underestimated by a factor of a few.

The results of the most recent DNS merger rate estimates of this kind [189] are summarised in the left panel of Figure 19. The combined Galactic merger rate, dominated by the double pulsar and J1906+0746 is found to be \(118_{- 79}^{+ 174}\,{\rm{My}}{{\rm{r}}^{- 1}}\), where the uncertainties reflect the 95% confidence level using the techniques summarised in Section 3.2.2. Extrapolating this number to include DNS binaries detectable by LIGO in other galaxies [297], the expected event rate is \(49_{- 33}^{+ 73} \times {10^{- 3}}\,{\rm{y}}{{\rm{r}}^{- 1}}\) for initial LIGO and \(265_{- 178}^{+ 390}\,{\rm{y}}{{\rm{r}}^{- 1}}\) for advanced LIGO. Future prospects for detecting gravitational wave emission from binary neutron star inspirals are therefore very encouraging, especially if the population of highly eccentric systems is significant [73]. Since much of the uncertainty in the rate estimates is due to our ignorance of the underlying distribution of double neutron star systems, future gravitational wave detection could ultimately constrain the properties of this exciting binary species.

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Although the double pulsar system J0737−3039 will not be important for ground-based detectors until its final coalescence in another 85 Myr, it may be a useful calibration source for the future space-based detector LISA [272]. It is calculated [175] that a 1-yr observation with LISA would detect (albeit with S/N ∼ 2) the continuous emission at a frequency of 0.2 mHz based on the current orbital parameters. Although there is the prospect of using LISA to detect similar systems through their continuous emission, current calculations [175] suggest that significant (S/N > 5) detections are not likely. Despite these limitations, it is likely that LISA observations will be able to place independent constraints on the Galactic DNS binary population after several years of operation.

3.5 Going further

Good starting points for further reading can be found in other review articles [300, 189]. Our coverage of compact object coalescence rates has concentrated on empirical methods. Two software packages are freely available which allow the user to synthesize and search radio pulsar populations [287, 394, 322]. An alternative approach is to undertake a full-blown Monte Carlo simulation of the most likely evolutionary scenarios described in Section 2.6. In this scheme, 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. Although selection effects are not always taken into account in this approach, the final census is usually normalized to the star formation rate.

Numerous population syntheses (most often to populations of binaries where one or both members are NSs) can be found in the literature [96, 324, 374, 303, 211, 27, 28]. A group in Moscow has made a web interface to their code [355]. 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 [176, 191, 177]. 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. Very recently [289] the results from empirical population constraints and full-blown binary population synthesis codes have been combined to constrain a variety of input parameters and physical conditions. The results of this work are promising, with stringent constraints being placed on the kick distributions, mass-loss fraction during mass transfer and common envelope assumptions.

4.1 Observing basics

Each pulsar is typically observed at least once or twice per month over the course of a year to establish its basic properties. Figure 20 summarises the essential steps involved in a “time-of-arrival” (TOA) measurement. Pulses from the neutron star traverse the interstellar medium before being received at the radio telescope where they are dedispersed and added to form a mean pulse profile.

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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; see [93]). The TOA is defined as the arrival time of some fiducial point on the integrated profile with respect to either the start or the midpoint of the observation. Since the profile has a stable form at any given observing frequency (see Section 2.3), the TOA can be accurately determined by cross-correlation of the observed profile with a high S/N “template” profile obtained from the addition of many observations at the particular observing frequency.

One of the main problems of employing large bandwidths is pulse dispersion. As discussed in Section 2.4, pulses emitted at lower radio frequencies travel slower and arrive later than those emitted at higher frequencies. This process has the effect of “stretching” the pulse across a finite receiver bandwidth, increasing W and therefore increasing ∊_TOA. For normal pulsars, dispersion can largely be compensated for by the incoherent dedispersion process outlined in Section 3.1.

The short periods of millisecond pulsars offer the ultimate in timing precision. In order to fully exploit this, a better 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 dedispersion [129, 226] the incoming signals are dedispersed 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 online either using high speed devices such as field programmable gate arrays [64, 291] or completely in software [343, 350]. Off-line data reduction, while disk-space limited, allows for more flexible analysis schemes [20].

The maximum time resolution obtainable via coherent dedispersion is the inverse of the total receiver bandwidth. The current state of the art is the detection [130] of features on nanosecond timescales in pulses from the 33-ms pulsar B0531+21 in the Crab nebula shown in Figure 21. Simple light travel-time arguments can be made to show that, in the absence of relativistic beaming effects [121], these incredibly bright bursts originate from regions less than 1 m in size.

