GEODETIC PRECESSION AND TIMING OF THE RELATIVISTIC BINARY PULSARS PSR B AND PSR B Maciej Konacki. Alex Wolszczan. and Ingrid H.

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1 The Astrophysical Journal, 589: , 2003 May 20 # The American Astronomical Society. All rights reserved. Printed in U.S.A. GEODETIC PRECESSION AND TIMING OF THE RELATIVISTIC BINARY PULSARS PSR B AND PSR B Maciej Konacki Department of Geological and Planetary Sciences, California Institute of Technology, MS , Pasadena, CA 91125; and Nicolaus Copernicus Astronomical Center, Rabiańska 8, Toruń, Poland; maciej@gps.caltech.edu Alex Wolszczan Department of Astronomy and Astrophysics, Pennsylvania State University, University Park, PA 16802; and Toruń Centre for Astronomy, Nicolaus Copernicus University, ul. Gagarina 11, Toruń, Poland; alex@astro.psu.edu and Ingrid H. Stairs Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada; stairs@astro.ubc.ca Received 2002 December 27; accepted 2003 January 30 ABSTRACT The pulsars B and B are two unique neutron star binaries exhibiting a wide range of relativistic phenomena that are impossible to detect in other systems. They constitute an exquisite observational ground on which theories can be tested. To date, the timing observations of B and B have been successfully used to test the strong field regime of relativistic gravity by measuring and then comparing with theory the evolution of the orbital elements of the pulsars. In this paper we develop a method that allows us to detect the timing signature of yet another relativistic phenomenon, the geodetic spin precession, and derive the misalignment angle between the orbital angular momentum and the spin vector of the pulsar, an important quantity that can be used to assess the degree of asymmetry of the supernova explosion that created the pulsar. Although we demonstrate that observations of PSR B using the Penn State Pulsar Machine and the Mark III system do not yet have a sufficient time span to detect precessional effects in the timing, we show that in about years we will be able to get a good grasp on the misalignment angle of this pulsar. This may seem a long time to wait but in fact is typical for timing relativistic binary pulsars and, as in the case of PSR B , patient observing will eventually turn out to be very rewarding. Subject headings: binaries: close pulsars: individual (B , B ) relativity stars: neutron 1. INTRODUCTION The radio pulsars B and B are relativistic binaries, each consisting of two neutron stars (NS), in which the observed pulsars exhibit rapid rotation. A general examination of motions occurring in such systems allows one to establish how each of the two bodies spin influences their orbital motion and how their motion in curved spacetime influences their spin (for a review see Damour & Taylor 1992). As it turns out, even for a rapidly rotating neutron star the spin-induced deviations from geodesic motion can be neglected (Will 1993). However, this is not the case for the effects belonging to the second category, the most prominent of which is the so-called geodetic precession. This phenomenon is a consequence of the curvature of space around a rotating body and can be described by the equation dx dt ¼ prec ^L µ X ; where X is the spin vector and ^L is a unit vector in the direction of the total angular momentum L. However, since the orbital angular momentum is much larger than the spin of both neutron stars, the vector of orbital angular momentum J is practically fixed in space and for our purposes can replace L in the above equation (see, e.g., Hamilton & ð1þ 495 Sarazin 1982). Thus, dx dt ^J prec µ X ; ð2þ where ^J is a unit vector in the direction of the orbital angular momentum and prec denotes the angular frequency of precession and is given by prec ¼ 3Gm 2ð1 þ m 1 =3MÞ ac 2 ð1 e 2 ÞP orb ; ð3þ where m 1 and m 2 are the pulsar and companion masses, M ¼ m 1 þ m 2, and a, e, andp orb are the orbital elements: semimajor axis, eccentricity, and period (all given in SI units), respectively (Barker & O Connell 1975a, 1975b). The spin vector