Abstract
Pulsars of very different types, including isolated objects and binaries (with short and longperiod orbits, and whitedwarf and neutronstar companions) provide the means to test both the predictions of general relativity and the viability of alternate theories of gravity. This article presents an overview of pulsars, then discusses the current status of and future prospects for tests of equivalenceprinciple violations and strongfield gravitational experiments.
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1 Introduction
Since their discovery in 1967 [60], radio pulsars have provided insights into physics on length scales covering the range from 1 m (giant pulses from the Crab pulsar [56]) to 10 km (neutron star) to kpc (Galactic) to hundreds of Mpc (cosmological). Pulsars present an extreme stellar environment, with matter at nuclear densities, magnetic fields of 10^{8} G to nearly 10^{14} G, and spin periods ranging from 1.5 ms to 8.5 s. The regular pulses received from a pulsar each correspond to a single rotation of the neutron star. It is by measuring the deviations from perfect observed regularity that information can be derived about the neutron star itself, the interstellar medium between it and the Earth, and effects due to gravitational interaction with binary companion stars.
In particular, pulsars have proved to be remarkably successful laboratories for tests of the predictions of general relativity (GR). The tests of GR that are possible through pulsar timing fall into two broad categories: setting limits on the magnitudes of parameters that describe violation of equivalence principles, often using an ensemble of pulsars, and verifying that the measured postKeplerian timing parameters of a given binary system match the predictions of strongfield GR better than those of other theories. Longterm millisecond pulsar timing can also be used to set limits on the stochastic gravitationalwave background (see, e.g., [73, 86, 65]), as can limits on orbital variability in binary pulsars for even lower wave frequencies (see, e.g., [20, 78]). However, these are not tests of the same type of precise prediction of GR and will not be discussed here. This review will present a brief overview of the properties of pulsars and the mechanics of deriving timing models, and will then proceed to describe the various types of tests of GR made possible by both single and binary pulsars.
2 Pulsars, Observations, and Timing
The properties and demographics of pulsars, as well as pulsar search and timing techniques, are thoroughly covered in the article by Lorimer in this series [87]. This section will present only an overview of the topics most important to understanding the application of pulsar observations to tests of GR.
2.1 Pulsar properties
Radio pulsars were firmly established to be neutron stars by the discovery of the pulsar in the Crab nebula [120]; its 33ms period was too fast for a pulsating or rotating white dwarf, leaving a rotating neutron star as the only surviving model [108, 53]. The 1982 discovery of a 1.5ms pulsar, PSR B1937+21 [12], led to the realization that, in addition to the “young” Crablike pulsars born in recent supernovae, there exists a separate class of older “millisecond” or “recycled” pulsars, which have been spun up to faster periods by accretion of matter and angular momentum from an evolving companion star. (See, for example, [21] and [109] for reviews of the evolution of such binary systems.) It is precisely these recycled pulsars that form the most valuable resource for tests of GR.
The exact mechanism by which a pulsar radiates the energy observed as radio pulses is still a subject of vigorous debate. The basic picture of a misaligned magnetic dipole, with coherent radiation from charged particles accelerated along the open field lines above the polar cap [55, 128], will serve adequately for the purposes of this article, in which pulsars are treated as a tool to probe other physics. While individual pulses fluctuate severely in both intensity and shape (see Figure 1), a profile “integrated” over several hundred or thousand pulses (i.e., a few minutes) yields a shape — a “standard profile” — that is reproducible for a given pulsar at a given frequency. (There is generally some evolution of pulse profiles with frequency, but this can usually be taken into account.) It is the reproducibility of timeaveraged profiles that permits highprecision timing.
Of some importance later in this article will be models of the pulse beam shape, the envelope function that forms the standard profile. The collection of pulse profile shapes and polarization properties have been used to formulate phenomenological descriptions of the pulse emission regions. At the simplest level (see, e.g., [112] and other papers in that series), the classifications can be broken down into Gaussianshaped “core” regions with little linear polarization and some circular polarization, and doublepeaked “cone” regions with stronger linear polarization and Sshaped position angle swings in accordance with the “Rotating Vector Model” (RVM; see [111]). while these models prove helpful for evaluating observed changes in the profiles of pulsars undergoing geodetic precession, there are ongoing disputes in the literature as to whether the core/cone split is physically meaningful, or whether both types of emission are simply due to the patchy strength of a single emission region (see, e.g., [90]).
2.2 Pulsar observations
A short description of pulsar observing techniques is in order. As pulsars have quite steep radio spectra (see, e.g., [93]), they are strongest at frequencies f_{0} of a few hundred MHz. At these frequencies, the propagation of the radio wave through the ionized interstellar medium (ISM) can have quite serious effects on the observed pulse. Multipath scattering will cause the profile to be convolved with an exponential tail, blurring the sharp profile edges needed for the best timing. Figure 2 shows an example of scattering; the effect decreases with sky frequency as roughly f ^{4}_{0} (see, e.g., [92]), and thus affects timing precision less at higher observing frequencies. A related effect is scintillation: Interference between the rays traveling along the different paths causes timeand frequencydependent peaks and valleys in the pulsar’s signal strength. The decorrelation bandwidth, across which the signal is observed to have roughly equal strength, is related to the scattering time and scales as f ^{4}_{0} (see, e.g., [92]). There is little any instrument can do to compensate for these effects; wide observing bandwidths at relatively high frequencies and generous observing time allocations are the only ways to combat these problems.
Another important effect induced by the ISM is the dispersion of the traveling pulses. Acting as a tenuous electron plasma, the ISM causes the wavenumber of a propagating wave to become frequencydependent. By calculating the group velocity of each frequency component, it is easy to show (see, e.g., [92]) that lower frequencies will arrive at the telescope later in time than the higherfrequency components, following a 1/f^{2} law. The magnitude of the delay is completely characterized by the dispersion measure (DM), the integrated electron content along the line of sight between the pulsar and the Earth. All lowfrequency pulsar observing instrumentation is required to address this dispersion problem if the goal is to obtain profiles suitable for timing. One standard approach is to split the observing bandpass into a multichannel “filterbank,” to detect the signal in each channel, and then to realign the channels following the 1/f^{2} law when integrating the pulse. This method is certainly adequate for slow pulsars and often for nearby millisecond pulsars. However, when the ratio of the pulse period to its DM becomes small, much sharper profiles can be obtained by sampling the voltage signals from the telescope prior to detection, then convolving the resulting time series with the inverse of the easily calculated frequencydependent filter imposed by the ISM. As a result, the pulse profile is perfectly aligned in frequency, without any residual dispersive smearing caused by finite channel bandwidths. In addition, fullStokes information can be obtained without significant increase in analysis time, allowing accurate polarization plots to be easily derived. This “coherent dedispersion” technique [57] is now in widespread use across normal observing bandwidths of several tens of MHz, thanks to the availability of inexpensive fast computing power (see, e.g., [10, 66, 123]). Some of the highestprecision experiments described below have used this approach to collect their data. Figure 3 illustrates the advantages of this technique.
2.3 Pulsar timing
Once dispersion has been removed, the resultant time series is typically folded modulo the expected pulse period, in order to build up the signal strength over several minutes and to obtain a stable timeaveraged profile. The pulse period may not be very easily predicted from the discovery period, especially if the pulsar happens to be in a binary system. The goal of pulsar timing is to develop a model of the pulse phase as a function of time, so that all future pulse arrival times can be predicted with a good degree of accuracy.
The profile accumulated over several minutes is compared by crosscorrelation with the “standard profile” for the pulsar at that observing frequency. A particularly efficient version of the crosscorrelation algorithm compares the two profiles in the frequency domain [130]. Once the phase shift of the observed profile relative to the standard profile is known, that offset is added to the start time of the observation in order to yield a “Time of Arrival” (TOA) that is representative of that fewminute integration. In practice, observers frequently use a time and phasestamp near the middle of the integration in order to minimize systematic errors due to a poorly known pulse period. As a rule, pulse timing precision is best for bright pulsars with short spin periods, narrow profiles with steep edges, and little if any profile corruption due to interstellar scattering.
With a collection of TOAs in hand, it becomes possible to fit a model of the pulsar’s timing behaviour, accounting for every rotation of the neutron star. Based on the magnetic dipole model [108, 53], the pulsar is expected to lose rotational energy and thus “spin down”. The primary component of the timing model is therefore a Taylor expansion of the pulse phase φ with time t:
where φ_{0} and t_{0} are a reference phase and time, respectively, and the pulse frequency ν is the time derivative of the pulse phase. Note that the fitted parameters ν and ν̇ and the magnetic dipole model can be used to derive an estimate of the surface magnetic field B sin α:
where α is the inclination angle between the pulsar spin axis and the magnetic dipole axis, R is the radius of the neutron star (about 10^{6} cm), and the moment of inertia is I≃10^{45} g cm^{2}. In turn, integration of the energy loss, along with the assumption that the pulsar was born with infinite spin frequency, yields a “characteristic age” τ_{c} for the pulsar:
2.3.1 Basic transformation
Equation (1) refers to pulse frequencies and times in a reference frame that is inertial relative to the pulsar. TOAs derived in the rest frame of a telescope on the Earth must therefore be translated to such a reference frame before Equation (1) can be applied. The best approximation available for an inertial reference frame is that of the Solar System Barycentre (SSB). Even this is not perfect; many of the tests of GR described below require correcting for the small relative accelerations of the SSB and the centreofmass frames of binary pulsar systems. But certainly for the majority of pulsars it is adequate. The required transformation between a TOA at the telescope τ and the emission time t from the pulsar is
Here D/f^{2} accounts for the dispersive delay in seconds of the observed pulse relative to infinite frequency; the parameter D is derived from the pulsar’s dispersion measure by D=DM/2.41×10^{4} Hz, with DM in units of pc cm^{3} and the observing frequency f in MHz. The Roemer term Δ_{R⊙} takes out the travel time across the solar system based on the relative positions of the pulsar and the telescope, including, if needed, the proper motion and parallax of the pulsar. The Einstein delay Δ_{E⊙} accounts for the time dilation and gravitational redshift due to the Sun and other masses in the solar system, while the Shapiro delay Δ_{S⊙} expresses the excess delay to the pulsar signal as it travels through the gravitational well of the Sun — a maximum delay of about 120 μs at the limb of the Sun; see [11] for a fuller discussion of these terms. The terms Δ_{R}, Δ_{E}, and Δ_{S} in Equation (4) account for similar “Roemer”, “Einstein”, and “Shapiro” delays within the pulsar binary system, if needed, and will be discussed in Section 2.3.2 below. Most observers accomplish the model fitting, accounting for these delay terms, using the program @@page11TEMPO [110]. The correction of TOAs to the reference frame of the SSB requires an accurate ephemeris for the solar system. The most commonly used ephemeris is the “DE200” standard from the Jet Propulsion Laboratory [127]. It is also clear that accurate timekeeping is of primary importance in pulsar modeling. General practice is to derive the timestamp on each observation from the Observatory’s local time standard — typically a Hydrogen maser — and to apply, retroactively, corrections to wellmaintained time standards such as UTC(BIPM), Universal Coordinated Time as maintained by the Bureau International des Poids et Mesures in Paris.
2.3.2 Binary pulsars
The terms Δ_{R}, Δ_{E}, and Δ_{S} in Equation (4), describe the “Roemer”, “Einstein”, and “Shapiro” delays within a pulsar binary system. The majority of binary pulsar orbits are adequately described by five Keplerian parameters: the orbital period P_{b}, the projected semimajor axis ϰ, the eccentricity e, and the longitude ω and epoch T_{0} of periastron. The angle ω is measured from the line of nodes Ω where the pulsar orbit intersects the plane of the sky. In many cases, one or more relativistic corrections to the Keplerian parameters must also be fit. Early relativistic timing models, developed in the first years after the discovery of PSR B1913+16, either did not provide a full description of the orbit (see, e.g., [22]), or else did not define the timing parameters, in a way that allowed deviations from GR to be easily identified (see, e.g., [49, 58]). The best modern timing model [33, 133, 43] incorporates a number of “postKeplerian” timing parameters which are included in the description of the three delay terms, and which can be fit in a completely phenomenological manner. The delays are defined primarily in terms of the phase of the orbit, defined by the eccentric anomaly u and true anomaly A_{e}(u), as well as ω, P_{b}, and their possible time derivatives. These are related by
where ω_{0} is the reference value of ω at time T_{0}. The delay terms then become:
Here γ represents the combined time dilation and gravitational redshift due to the pulsar’s orbit, and r and s are, respectively, the range and shape of the Shapiro delay. Together with the orbital period derivative Ṗ_{b} and the advance of periastron ω̇, they make up the postKeplerian timing parameters that can be fit for various pulsar binaries. A fuller description of the timing model also includes the aberration parameters δ_{r} and δ_{θ}, but these parameters are not in general separately measurable. The interpretation of the measured postKeplerian timing parameters will be discussed in the context of doubleneutronstar tests of GR in Section 4.
