The binary pulsar and general relativity

5.1 The binary pulsar and general relativity

The 1974 discovery of the binary pulsar B1913+16 by Joseph Taylor and Russell Hulse during a routine search for new pulsars provided the first possibility of probing new aspects of gravitational theory: the effects of strong relativistic internal gravitational fields on orbital dynamics, and the effects of gravitational radiation reaction. For reviews of the discovery, see the published Nobel Prize lectures by Hulse and Taylor [132 , 252 ]. For reviews of the current status of pulsars, including binary and millisecond pulsars, see [173 , 243

Table 6:

Parameters of the binary pulsar B1913+16. The numbers in parentheses denote errors in the last digit. Data taken from [18

, 272

]. Note that $γ ′$ is not the same as the PPN parameter $γ$ (see Equations (78

)).

Parameter	Symbol	Value
	(units)
(i) “Physical” parameters:

Right Ascension	$α$	19^h15^m27.^s 99999(2)
Declination	$δ$	16°06’27.”4034(4)
Pulsar period	$Pp$ (ms)	59.0299983444181(5)
Derivative of period	$P˙p$	8.62713(8) $×$ 10^–18

(ii) “Keplerian” parameters:

Projected semimajor axis	$ap sin i$ (s)	2.341774(1)
Eccentricity	$e$	0.6171338(4)
Orbital period	$Pb$ (day)	0.322997462727(5)
Longitude of periastron	$ω0$ (°)	226.57518(4)
Julian date of periastron	$T0$ (MJD)	46443.99588317(3)

(iii) “Post-Keplerian” parameters:

Mean rate of periastron advance	$⟨˙ω⟩$ (° yr^–1)	4.226595(5)
Redshift/time dilation	$γ′$ (ms)	4.2919(8)
Orbital period derivative	$P˙b$ (10^–12)	–2.4184(9)

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 (see Table 6). Because the orbit is so close ( $≈ 1R ⊙$ ) and because there is no evidence of an eclipse of the pulsar signal or of mass transfer from the companion, it is generally agreed that the companion is compact. Evolutionary arguments suggest that it is most likely a dead pulsar, while B1913+16 is a “recycled” 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 GPS, the observers can keep track of pulse time-of-arrival with an accuracy of 13 $μ$ s, despite extended gaps between observing sessions (including a several-year gap in the middle 1990s for an upgrade of the Arecibo radio telescope). The pulsar has shown no evidence of “glitches” in its pulse period.

Three factors make this system an arena where relativistic celestial mechanics must be used: the relatively large size of relativistic effects [ $vorbit ≈ (m ∕r)1∕2 ≈ 10−3$ ], 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:

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;
five “Keplerian” parameters, most closely related to those appropriate for standard Newtonian binary systems, such as the eccentricity e, the orbital period P_b, and the semi-major axis of the pulsar projected along the line of sight, a_p sin i; and
five “post-Keplerian” parameters.

The five post-Keplerian parameters are: $⟨ω˙⟩$ , the average rate of periastron advance; $γ′$ , 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; $˙ Pb$ , the rate of change of orbital period, caused predominantly by gravitational radiation damping; and r and s = sin i, 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. An additional 14 relativistic parameters are measurable in principle [88 ].

In GR, the five 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) [281 ])

⟨ω˙⟩ = 6πfb(2πmfb )2∕3(1 − e2)−1, m ( m ) γ′ = e(2πfb)−1(2πmfb )2∕3--2 1 + --2 , m m P˙b = − 192π-(2 πℳfb )5∕3F (e), (78 ) 5 s = sin i, r = m2,

where $m1$ and $m2$ denote the pulsar and companion masses, respectively. The formula for $⟨ω˙⟩$ 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) [281

]). The formula for $P˙b$ includes only quadrupole gravitational radiation; it ignores other sources of energy loss, such as tidal dissipation (TEGP 12.1 (f) [281

]). Notice that, by virtue of Kepler’s third law, $(2πfb )2 = m ∕a3$ , $(2πmfb )2∕3 = m ∕a ∼ ε$ , thus the first two post-Keplerian parameters can be seen as $𝒪 (ε)$ , or 1PN corrections to the underlying variable, while the third is an $𝒪 (ε5∕2)$ , 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 B1913+16 because the orbit’s 47°inclination does not lead to a substantial Shapiro delay.

Because $fb$ 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, $P˙b$ measures the chirp mass, and $γ′$ measures a complicated function of the masses. GR passes the test if it provides a consistent solution to these constraints, within the measurement errors.

Figure 6:

Constraints on masses of the pulsar and its companion from data on B1913+16, assuming GR to be valid. The width of each strip in the plane reflects observational accuracy, shown as a percentage. An inset shows the three constraints on the full mass plane; the intersection region (a) has been magnified 400 times for the full figure.

From the intersection of the $⟨ω˙⟩$ and $′ γ$ constraints we obtain the values $m1 = 1.4414 ± 0.0002M ⊙$ and $m2 = 1.3867 ± 0.0002M ⊙$ . The third of Equations (78 ) then predicts the value $P˙b = − 2.40242 ± 0.00002 × 10−12$ . In order to compare the predicted value for $˙Pb$ 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 $˙ Pb$ was known only to 10 percent accuracy. Damour and Taylor [87 ] 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; the current value of this effect is $P˙gal≃ − (0.0128 ± 0.0050) × 10−12 b$ . Subtracting this from the observed $P˙ b$ (see Table 6) gives the corrected $˙corr − 12 Pb = − (2.4056 ± 0.0051) × 10$ , 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. Uncertainties in the parameters that go into the galactic correction are now the limiting factor in the accuracy of the test of gravitational damping.

Figure 7:

Plot of the cumulative shift of the periastron time from 1975 – 2005. The points are data, the curve is the GR prediction. The gap during the middle 1990s was caused by a closure of Arecibo for upgrading [272

A third way to display the agreement with GR is by comparing the observed phase of the orbit with a theoretical template phase as a function of time. If $fb$ varies slowly in time, then to first order in a Taylor expansion, the orbital phase is given by $Φb(t) = 2πfb0t + πf˙b0t2$ . The time of periastron passage $tP$ is given by $Φ (tP ) = 2πN$ , where $N$ is an integer, and consequently, the periastron time will not grow linearly with $N$ . Thus the cumulative difference between periastron time $tP$ and $N ∕fb0$ , the quantities actually measured in practice, should vary according to $tP − N ∕fb0 = − ˙fb0N 2∕2f3b0 ≈ − (f ˙b0∕2fb0)t2$ . 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 $f˙ b0$ [272 ].

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 SEP of GR.

Recent observations [157 , 271 ] indicate variations in the pulse profile, which suggests that the pulsar is undergoing geodetic precession on a 300-year timescale as it moves through the curved spacetime generated by its companion (see Section 3.7.2). 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 2025.