X precesses around J with the frequency prec as long as X is not parallel to J, i.e.,, the misalignment angle between X and J, is different from zero (Fig. 1). The misalignment angle is a conserved quantity in this motion. In the absence of spin precession the magnetic moment of the pulsar, l, exhibits simple rotation (Fig. 2). According to the rotating vector model (Radhakrishnan & Cooke 1969) appropriate for dipole magnetic fields, the observed pulsed emission occurs when the pulsar beam sweeps past the line of sight (Fig. 3). The geodetic precession induces additional motion of the magnetic moment that changes the latitude

2 496 KONACKI, WOLSZCZAN, & STAIRS Vol. 589 Fig. 3. Geometry of pulse emission; X is the pulsar s spin vector, l is the magnetic moment vector pointing in the direction of the observable magnetic pole, n 0 is the direction to the observer, is the angle between X and l, is the angle between n 0 and X, is the impact parameter and is the opening semiangle of the beam. A schematic cone and core beam cross section and the resulting pulse profile are also shown. Fig. 1. Geometry of geodetic precession in the inertial reference frame O ¼fi; j; kg defined by the orbital angular momentum vector J; n is the direction to the observer, X is the pulsar s spin vector, i denotes the orbital inclination defined here as the angle between J and n, is the misalignment angle between J and X and ðtþ is the precessional phase. and longitude at which the observer s line of sight crosses the beam. Consequently, for highly relativistic binaries, one expects variations in intensity and separation of pulse components and in pulse polarization characteristics (Damour & Ruffini 1974). Obviously, detecting such variations is not easy, as the expected precession periods for such NS-NS systems are on the order of hundreds of years (Barker & O Connell 1975b). However, once detected, the geodetic precession induced profile variations can be used to estimate the misalignment angle and hence provide an Fig. 2. Geometry of the spin and the magnetic moment vectors in the reference frame O 0 ¼fi 0 ; j 0 ; k 0 g defined by the spin vector X; is the angle between X and the magnetic moment vector l (l points in the direction of the observable magnetic pole), ðtþ is the pulsar s rotational phase. important constraint on properties of the progenitor system and on the degree of asymmetry of the second supernova explosion (see, e.g., Hughes & Bailes 1999; Wex, Kalogera, & Kramer 2000). Secular changes in the intensity ratio of the pulse components of PSR B that could be attributed to geodetic precession were first reported by Weisberg, Romani, & Taylor (1989), following the predictions by Damour & Ruffini (1974) and Esposito & Harrison (1975) and the solution of the gravitational two-body problem with spin by Barker & O Connell (1975a, 1975b). However, subsequent attempts by Cordes, Wasserman, & Blaskiewicz (1990) to detect precession-induced variations in the polarization signatures of the pulsar were unsuccessful. The misalignment angle for PSR B was determined to be ¼ 22 þ3 8 deg by Kramer (1998) based on the detection of a significant change in the separation between pulse components. Encouragingly, the anticipated variations in the intensity ratio of the pulse components of PSR B have been measured by Arzoumanian, Taylor, & Wolszczan (1999) and Stairs et al. (2000c). The pulse-profile related indicators are not the only available experimental means to detect geodetic precession and to subsequently determine the misalignment angle. Since the effect manifests itself as a periodic rotation of the pulsar beam, its signature in the pulse timing provides the most fundamental precession indicator. In fact, the first highly probable case of a free precession of a neutron star was initially detected in its timing observations (Stairs, Lyne, & Shemar 2000a). In this paper, we present a theoretical approach to the problem of a geodetic precession signature in the pulse timing observations that span only a fraction of the entire precession period. We show that this signature can be parameterized in terms of higher order derivatives of the pulsar spin period (see also Kopeikin, Doroshenko, & Getino 1996) and possibly measured on a year timescale that is obviously much more practical than the precession periods for PSR B and PSR B , predicted to be 700 and 300 yr, respectively. In x 2 we derive suitable formulae describing the timing variations generated by geodetic precession and show how to incorporate them in the timing model to obtain information on the pulsar s geodetic

