The system consists of a pulsar of nominal period 59 ms in a close binary orbit with an as yet unseen companion. The orbital period is about 7.75 hours, and the eccentricity is 0.617. From detailed analyses of the arrival times of pulses (which amounts to an integrated version of the Doppler-shift methods used in spectroscopic binary systems), extremely accurate orbital and physical parameters for the system have been obtained (Table 6). Because the orbit is so close () and because there is no evidence of an eclipse of the pulsar signal or of mass transfer from the companion, it is generally believed that the companion is compact: Evolutionary arguments suggest that it is most likely a dead pulsar. Thus the orbital motion is very clean, free from tidal or other complicating effects. Furthermore, the data acquisition is ``clean'' in the sense that by exploiting the intrinsic stability of the pulsar clock combined with the ability to maintain and transfer atomic time accurately using such devices as the Global Positioning System, the observers can keep track of the pulsar phase with an accuracy of 15 s, despite extended gaps between observing sessions (including a several-year gap during the middle 1990s upgrade of the Arecibo radio telescope). The pulsar has shown no evidence of ``glitches'' in its pulse period.
Table 6: Parameters of the binary pulsar PSR 1913+16. The numbers in parentheses denote errors in the last digit. Data taken from an online catalogue of pulsars maintained by Stephen Thorsett of the University of California, Santa Cruz, see .
Three factors make this system an arena where relativistic celestial mechanics must be used: the relatively large size of relativistic effects , a factor of 10 larger than the corresponding values for solar-system orbits; the short orbital period, allowing secular effects to build up rapidly; and the cleanliness of the system, allowing accurate determinations of small effects. Because the orbital separation is large compared to the neutron stars' compact size, tidal effects can be ignored. Just as Newtonian gravity is used as a tool for measuring astrophysical parameters of ordinary binary systems, so GR is used as a tool for measuring astrophysical parameters in the binary pulsar.
The observational parameters that are obtained from a least-squares solution of the arrival-time data fall into three groups: (i) non-orbital parameters, such as the pulsar period and its rate of change (defined at a given epoch), and the position of the pulsar on the sky; (ii) five ``Keplerian'' parameters, most closely related to those appropriate for standard Newtonian binary systems, such as the eccentricity e and the orbital period ; and (iii) five ``post-Keplerian'' parameters. The five post-Keplerian parameters are: , the average rate of periastron advance; , the amplitude of delays in arrival of pulses caused by the varying effects of the gravitational redshift and time dilation as the pulsar moves in its elliptical orbit at varying distances from the companion and with varying speeds; , the rate of change of orbital period, caused predominantly by gravitational radiation damping; and r and , respectively the ``range'' and ``shape'' of the Shapiro time delay of the pulsar signal as it propagates through the curved spacetime region near the companion, where i is the angle of inclination of the orbit relative to the plane of the sky.
In GR, these post-Keplerian parameters can be related to the masses of the two bodies and to measured Keplerian parameters by the equations (TEGP 12.1, 14.6 (a) )
where and denote the pulsar and companion masses, respectively. The formula for ignores possible non-relativistic contributions to the periastron shift, such as tidally or rotationally induced effects caused by the companion (for discussion of these effects, see TEGP 12.1 (c) ). The formula for includes only quadrupole gravitational radiation; it ignores other sources of energy loss, such as tidal dissipation (TEGP 12.1 (f) ). Notice that, by virtue of Kepler's third law, , , thus the first two post-Keplerian parameters can be seen as , or 1PN corrections to the underlying variable, while the third is an , or 2.5PN correction. The current observed values for the Keplerian and post-Keplerian parameters are shown in Table 6 . The parameters r and s are not separately measurable with interesting accuracy for PSR 1913+16 because the orbit's inclination does not lead to a substantial Shapiro delay.
Because and e are separately measured parameters, the measurement of the three post-Keplerian parameters provide three constraints on the two unknown masses. The periastron shift measures the total mass of the system, measures the chirp mass, and measures a complicated function of the masses. GR passes the test if it provides a consistent solution to these constraints, within the measurement errors.
From the intersection of the and constraints we obtain the values and . The third of Eqs. (60) then predicts the value . In order to compare the predicted value for with the observed value of Table 6, it is necessary to take into account the small effect of a relative acceleration between the binary pulsar system and the solar system caused by the differential rotation of the galaxy. This effect was previously considered unimportant when was known only to 10 percent accuracy. Damour and Taylor  carried out a careful estimate of this effect using data on the location and proper motion of the pulsar, combined with the best information available on galactic rotation, and found
Subtracting this from the observed (Table 6) gives the residual
which agrees with the prediction within the errors. In other words,
The consistency among the measurements is displayed in Figure 6, in which the regions allowed by the three most precise constraints have a single common overlap.
A third way to display the agreement with general relativity is by comparing the observed phase of the orbit with a theoretical template phase as a function of time. If varies slowly in time, then to first order in a Taylor expansion, the orbital phase is given by . The time of periastron passage is given by , where N is an integer, and consequently, the periastron time will not grow linearly with N . Thus the cumulative difference between periastron time and , the quantities actually measured in practice, should vary according to . Figure 7 shows the results: the dots are the data points, while the curve is the predicted difference using the measured masses and the quadrupole formula for .
The consistency among the constraints provides a test of the assumption that the two bodies behave as ``point'' masses, without complicated tidal effects, obeying the general relativistic equations of motion including gravitational radiation. It is also a test of strong gravity, in that the highly relativistic internal structure of the neutron stars does not influence their orbital motion, as predicted by the strong equivalence principle of GR.
Recent observations [84, 138] indicate variations in the pulse profile, which suggests that the pulsar is undergoing precession as it moves through the curved spacetime generated by its companion, an effect known as geodetic precession. The amount is consistent with GR, assuming that the pulsar's spin is suitably misaligned with the orbital angular momentum. Unfortunately, the evidence suggests that the pulsar beam may precess out of our line of sight by 2020.
|The Confrontation between General Relativity and
Clifford M. Will
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