3 Tests of GR — Equivalence Principle Violations
Equivalence principles are fundamental to gravitational theory; for full descriptions, see, e.g., [94] or [152]. Newton formulated what may be considered the earliest such principle, now called the “Weak Equivalence Principle” (WEP). It states that in an external gravitational field, objects of different compositions and masses will experience the same acceleration. The Einstein Equivalence Principle (EEP) includes this concept as well as those of Lorentz invariance (nonexistence of preferred reference frames) and positional invariance (nonexistence of preferred locations) for nongravitational experiments. This principle leads directly to the conclusion that nongravitational experiments will have the same outcomes in inertial and in freelyfalling reference frames. The Strong Equivalence Principle (SEP) adds Lorentz and positional invariance for gravitational experiments, thus including experiments on objects with strong selfgravitation. As GR incorporates the SEP, and other theories of gravity may violate all or parts of it, it is useful to define a formalism that allows immediate identifications of such violations.
The parametrized postNewtonian (PPN) formalism was developed [150] to provide a uniform description of the weakgravitationalfield limit, and to facilitate comparisons of rival theories in this limit. This formalism requires 10 parameters (γ_{PPN}, β, ξ, α_{1}, α_{2}, α_{3}, ζ_{1}, ζ_{2}, ζ_{3}, and ζ_{4}), which are fully described in the article by Will in this series [147], and whose physical meanings are nicely summarized in Table 2 of that article. (Note that γ_{PPN} is not the same as the PostKeplerian pulsar timing parameter γ) Damour and EspositoFarèse [38, 36] extended this formalism to include strongfield effects for generalized tensormultiscalar gravitational theories. This allows a better understanding of limits imposed by systems including pulsars and white dwarfs, for which the amounts of selfgravitation are very different. Here, for instance, α_{1} becomes \({\hat \alpha _1} = {\alpha _1} + {\alpha '_1}({c_1} + {c_2}) + \ldots ,\), where c_{i} describes the “compactness” of mass m_{i}. The compactness can be written
where G is Newton’s constant and E ^{grav}_{ i} is the gravitational selfenergy of mass m_{i}, about 0.2 for a neutron star (NS) and 10^{4} for a white dwarf (WD). Pulsar timing has the ability to set limits on α̂_{1}, which tests for the existence of preferredframe effects (violations of Lorentz invariance); α̂_{3}, which, in addition to testing for preferredframe effects, also implies nonconservation of momentum if nonzero; and ζ_{2}, which is also a nonconservative parameter. Pulsars can also be used to set limits on other SEPviolation effects that constrain combinations of the PPN parameters: the Nordtvedt (“gravitational Stark”) effect, dipolar gravitational radiation, and variation of Newton’s constant. The current pulsar timing limits on each of these will be discussed in the next sections. Table 1 summarizes the PPN and other testable parameters, giving the best pulsar and solarsystem limits.
3.1 Strong Equivalence Principle: Nordtvedt effect
The possibility of direct tests of the SEP through Lunar Laser Ranging (LLR) experiments was first pointed out by Nordtvedt [104]. As the masses of Earth and the Moon contain different fractional contributions from selfgravitation, a violation of the SEP would cause them to fall differently in the Sun’s gravitational field. This would result in a “polarization” of the orbit in the direction of the Sun. LLR tests have set a limit of η<0.001 (see, e.g., [45, 147]), where η is a combination of PPN parameters:
The strongfield formalism instead uses the parameter Δ_{i} [41], which for object “i” may be written as
Pulsarwhitedwarf systems then constrain Δ_{net}=Δ_{pulsar}−Δ_{companion} [41]. If the SEP is violated, the equations of motion for such a system will contain an extra acceleration Δ_{net}g, where g is the gravitational field of the Galaxy. As the pulsar and the white dwarf fall differently in this field, this Δ_{net}g term will influence the evolution of the orbit of the system. For loweccentricity orbits, by far the largest effect will be a longterm forcing of the eccentricity toward alignment with the projection of g onto the orbital plane of the system. Thus, the time evolution of the eccentricity vector will not only depend on the usual GRpredicted relativistic advance of periastron (ω̇), but will also include a constant term. Damour and Schäfer [41] write the timedependent eccentricity vector as
where e_{R}(t) is the ω̇induced rotating eccentricity vector, and e_{F} is the forced component. In terms of Δ_{net}, the magnitude of _{e}_{F} may be written as [41, 145]
where g_{⊥} is the projection of the gravitational field onto the orbital plane, and a=x/sin i is the semimajor axis of the orbit. For smalleccentricity systems, this reduces to
where M is the total mass of the system, and, in GR, F=1 and G is Newton’s constant.
Clearly, the primary criterion for selecting pulsars to test the SEP is for the orbital system to have a large value of P ^{2}_{b} /e, greater than or equal to 10^{7} days^{2} [145]. However, as pointed out by Damour and Schäfer [41] and Wex [145], two agerelated restrictions are also needed. First of all, the pulsar must be sufficiently old that the ω̇induced rotation of e has completed many turns and e_{R}(t) can be assumed to be randomly oriented. This requires that the characteristic age τ_{c} be ≫2π/ω̇, and thus young pulsars cannot be used. Secondly, ω̇ itself must be larger than the rate of Galactic rotation, so that the projection of g onto the orbit can be assumed to be constant. According to Wex [145], this holds true for pulsars with orbital periods of less than about 1000 days.
Converting Equation (16) to a limit on Δ_{net} requires some statistical arguments to deal with the unknowns in the problem. First is the actual component of the observed eccentricity vector (or upper limit) along a given direction. Damour and Schäfer [41] assume the worst case of possible cancellation between the two components of e, namely that e_{F}≃e_{R}. With an angle θ between g_{⊥} and e_{R} (see Figure 4), they write e_{F}≤e/(2sin(θ/2)). Wex [145, 146] corrects this slightly and uses the inequality
where e=e. In both cases, θ is assumed to have a uniform probability distribution between 0 and 2π.
Next comes the task of estimating the projection of g onto the orbital plane. The projection can be written as
where i is the inclination angle of the orbital plane relative to the line of sight, Ω is the line of nodes, and λ is the angle between the line of sight to the pulsar and g [41]. The values of λ and g can be determined from models of the Galactic potential (see, e.g., [83, 1]). The inclination angle i can be estimated if even crude estimates of the neutron star and companion masses are available, from statistics of NS masses (see, e.g., [136]) and/or a relation between the size of the orbit and the WD companion mass (see, e.g., [114]). However, the angle Ω is also usually unknown and also must be assumed to be uniformly distributed between 0 and 2π.
Damour and Schäfer [41] use the PSR B1953+29 system and integrate over the angles θ and Ω to determine a 90% confidence upper limit of Δ_{net}<1.1×10^{2}. Wex [145] uses an ensemble of pulsars, calculating for each system the probability (fractional area in θΩ space) that Δ_{net} is less than a given value, and then deriving a cumulative probability for each value of Δ_{net}. In this way he derives Δ_{net}<5×10^{3} at 95% confidence. However, this method may be vulnerable to selection effects; perhaps the observed systems are not representative of the true population. Wex [146] later overcomes this problem by inverting the question. Given a value of Δ_{net}, an upper limit on θ is obtained from Equation (17). A Monte Carlo simulation of the expected pulsar population (assuming a range of masses based on evolutionary models and a random orientation of (Ω) then yields a certain fraction of the population that agree with this limit on θ. The collection of pulsars ultimately gives a limit of Δ_{net}<9×10^{3} at 95% confidence. This is slightly weaker than Wex’s previous limit but derived in a more rigorous manner.
Prospects for improving the limits come from the discovery of new suitable pulsars, and from better limits on eccentricity from longterm timing of the current set of pulsars. In principle, measurement of the full orbital orientation (i.e., Ω and i) for certain systems could reduce the dependence on statistical arguments. However, the possibility of cancellation between e_{F} and e_{R} will always remain. Thus, even though the required angles have in fact been measured for the millisecond pulsar J04374715 [139], its comparatively large observed eccentricity of ∼2×10^{5} and short orbital period mean it will not significantly affect the current limits.
3.2 Preferredframe effects and nonconservation of momentum
3.2.1 Limits on α̂_{1}
A nonzero α̂_{1} implies that the velocity w of a binary pulsar system (relative to a “universal” background reference frame given by the Cosmic Microwave Background, or CMB) will affect its orbital evolution. In a manner similar to the effects of a nonzero Δ_{net}, the time evolution of the eccentricity will depend on both ω̇ and a term that tries to force the semimajor axis of the orbit to align with the projection of the system velocity onto the orbital plane.
The analysis proceeds in a similar fashion to that for Δ_{net}, except that the magnitude of e_{F} is now written as [34, 18]
where w_{⊥} is the projection of the system velocity onto the orbital plane. The angle λ, used in determining this projection in a manner similar to that of Equation (18), is now the angle between the line of sight to the pulsar and the absolute velocity of the binary system.
The figure of merit for systems used to test α̂_{1} is P ^{1/3}_{b} /e. As for the Δ_{net} test, the systems must be old, so that τ_{c}≫2π/ω̇, and ω̇ must be larger than the rate of Galactic rotation. Examples of suitable systems are PSR J2317+1439 [27, 18] with a last published value of e<1.2×10^{6} in 1996 [28], and PSR J1012+5307, with e<8×10^{7}[84]. This latter system is especially valuable because observations of its whitedwarf component yield a radial velocity measurement [24], eliminating the need to find a lower limit on an unknown quantity. The analysis of Wex [146] yields a limit of α̂_{1}<1.4×10^{4}. This is comparable in magnitude to the weakfield results from lunar laser ranging, but incorporates strong field effects as well.
3.2.2 Limits on α̂_{3}
Tests of α̂_{3} can be derived from both binary and single pulsars, using slightly different techniques. A nonzero α̂_{3}, which implies both a violation of local Lorentz invariance and nonconservation of momentum, will cause a rotating body to experience a selfacceleration a_{self} in a direction orthogonal to both its spin Ω_{S} and its absolute velocity w [107]:
Thus, the selfacceleration depends strongly on the compactness of the object, as discussed in Section 3 above.
An ensemble of single (isolated) pulsars can be used to set a limit on α̂_{3} in the following manner. For any given pulsar, it is likely that some fraction of the selfacceleration will be directed along the line of sight to the Earth. Such an acceleration will contribute to the observed period derivative Ṗ via the Doppler effect, by an amount
where n̂ is a unit vector in the direction from the pulsar to the Earth. The analysis of Will [152] assumes random orientations of both the pulsar spin axes and velocities, and finds that, on average, \(\left {{{\dot P}_{{{\hat \alpha }_3}}}} \right \simeq 5 \times {10^{  5}}\left {{{\hat \alpha }_3}} \right\), independent of the pulse period. The sign of the α̂_{3} contribution to Ṗ, however, may be positive or negative for any individual pulsar; thus, if there were a large contribution on average, one would expect to observe pulsars with both positive and negative total period derivatives. Young pulsars in the field of the Galaxy (pulsars in globular clusters suffer from unknown accelerations from the cluster gravitational potential and do not count toward this analysis) all show positive period derivatives, typically around 10^{14} s/s. Thus, the maximum possible contribution from α̂_{3} must also be considered to be of this size, and the limit is given by α̂_{3}<2×10^{10} [152].
Bell [16] applies this test to a set of millisecond pulsars; these have much smaller period derivatives, on the order of 10^{20}s/s. Here, it is also necessary to account for the “Shklovskii effect” [119] in which a similar Dopplershift addition to the period derivative results from the transverse motion of the pulsar on the sky:
where μ is the proper motion of the pulsar and d is the distance between the Earth and the pulsar. The distance is usually poorly determined, with uncertainties of typically 30% resulting from models of the dispersive free electron density in the Galaxy [132, 30]. Nevertheless, once this correction (which is always positive) is applied to the observed period derivatives for isolated millisecond pulsars, a limit on α̂_{3} on the order of 10^{15} results [16, 19].