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4.2 The timing model

A set of timing residuals for the nearby pulsar B1133+16 spanning almost 10 years is shown in Figure 22. Ideally, the residuals should have a zero mean and be free from any systematic trends (see Panel a of Figure 22). To reach this point, however, the model needs to be refined in a bootstrap fashion. Early sets of residuals will exhibit a number of trends indicating a systematic error in one or more of the model parameters, or a parameter not incorporated into the model.

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From Equation (10), an error in the assumed Ω₀ results in a linear slope with time. A parabolic trend results from an error in \({{\dot \Omega}_0}\) (see Panel b of Figure 22). Additional effects will arise if the assumed position of the pulsar (the unit vector \(\underline {\hat s}\) in Equation (9)) used in the barycentric time calculation is incorrect. A position error results in an annual sinusoid (see Panel c of Figure 22). A proper motion produces an annual sinusoid of linearly increasing magnitude (see Panel d of Figure 22).

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 with a precision which improves as the length of the data span increases. For example, 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 [185, 181] defined at midnight UT on December 5, 1988! Measurements of other parameters are no less impressive, with astrometric errors of ∼ 3 μarcsec being presently possible for the bright millisecond pulsar J0437−4715 [388].

4.3 Timing stability

Ideally, after correctly applying a timing model, we would expect a set of uncorrelated timing residuals with a zero mean and a Gaussian scatter with a standard deviation consistent with the measurement uncertainties. As can be seen in Figure 23, this is not always the case; the residuals of many pulsars exhibit a quasi-periodic wandering with time.

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Such “timing noise” is most prominent in the youngest of the normal pulsars [253, 83] and present at a lower level in the much older millisecond pulsars [185, 9]. While the physical processes of this phenomenon are not well understood, it seems likely that they may be connected to super-fluid processes and temperature changes in the interior of the neutron star [3], or to processes in the magnetosphere [75, 74].

The relative dearth of timing noise for the older pulsars is a very important finding. It implies that the measurement precision presently depends primarily on the particular hardware constraints of the observing system. Consequently, a large effort in hardware development is now being made to improve the precision of these observations using, in particular, coherent dedispersion outlined in Section 4.1. Much progress in this area has been made by groups at Princeton [306], Berkeley [377], Jodrell Bank [379], UBC [376], Swinburne [359] and ATNF [13]. From high quality observations spanning over a decade [327, 328, 185], these groups have demonstrated that the timing stability of millisecond pulsars over such timescales is comparable to terrestrial atomic clocks.

This phenomenal stability is demonstrated in Figure 24 which shows σ _z [254, 380], a parameter closely resembling the Allan variance used by the clock community to estimate the stability of atomic clocks [362, 1]. Both PSRs B1937+21 and B1855+09 seem to be limited by a power law component which produces a minimum in σ _z after 2 yr and 5 yr respectively. This is most likely a result of a small amount of intrinsic timing noise [185]. The σ _z based on timing observations [388] of the bright millisecond pulsar J0437−4715 is now 1–2 orders of magnitude smaller than the other two pulsars or the atomic clock!

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4.4 Timing binary pulsars

For binary pulsars, the timing model introduced in Section 4.2 needs to be extended to incorporate the additional motion of the pulsar as it orbits the common centre-of-mass of the binary system. Describing the binary orbit using Kepler’s laws to refer the TOAs to the binary barycentre requires five additional model parameters: the orbital period P_b, projected semi-major orbital axis x, 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. Analogous to the radial velocity curve in a spectroscopic binary, for binary pulsars the orbit is described by the apparent pulse period against time. An example of this is shown in Panel a of Figure 25. Alternatively, when radial accelerations can be measured, the orbit can also be visualised in a plot of acceleration versus period as shown in Panel b of Figure 25. This method is particularly useful for determining binary pulsar orbits from sparsely sampled data [115].

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4.5 Testing general relativity

The key point in the PK definitions introduced in the previous section is that, given the precisely measured Keplerian parameters, the only two unknowns are the masses of the pulsar and its companion, m_p and m_c. Hence, from a measurement of just two PK parameters (e.g., \({\dot \omega}\) and γ) one can solve for the two masses and, using Equation (11), find the orbital inclination angle i. If three (or more) PK parameters are measured, the system is “overdetermined” and can be used to test GR (or, more generally, any other theory of gravity) by comparing the third PK parameter with the predicted value based on the masses determined from the other two.