3 No. 1, 2003 RELATIVISTIC BINARY PULSARS 497 precession and misalignment angle. In x 3, we present an updated timing model for PSR B and use it in x 4, together with the published data for PSR B , to constrain the geometry of geodetic precession of these two neutron stars. 2. THE EFFECT OF GEODETIC PRECESSION ON THE PULSE TIMING In the reference frame O 0 ¼fi 0 ; j 0 ; k 0 g defined by the pulsar s spin vector X (Fig. 2), the motion of the magnetic moment l, which points in the direction of the observable magnetic pole, is given by lðtþ¼ðsin cos ðtþ; sin sin ðtþ; cos Þ ; ðtþ¼ psr ðt t 0 Þþ 0 ; where ðtþ is the rotational phase, psr ¼ 2 (where ¼ 1=P psr ), P psr is the pulsar s period, and 0 is the rotational phase at the epoch t 0. The angle is the angle between the spin and magnetic moment vectors. In the reference frame O ¼fi; j; kg, defined by the orbital angular momentum vector J (Fig. 1), the spin vector X precesses around J. We can also assume that the direction toward the observer n is in the plane defined by the unit vectors i and k. Thus ð4þ n ¼ sin i i þ cos i k ¼ ðsin i; 0; cos iþ ; ð5þ where i is the orbital inclination defined as the angle between J and n. In the frame O, the motion of the magnetic moment l is a superposition of its rotational motion in the frame O 0 and the rotation of that frame with respect to the inertial frame O. The problem of finding timing variations due to precession has been analyzed in the case of free precession (e.g., Nelson, Finn, & Wasserman 1990; Jones & Andersson 2001). The approach presented there assumes that a pulse is identified in the frame O whenever the magnetic moment l crosses the plane containing the direction toward the observer and the orbital angular momentum vector that is, whenever the azimuthal angle of the vector l equals that of the observer (in the frame O). Clearly, this is neither a necessary nor a sufficient condition for observing a pulse. First of all, in order to detect a pulse it is necessary for the angle between the magnetic moment l and the direction to the observer n to be less than or equal to the opening semiangle of the beam, (Fig. 3). It does not suffice that both vectors (l and n) lie in the same semiplane. Second, a pulse can be observed even though the magnetic moment vector does not cross the plane that contains the vectors n and J. In other words, one can imagine an orientation of the spin vector X in the inertial reference frame O for which the beam crosses the plane containing the vectors n and J (and thus a pulse is observed) but the vector l does not. One can avoid these issues by analyzing the problem in the reference frame O 0 which is corotating with the precessional motion. Namely, in the frame O 0 the direction to the observer is given by n 0 ¼ðn 0 x; n 0 y; n 0 zþ; n 0 x ¼ cos ðtþ cos sin i sin cos i ; n 0 y ¼ sin ðtþ sin i; n 0 z ¼ cos ðtþ sin sin i þ cos cos i ; ðtþ ¼ prec ðt t 0 Þþ 0 ; ð6þ where ðtþ and 0 are the precession phases at the respective moment t and the initial epoch t 0, prec is the angular precession rate, and is the misalignment angle (Fig. 1). The above relations are obtained by simply rotating the vector n ¼ðsin i; 0; cos iþ first around the axis k by the angle ðtþ and then around the axis j by the angle. This way, a pulse is identified when the magnetic moment l crosses the plane defined by the vectors n 0 and X and, at the same time, the angle between n 0 and l does not exceed the opening semiangle. Clearly, the direction toward the observer n 0 (i.e., as seen from O 0 ) is not constant and is responsible