In the case of a binarypulsar—whitedwarf system, both bodies experience a selfacceleration. The combined accelerations affect both the velocity of the centre of mass of the system (an effect which may not be readily observable) and the relative motion of the two bodies [19]. The relativemotion effects break down into a term involving the coupling of the spins to the absolute motion of the centre of mass, and a second term which couples the spins to the orbital velocities of the stars. The second term induces only a very small, unobservable correction to P_{b} and ω̇ [19]. The first term, however, can lead to a significant test of α̂_{3}. Both the compactness and the spin of the pulsar will completely dominate those of the white dwarf, making the net acceleration of the two bodies effectively
where c_{p} and Ω_{Sp} denote the compactness and spin angular frequency of the pulsar, respectively, and w is the velocity of the system. For evolutionary reasons (see, e.g., [21]), the spin axis of the pulsar may be assumed to be aligned with the orbital angular momentum of the system, hence the net effect of the acceleration will be to induce a polarization of the eccentricity vector within the orbital plane. The forced eccentricity term may be written as
where β is the (unknown) angle between w and Ω_{Sp}, and P is, as usual, the spin period of the pulsar: P=2π/ω_{Sp}.
The figure of merit for systems used to test α̂_{3} is P ^{2}_{b} /(eP). The additional requirements of τ_{c}≫2π/ω̇ and ω̇ being larger than the rate of Galactic rotation also hold. The 95% confidence limit derived by Wex [146] for an ensemble of binary pulsars is α̂_{3}<1.5×10^{19}, much more stringent than for the singlepulsar case.
3.2.3 Limits on ζ_{2}
Another PPN parameter that predicts the nonconservation of momentum is ζ_{2}. It will contribute, along with α_{3}, to an acceleration of the centre of mass of a binary system [149, 152]
where n_{p} is a unit vector from the centre of mass to the periastron of m_{1}. This acceleration produces the same type of Dopplereffect contribution to a binary pulsar’s Ṗ as described in Section 3.2.2. In a smalleccentricity system, this contribution would not be separable from the Ṗ intrinsic to the pulsar. However, in a highly eccentric binary such as PSR B1913+16, the longitude of periastron advances significantly — for PSR B1913+16, it has advanced nearly 120° since the pulsar’s discovery. In this case, the projection of a_{cm} along the line of sight to the Earth will change considerably over the long term, producing an effective second derivative of the pulse period. This P̈ is given by [149, 152]
where \(X = {m_1}/{m_2}\) is the mass ratio of the two stars and an average value of cosω is chosen. As of 1992, the 95% confidence upper limit on P̈ was 4×10^{30} s^{1} [133, 149]. This leads to an upper limit on (α_{3}+ζ_{2}) of 4×10^{5} [149]. As α_{3} is orders of magnitude smaller than this (see Section 3.2.2), this can be interpreted as a limit on ζ_{2} alone. Although PSR B1913+16 is of course still observed, the infrequent campaign nature of the observations makes it difficult to set a much better limit on P̈ (J. Taylor, private communication, as cited in [75]). The other wellstudied doubleneutronstar binary, PSR B1534+12, yields a weaker test due to its orbital parameters and very similar component masses. A complication for this test is that an observed P̈ could also be interpreted as timing noise (sometimes seen in recycled pulsars [73]) or else a manifestation of profile changes due to geodetic precession [79, 75].
3.3 Strong Equivalence Principle: Dipolar gravitational radiation
General relativity predicts gravitational radiation from the timevarying mass quadrupole of a binary pulsar system. The spectacular confirmation of this prediction will be discussed in Section 4 below. GR does not, however, predict dipolar gravitational radiation, though many theories that violate the SEP do. In these theories, dipolar gravitational radiation results from the difference in gravitational binding energy of the two components of a binary. For this reason, neutronstar—whitedwarf binaries are the ideal laboratories to test the strength of such dipolar emission. The expected rate of change of the period of a circular orbit due to dipolar emission can be written as [152, 35]
where G_{*}=G in GR, and \({\alpha _{{c_i}}}\) is the coupling strength of body “i” to a scalar gravitational field [35]. (Similar expressions can be derived when casting Ṗ_{b dipole} in terms of the parameters of specific tensorscalar theories, such as BransDicke theory [23]. Equation (27), however, tests a more general class of theories.) Of course, the best test systems here are pulsar—whitedwarf binaries with short orbital periods, such as PSR B0655+64 and PSR J1012+5307, where \({\alpha _{{c_1}}} \gg {\alpha _{{c_2}}}\) so that a strong limit can be set on the coupling of the pulsar itself. For PSR B0655+64, Damour and EspositoFarèse [35] used the observed limit of \( {\dot P_{\rm{b}}} = \left( {1 \pm 4} \right) \times {10^{  13}} \) [5] to derive \({\left( {{\alpha _{{c_1}}}  {\alpha _0}} \right)^2} < 3 \times {10^{  4}}\left( {1  \sigma } \right)\), where α_{0} is a reference value of the coupling at infinity. More recently, Arzoumanian [6] has set a somewhat tighter 2σ upper limit of \(\left {{{\dot P}_{\rm{b}}}/{P_{\rm{b}}}} \right < 1 \times {10^{  10}}\ {\rm{y}}{{\rm{r}}^{  1}}\), or \(\left {{{\dot P}_{\rm{b}}}} \right < 2.7 \times {10^{  13}}\), which yields \({\left( {{\alpha _{{c_1}}}  {\alpha _0}} \right)^2} < 2.7 \times {10^{  4}}\). For PSR J1012+5307, a “Shklovskii” correction (see [119] and Section 3.2.2) for the transverse motion of the system and a correction for the (small) predicted amount of quadrupolar radiation must first be subtracted from the observed upper limit to arrive at \({\dot P_{\rm{b}}} = \left( {  0.6 \pm 1.1} \right) \times {10^{  13}}\) and \({\left( {{\alpha _{{c_1}}}  {\alpha _0}} \right)^2} < 4 \times {10^{  4}}\) at 95% confidence [84]. It should be noted that both these limits depend on estimates of the masses of the two stars and do not address the (unknown) equation of state of the neutron stars.
Limits may also be derived from doubleneutronstar systems (see, e.g., [148, 151]), although here the difference in the coupling constants is small and so the expected amount of dipolar radiation is also small compared to the quadrupole emission. However, certain alternative gravitational theories in which the quadrupolar radiation predicts a positive orbital period derivative independently of the strength of the dipolar term (see, e.g., [117, 99, 85]) are ruled out by the observed decreasing orbital period in these systems [142].
Other pulsar—whitedwarf systems with short orbital periods are mostly found in globular clusters, where the cluster potential will also contribute to the observed Ṗ_{b}, or in interacting systems, where tidal effects or magnetic braking may affect the orbital evolution (see, e.g., [4, 50, 100]). However, one system that offers interesting prospects is the recently discovered PSR J11416545 [72], which is a young pulsar with whitedwarf companion in a 4.75hour orbit. In this case, though, the pulsar was formed after the white dwarf, instead of being recycled by the whitedwarf progenitor, and so the orbit is still highly eccentric. This system is therefore expected both to emit sizable amounts of quadrupolar radiation — Ṗ_{b} could be measurable as soon as 2004 [72] — and to be a good test candidate for dipolar emission [52].
3.4 Preferredlocation effects: Variation of Newton’s constant
Theories that violate the SEP by allowing for preferred locations (in time as well as space) may permit Newton’s constant G to vary. In general, variations in G are expected to occur on the timescale of the age of the Universe, such that \(\dot G/G \sim {H_0} \sim 0.7 \times {10^{  10}}\ {\rm{y}}{{\rm{r}}^{  1}}\), where H_{0} is the Hubble constant. Three different pulsarderived tests can be applied to these predictions, as a SEPviolating timevariable G would be expected to alter the properties of neutron stars and white dwarfs, and to affect binary orbits.
3.4.1 Spin tests
By affecting the gravitational binding of neutron stars, a nonzero Ġ would reasonably be expected to alter the moment of inertia of the star and hence change its spin on the same timescale [32]. Goldman [54] writes
where I is the moment of inertia of the neutron star, about 10^{45} g cm^{2}, and N is the (conserved) total number of baryons in the star. By assuming that this represents the only contribution to the observed Ṗ of PSR B0655+64, in a manner reminiscent of the test of α̂_{3} described above, Goldman then derives an upper limit of \(\left {\dot G/G} \right \le \left( {2.2  5.5} \right) \times {10^{  11}}\;{\rm{y}}{{\rm{r}}^{  1}}\), depending on the stiffness of the neutron star equation of state. Arzoumanian [5] applies similar reasoning to PSR J2019+2425 [103], which has a characteristic age of 27 Gyr once the “Shklovskii” correction is applied [102]. Again, depending on the equation of state, the upper limits from this pulsar are \(\left {\dot G/G} \right \le \left( {1.4  3.2} \right) \times {10^{  11}}\;{\rm{y}}{{\rm{r}}^{  1}}\) [5]. These values are similar to those obtained by solarsystem experiments such as laser ranging to the Viking Lander on Mars (see, e.g., [115, 59]). Several other millisecond pulsars, once “Shklovskii” and Galacticacceleration corrections are taken into account, have similarly large characteristic ages (see, e.g., [28, 137]).
3.4.2 Orbital decay tests
The effects on the orbital period of a binary system of a varying G were first considered by Damour, Gibbons, and Taylor [39], who expected
Applying this equation to the limit on the deviation from GR of the Ṗ_{b} for PSR 1913+16, they found a value of \(\dot G/G = (1.0 \pm 2.3) \times {10^{  11}}\;{\rm{y}}{{\rm{r}}^{  1}}\). Nordtvedt [106] took into account the effects of Ġ on neutronstar structure, realizing that the total mass and angular momentum of the binary system would also change. The corrected expression for Ṗ_{b} incorporates the compactness parameter c_{i} and is
(Note that there is a difference of a factor of 2 in Nordtvedt’s definition of c_{i} versus the Damour definition used throughout this article.) Nordtvedt’s corrected limit for PSR B1913+16 is therefore slightly weaker. A better limit actually comes from the neutronstar—whitedwarf system PSR B1855+09, with a measured limit on Ṗ_{b} of (0:6±1.2)×10^{12} [73]. Using Equation (29), this leads to a bound of \(\dot G/G = (  9 \pm 18) \times {10^{  12}}\;{\rm{y}}{{\rm{r}}^{  1}}\), which Arzoumanian [5] corrects using Equation (30) and an estimate of NS compactness to \(\dot G/G = (  1.3 \pm 2.7) \times {10^{  11}}\;{\rm{y}}{{\rm{r}}^{  1}}\). Prospects for improvement come directly from improvements to the limit on Ṗ_{b}. Even though PSR J1012+5307 has a tighter limit on Ṗ_{b} [84], its shorter orbital period means that the Ġ limit it sets is a factor of 2 weaker than obtained with PSR B1855+09.
3.4.3 Changes in the Chandrasekhar mass
The Chandrasekhar mass, M_{Ch}, is the maximum mass possible for a white dwarf supported against gravitational collapse by electron degeneracy pressure [29]. Its value — about 1.4M_{⊙} — comes directly from Newton’s constant: \({M_{{\rm{Ch}}}} \sim {\left( {\hbar c/G} \right)^{3/2}}/m_{\rm{n}}^2\), where ħ is Planck’s constant and m_{n} is the neutron mass. All measured and constrained pulsar masses are consistent with a narrow distribution centred very close to M_{Ch}: 1.35±0.04M_{⊙} [136]. Thus, it is reasonable to assume that M_{Ch} sets the typical neutron star mass, and to check for any changes in the average neutron star mass over the lifetime of the Universe. Thorsett [135] compiles a list of measured and average masses from 5 doubleneutronstar binaries with ages ranging from 0.1 Gyr to 12 or 13 Gyr in the case of the globularcluster binary B2127+11C. Using a Bayesian analysis, he finds a limit of \(\dot G/G = (  0.6 \pm 4.2) \times {10^{  12}}\;{\rm{y}}{{\rm{r}}^{  1}}\) at the 95% confidence level, the strongest limit on record. Figure 5 illustrates the logic applied.
While some cancellation of “observed” mass changes might be expected from the changes in neutronstar binding energy (cf. Section 3.4.2 above), these will be smaller than the M_{Ch} changes by a factor of order the compactness and can be neglected. Also, the claimed variations of the fine structure constant of order \(\Delta \alpha /\alpha \simeq  0.72 \pm 0.18 \times {10^{  5}}\) [140] over the redshift range 0.5<z<3.5 could introduce a maximum derivative of 1/(ħc)·d(ħc)/dt of about 5×10^{16} yr^{1} and hence cannot influencee the Chandrasekhar mass at the same level as the hypothesized changes in G.