The first binary pulsar used to test GR in this way was PSR B1913+16 discovered by Hulse & Taylor in 1974 [150]. Measurements of three PK parameters (\({\dot \omega}\), γ and Ṗ_b) were obtained from long-term timing observations at Arecibo [366, 367, 391]. The measurement of orbital decay, which corresponds to a shrinkage of about 3.2 mm per orbit, is seen most dramatically as the gradually increasing shift in orbital phase for periastron passages with respect to a non-decaying orbit shown in Figure 26. This figure includes recent Arecibo data taken since the upgrade of the telescope in the mid 1990s.

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The measurement of orbital decay in the B1913+16 system, obtained from observations spanning a 30-yr baseline [391], is within 0.2% of the GR prediction and provided the first indirect evidence for the existence of gravitational waves. Hulse and Taylor were awarded the 1993 Nobel Physics prize [368, 149, 363] in recognition of their discovery of this remarkable laboratory for testing GR. A similar, though less precise, test of GR from this combination of PK parameters has recently been performed in the double neutron star binary PSR B2127+11C in the globular cluster M15 [158]. For this system, the measurement uncertainties permit a test of GR to 3% precision which is unlikely to improve due to various kinematic contaminations including the acceleration of the binary system in the cluster’s gravitational potential.

Five PK parameters have been measured for the double pulsar discussed in detail below and for the double neutron star system PSR B1534+12 [351] where the test of GR comes from measurements of \({\dot \omega}\), γ and s. In this system, the agreement between theory and observation is within 0.7% [351]. This test will improve in the future as the timing baseline extends and a more significant measurement of r can be made.

Currently the best binary pulsar system for testing GR in the strong-field regime is the double pulsar J0737−3039. In this system, where two independent pulsar clocks can be timed, five PK parameters of the 22.7-ms pulsar “A” have been measured as well as two additional constraints from the measured mass function and projected semi-major axis of the 2.7-s pulsar “B”. A useful means of summarising the limits so far is Figure 27 which shows the allowed regions of parameter space in terms of the masses of the two pulsars. The shaded regions are excluded by the requirement that sin i < 1. Further constraints are shown as pairs of lines enclosing permitted regions as predicted by GR. The measurement [201] of \(\dot \omega = 16.899 \pm 0.001\,\deg \,{\rm{y}}{{\rm{r}}^{- 1}}\) gives the total system mass M = 2.5871 ± 0.0002 M_⊙.

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The measurement of the projected semi-major axes of both orbits gives the mass ratio R = 1.071 ± 0.001. The mass ratio measurement is unique to the double pulsar system and rests on the basic assumption that momentum is conserved. This constraint should apply to any reasonable theory of gravity. The intersection between the lines for \({\dot \omega}\) and R yield the masses of A and B as m_A = 1.3381 ± 0.007 M_⊙ and m_B = 1.2489 ± 0.0007 M_⊙. From these values, using Equations (13–16) the expected values of γ, Ṗ_b, r and s may be calculated and compared with the observed values. These four tests of GR all agree with the theory to within the uncertainties. Currently the tightest constraint is the Shapiro delay parameter s where the observed value is in agreement with GR at the 0.05% level [201].

Another unique feature of the double pulsar system is the interactions between the two pulsars’ radiation beams. Specifically, the signal from A is eclipsed for 30 s each orbit by the magnetosphere of B [239, 184, 260] and the radio pulses from B are modulated by the relativistic wind from A during two phases of the orbit [258]. These provide unique insights into plasma physics [125] and, as shown shown in Figure 27, a measurement of relativistic spin-orbit coupling in the binary system and, hence, another constraint on GR. By careful modeling of the change in eclipse profiles of A over a four-year baseline, Breton et al. [44] have been able to fit a remarkably simple model³ and determine the precession of B’s spin axis about the orbital angular momentum vector. This remarkable measurement agrees, within the 13% measurement uncertainty, to the GR prediction.

It took only two years for the double pulsar system to surpass the tests of GR possible from three decades of monitoring PSR B1913+16 and over a decade of timing PSR B1534+12. Ongoing precision timing measurements of the double pulsar system should soon provide even more stringent and new tests of GR. Crucial to these measurements will be the timing of the 2.7-s pulsar B, where the observed profile is significantly affected by A’s relativistic wind [239, 258]. A careful decoupling of these profile variations is required to accurately measure TOAs for this pulsar and determine the extent to which the theory-independent mass ratio R can be measured. This task is compounded by the fact that pulsar B’s signal is now [44] significantly weaker compared to the discovery observations five years ago [239]. This appears to be a direct result of its beam precessing out of our line of sight.

