L29, 1997 September 1 We present a study of the gravitational time delay of arrival of signals from binary pulsar systems with rotating black hole companions. In particular, we investigate the strength of this effect (Shapiro delay) as a function of the inclination, eccentricity, and period of the orbit, as well as the mass and angular momentum of the black hole. This study is based on direct numerical integration of null geodesics in a Kerr background geometry. We find that, for binaries with sufficiently high orbital inclinations (>89°) and compact companion masses greater than 10 M
, the effect arising from the rotation of the black hole in the system amounts to a microsecond-level variation of the arrival times of the pulsar pulses. If measurable, this variation could provide a unique signature for the presence of a rotating black hole in a binary pulsar system.
Subject headings: pulsars: general
relativity
Binary pulsars are excellent laboratories for testing general relativity (GR) and other theories of gravity. Long-term timing observations of relativistic binaries, such as PSR B1913+16 (Hulse & Taylor 1975), yield highly precise measurements of post-Keplerian orbital parameters of the pulsar. In particular, observations of the orbital decay of PSR B1913+16 induced by gravitational radiative effects have verified the validity of GR to within 0.5% (Taylor & Weisberg 1989). Recently, pulsar surveys have unveiled additional relativistic binary pulsars (for a recent review see Wolszczan 1997). Of particular interest is PSR B1534+12 (Wolszczan 1991; Taylor et al. 1992), in which case the presence of post-Keplerian orbital effects and proximity of the pulsar to Earth give this object a potential to become an exceptionally precise probe of relativistic gravity.
One measurable relativistic effect in the timing of binary pulsars is the time delay that pulses experience as a result of the spacetime curvature induced by the pulsar's companion. This is precisely the effect proposed by Shapiro (1964) as a fourth test of GR, in addition to the tests to measure perihelion shift, deflection of light, and gravitational redshift. The Shapiro delay is measurable in binary pulsars with high orbital inclinations. So far, it has been detected in two low-mass binaries, PSR B1855+09 (Kaspi, Taylor, & Ryba 1994; i 
87
7) and PSR J1713+07 (Camilo, Foster, & Wolszczan 1994; i 785), and in the high-mass PSR B1534+12 system mentioned above (i 784). Shapiro delay has also been marginally detected in PSR B1913+16 (Taylor et al. 1992). The maximum amplitude of the pulse-timing residuals from the Shapiro delay for these binary pulsars is approximately 15.0, 7.5, and 55.0 
s, respectively. Two other effects that are measurable in principle are the spin-orbit coupling (geodetic precession; see Weisberg, Romani, & Taylor 1989) and the gravitational deflection of light (Doroshenko & Kopeikin 1995).
The relativistic orbital effects in compact neutron starneutron star (NS-NS) or neutron starwhite dwarf binaries are accounted for in a theory-independent timing formula developed by Damour & Deruelle (1985, 1986). Since any gravitational effects caused by the rotation of companions to pulsars in such binaries are practically negligible, this timing model does not take them into account (a rotational effect that the Damour and Deruelle timing model does include is that of aberration induced by rotation of the pulsar). Nonetheless, future pulsar detections could yield binary pulsars with black hole (BH) companions, thereby creating favorable conditions to measure the impact of the companion's rotation on the timing of pulses. In particular, Monte Carlo simulations by Lipunov et al. (1994) indicate that, in the best-case scenario, the number of PSR-BH binaries is expected to be about 1 per 700 pulsars, with BH masses of approximately 20 M. If rotational effects are included, then, as shown below, the inertial frame drag by the gravitational field produces both an extra time delay and an advance of signals, depending on the relative orientation of the orbital angular momentum of the pulsar and the rotational angular momentum of the companion.
In this Letter, we investigate the conditions necessary to detect the effects of companion rotation on the pulse arrival times from a binary pulsar. In particular, we are concerned with a possible application of such measurements to gather additional information regarding the nature of the pulsar's companion. Although it is possible to derive a timing formula for the Shapiro delay without a specific theory of gravity in mind, our results are obtained using the theory of GR. We start by considering the spacetime curvature in the vicinity of the companion. If rotation is ignored, Birkhoff's theorem (Hawking & Ellis 1973) guarantees that the only solution describing the exterior gravitational field of an object is the Schwarzschild metric. There is, however, no such theorem for the spacetime geometry exterior to a rotating object. Nonetheless, it is expected that for a 
clean
binary pulsar, given enough time, the gravitational field of a rotating companion will settle down to the Kerr family of stationary and axisymmetric spacetimes (Wald 1984). Our study then assumes that the propagation of the pulses takes place in a Kerr spacetime background; that is, we consider PSR-BH binary systems.