One of the five systems used by Thorsett has since been shown to have a whitedwarf companion [138], but as this is one of the youngest systems, this will not change the results appreciably. The recently discovered PSR J18111736 [89], a doubleneutronstar binary, has a characteristic age of only τ_{c}∼1 Gyr and, therefore, will also not significantly strengthen the limit. Ongoing searches for pulsars in globular clusters stand the best chance of discovering old doubleneutronstar binaries for which the component masses can eventually be measured.
4 Tests of GR — StrongField Gravity
The bestknown uses of pulsars for testing the predictions of gravitational theories are those in which the predicted strongfield effects are compared directly against observations. As essentially pointlike objects in strong gravitational fields, neutron stars in binary systems provide extraordinarily clean tests of these predictions. This section will cover the relation between the “postKeplerian” timing parameters and strongfield effects, and then discuss the three binary systems that yield complementary highprecision tests.
4.1 PostKeplerian timing parameters
In any given theory of gravity, the postKeplerian (PK) parameters can be written as functions of the pulsar and companion star masses and the Keplerian parameters. As the two stellar masses are the only unknowns in the description of the orbit, it follows that measurement of any two PK parameters will yield the two masses, and that measurement of three or more PK parameters will overdetermine the problem and allow for selfconsistency checks. It is this test for internal consistency among the PK parameters that forms the basis of the classic tests of strongfield gravity. It should be noted that the basic Keplerian orbital parameters are wellmeasured and can effectively be treated as constants here.
In general relativity, the equations describing the PK parameters in terms of the stellar masses are (see [33, 133, 43]):
where s≡sin i, \(M = {m_1} + {m_2}\) and \({T_ \odot } \equiv G{M_ \odot }/{c^3} = 4.925490947\;\mu {\rm{s}}\). Other theories of gravity, such as those with one or more scalar parameters in addition to a tensor component, will have somewhat different mass dependencies for these parameters. Some specific examples will be discussed in Section 4.4 below.
4.2 The original system: PSR B1913+16
The prototypical doubleneutronstar binary, PSR B1913+16, was discovered at the Arecibo Observatory [96] in 1974 [62]. Over nearly 30 years of timing, its system parameters have shown a remarkable agreement with the predictions of GR, and in 1993 Hulse and Taylor received the Nobel Prize in Physics for its discovery [61, 131]. In the highly eccentric 7.75hour orbit, the two neutron stars are separated by only 3.3 lightseconds and have velocities up to 400 km/s. This provides an ideal laboratory for investigating strongfield gravity.
For PSR B1913+16, three PK parameters are well measured: the combined gravitational redshift and time dilation parameter γ the advance of periastron ω̇, and the derivative of the orbital period, Ṗ_{b}. The orbital parameters for this pulsar, measured in the theoryindependent “DD” system, are listed in Table 2 [133, 144].
The task is now to judge the agreement of these parameters with GR. A second useful timing formalism is “DDGR” [33, 43], which assumes GR to be the true theory of gravity and fits for the total and companion masses in the system, using these quantities to calculate “theoretical” values of the PK parameters. Thus, one can make a direct comparison between the measured DD PK parameters and the values predicted by DDGR using the same data set; the parameters for PSR B1913+16 agree with their predicted values to better than 0.5% [133]. The classic demonstration of this agreement is shown in Figure 6 [144], in which the observed accumulated shift of periastron is compared to the predicted amount.
In order to check the selfconsistency of the overdetermined set of equations relating the PK parameters to the neutron star masses, it is helpful to plot the allowed m_{1}−m_{2} curves for each parameter and to verify that they intersect at a common point. Figure 7 displays the ω̇ and γ curves for PSR B1913+16; it is clear that the curves do intersect, at the point derived from the DDGR mass predictions.
Clearly, any theory of gravity that does not pass such a selfconsistency test can be ruled out. However, it is possible to construct alternate theories of gravity that, while producing very different curves in the m_{1}−m_{2} plane, do pass the PSR B1913+16 test and possibly weakfield tests as well [38]. Such theories are best dealt with by combining data from multiple pulsars as well as solarsystem experiments (see Section 4.4).
A couple of practical points are worth mentioning. The first is that the unknown radial velocity of the binary system relative to the SSB will necessarily induce a Doppler shift in the orbital and neutronstar spin periods. This will change the observed stellar masses by a small fraction but will cancel out of the calculations of the PK parameters [33]. The second is that the measured value of the orbital period derivative Ṗ_{b} is contaminated by several external contributions. Damour and Taylor [42] consider the full range of possible contributions to Ṗ_{b} and calculate values for the two most important: the acceleration of the pulsar binary centreofmass relative to the SSB in the Galactic potential, and the “Shklovskii” v^{2}/r effect due to the transverse proper motion of the pulsar (cf. Section 3.2.2). Both of these contributions have been subtracted from the measured value of Ṗ_{b} before it is compared with the GR prediction. It is our current imperfect knowledge of the Galactic potential and the resulting models of Galactic acceleration (see, e.g., [83, 1]) which now limits the precision of the test of GR resulting from this system.
4.3 PSR B1534+12 and other binary pulsars
A second doubleneutronstar binary, PSR B1534+12, was discovered during a driftscan survey at Arecibo Observatory in 1990 [153]. This system is quite similar to PSR B1913+16: It also has a short (10.1hour) orbit, though it is slightly wider and less eccentric. PSR B1534+12 does possess some notable advantages relative to its more famous cousin: It is closer to the Earth and therefore brighter; its pulse period is shorter and its profile narrower, permitting better timing precision; and, most importantly, it is inclined nearly edgeon to the line of sight from the Earth, allowing the measurement of Shapiro delay as well as the 3 PK parameters measurable for PSR B1913+16. The orbital parameters for PSR B1534+12 are given in Table 3 [125].
As for PSR B1913+16, a graphical version of the internal consistency test is a helpful way to understand the agreement of the measured PK parameters with the predictions of GR. This is presented in Figure 8. It is obvious that the allowedmass region derived from the observed value of Ṗ_{b} does not in fact intersect those from the other PK parameters. This is a consequence of the proximity of the binary to the Earth, which makes the “Shklovskii” contribution to the observed Ṗ_{b} much larger than for PSR B1913+16. The magnitude of this contribution depends directly on the poorly known distance to the pulsar. At present, the best independent measurement of the distance comes from the pulsar’s dispersion measure and a model of the free electron content of the Galaxy [132], which together yield a value of 0.7±0.2 kpc. If GR is the correct theory of gravity, then the correction derived from this distance is inadequate, and the true distance can be found by inverting the problem [17, 121]. The most recent value of the distance derived in this manner is 1.02±0.05 kpc [125]. (Note that the newer “NE2001” Galactic model [30] incorporates the GRderived distance to this pulsar and hence cannot be used in this case.) It is possible that, in the long term, a timing or interferometric parallax may be found for this pulsar; this would alleviate the Ṗ_{b} discrepancy. The GRderived distance is in itself interesting, as it has led to revisions of the predicted merger rate of doubleneutronstar systems visible to gravitationalwave detectors such as LIGO (see, e.g., [121, 7, 71]) — although recent calculations of merger rates determine the most likely merger rates for particular population models and hence are less vulnerable to distance uncertainties in any one system [74].
Despite the problematic correction to Ṗ_{b}, the other PK parameters for PSR B1534+12 are in excellent agreement with each other and with the values predicted from the DDGRderived masses. An important point is that the three parameters ω̇, γ, and s (shape of Shapiro delay) together yield a test of GR to better than 1%, and that this particular test incorporates only “quasistatic” strongfield effects. This provides a valuable complement to the mixed quasistatic and radiative test derived from PSR B1913+16, as it separates the two sectors of the theory.
There are three other confirmed doubleneutronstar binaries at the time of writing. PSR B2127+11C [2, 3] is in the globular cluster M15. While its orbital period derivative has been measured [44], this parameter is affected by acceleration in the cluster potential, and the system has not yet proved very useful for tests of GR, though longterm observations may demonstrate otherwise. The two binaries PSRs J1518+4904 [101] and J18111736 [89] have such wide orbits that, although ω̇ is measured in each case, prospects for measuring further PK parameters are dim. In several circular pulsar—whitedwarf binaries, one or two PK parameters have been measured — typically ω̇ or the Shapiro delay parameters — but these do not overconstrain the unknown masses. The existing system that provides the most optimistic outlook is again the pulsar—whitedwarf binary PSR J11416545 [72], for which multiple PK parameters should be measurable within a few years — although one may need to consider the possibility of classical contributions to the measured ω̇ from a mass quadrupole of the companion.
4.4 Combined binarypulsar tests
Because of their different orbital parameters and inclinations, the doubleneutronstar systems PSRs B1913+16 and B1534+12 provide somewhat different constraints on alternative theories of gravity. Taken together with some of the limits on SEP violation discussed above, and with solarsystem experiments, they can be used to disallow certain regions of the parameter space of these alternate theories. This approach was pioneered by Taylor et al. [134], who combined PKparameter information from PSRs B1913+16 and B1534+12 and the Damour and Schäfer result on SEP violation by PSR B1855+09 [41] to set limits on the parameters β′ and β″ of a class of tensorbiscalar theories discussed in [38] (Figure 9). In this class of theories, gravity is mediated by two scalar fields as well as the standard tensor, but the theories can satisfy the weakfield solarsystem tests. Strongfield deviations from GR would be expected for nonzero values of β′ and β″, but the theories approach the limit of GR as the parameters β′ and β″ approach zero.
A different class of theories, allowing a nonlinear coupling between matter and a scalar field, was later studied by Damour and EspositoFarèse [35, 37]. The function coupling the scalar field φ to matter is given by \(A\left( \phi \right) = \exp \left( {{\tfrac{1}{2}}{\beta _0}{\phi ^2}} \right)\), and the theories are described by the parameters β_{0} and α_{0}=β_{0}φ_{0}, where φ_{0} is the value that φ approaches at spatial infinity (cf. Section 3.3). These theories allow significant strongfield effects when β_{0} is negative, even if the weakfield limit is small. They are best tested by combining results from PSRs B1913+16, B1534+12 (which contributes little to this test), B0655+64 (limits on dipolar gravitational radiation), and solarsystem experiments (Lunar laser ranging, Shapiro delay measured by Viking [116], and the perihelion advance of Mercury [118]). The allowed parameter space from the combined tests is shown graphically in Figure 10 [37]. Currently, for most neutronstar equations of state, the solarsystem tests set a limit on α_{0} (α ^{2}_{0} <10^{3}) that is a few times more stringent than those set by PSRs B1913+16 and B0655+64, although the pulsar observations do eliminate large negative values of β_{0}. With the limits from the pulsar observations improving only very slowly with time, it appears that solarsystem tests will continue to set the strongest limits on α_{0} in this class of theories, unless a pulsar—blackhole system is discovered. If such a system were found with a ∼ 10M_{⊙} black hole and an orbital period similar to that of PSR B1913+16 (∼8 hours), the limit on α_{0} derived from this system would be about 50 times tighter than that set by current solarsystem tests, and 10 times better than is likely to be achieved by the Gravity Probe B experiment [37].
4.5 Independent geometrical information: PSR J04374715
A different and complementary test of GR has recently been permitted by the millisecond pulsar PSR J04374715 [139]. At a distance of only 140 pc, it is the closest millisecond pulsar to the Earth [69], and is also extremely bright, allowing rootmeansquare timing residuals of 35 ns with the 64m Parkes telescope [9], comparable to or better than the best millisecond pulsars observed with current instruments at the 300m Arecibo telescope [96].
The proximity of this system means that the orbital motion of the Earth changes the apparent inclination angle i of the pulsar orbit on the sky, an effect known as the annualorbital parallax [76]. This results in a periodic change of the projected semimajor axis x of the pulsar’s orbit, written as
where r_{⊕}(t) is the timedependent vector from the centre of the Earth to the SSB, and Ω′ is a vector on the plane of the sky perpendicular to the line of nodes. A second contribution to the observed i and hence x comes from the pulsar system’s transverse motion in the plane of the sky [77]:
where μ is the proper motion vector. By including both these effects in the model of the pulse arrival times, both the inclination angle i and the longitude of the ascending node Ω can be determined [139]. As sin i is equivalent to the shape of the Shapiro delay in GR (PK parameter s), the effect of the Shapiro delay on the timing residuals can then easily be computed for a range of possible companion masses (equivalent to the PK parameter r in GR). The variations in the timing residuals are well explained by a companion mass of 0.236±0.017M⊙ (Figure 11). The measured value of ω̇, together with i, also provide an estimate of the companion mass, 0.23±0.14M_{⊙}, which is consistent with the Shapirodelay value.