The relativistic effects observed in the double pulsar system are so large that corrections to higher post-Newtonian order may soon need to be considered. For example, \(\dot \omega\) may be measured precisely enough to require terms of second post-Newtonian order to be included in the computations [91]. In addition, in contrast to Newtonian physics, GR predicts that the spins of the neutron stars affect their orbital motion via spin-orbit coupling. This effect would most clearly be visible as a contribution to the observed \({\dot \omega}\) in a secular [26] and periodic fashion [395]. For the J0737−3039 system, the expected contribution is about an order of magnitude larger than for PSR B1913+16 [239]. As the exact value depends on the pulsars’ moment of inertia, a potential measurement of this effect would allow the moment of inertia of a neutron star to be determined for the first time [91]. Such a measurement would be invaluable for studies of the neutron star equation of state and our understanding of matter at extreme pressure and densities [206].

Finally, in the neutron star-white dwarf binary J1141−6545, the measurement of Ṗ _b , \({\dot \omega}\), γ and s permit a 6% test of GR [35]. Despite this being nominally only a relatively weak constraint for GR, the significantly different self gravities of the neutron star and white dwarf permits constraints on two coupling parameters between matter and the scalar gravitational field. These parameters are non-existent in GR, which describes gravity in terms of a tensor field, but are predicted by some alternative theories of gravity which consider tensor and scalar field components. The constraints provided by PSR J1141−6545 for the strongly non-linear coupling parameter are currently the most stringent to date and are set to improve dramatically over the next five years of timing observations.

4.6 Pulsar timing and neutron star masses

Multiple PK parameters have now been measured for a number of binary pulsars which provide very precise measurements of the neutron star masses [371, 345]. Figure 28 shows the distribution of masses following updates to a recent compilation [345]. While the young pulsars and the double neutron star binaries are consistent with, or just below, the canonical 1.4 M_⊙, we note that the millisecond pulsars in binary systems have, on average, significantly larger masses. This provides strong support for their formation through an extended period of accretion in the past, as discussed in Section 2.6.

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Also shown on this diagram are several eccentric binary systems in globular clusters which have their masses constrained from measurements of the relativistic advance of periastron and the Keplerian mass function. In these cases, the condition sin i < 1 sets a lower limit on the companion mass m_c > (f_massM²)¹/³ and a corresponding upper limit on the pulsar mass. Probability density functions for both m_p and m_c can also be estimated in a statistical sense by assuming a random distribution of orbital inclination angles. An example of this is shown in Figure 29 for the eccentric binary millisecond pulsar in M5 [118] where the nominal pulsar mass is 2.08 ± 0.19 M_⊙! This is a potentially outstanding result. If confirmed by the measurement of other relativistic parameters, these supermassive neutron stars will have important constraints on the equation of state of superdense matter.

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Currently the largest measurement of a radio pulsar mass through multiple PK parameters is the eccentric millisecond pulsar binary system J1903+0327 [70] (see also Section 2.9). Thus, in addition to challenging models of millisecond pulsar formation, this new discovery has important implications for fundamental physics. When placed on the mass-radius diagram for neutron stars [206], this 1.74 M_⊙ pulsar appears to be incompatible with at least four equations of state for superdense matter. Future timing measurements are required to consolidate and verify this potentially very exciting result.

4.7 Pulsar timing and gravitational wave detection

We have seen in the Section 4.4 how pulsar timing can be used to provide indirect evidence for the existence of gravitational waves from coalescing stellar-mass binaries. In this final section, we look at how pulsar timing might soon be used to detect gravitational radiation directly. The idea to use pulsars as natural gravitational wave detectors was first explored in the late 1970s [330, 95]. 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 timescale T. The effect of a passing gravitational wave would be to perturb the local spacetime metric and cause a change in the observed rotational frequency of the pulsar. For regular pulsar timing observations with typical TOA uncertainties of ϵ_TOA, this “detector” would be sensitive to waves with dimensionless amplitudes h ≲ ∊_TOA/T and frequencies f ∼ 1/T [33, 41].

4.7.2 Constraints on massive black hole binaries

In addition to probing the GWB, pulsar timing is beginning to place interesting constraints on the existence of massive black hole binaries. Arecibo data for PSRs B1937+21 and J1713+0747 already make the existence of an equal-mass black hole binary in Sagittarius A* unlikely [214]. More recently, timing data from B1855+09 have been used to virtually rule out the existence of a proposed supermassive black hole as the explanation for the periodic motion seen at the centre of the radio galaxy 3C66B [358].

