4.4 Timing binary pulsars

For binary pulsars, the simple timing model introduced in Section 4.2 needs to be extended to incorporate the additional motion of the pulsar as it orbits the common centre-of-mass of the binary system. Treating the binary orbit using Kepler’s laws to refer the TOAs to the binary barycentre requires five additional model parameters: the orbital period $P b$ , projected semi-major orbital axis $x$ , orbital eccentricity $e$ , longitude of periastron $w$ and the epoch of periastron passage $T0$ . This description, using five “Keplerian parameters”, is identical to that used for spectroscopic binary stars. Analogous to the radial velocity curve in a spectroscopic binary, for binary pulsars the orbit is described by the apparent pulse period against time. An example of this is shown in Panel a of Figure 25

. Alternatively, when radial accelerations can be measured, the orbit can also be visualised in a plot of acceleration versus period as shown in Panel b of Figure 25

. This method is particularly useful for determining binary pulsar orbits from sparsely sampled data [102

Figure 25:

Panel a: Keplerian orbital fit to the $669 -day$ binary pulsar J0407+1607 [190

]. Panel b: Orbital fit in the period-acceleration plane for the globular cluster pulsar 47 Tuc S [102 ].

Constraints on the masses of the pulsar $mp$ and the orbiting companion $mc$ can be placed by combining $x$ and $Pb$ to obtain the mass function

where $G$ is Newton’s gravitational constant and $i$ is the (initially unknown) angle between the orbital plane and the plane of the sky (i.e. an orbit viewed edge-on corresponds to $i = 90o$ ). In the absence of further information, the standard practice is to consider a random distribution of inclination angles. Since the probability that $i$ is less than some value $i 0$ is $p(< i ) = 1 - cos(i) 0 0$ , the 90% confidence interval for $i$ is $o o 26 < i < 90$ . For an assumed pulsar mass, the 90% confidence interval for $mc$ can be obtained by solving Equation (11

) for $o i = 26$ and $o 90$ . If the sum of the masses $M = mp + mc$ can be determined (e.g., through a measurement of relativistic periastron advance described below), then the condition $sini < 1$ sets a lower limit on the companion mass $m > (f M 2)1/3 c mass$ and a corresponding upper limit on the pulsar mass.

Although most of the presently known binary pulsar systems can be adequately timed using Kepler’s laws, there are a number which require an additional set of “post-Keplerian” (PK) parameters which have a distinct functional form for a given relativistic theory of gravity. In general relativity (GR) the PK formalism gives the relativistic advance of periastron

the time dilation and gravitational redshift parameter

the rate of orbital decay due to gravitational radiation

and the two Shapiro delay parameters

and

which describe the delay in the pulses around superior conjunction where the pulsar radiation traverses the gravitational well of its companion. In the above expressions, all masses are in solar units, $M =_ mp + mc$ , $x =_ apsini/c$ , $s =_ sin i$ and $3 T o. =_ GMo. /c = 4.925490947 ms$ . Some combinations, or all, of the PK parameters have now been measured for a number of binary pulsar systems. Further PK parameters due to aberration and relativistic deformation [80 ] are not listed here but may soon be important for the double pulsar [185

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

Figure 26:

Orbital decay in the binary pulsar B1913+16 system demonstrated as an increasing orbital phase shift for periastron passages with time. The GR prediction due entirely to the emission of gravitational radiation is shown by the parabola. Figure provided by Joel Weisberg [336

The first binary pulsar used to test GR in this way was PSR B1913+16 discovered by Hulse & Taylor in 1974 [129 ]. Measurements of three PK parameters ( $w$ , $g$ and $Pb$ ) were obtained from long-term timing observations at Arecibo [312

, 313

]. The measurement of orbital decay, which corresponds to a shrinkage of about $3.2 mm$ per orbit, is seen most dramatically as the gradually increasing shift in orbital phase for periastron passages with respect to a non-decaying orbit shown in Figure 26

. This figure includes recent Arecibo data taken since the upgrade of the telescope in the mid 1990s. The measurement of orbital decay, now spanning a $30-yr$ baseline [336 ], is within 0.2% of the GR prediction and provided the first indirect evidence for the existence of gravitational waves. Hulse and Taylor were awarded the 1993 Nobel Physics prize [314 , 128 , 309 ] in recognition of their discovery of this remarkable laboratory for testing GR.

More recently, five PK parameters have been measured for PSRs B1534+12 [298 ] and J0737 $-$ 3039A [161 ]. For PSR B1534+12, the test of GR comes from measurements of $w$ , $g$ and $s$ , where the agreement between theory and observation is within 0.7% [298 ]. This test will improve in the future as the timing baseline extends and a more significant measurement of $r$ can be made. Although a significant measurement of $Pb$ exists, it is known to be contaminated by kinematic effects which depend on the assumed distance to the pulsar [293 ]. Assuming GR to be correct, the observed and theoretical $Pb$ values can be reconciled to provide a “relativistic measurement” of the distance $d = 1.04 ± 0.03 kpc$ [288 ]. Prospects for independent parallax measurements of the distance to this pulsar using radio interferometry await more sensitive telescopes [289 ].