While this result does not include a true selfconsistency check in the manner of the doubleneutronstar tests, it is nevertheless important, as it represents the only case in which an independent, purely geometric determination of the inclination angle of a binary orbit predicts the shape of the Shapiro delay. It can thus be considered to provide an independent test of the predictions of GR.
4.6 Spinorbit coupling and geodetic precession
A complete discussion of GR effects in pulsar observations must mention geodetic precession, though these results are somewhat qualitative and do not (yet) provide a modelfree test of GR. In standard evolutionary scenarios for doubleneutronstar binaries (see, e.g., [21, 109]), both stellar spins are expected to be aligned with the orbital angular momentum just before the second supernova explosion. After this event, however, the observed pulsar’s spin is likely to be misaligned with the orbital angular momentum, by an angle of the order of 20° [13]. A similar misalignment may be expected when the observed pulsar is the secondformed degenerate object, as in PSR J11416545. As a result, both vectors will precess about the total angular momentum of the system (in practice the total angular momentum is completely dominated by the orbital angular momentum). The evolution of the pulsar spin axis S_{1} can be written as [40, 14]
where the vector Ω ^{spin}_{1} is aligned with the orbital angular momentum. Its magnitude is given by
where \({T_ \odot } \equiv G{M_ \odot }/{c^3} = 4.905490947\;\mu {\rm{s}}\), as in Section 4.1. This predicted rate of precession is small; the three systems with the highest Ω ^{spin}_{1} values are:

PSR J11416545 at 1.35° yr^{1},

PSR B1913+16 at 1.21° yr^{1},

PSR B1534+12 at 0.52° yr^{1}.
The primary manifestation of this precession is expected to be a slow change in the shape of the pulse profile, as different regions of the pulse emission beam move into the observable region.
Evidence for longterm profile shape changes is in fact seen in PSRs B1913+16 and B1534+12. For PSR B1913+16, profile shape changes were first reported in the 1980s [141], with a clear change in the relative heights of the two profile peaks over several years (Figure 12). No similar changes were found in the polarization of the pulsar [31]. Interestingly, although a simple picture of a coneshaped beam might lead to an expectation of a change in the separation of the peaks with time, no evidence for this was seen until the late 1990s, at the Effelsberg 100m telescope [80], by which point the two peaks had begun to move closer together at a rather fast rate. Kramer [80] used this changing peak separation, along with the predicted precession rate and a simple conal model of the pulse beam, to estimate a spinorbit misalignment angle of about 22° and to predict that the pulsar will disappear from view in about 2025 (see Figure 13), in good agreement with an earlier prediction by Istomin [64] made before the peak separation began to change. Recent results from Arecibo [143] confirm the gist of Kramer’s results, with a misalignment angle of about 21°. Both sets of authors find there are four degenerate solutions that can fit the profile separation data; two can be discarded as they predict an unreasonably large misalignment angle of ∼ 180°22°=158° [13], and a third is eliminated because it predicts the wrong direction of the position angle swing under the Rotating Vector Model [111]. The main area of dispute is the actual shape of the emission region; while Weisberg and Taylor find an hourglassshaped beam (see Figure 14), Kramer maintains that a nearly circular cone plus an offset core is adequate (see Figure 15). In any event, it is clear that the interpretation of the profile changes requires some kind of model of the beam shape. Kramer [81, 82] lets the rate of precession vary as another free parameter in the pulseshape fit, and finds a value of 1.2°±0.2°. This is consistent with the GR prediction but still depends on the beamshape model and is therefore not a true test of the precession rate.
PSR B1534+12, despite the disadvantages of a more recent discovery and a much longer precession period, also provides clear evidence of longterm profile shape changes. These were first noticed at 1400 MHz by Arzoumanian [5, 8] and have become more obvious at this frequency and at 430 MHz in the postupgrade period at Arecibo [124]. The principal effect is a change in the lowlevel emission near to the main pulse (Figure 16), though related changes in polarization are now also seen. As this pulsar shows polarized emission through most of its pulse period, it should be possible to form a better picture of the overall geometry than for PSR B1913+16; this may make it easier to derive an accurate model of the pulse beam shape.
As for other tests of GR, the pulsar—whitedwarf binary PSR J11416545 promises interesting results. As noted by the discoverers [72], the region of sky containing this pulsar had been observed at the same frequency in an earlier survey [70], but the pulsar was not seen, even though it is now very strong. It is possible that interference corrupted that original survey pointing, or that a software error prevented its detection, but it is also plausible that the observed pulsar beam is evolving so rapidly that the visible beam precessed into view during the 1990s. Clearly, careful monitoring of this pulsar’s profile is in order.
5 Conclusions and Future Prospects
The tremendous success to date of pulsars in testing different aspects of gravitational theory leads naturally to the question of what can be expected in the future. Improvements to the equivalenceprinciple violation tests will come from both refining the timing parameters of known pulsars (in particular, limits on eccentricities and orbital period derivatives) and the discovery of further pulsar—whitedwarf systems. Potentially coalescing pulsar—whitedwarf binaries, such as PSRs J11416545, J0751+1807 [88], and 17575322 [46], bear watching from the point of view of limits on dipolar gravitational radiation. Another worthy, though difficult, goal is to attempt to derive the full orbital geometry for ultraloweccentricity systems, as has been done for PSR J04374715 [139]; this would quickly lead to significant improvements in the eccentricitydependent tests.
The orbitalperiodderivative measurements of doubleneutronstar binaries are already limited more by systematics (Galactic acceleration models for PSR B1913+16, and poorly known distance for PSR B1534+12) than by pulsar timing precision. However, with improved Galactic modeling and a realistic expectation of an interferometric (VLBI) parallax for PSR B1534+12, there is still hope for testing more carefully the prediction of quadrupolar gravitational radiation from these systems. The other timing parameters, equally important for tests of the quasistatic regime, can be expected to improve with time and better instrumentation, such as the widerbandwidth coherent dedispersion systems now being installed at many observatories (see, e.g., [68, 129]). Especially exciting would be a measurement of the elusive Shapiro delay in PSR B1913+16; the longitude of periastron is now precessing into an angular range where it may facilitate such a measurement [144].
In the last few years, surveys of the Galactic Plane and anking regions, using the 64m Parkes telescope in Australia [9], have discovered several hundred new pulsars (see, e.g., [91, 48]), including several new circularorbit pulsar—whitedwarf systems [46, 47, 26] and the eccentric pulsar—whitedwarf binary PSR J11416545 [72]. A complete reprocessing of the Galactic Plane survey with improved interference filtering is in progress; thus there is still hope that a truly new system such as a pulsar—blackhole binary may emerge from this large survey. Several ongoing smaller surveys of small regions and globular clusters (see, e.g., [25, 113]) are also finding a number of new and exotic binaries, some of which may eventually turn out to be useful for tests of GR. The possible recent appearance of PSR J11416545 and the predicted disappearance of PSR B1913+16 due to geodetic precession make it worthwhile to periodically revisit previously surveyed parts of the sky in order to check for newlyvisible exotic binaries. Over the next several years, largescale surveys are planned at Arecibo [97] and the new 100m Green Bank Telescope [98], offering the promise of over 1000 new pulsars including interesting binary systems. The sensitivity of these surveys will of course be dwarfed by the potential of the proposed Square Kilometre Array radio telescope [63], which will be sensitive to pulsars clear through our Galaxy and into neighbouring galaxies such as M31. The next 10 or 20 years promise to be exciting times for pulsar searchers and for those looking to set evermorestringent limits on deviations from general relativity.
References
Allen, C., and Martos, M.A., “The galactic orbits and tidal radii of selected star clusters”, Rev. Mex. Astron. Astr., 16, 25–36, (1988). For a related online version see: C. Allen, et al., “The galactic orbits and tidal radii of selected star clusters”, [ADS Astronomy Abstract Service]: cited on 27 November 2002, http://adsabs.harvard.edu/cgibin/nphbib_query?bibcode=1988RMxAA..16...25A&db_key=AST. 3.1, 4.2
Anderson, S., Kulkarni, S., Prince, T., and Wolszczan, A., “Timing Solution of PSR 2127+11 C”, in Backer, D., ed., Impact of Pulsar Timing on Relativity and Cosmology, r1r14, (Center for Particle Astrophysics, Berkeley, CA, U.S.A, 1990). 4.3
Anderson, S.B., A study of recycled pulsars in globular clusters, PhD Thesis, (California Institute of Technology, Pasadena, CA, U.S.A., 1992). 4.3
Applegate, J.H., and Shaham, J., “Orbital Period Variability in the Eclipsing Pulsar Binary PSR B1957+20: Evidence for a Tidally Powered Star”, Astrophys. J., 436, 312–318, (1994). 3.3
Arzoumanian, Z., Radio Observations of Binary Pulsars: Clues to Binary Evolution and Tests of General Relativity, PhD Thesis, (Princeton University, Princeton, N.J., U.S.A., 1995). 3.3, 3.4.1, 3.4.2, 4.6
Arzoumanian, Z., “Improved Bounds on Violation of the Strong Equivalence Principle”, in Bailes, M., Nice, D.J., and Thorsett, S.E., eds., Radio Pulsars: In celebration of the contributions of Andrew Lyne, Dick Manchester and Joe Taylor — A Festschrift honoring their 60th Birthdays, volume 302 of ASP Conference Proceedings, (Astronomical Society of the Pacific, San Francisco, CA, U.S.A., 2003). For a related online version see: Z. Arzoumanian, “Improved Bounds on Violation of the Strong Equivalence Principle”, (December, 2002), [Online Los Alamos Archive Preprint]: cited on 27 November 2002, http://xxx.arxiv.org/abs/astroph/0212180. In press; available September 2003. 1, 3.3