A simulation of the expected modulations of the timing residuals for the putative binary system, with a total mass of 5.4 × 10¹⁰ M_≲, is shown along with the observed timing residuals in Figure 30. Although the exact signature depends on the orientation and eccentricity of the binary system, Monte Carlo simulations show that the existence of such a massive black hole binary is ruled out with at least 95% confidence [161].

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4.7.3 A millisecond pulsar timing array

A natural extension of the single-arm detector concept discussed above is the idea of using timing data for a number of pulsars distributed over the whole sky to detect gravitational waves [137]. Such a “timing array” would have the advantage over a single arm in that, through a cross-correlation analysis of the residuals for pairs of pulsars distributed over the sky, it should be possible to separate the timing noise of each pulsar from the signature of the GWB common to all pulsars in the array. To illustrate this, consider the fractional frequency shift of the ith pulsar in an array

$${{\delta {\nu _i}} \over {{\nu _i}}} = {\alpha _i}{\mathcal A}(t) + {{\mathcal N}_i}(t),$$

(21)

where α _i is a geometric factor dependent on the line-of-sight direction to the pulsar and the propagation and polarisation vectors of the gravitational wave of dimensionless amplitude A. The timing noise intrinsic to the pulsar is characterised by the function \({{\mathcal N}_i}\). The result of a cross-correlation between pulsars i and j is then

$${\alpha _i}{\alpha _j}\langle {{\mathcal A}^2}\rangle + {\alpha _i}\langle {\mathcal A}{{\mathcal N}_j}\rangle + {\alpha _j}\langle {\mathcal A}{{\mathcal N}_i}\rangle + \langle {{\mathcal N}_i}{{\mathcal N}_j}\rangle ,$$

(22)

where the bracketed terms indicate cross-correlations. Since the wave function and the noise contributions from the two pulsars are independent, the cross correlation tends to \({\alpha _i}{\alpha _j}\langle {{\mathcal A}^2}\rangle\) as the number of residuals becomes large. Summing the cross-correlation functions over a large number of pulsar pairs provides additional information on this term as a function of the angle on the sky [136] and allows the separation of the effects of clock and ephemeris errors from the GWB [112].

A recent analysis [160] applying the timing array concept to data for seven millisecond pulsars has reduced the energy density limit to Ω_gh² < 1.9 × 10⁻⁸ for a background of supermassive black hole sources. The corresponding limits on the background of relic gravitational waves and cosmic strings are 2.0 × 10⁻⁸ and 1.9 × 10⁻⁸ respectively. These limits can be used to constrain the merger rate of supermassive black hole binaries at high redshift, investigate inflationary parameters and place limits on the tension of currently proposed cosmic string scenarios.

The region of the gravitational wave energy density spectrum probed by the current pulsar timing array is shown in Figure 31 where it can be seen that the pulsar timing regime is complementary to the higher frequency bands of LISA and LIGO.

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A number of long-term timing projects are now underway to make a large-scale pulsar timing array a reality. The Parkes pulsar timing array [11, 143] observes twenty millisecond pulsars twice a month. The European Pulsar Timing Array [10] uses the Lovell, Westerbork, Effelsberg and Nancay radio telescopes to regularly observe a similar number. Finally, at Arecibo and Green Bank, regular timing of a dozen or more millisecond pulsars is carried out by a North American consortium [267]. It is expected that a combined analysis of all these efforts could reach a limits of Ω_gh² < 10⁻¹⁰ before 2010 [143]. Looking further ahead, the increase in sensitivity provided by the Square Kilometre Array [154, 198] should further improve the limits of the spectrum probed by pulsar timing. As Figure 31 shows, the SKA could provide up to two orders of magnitude improvement over current capabilities.

4.8 Going further

For further details on the techniques and prospects of pulsar timing, a number of excellent reviews are available [17, 362, 30, 344, 226]. Two audio files and slides from lectures at the centennial meeting of the American Physical Society are also available on-line [15, 399]. The tests of general relativity discussed in Section 4.4, along with further effects, are reviewed elsewhere in this journal by Will [400] and Stairs [344]. A recent review by Kramer & Stairs [200] provides a detailed account of the double pulsar. Further discussions on the realities of using pulsars as gravity wave detectors can be found in other review articles [323, 14, 143].