for the motion of the plane of the observer in the reference frame O 0. Because of that, the times of observed pulses will be systematically altered. This effect can be quantified by the azimuthal angle of n 0, given by tan ðtþ ¼n 0 yðtþ=n 0 xðtþ : Now, assume that at the epoch t 0 a pulse was observed, so that 0 ¼ ðt 0 Þ¼ 0 ¼ ðt 0 Þ. If a subsequent pulse was observed at the moment t, then this pulse was delayed/ advanced by the amount of ðtþ=2 ¼ ½ðtÞ 0 Š=2 of the pulsar period, because ðtþ is simply the additional fraction of the pulsar s turn that is necessary for the magnetic moment to cross the observer plane. It follows that ðtþ is the effective timing signature of the geodetic precession. Given the time span of available timing data for B (10 yr; Stairs et al. 2002) and B (30 yr; Taylor & Weisberg 1989), we are only interested in the behavior of ðtþ on short timescales. Specifically, we have D prec ðt t 0 Þ5 1 for the timescales ðt t 0 Þ in question, and thus ðtþ can be expanded in the Taylor series with respect to D: dðd ¼ 0Þ ðtþ prec ðt t 0 Þ þ 1 d 2 ðd ¼ 0Þ prec ðt t 0 Þ 2 þ 1 d 3 ðd ¼ 0Þ prec ðt t 0 Þ 3 : ð8þ Since for the millisecond pulsars the rotational phase is very well determined by a simple spin-down model, the evolution of a rotational phase for the precessing pulsar can be modeled as ðtþ ¼ 0 þ psr ðt t 0 Þþ 1 d psr ðt t 0 Þ 2 þ ðtþ ; ð9þ 2 dt where d psr =dt is the first derivative of the pulsar s rotational frequency and ðtþ is given by equation (8). Consequently the quantities d dt dðd ¼ 0Þ prec ; d 2 dt 2 d2 ðd ¼ 0Þ 2 ð7þ 2 prec ; ð10þ make small contributions to the rotational frequency of the pulsar and its first derivative and hence cannot be separately determined. However, the parameter d 3 dt 3 d3 ðd ¼ 0Þ 3 3 prec ð11þ can in principle be measured as a second derivative of the rotational frequency that is, as the effect that is changing

4 498 KONACKI, WOLSZCZAN, & STAIRS Vol. 589 with time as ðt t 0 Þ 3. Subsequently the value of d 3 =dt 3 can be used to constrain the misalignment angle. Note that in order to compute the theoretical values of d=dt; d 2 =dt 2 ; and d 3 =dt 3 it is more convenient to use the relations qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos ðtþ ¼n 0 x= 1 ðn 0 zþ 2 ; sin ðtþ ¼n 0 y= 1 ðn 0 zþ 2 ð12þ rather than equation (7). For example, the set of equations d sin d ¼ cos ; d 2 sin 2 ¼ sin d 2 þ cos d2 2 ; d 3 sin 3 ¼ cos d 3 3sin d d 2 2 þ cos d3 3 ð13þ can be used to solve for the derivatives of over D and to find d=dt; d 2 =dt 2 ; and d 3 =dt 3. Finally, also note that for two specific configurations, ¼ i; 0 ¼ 0and ¼ i; 0 ¼, the parameters d=dt; d 2 =dt 2 ; and d 3 =dt 3 are not defined. This happens when n 0 crosses X (i.e., it goes through the point n 0 x ¼ n 0 y ¼ 0) and ðtþ discontinuously changes its value. From the above considerations it follows that d=dt; d 2 =dt 2 ; and d 3 =dt 3 are functions of i,, and 0 the orbital inclination, the misalignment angle, and the precessional phase at the epoch t 0. However, in order to observe a pulse, the angle between the magnetic moment l and the direction to the observer n 0 cannot be larger than the opening semiangle,, of the beam. In other words, if we define the impact parameter (the angle between l and n 0 at their closest approach; Everett & Weisberg 2001) as ¼ ; ð14þ where is the angle between X and n 0 (Fig. 3), a pulse can be observed if jj. Note that for not precessing (and observed) pulsars this condition is always satisfied. However, for a precessing pulsar the value of the impact parameter will change with time and at some point the pulsar may become unobservable (Kramer 1998). Effectively the angle constrains (through the impact parameter) the allowed values of the parameters d=dt; d 2 =dt 2 ; and d 3 =dt THE TIMING MODEL OF PSR B PSR B has been timed with the Arecibo telescope since its discovery in 1990 (Wolszczan 1991). Before the Arecibo upgrade ( ) the pulsar had been timed at 430 and 1400 MHz with the Princeton Mark III back end, which utilized two 32 channel filter banks to dedisperse the incoming