Arzoumanian, Z., Cordes, J.M., and Wasserman, I., “Pulsar Spin Evolution, Kinematics, and the Birthrate of Neutron Star Binaries”, Astrophys. J., 520, 696–705, (1999). 4.3
Arzoumanian, Z., Taylor, J.H., and Wolszczan, A., “Evidence for Relativistic Precession in the Pulsar Binary B1534+12”, in Arzoumanian, Z., van der Hooft, F., and van den Heuvel, E.P.J., eds., Pulsar Timing, General Relativity, and the Internal Structure of Neutron Stars, 85–88, (North Holland, Amsterdam, Netherlands, 1999). 4.6
Australia Telescope National Facility, “The Parkes Oservatory Home Page”, (August, 2000), [Online Resources]: cited on 27 November 2002, http://www.parkes.atnf.csiro.au. 2, 4.5, 5
Backer, D.C., Dexter, M.R., Zepka, A., Ng, D., Werthimer, D.J., Ray, P.S., and Foster, R.S., “A Programmable 36 MHz Digital Filter Bank for Radio Science”, Publ. Astron. Soc. Pac., 109, 61–68, (1997). 2.2
Backer, D.C., and Hellings, R.W., “Pulsar Timing and General Relativity”, Annu. Rev. Astron. Astrophys., 24, 537–575, (1986). 2.3.1
Backer, D.C., Kulkarni, S.R., Heiles, C., Davis, M.M., and Goss, W.M., “A Millisecond Pulsar”, Nature, 300, 615–618, (1982). 2.1
Bailes, M., “Geodetic precession in binary pulsars”, Astron. Astrophys., 202, 109–112, (1988). 4.6, 4.6
Barker, B.M., and O’Connell, R.F., “Relativistic Effects in the binary pulsar PSR 1913+16”, Astrophys. J., 199, L25–L26, (1975). 4.6
Bartlett, D.F., and van Buren, D., “Equivalence of active and passive gravitational mass using the moon”, Phys. Rev. Lett., 57, 21–24, (1986). 1
Bell, J.F., “A tighter constraint on postNewtonian gravity using millisecond pulsars”, Astrophys. J., 462, 287–288, (1996). 3.2.2, 3.2.2
Bell, J.F., and Bailes, M., “A New Method for Obtaining Binary Pulsar Distances and its Implications for Tests of General Relativity”, Astrophys. J., 456, L33–L36, (1996). 4.3
Bell, J.F., Camilo, F., and Damour, T., “A tighter test of local Lorentz invariance using PSR J2317+1439”, Astrophys. J., 464, 857–858, (1996). 3.2.1, 3.2.1
Bell, J.F., and Damour, T., “A new test of conservation laws and Lorentz invariance in relativistic gravity”, Class. Quantum Grav., 13, 3121–3127, (1996). 3.2.2
Bertotti, B., Carr, B.J., and Rees, M.J., “Limits from the timing of pulsars on the cosmic gravitational wave background”, Mon. Not. R. Astron. Soc., 203, 945–954, (1983). 1
Bhattacharya, D., and van den Heuvel, E.P.J., “Formation and evolution of binary and millisecond radio pulsars”, Phys. Rep., 203, 1–124, (1991). 2.1, 3.2.2, 4.6
Blandford, R., and Teukolsky, S.A., “Arrivaltime analysis for a pulsar in a binary system”, Astrophys. J., 205, 580–591, (1976). 2.3.2
Brans, C., and Dicke, R.H., “Mach’s Principle and a Relativistic Theory of Gravitation”, Phys. Rev., 124, 925–935, (1961). 3.3
Callanan, P.J., Garnavich, P.M., and Koester, D., “The mass of the neutron star in the binary millisecond pulsar PSR J1012+5307”, Mon. Not. R. Astron. Soc., 298, 207–211, (1998). For a related online version see: P.J. Callanan, et al., “The mass of the neutron star in the binary millisecond pulsar PSR J1012+5307”, [ADS Astronomy Abstract Service]: cited on 27 November 2002, http://adsabs.harvard.edu/cgibin/nphbib_query?bibcode=1998MNRAS.298..207C&db_key=AST. 3.2.1
Camilo, F., Lorimer, D.R., Freire, P., Lyne, A.G., and Manchester, R.N., “Observations of 20 millisecond pulsars in 47 Tucanae at 20 cm”, Astrophys. J., 535, 975–990, (2000). 5
Camilo, F., Lyne, A.G., Manchester, R.N., Bell, J.F., Stairs, I.H., D’Amico, N., Kaspi, V.M., Possenti, A., Crawford, F., and McKay, N.P.F., “Discovery of five binary radio pulsars”, Astrophys. J., 548, L187–L191, (2001). 5
Camilo, F., Nice, D.J., and Taylor, J.H., “Discovery of two fastrotating pulsars”, Astrophys. J., 412, L37–L40, (1993). 3.2.1
Camilo, F., Nice, D.J., and Taylor, J.H., “A search for millisecond pulsars at galactic latitudes 50°<b<20°”, Astrophys. J., 461, 812–819, (1996). 3.2.1, 3.4.1
Chandrasekhar, S., “The Maximum Mass of Ideal White Dwarfs”, Astrophys. J., 74, 81–82, (1931). 3.4.3
Cordes, J.M., and Lazio, T.J.W., “NE2001. I. A New Model for the Galactic Distribution of Free Electrons and its Fluctuations”, (July, 2002), [Online Los Alamos Archive Preprint]: cited on 26 November 2002, http://xxx.arxiv.org/abs/astroph/0207156. 3.2.2, 4.3
Cordes, J.M., Wasserman, I., and Blaskiewicz, M., “Polarization of the binary radio pulsar 1913+16: Constraints on geodetic precession”, Astrophys. J., 349, 546–552, (1990). 4.6
Counselman, C.C., and Shapiro, I.I., “Scientific Uses of Pulsars”, Science, 162, 352–354, (1968). 3.4.1
Damour, T., and Deruelle, N., “General Relativistic Celestial Mechanics of Binary Systems. II. The PostNewtonian Timing Formula”, Ann. Inst. Henri Poincare, 44, 263–292, (1986). 2.3.2, 4.1, 4.2, 4.2
Damour, T., and EspositoFarèse, G., “Testing local Lorentz invariance of gravity with binary pulsar data”, Phys. Rev. D, 46, 4128–4132, (1992). 3.2.1
Damour, T., and EspositoFarèse, G., “Tensorscalar gravity and binarypulsar experiments”, Phys. Rev. D, 54, 1474–1491, (1996). For a related online version see: T. Damour, et al., “Tensorscalar gravity and binarypulsar experiments”, [ADS Astronomy Abstract Service]: cited on 27 November 2002, http://adsabs.harvard.edu/cgibin/nphbib_query?bibcode=1996PhRvD..54.1474D&db_key=PHY. 3.3, 3.3, 4.4
Damour, T., and EspositoFarèse, G., “Testing gravity to second postNewtonian order: A fieldtheory approach”, Phys. Rev. D, 53, 5541–5578, (1996). For a related online version see: T. Damour, et al., “Testing gravity to second postNewtonian order: A fieldtheory approach”, [ADS Astronomy Abstract Service]: cited on 27 November 2002, http://adsabs.harvard.edu/cgibin/nphbib_query?bibcode=1996PhRvD..53.5541D&db_key=PHY. 3
Damour, T., and EspositoFarèse, G., “Gravitationalwave versus binarypulsar tests of strongfield gravity”, Phys. Rev. D, 58, 042001–1–042001–12, (1998). 4.4, 10
Damour, T., and G., EspositoFarèse., “Tensormultiscalar theories of gravitation”, Class. Quantum Grav., 9, 2093–2176, (1992). 3, 4.2, 4.4
Damour, T., Gibbons, G.W., and Taylor, J.H., “Limits on the Variability of G Using BinaryPulsar Data”, Phys. Rev. Lett., 61, 1151–1154, (1988). 3.4.2
Damour, T., and Ruffini, R., “Sur certaines vérifications nouvelles de la Relativité génerale rendues possibles par la découverte d’un pulsar membre d’un systéme binaire (Certain new verifications of general relativity made possible by the discovery of a pulsar belonging to a binary system)”, Comptes Rendus Acad. Sci. Ser. A, 279, 971–973, (1974). For a related online version see: T. Damour, et al., “Certain new verifications of general relativity made possible by the discovery of a pulsar belonging to a binary system”, [ADS Astronomy Abstract Service]: cited on 27 November 2002, http://adsabs.harvard.edu/cgibin/nphbib_query?bibcode=1974CRASM.279..971D&db_key=AST. 4.6
Damour, T., and Schäfer, G., “New Tests of the Strong Equivalence Principle Using BinaryPulsar Data”, Phys. Rev. Lett., 66, 2549–2552, (1991). 3.1, 3.1, 3.1, 3.1, 3.1, 3.1, 4.4, 9
Damour, T., and Taylor, J.H., “On the Orbital Period Change of the Binary Pulsar PSR 1913+16”, Astrophys. J., 366, 501–511, (1991). 4.2
Damour, T., and Taylor, J.H., “StrongField Tests of Relativistic Gravity and Binary Pulsars”, Phys. Rev. D, 45, 1840–1868, (1992). 2.3.2, 4.1, 4.2
Deich, W.T.S., and Kulkarni, S.R., “The Masses of the Neutron Stars in M15C”, in van Paradijs, J., van del Heuvel, E.P.J., and Kuulkers, E., eds., Proceedings of the 165th Symposium of the International Astronomical Union: Compact Stars in Binaries, 279–285, (Kluwer, Dordrecht, Netherlands, 1996). 4.3
Dickey, J.O., Bender, P.L., Faller, J.E., Newhall, X X, Ricklefs, R.L., Ries, J.G., Shelus, P.J., Veillet, C., Whipple, A.L., Wiant, J.R., Williams, J.G., and Yoder, C.F., “Lunar Laser Ranging: A Continuing Legacy of the Apollo Program”, Science, 265, 482–490, (1994). 1, 3.1
Edwards, R.T., and Bailes, M., “Discovery of Two Relativistic Neutron Star — White Dwarf Binaries”, Astrophys. J. Lett., 547, L37–L40, (2001). 5
Edwards, R.T., and Bailes, M., “Recycled Pulsars Discovered at High Radio Frequency”, Astrophys. J., 553, 801–808, (2001). 5
Edwards, R.T., Bailes, M., van Straten, W., and Britton, M.C., “The Swinburne intermediatelatitude pulsar survey”, Mon. Not. R. Astron. Soc., 326, 358–374, (2001). 5
Epstein, R., “The Binary Pulsar: Post Newtonian Timing Effects”, Astrophys. J., 216, 92–100, (1977). 2.3.2
Ergma, E., and Sarna, M.J., “The eclipsing binary millisecond pulsar PSR B174424 A — possible test for a magnetic braking mechanism”, Astron. Astrophys., 363, 657–659, (2000). 3.3
Eubanks, T.M., Martin, J.O., Archinal, B. A, Josties, F.J., Klioner, S.A., Shapiro, S., and Shapiro, I.I., “Advances in solar system tests of gravity”, (August, 1999), [Online Preprint]: cited on 26 November 2002, ftp://casa.usno.navy.mil/navnet/postscript/prd_15.ps. 1
Gérard, J.M., and Wiaux, Y., “Gravitational dipole radiations from binary systems”, Phys. Rev. D, 66, 24040–1–24040–9, (2002). For a related online version see: J.M. Gérard, et al., “Gravitational dipole radiations from binary systems”, [ADS Astronomy Abstract Service]: cited on 27 November 2002, http://adsabs.harvard.edu/cgibin/nphbib_query?bibcode=2002PhRvD..66b4040G&db_key=PHY. 3.3
Gold, T., “Rotating Neutron Stars as the Origin of the Pulsating Radio Sources”, Nature, 218, 731–732, (1968). 2.1, 2.3
Goldman, I., “Upper Limit on G Variability Derived from the SpinDown of PSR 0655+64”, Mon. Not. R. Astron. Soc., 244, 184–187, (1990). 3.4.1
Goldreich, P., and Julian, W.H., “Pulsar electrodynamics”, Astrophys. J., 157, 869–880, (1969). 2.1
Hankins, T.H., Kern, J.S., Weatherall, J.C., and Eilek, J.A., “Nanosecond radio bursts from strong plasma turbulence in the Crab pulsar”, Nature, 422, 141–143, (2003). 1
Hankins, T.H., and Rickett, B.J., “Pulsar Signal Processing”, in Methods in Computational Physics Volume 14 — Radio Astronomy, 55–129, (Academic Press, New York, N.Y., U.S.A., 1975). 2.2
Haugan, M.P., “PostNewtonian Arrivaltime Analysis for a Pulsar in a Binary System”, Astrophys. J., 296, 1–12, (1985). 2.3.2
Hellings, R.W., Adams, P.J., Anderson, J.D., Keesey, M.S., Lau, E.L., Standish, E.M., Canuto, V.M., and Goldman, I., “Experimental Test of the Variability of G Using Viking Lander Ranging Data”, Phys. Rev. Lett., 51, 1609–1612, (1983). 1, 3.4.1
Hewish, A., Bell, S.J., Pilkington, J.D.H., Scott, P.F., and Collins, R.A., “Observation of a Rapidly Pulsating Radio Source”, Nature, 217, 709–713, (1968). 1