Before addressing the problem in its full generality, we consider, as an illustration, the case of a gravitational time delay of signals propagating in the equatorial plane of a rotating BH. From integrating the null geodesic equation for the Kerr metric, one obtains that the time required for light to travel in the equatorial plane from an arbitrary point r to the distance d of closest approach (d 
r) is given by (Dymnikova 1984, 1986)
with Mh and a being the mass and angular momentum per unit mass of the BH, respectively. In the last term, the upper (lower) sign is chosen if the orbital angular momentum of the photon is (anti-)parallel to the angular momentum of the BH. Because of this relative sign, the photon is accelerated, if its motion is along the direction of the BH rotation and it is delayed otherwise. The first term in equation (1) corresponds to signal propagation in a flat spacetime. The second and third terms encapsulate the Shapiro delay as originally proposed (Shapiro 1964). The fourth term in equation (1) is the second-order correction in Mh to the Shapiro time delay. Finally, the last term in this equation describes the first order in a effect from rotation; this effect can be traced back to the dragging of inertial frames.
From equation (1), the relative time delay of two signals, emitted at point re, traveling in opposite directions around the BH and arriving at point ro is given by
To obtain an order-of-magnitude estimate of the relative time delay for photon orbits slightly off the equatorial plane, we approximate the impact parameter by d ap(1 - e) cos i, with ap, e, and i being the semimajor axis, eccentricity, and inclination of the pulsar's orbit, respectively. Equation (2) then yields
This order-of-magnitude estimate indicates that only orbital inclinations i 
89° and companion masses Mh > 10 M would yield a potentially detectable effect (
ts 
few s). BH masses of about 10 M are believed to be typical for NS-BH binaries (Phinney 1991; Narayan, Piran, & Shemi 1991).
Another possibility to increase the detectability of this effect is to decrease the characteristic orbital separation ap. However, this decreases the lifetime for the binary to spiral in and merge because of the emission of gravitational radiation. The lifetime to merging, 
m, is approximately given by
with
Mp being the pulsar's mass, and Mt = Mp + Mh. Thus, because of the m 
a
dependence, any decrease of ap or increase of e, to bring up ts in equation (3) above the detectability level, will make the binary unlikely to be observable at that separation.
Our study was carried out under the following assumptions, which favor detectability: (1) high orbital inclinations, and the line of sight is perpendicular to the rotational axis of the companion, so the rotational contribution to gravitational time delay is maximized; (2) the longitude of periastron is 
= 180° (superior conjunction) to yield a complete antisymmetric (delay-acceleration) effect; and (3) orbital parameters are selected such that the lifetime to merging never becomes shorter than 107 yr. Time delay differences, ts, are then computed by means of the ray-tracing technique. A bundle of photons is integrated back from the point of observation, ro, toward the binary orbital plane using a fourth-order Runge-Kutta integrator with variable time step. Photons passing within a distance h 
10-4ap of the pulsar's orbit are tagged, and their time delay is computed by comparing their propagation time with that from a flat spacetime trajectory. Approximately 20,000 photons are required to obtain a sufficiently complete sample of time delays throughout the orbital phase. In order to guarantee that photons in the bundle passing within a distance h are recorded, the integration step in the vicinity of the orbital plane is chosen to be less than h. Once the time delays are recorded, the data are interpolated to a uniform orbital phase grid. Finally, differences in time delays are computed by subtracting the time delays of a pair of points along the pulsar's orbit that otherwise would have identical travel time in flat space. That is, the data are folded about the orbital phase of closest approach and then subtracted. For orbits in the equatorial plane, the above procedure is entirely equivalent to measuring the last term in equation (1). Furthermore, since this is a differential measurement, we have verified that the value measured of ts does not depend on the distance to the observer ro provided that ro 
ap (see eq. [1]).
To verify the validity of our approach, we conducted, in addition to standard convergence tests, a null test with a nonrotating BH and a test with orbital inclination i = 90°. The latter test produced results that differed by 2% from those obtained with equation (3). Table 1 summarizes the results for our best-case scenarios, with Mp = 1.4 M, Mh = 20 M, and a = Mh. The last column reports the maximum time delay ts when ap, e, and i are varied. A natural consequence of considering high orbital inclinations is that the scalings of ts with a
Mh and Mh, implied by equation (3), are to some extent preserved. We have conducted a series of simulations varying Mh and a that suggest that the scaling of equation (3) with respect to those quantities is indeed present for sufficiently large orbital inclinations. Regarding ap, e, and i, the situation is not as clear. These parameters determine the photon distance (d) of closest approach to the BH; thus, differences between the results in Table 1 and those obtained from equation (3) provide an indication of the extent to which the off the equatorial plane approximation breaks down, namely, d ap(1 - e) cos i. In particular, the scaling is absent for the cases with very large inclinations greater than 895 andor small periastron separations less than R, for which higher order effects in a become important. Figure 1 depicts the typical outcomes of our simulations. They correspond to cases 1, 2, 7, and 8 in Table 1. In Figure 1, positive and negative values imply delays and accelerations, respectively.