Figure 27:

‘Mass-mass’ diagram showing the observational constraints on the masses of the neutron stars in the double pulsar system J0737 $-$ 3039. Inset is an enlarged view of the small square encompassing the intersection of the tightest constraints. Figure provided by Michael Kramer [161 ].

For PSR J0737 $-$ 3039, where two independent pulsar clocks can be timed, five PK parameters of the $22.7-ms$ pulsar “A” have been measured as well as two additional constraints from the measured mass function and projected semi-major axis of the $2.7 -s$ pulsar “B”. In terms of a laboratory for GR, then, J0737 $-$ 3039 promises to go well beyond the results possible from PSRs B1913+16 and B1534+12. A useful means of summarising the limits so far is Figure 27

which shows the allowed regions of parameter space in terms of the masses of the two pulsars. The shaded regions are excluded by the requirement that $sin i < 1$ . Further constraints are shown as pairs of lines enclosing permitted regions as predicted by GR. The measurement of $w = 16.899 ± 0.001 deg yr-1$ gives the total system mass $M = 2.587 ± 0.001M o.$ . The measurement of the projected semi-major axes of both orbits gives the mass ratio $R = 1.071 ± 0.001$ . The mass ratio measurement is unique to the double pulsar system and rests on the basic assumption that momentum is conserved. This constraint should apply to any reasonable theory of gravity. The intersection between the lines for $w$ and $R$ yield the masses of A and B as $mA = 1.338 ± 0.001 Mo .$ and $m = 1.249 ± 0.001 M B o .$ . From these values, using Equations (13

-16

) the expected values of $g$ , $P b$ , $r$ and $s$ may be calculated and compared with the observed values. These four tests of GR all agree with the theory to within the uncertainties. Currently the tightest constraint is the Shapiro delay parameter $s$ where the observed value is in agreement with GR at 0.1% level.

Less than two years after its discovery, the double pulsar system has already surpassed the three decades of monitoring PSR B1913+16 and over a decade of timing PSR B1534+12 as a precision test of GR. On-going precision timing measurements of the double pulsar system should soon provide even more stringent and new tests of GR. Crucial to these measurements will be the timing of the $2.7-s$ pulsar B, where the observed profile is significantly affected by A’s relativistic wind [198 , 216 ]. A careful decoupling of these profile variations is required to accurately measure TOAs for this pulsar and determine the extent to which the theory-independent mass ratio $R$ can be measured.

The relativistic effects observed in the double pulsar system are so large that corrections to higher post-Newtonian order may soon need to be considered. For example, $w$ may be measured precisely enough to require terms of second post-Newtonian order to be included in the computations [81 ]. In addition, in contrast to Newtonian physics, GR predicts that the spins of the neutron stars affect their orbital motion via spin-orbit coupling. This effect would most clearly be visible as a contribution to the observed $w$ in a secular [26 ] and periodic fashion [337 ]. For the J0737 $-$ 3039 system, the expected contribution is about an order of magnitude larger than for PSR B1913+16 [198 ]. As the exact value depends on the pulsars’ moment of inertia, a potential measurement of this effect allows the moment of inertia of a neutron star to be determined for the first time [81 ]. Such a measurement would be invaluable for studies of the neutron star equation of state and our understanding of matter at extreme pressure and densities [168 ].

The systems discussed above are all double neutron star binaries. A further self-consistency test of GR has recently been made in the $4.7-hr$ relativistic binary J1141 $-$ 6545, where the measurement [25 ] of $w$ , $g$ and $Pb$ yield a pulsar mass of $1.30± 0.02 Mo .$ and a companion mass of $0.99 ± 0.02M o.$ . Since the mass of the companion is some seven standard deviations below the mean neutron star mass (see Figure 28 ), it is most likely a white dwarf. The observed $Pb = -(4 ± 1) × 10- 13$ is consistent, albeit with limited precision, with the predicted value from GR ( $- 3.8× 10-13$ ). Continued timing should reduce the relative error in $Pb$ down to 1% by 2010 [25 ].

Figure 28:

Distribution of neutron star masses as inferred from timing observations of binary pulsars [292

]. The vertical dotted line shows the canonical neutron star mass of $1.4Mo .$ . Figure provided by Ingrid Stairs.

PK parameters have now been measured for a number of other binary pulsars which provide interesting constraints on neutron star masses [318 , 292

]. Figure 28

shows the distribution taken from a recent compilation [292 ]. While the young pulsars and the double neutron star binaries are consistent with, or just below, the canonical $1.4 Mo .$ , we note that the millisecond pulsars in binary systems have, on average, significantly larger masses. This provides strong support for their formation through an extended period of accretion in the past, as discussed in Section 2.6.