Hulse, R.A., “The discovery of the binary pulsar”, Rev. Mod. Phys., 66, 699–710, (1994). 4.2
Hulse, R.A., and Taylor, J.H., “Discovery of a pulsar in a binary system”, Astrophys. J., 195, L51–L53, (1975). 4.2
International Square Kilometre Array Steering Committee, “SKA Home Page”, (November, 2002), [Online Resources]: cited on 27 November 2002, http://www.ras.ucalgary.ca/SKA/index.html. 5
Istomin, Ya. N., “Precession of the axis of rotation of the pulsar PSR B1913+16”, Sov. Astron., 17, 711–718, (1991). 4.6
Jaffe, A.H., and Backer, D.C., “Gravitational Waves Probe the Coalescence Rate of Massive Black Hole Binaries”, Astrophys. J., 583, 616–631, (2003). 1
Jenet, F.A., Cook, W.R., Prince, T.A., and Unwin, S.C., “A Wide Bandwidth Digital Recording System for Radio Pulsar Astronomy”, Publ. Astron. Soc. Pac., 109, 707–718, (1997). 2.2
Jodrell Bank Observatory, “Jodrell Bank Observatory”, (November, 2002), [Online Resources]: cited on 27 November 2002, http://www.jb.man.ac.uk. 3
Jodrell Bank Observatory Pulsar Group, “COBRA: Pulsar Documentation”, (November, 2001), [Online Resources]: cited on 27 November 2002, http://www.jb.man.ac.uk/~pulsar/cobra. 5
Johnston, S., Lorimer, D.R., Harrison, P.A., Bailes, M., Lyne, A.G., Bell, J.F., Kaspi, V.M., Manchester, R. N., D’Amico, N., Nicastro, L., and Jin, S.Z., “Discovery of a very bright, nearby binary millisecond pulsar”, Nature, 361, 613–615, (1993). 4.5
Johnston, S., Lyne, A.G., Manchester, R.N., Kniffen, D.A., D’Amico, N., Lim, J., and Ashworth, M., “A High Frequency Survey of the Southern Galactic Plane for Pulsars”, Mon. Not. R. Astron. Soc., 255, 401–411, (1992). 4.6
Kalogera, V., Narayan, R., Spergel, D.N., and Taylor, J.H., “The Coalescence Rate of Double Neutron Star Systems”, Astrophys. J., 556, 340–356, (2001). 4.3
Kaspi, V.M., Lyne, A.G., Manchester, R.N., Crawford, F., Camilo, F., Bell, J.F., D’Amico, N., Stairs, I.H., McKay, N.P.F., Morris, D.J., and Possenti, A., “Discovery of a Young Radio Pulsar in a Relativistic Binary Orbit”, Astrophys. J., 543, 321–327, (2000). 3.3, 4.3, 4.6, 5
Kaspi, V.M., Taylor, J.H., and Ryba, M., “HighPrecision Timing of Millisecond Pulsars. III. LongTerm Monitoring of PSRs B1855+09 and B1937+21”, Astrophys. J., 428, 713–728, (1994). 1, 3.2.3, 3.4.2
Kim, C., Kalogera, V., and Lorimer, D.R., “The Probability Distribution of Binary Pulsar Coalescence Rates. I. Double Neutron Star Systems in the Galactic Field”, Astrophys. J., 584, 985–995, (2003). 4.3
Konacki, M., Wolszczan, A., and Stairs, I.H., “Geodetic Precession and Timing of the Relativistic Binary Pulsars PSR B1534+12 and PSR B1913+16”, Astrophys. J., 589, 495–502, (2003). 3.2.3
Kopeikin, S.M., “On possible implications of orbital parallaxes of wide orbit binary pulsars and their measurability”, Astrophys. J., 439, L5–L8, (1995). 4.5
Kopeikin, S.M., “Proper motion of binary pulsars as a source of secular variation of orbital parameters”, Astrophys. J., 467, L93–L95, (1996). 4.5
Kopeikin, S.M., “Binary pulsars as detectors of ultralowfrequency graviational waves”, Phys. Rev. D, 56, 4455–4469, (1997). 1
Kopeikin, S.M., Doroshenko, O.V., and Getino, J., “Strong Field Test of the Relativistic SpinOrbit Coupling in Binary Pulsars”, in Lopez Garcia, A., Yagudina, E.I., Martinez Uso, M.J., and Cordero Barbero, A., eds., Proceedings of the 3rd International Workshop on Positional Astronomy and Celestial Mechanics, held in Cuenca, Spain, October 17–21, 1994, 555, (Universitat, Observatorio Astronomico, Valencia, Spain, 1996). 3.2.3
Kramer, M., “Determination of the Geometry of the PSR B1913+16 System by Geodetic Precession”, Astrophys. J., 509, 856–860, (1998). 4.6, 13
Kramer, M., “Geodetic Precession in Binary Neutron Stars”, in Gurzadyan, V.G., Jantzen, R.T., and Ruffini, R., eds., The Ninth Marcel Grossmann Meeting: On Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories: Proceedings of the MGIXMM meeting held at the University of Rome “La Sapienza”, 2–8 July 2000, 219–238, (World Scientific, Singapore, 2002). For a related online version see: M. Kramer, “Geodetic Precession in Binary Neutron Stars”, (May, 2001), [Online Los Alamos Archive Preprint]: cited on 27 November 2002, http://arxiv.org/abs/astroph/0105089. 4.6, 15
Kramer, M., Loehmer, O., and Karastergiou, A., “Geodetic Precession in PSR B1913+16”, in Bailes, M., Nice, D.J., and Thorsett, S.E., eds., Radio Pulsars: In celebration of the contributions of Andrew Lyne, Dick Manchester and Joe Taylor — A Festschrift honoring their 60th Birthdays, volume 302 of ASP Conference Proceedings, (Astronomical Society of the Pacific, San Francisco, CA, U.S.A., 2003). In press; available September 2003. 4.6
Kuijken, K., and Gilmore, G., “The mass distribution in the Galactic disc. I — A technique to determine the integral surface mass density of the disc near the Sun”, Mon. Not. R. Astron. Soc., 239, 571, (1989). 3.1, 4.2
Lange, C., Camilo, F., Wex, N., Kramer, M., Backer, D.C., Lyne, A.G., and Doroshenko, O., “Precision Timing of PSR J1012+5307”, Mon. Not. R. Astron. Soc., 326, 274–282, (2001). For a related online version see: C. Lange, et al., “Precision timing measurements of PSR J1012+5307”, [ADS Astronomy Abstract Service]: cited on 27 November 2002, http://adsabs.harvard.edu/cgibin/nphbib_query?bibcode=2001MNRAS.326..274L&db_key=AST. 3.2.1, 3.3, 3.4.2
Lightman, A.P., and Lee, D.L., “New TwoMetric Theory of Gravity with Prior Geometry”, Phys. Rev. D, 8, 3293–3302, (1973). 3.3
Lommen, A.N., Precision MultiTelescope Timing of Millisecond Pulsars: New Limits on the Gravitational Wave Background and Other Results from the Pulsar Timing Array, PhD Thesis, (University of California at Berkeley, Berkeley, CA, U.S.A., 1995). 1
Lorimer, D.R., “Binary and Millisecond Pulsars at the New Millenium”, Living Rev. Relativity, 4, lrr20015, (June, 2001), [Online Article]: cited on 27 November 2002, http://www.livingreviews.org/lrr20015. 2
Lundgren, S.C., Zepka, A.F., and Cordes, J.M., “A millisecond pulsar in a six hour orbit:PSR J0751+1807”, Astrophys. J., 453, 419–423, (1995). 5
Lyne, A.G., Camilo, F., Manchester, R.N., Bell, J.F., Kaspi, V.M., D’Amico, N., McKay, N.P.F., Crawford, F., Morris, D.J., Sheppard, D.C., and Stairs, I.H., “The Parkes multibeam survey: PSR J18111736 — a pulsar in a highly eccentric binary system”, Mon. Not. R. Astron. Soc., 312, 698–702, (2000). 3.4.3, 4.3
Lyne, A.G., and Manchester, R.N., “The shape of pulsar radio beams”, Mon. Not. R. Astron. Soc., 234, 477–508, (1988). 2.1
Manchester, R.N., Lyne, A.G., Camilo, F., Bell, J.F., Kaspi, V.M., D’Amico, N., McKay, N.P.F., Crawford, F., Stairs, I.H., Possenti, A., Morris, D.J., and Sheppard, D.C., “The Parkes multibeam pulsar survey  I. Observing and data analysis systems, discovery and timing of 100 pulsars”, Mon. Not. R. Astron. Soc., 328, 17–35, (2001). 5
Manchester, R.N., and Taylor, J.H., Pulsars, (W.H. Freeman, San Francisco, CA, U.S.A., 1977). 2.2, 2.2
Maron, O., Kijak, J., Kramer, M., and Wielebinski, R., “Pulsar spectra of radio emission”, Astron. Astrophys. Suppl., 147, 195–203, (2000). 2.2
Misner, C.W., Thorne, K.S., and Wheeler, J.A., Gravitation, (W.H. Freeman, San Francisco, CA, U.S.A., 1973). 3
Müller, J., Nordtvedt, K., and Vokrouhlický, D., “Improved constraint on the α_{1} parameter from lunar motion”, Phys. Rev. D, 54, R5927–R5930, (1996). 1
National Astronomy and Ionosphere Center, “Arecibo Observatory”, (December, 2000), [Online Resources]: cited on 27 November 2002, http://www.naic.edu. 4.2, 4.5, 16
National Astronomy and Ionosphere Center, “Arecibo LBand Feed Array”, (November, 2002), [Online Resources]: cited on 27 November 2002, http://alfa.naic.edu. 5
National Radio Astronomy Observatory, “Green Bank Telescope”, (November, 2002), [Online Resources]: cited on 27 November 2002, http://www.gb.nrao.edu/GBT/GBT.html. 1, 2, 5
Ni, W., “A New Theory of Gravity”, Phys. Rev. D, 7, 2880–2883, (1973). 3.3
Nice, D.J., Arzoumanian, Z., and Thorsett, S.E., “Binary Eclipsing Millisecond Pulsars: A Decade of Timing”, in Kramer, M., Wex, N., and Wielebinski, R., eds., Pulsar Astronomy — 2000 and Beyond: Proceedings of the 177th Colloquium of the IAU held in Bonn, Germany, 30 August–3 September 1999, volume 202 of ASP Conference Series, 67–72, (Astronomical Society of the Pacific, San Francisco, CA, U.S.A., 2000). 3.3
Nice, D.J., Sayer, R.W., and Taylor, J.H., “PSR J1518+4904: A Mildly Relativistic Binary Pulsar System”, Astrophys. J., 466, L87–L90, (1996). 4.3
Nice, D.J., and Taylor, J.H., “PSRs J2019+2425 and J2322+2057 and the Proper Motions of Millisecond Pulsars”, Astrophys. J., 441, 429–435, (1995). 3.4.1
Nice, D.J., Taylor, J.H., and Fruchter, A.S., “Two Newly Discovered Millisecond Pulsars”, Astrophys. J. Lett., 402, L49–L52, (1993). 3.4.1
Nordtvedt, K., “Testing Relativity with Laser Ranging to the Moon”, Phys. Rev., 170, 1186–1187, (1968). 3.1
Nordtvedt, K., “Probing gravity to the second postNewtonian order and to one part in 10 to the 7th using the spin axis of the sun”, Astrophys. J., 320, 871–874, (1987). 1
Nordtvedt, K., “Ġ/G and a Cosmological Acceleration of Gravitationally Compact Bodies”, Phys. Rev. Lett., 65, 953–956, (1990). 3.4.2
Nordtvedt, K.J., and Will, C.M., “Conservation Laws and Preferred Frames in Relativistic Gravity. II. Experimental Evidence to Rule Out PreferredFrame Theories of Gravity”, Astrophys. J., 177, 775–792, (1972). For a related online version see: K.J. Nordtvedt, et al., “Conservation Laws and Preferred Frames in Relativistic Gravity. II. Experimental Evidence to Rule Out PreferredFrame Theories of Gravity”, [ADS Astronomy Abstract Service]: cited on 27 November 2002, http://adsabs.harvard.edu/cgibin/nphbib_query?bibcode=1972ApJ...177..775N&db_key=AST. 3.2.2
Pacini, F., “Rotating Neutron Stars, Pulsars, and Supernova Remnants”, Nature, 219, 145–146, (1968). 2.1, 2.3
Phinney, E.S., and Kulkarni, S.R., “Binary and millisecond pulsars”, Annu. Rev. Astron. Astrophys., 32, 591–639, (1994). 2.1, 4.6
Princeton University Pulsar Lab, “Tempo”, (May, 2000), [Online Resources]: cited on 27 November 2002, http://pulsar.princeton.edu/tempo/index.html. 2.3.1
Radhakrishnan, V., and Cooke, D.J., “Magnetic poles and the polarization structure of pulsar radiation”, Astrophys. Lett., 3, 225–229, (1969). 2.1, 4.6
Rankin, J.M., “Toward an empirical theory of pulsar emission. I. Morphological taxonomy”, Astrophys. J., 274, 333–358, (1983). 2.1