Fig. 1
Our calculations demonstrate that, for reasonably compact and eccentric orbits, rotation of a sufficiently massive BH companion to the orbiting pulsar leads to approximately microsecond-order departures from pulse arrival times predicted for a Shapiro delay in the case of a nonrotating body. Clearly, it is not practical to consider tighter orbits, because, even if they were plausible on the grounds of stellar evolution, their lifetimes against the gravitational radiation would be extremely short.
A detectability of approximately microsecond-level effects in pulse timing residuals has been demonstrated for several millisecond pulsars (e.g., Kaspi, Taylor, & Ryba 1994; Camilo, Foster, & Wolszczan 1994). Unfortunately, these very rapidly rotating neutron stars are the products of the kind of binary evolution that is expected not to yield BH companions (Phinney & Kulkarni 1994). Recent calculations (Lipunov et al. 1994) indicate that the most typical NS-BH binaries, resulting from the evolution of massive stars, would contain normal slow pulsars, in which case the practical timing accuracy is rarely better than 0.21.0 ms. In addition, such pulsars are short lived (107 yr) and are expected to move in relatively wide, long-period orbits.
Less probable, but not implausible, evolutionary scenarios that could produce compact NS-BH binaries, possibly containing rapidly spinning recycled pulsars, involve an accretion-induced collapse of one of the stellar binary companions to a BH (Narayan et al. 1991; Lipunov et al. 1994). For such systems, sufficiently extreme orbital parameters (Table 1), and a possible microsecond-level precision of the pulse timing measurements, could create more favorable conditions for a detection of the effects discussed in this Letter. For example, PSR B1534+12, one of the NS-NS binaries mentioned above, which has been timed for more than 4 years, allows a detection of orbital phase-dependent effects in timing residuals with an approximately 1 s accuracy (Arzoumanian 1995). For NS-BH binaries timable with such precision, our study implies that, among other applications, a measurement of the rotational contribution to the Shapiro delay could be used to determine the angular momentum of the BH companion.
This research was supported in part through NSF grants PHY-9601413 and PHY-9357219 (NYI) to P. L. and AST-9619552 to A. W.
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Full image (24kb) | Discussion in textDifferences in arrival times due to the companion rotation, ts(s), for PSR-BH systems with Mh = 20 M, Mp = 1.4 M, a = Mh, and ap = 5 R. Values in parentheses denote the respective eccentricities and orbital inclinations corresponding to cases 1, 2, 7, and 8 in Table 1.
| Case | i (deg) |
ap (R ) |
Pb (days) |
e | m (108 yr) |
ts ( s) |
| 1... | 89.0 | 5 | 0.2803 | 0.0 | 4.19 | 0.203 |
| 2... | 89.0 | 5 | 0.2803 | 0.4 | 1.52 | 0.358 |
| 3... | 89.0 | 5 | 0.2803 | 0.8 | 0.04 | 1.556 |
| 4... | 89.0 | 10 | 0.7928 | 0.0 | 67.1 | 0.099 |
| 5... | 89.0 | 10 | 0.7928 | 0.4 | 24.4 | 0.168 |
| 6... | 89.0 | 10 | 0.7928 | 0.8 | 0.61 | 0.564 |
| 7... | 89.5 | 5 | 0.2803 | 0.0 | 4.19 | 0.535 |
| 8... | 89.5 | 5 | 0.2803 | 0.4 | 1.52 | 1.396 |
| 9... | 89.5 | 5 | 0.2803 | 0.8 | 0.04 | 2.511 a |
| 10... | 89.5 | 10 | 0.7928 | 0.0 | 67.1 | 0.219 |
| 11... | 89.5 | 10 | 0.7928 | 0.4 | 24.4 | 0.407 |
| 12... | 89.5 | 10 | 0.7928 | 0.8 | 0.61 | 2.091 a |
The table shows arrival time delay differences of pulses from binary pulsars with rotating BH companions for the pulsar mass Mp = 1.4 M, the BH mass Mh = 20 M, and the BH angular momentum per unit mass a = Mh.1 from superior conjunction.