Ransom, S.M., Hessels, J.W.T., Stairs, I.H., Kaspi, V.M., Backer, D.C., Greenhill, L.J., and Lorimer, D.R., “Recent Globular Cluster Searches with Arecibo and the Green Bank Telescope”, in Bailes, M., Nice, D.J., and Thorsett, S.E., eds., Radio Pulsars: In celebration of the contributions of Andrew Lyne, Dick Manchester and Joe Taylor — A Festschrift honoring their 60th Birthdays, volume 302 of ASP Conference Proceedings, (Astronomical Society of the Pacific, San Francisco, CA, U.S.A., 2003). For a related online version see: S.M. Ransom, et al., “Recent Globular Cluster Searches with Arecibo and the Green Bank Telescope”, (November, 2002), [Online Los Alamos Archive Preprint]: cited on 27 November 2002, http://xxx.arxiv.org/abs/astroph/0211160. In press; available September 2003. 5
Rappaport, S., Podsiadlowski, P., Joss, P.C., DiStefano, R., and Han, Z., “The relation between white dwarfs mass and orbital period in wide binary radio pulsars”, Mon. Not. R. Astron. Soc., 273, 731–741, (1995). 3.1
Reasenberg, R.D., “The constancy of G and other gravitational experiments”, Philos. Trans. R. Soc. London, Ser. A, 310, 227, (1983). 1, 3.4.1
Reasenberg, R.D., Shapiro, I.I., MacNeil, P.E., Goldstein, R.B., Breidenthal, J.C., Brenkle, J.P., Cain, D.L., Kaufman, T.M., Komarek, T.A., and Zygielbaum, A.I., “Viking Relativity Experiment: Verification of Signal Retardation by Solar Gravity”, Astrophys. J., 234, L219–L221, (1979). 4.4
Rosen, N., “A Bimetric theory of gravitation”, Gen. Relativ. Gravit., 4, 435–447, (1973). 3.3
Shapiro, I.I., “Solar System Tests of General Relativity: Recent Results and Present Plans”, in Ashby, N., Bartlett, D.F., and Wyss, W., eds., General Relativity and Gravitation, 313–330, (Cambridge University Press, Cambridge, U.K., 1990). 1, 4.4
Shklovskii, I.S., “Possible causes of the secular increase in pulsar periods”, Sov. Astron., 13, 562–565, (1970). 3.2.2, 3.3
Staelin, D.H., and Reifenstein III., E.C., “Pulsating radio sources near the Crab Nebula”, Science, 162, 1481–1483, (1968). 2.1
Stairs, I.H., Arzoumanian, Z., Camilo, F., Lyne, A.G., Nice, D.J., Taylor, J.H., Thorsett, S.E., and Wolszczan, A., “Measurement of Relativistic Orbital Decay in the PSR B1534+12 Binary System”, Astrophys. J., 505, 352–357, (1998). 4.3
Stairs, I.H., Manchester, R.N., Lyne, A G., Kaspi, V.M., Camilo, F., Bell, J.F., D’Amico, N., Kramer, M., Crawford, F., Morris, D.J., McKay, N.P.F., Lumsden, S.L., TacconiGarman, L.E., Cannon, R.D., Hambly, N.C., and Wood, P.R., “PSR J17403052 — a Pulsar with a Massive Companion”, Mon. Not. R. Astron. Soc., 325, 979–988, (2001). 2
Stairs, I.H., Splaver, E.M., Thorsett, S.E., Nice, D.J., and Taylor, J.H., “A Baseband Recorder for Radio Pulsar Observations”, Mon. Not. R. Astron. Soc., 314, 459–467, (2000). 3, 2.2
Stairs, I.H., Thorsett, S.E., Taylor, J.H., and Arzoumanian, Z., “Geodetic Precession in PSR B1534+12”, in Kramer, M., Wex, N., and Wielebinski, R., eds., Pulsar Astronomy — 2000 and Beyond: Proceedings of the 177th Colloquium of the IAU held in Bonn, Germany, 30 August–3 September 1999, volume 202 of ASP Conference Series, 121–124, (Astronomical Society of the Pacific, San Francisco, CA, U.S.A., 2000). 4.6
Stairs, I.H., Thorsett, S.E., Taylor, J.H., and Wolszczan, A., “Studies of the Relativistic Binary Pulsar PSR B1534+12: I. Timing Analysis”, Astrophys. J., 581, 501–508, (2002). 4.3, 3, 4.3, 8
Stairs, I.H. and Thorsett, S.E., and Camilo, F., “CoherentlyDedispersed Polarimetry of Millisecond Pulsars”, Astrophys. J. Suppl. Ser., 123, 627–638, (1999). 3
Standish, E.M., “Orientation of the JPL ephemerides, DE200/LE200, to the dynamical equinox of J 2000”, Astron. Astrophys., 114, 297–302, (1982). 2.3.1
Sturrock, P.A., “A model of pulsars”, Astrophys. J., 164, 529–556, (1971). 2.1
Swinburne Pulsar Group, “The Caltech, Parkes, Swinburne Recorder Mk II”, (November, 2002), [Online Resources]: cited on 27 November 2002, http://astronomy.swin.edu.au/pulsar/. 5
Taylor, J.H., “Pulsar Timing and Relativistic Gravity”, Philos. Trans. R. Soc. London, Ser. A, 341, 117–134, (1992). 2.3
Taylor, J.H., “Binary Pulsars and Relativistic Gravity”, in Frangsmyr, T., ed., Les Prix Nobel 1993: Nobel Prizes, Presentations, Biographies, and Lectures, 80–101, (Norstedts Tryckeri, Stockholm, Sweden, 1994). For a related online version see: J.H. Taylor, “Binary Pulsars and Relativistic Gravity”, [Online Resource]: cited on 27 November 2002, http://www.nobel.se/physics/laureates/1993/taylorlecture.pdf. 4.2
Taylor, J.H., and Cordes, J.M., “Pulsar Distances and the Galactic Distribution of Free Electrons”, Astrophys. J., 411, 674–684, (1993). 3.2.2, 4.3
Taylor, J.H., and Weisberg, J.M., “Further experimental tests of relativistic gravity using the binary pulsar PSR 1913+16”, Astrophys. J., 345, 434–450, (1989). 2.3.2, 3.2.3, 4.1, 4.2, 4.2, 12
Taylor, J.H., Wolszczan, A., Damour, T., and Weisberg, J.M., “Experimental constraints on strongfield relativistic gravity”, Nature, 355, 132–136, (1992). 4.4, 9
Thorsett, S.E., “The Gravitational Constant, the Chandrasekhar Limit, and Neutron Star Masses”, Phys. Rev. Lett., 77, 1432–1435, (1996). 1, 3.4.3, 5
Thorsett, S.E., and Chakrabarty, D., “Neutron Star Mass Measurements. I. Radio Pulsars”, Astrophys. J., 512, 288–299, (1999). 3.1, 3.4.3
Toscano, M., Sandhu, J.S., Bailes, M., Manchester, R.N., Britton, M.C., Kulkarni, S.R., Anderson, S.B., and Stappers, B.W., “Millisecond pulsar velocities”, Mon. Not. R. Astron. Soc., 307, 925–933, (August, 1999). 3.4.1
van Kerkwijk, M., and Kulkarni, S.R., “A Massive White Dwarf Companion to the Eccentric Binary Pulsar System PSR B2303+46”, Astrophys. J., 516, L25–L28, (1999). 3.4.3
van Straten, W., Bailes, M., Britton, M., Kulkarni, S.R., Anderson, S.B., Manchester, R.N., and Sarkissian, J., “A test of general relativity from the threedimensional orbital geometry of a binary pulsar”, Nature, 412, 158–160. (vbb01) year = 2001,. 3.1, 4.5, 4.5, 11, 5
Webb, J.K., Murphy, M.T., Flambaum, V.V., Dzuba, V.A., Barrow, J.D., Churchill, C.W., Prochaska, J.X., and Wolfe, A.M., “Further Evidence for Cosmological Evolution of the Fine Structure Constant”, Phys. Rev. Lett., 87, 91301–1–91301–4, (2001). 3.4.3
Weisberg, J.M., Romani, R.W., and Taylor, J.H., “Evidence for geodetic spin precession in the binary pulsar 1913+16”, Astrophys. J., 347, 1030–1033, (1989). 4.6, 13
Weisberg, J.M., and Taylor, J.H., “Gravitational Radiation from an Orbiting Pulsar”, Gen. Relativ. Gravit., 13, 1–6, (1981). 3.3
Weisberg, J.M., and Taylor, J.H., “General Relativistic Geodetic Spin Precession in Binary Pulsar B1913+16: Mapping the Emission Beam in Two Dimensions”, Astrophys. J., 576, 942–949, (2002). 4.6, 14
Weisberg, J.M., and Taylor, J.H., “The Relativistic Binary Pulsar B1913+16”, in Bailes, M., Nice, D.J., and Thorsett, S.E., eds., Radio Pulsars: In celebration of the contributions of Andrew Lyne, Dick Manchester and Joe Taylor — A Festschrift honoring their 60th Birthdays, volume 302 of ASP Conference Proceedings, (Astronomical Society of the Pacific, San Francisco, CA, U.S.A., 2003). For a related online version see: J.M. Weisberg, et al., “The Relativistic Binary Pulsar B1913+16”, (November, 2002), [Online Los Alamos Archive Preprint]: cited on 27 November 2002, http://xxx.arxiv.org/abs/astroph/0211217. In press; available September 2003. 4.2, 2, 6, 4.2, 7, 5
Wex, N., “New limits on the violation of the Strong Equivalence Principle in strong field regimes”, Astron. Astrophys., 317, 976–980, (1997). 3.1, 3.1, 4, 3.1, 3.1
Wex, N., “Smalleccentricity binary pulsars and relativistic gravity”, in Kramer, M., Wex, N., and Wielebinski, R., eds., Pulsar Astronomy — 2000 and Beyond: Proceedings of the 177th Colloquium of the IAU held in Bonn, Germany, 30 August–3 September 1999, volume 202 of ASP Conference Series, 113–116, (Astronomical Society of the Pacific, San Francisco, CA, U.S.A., 2000). 1, 3.1, 3.1, 3.2.1, 3.2.2
Will, C.M., “The Confrontation Between General Relativity and Experiment”, Living Rev. Relativity, 4, lrr20014, (May, 2001), [Online Article]: cited on 27 November 2002, http://www.livingreviews.org/lrr20014. 3, 1, 3.1
Will, C.M., “Gravitational radiation from binary systems in alternative metric theories of gravity — Dipole radiation and the binary pulsar”, Astrophys. J., 214, 826–839, (1977). For a related online version see: C.M. Will, “Gravitational radiation from binary systems in alternative metric theories of gravity — Dipole radiation and the binary pulsar”, [ADS Astronomy Abstract Service]: cited on 27 November 2002, http://adsabs.harvard.edu/cgibin/nphbib_query?bibcode=1977ApJ...214..826W&db_key=AST. 3.3
Will, C.M., “Is momentum conserved? A test in the binary system PSR 1913+16”, Astrophys. J., 393, L59–L61, (1992). 1, 3.2.3, 3.2.3, 3.2.3
Will, C.M., and Nordtvedt, K.J., “Conservation Laws and Preferred Frames in Relativistic Gravity. I. PreferredFrame Theories and an Extended PPN Formalism”, Astrophys. J., 177, 757–774, (1972). For a related online version see: C.M. Will, et al., “Conservation Laws and Preferred Frames in Relativistic Gravity. I. PreferredFrame Theories and an Extended PPN Formalism”, [ADS Astronomy Abstract Service]: cited on 27 November 2002, http://adsabs.harvard.edu/cgibin/nphbib_query?bibcode=1972ApJ...177..757W&db_key=AST. 3
Will, C.M., and Zaglauer, H.W., “Gravitational radiation, close binary systems, and the BransDicke theory of gravity”, Astrophys. J., 346, 366–377, (1989). 3.3
Will, C.W., Theory and Experiment in Gravitational Physics, (Cambridge University Press, Cambridge, U.K., 1993). 3, 1, 3.2.2, 3.2.3, 3.2.3, 3.3
Wolszczan, A., “A nearby 37.9 ms radio pulsar in a relativistic binary system”, Nature, 350, 688–690, (1991). 4.3
Acknowledgements
The author holds an NSERC University Faculty Award and is supported by a Discovery Grant. She thanks Michael Kramer, George Hobbs, and Zaven Arzoumanian for careful readings of the manuscript, Duncan Lorimer for generously sharing his keyworded reference list, and Gilles EspositoFarèse, Michael Kramer, Joe Taylor, Steve Thorsett, Willem van Straten, and Joel Weisberg for allowing reproduction of figures from their work. The Arecibo Observatory, a facility of the National Astronomy and Ionosphere Center, is operated by Cornell University under a cooperative agreement with the National Science Foundation. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The Parkes radio telescope is part of the Australia Telescope which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.
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Stairs, I.H. Testing General Relativity with Pulsar Timing. Living Rev. Relativ. 6, 5 (2003). https://doi.org/10.12942/lrr20035
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DOI: https://doi.org/10.12942/lrr20035