signal (Stinebring et al. 1992). After the upgrade (1997 November), the PSR B timing resumed with two new back ends: the Princeton Mark IV coherent dedisperser (Stairs et al. 2000b) and the 128 channel filter bank based Penn State Pulsar Machine (PSPM; Cadwell 1997). The most recent timing analysis of the Mark III and Mark IV data, a discussion of the relativistic effects in this binary system, and the resulting updated tests of general relativity have been presented by Stairs et al. (2002). Details of the time-of-arrival (TOA) measurement and modeling procedures with the TEMPO software package 1 applied in our analysis are also included in this paper and will not be repeated here. A description of the data acquisition with the PSPM back end can be found in Wolszczan et al. (2000). In order to use the existing data to detect or set useful limits to any TOA variations that have not been accounted for in the relativistic timing model for PSR B , one has to achieve a precise phase-up of the pre- and post-upgrade TOA measurements. Ideally, one would need to correct the observed TOAs for any phase offsets resulting from the utilization of different data-acquisition back ends during these two periods and to properly account for a pulse phase change between the beginning of 1994 and the end of 1997 caused by a systematic, long-term decline of the pulsar s dispersion measure (DM; Stairs et al. 2002). To obtain a properly phased, two-frequency TOA set spanning the entire observing period, we have proceeded as follows. Because the initial PSPM timing data for PSR B were taken already in 1994, in parallel with the Mark III back end, we were in position to phase-connect the preupgrade Mark III data and the post-upgrade PSPM measurements at 430 MHz. Since the measured +22 ls offset between the 1994 and 1997 PSPM data is consistent with the observed pc cm 3 yr 1 decrease of the dispersion measure of the pulsar, we have assumed that any instrumental contribution to the observed phase offset should be small enough to be ignorable. The corresponding 1400 MHz TOA measurements were phased up by experimentally determining the offset between the pre- and the post-upgrade data sets and by compensating for it in the TOA modeling process. We have determined the dispersion measure of PSR B using the three-frequency TOA measurements obtained with the PSPM in at 430, 1130, and 1400 MHz. To compensate for offsets caused by the instrumental broadening and the intrinsic frequency dependence of the template pulse profiles, we have aligned them by iterating the DM fit to TOAs at the three pairs of observing frequencies until the successive template alignments yielded a DM value that agreed for the three frequency pairs within the measurement error. An appropriate template profile alignment for the pre-upgrade Mark III observations at 430 and 1400 MHz was achieved by making all the relevant phase offset measurements relative to the initial phase of the post-upgrade PSPM TOAs. We have least-squares fitted the relativistic, theoryindependent timing model of Damour & Deruelle (1986) to the phase-adjusted, two-frequency TOA set described above, using the TEMPO analysis code (see also Stairs et al. 2002). The model fitting was performed in a series of iterations starting with a fit for DM with all other model parameters held fixed at their initial values. In the next step, all model parameters were fitted for with a fixed DM obtained from the initial iteration. This procedure was then repeated until variations of the best-fit parameter values became insignificant. The final model including the spin and the astrometric parameters of the pulsar, as well as the five 1 TEMPO is available at:

5 No. 1, 2003 RELATIVISTIC BINARY PULSARS 499 TABLE 1 Observed Parameters for PSR B Parameter Value Right ascension, (J2000) h 37 m (3) Declination, (J2000)... 11= >55186(7) Proper motion in, l (1) mas yr 1 Proper motion in, l (2) mas yr 1 Period, P (3) ms Period derivative, _P (10 15 ) (5) Epoch (MJD) Orbital elements, DD model: Projected semimajor axis, x (8) s Eccentricity, e (1) Epoch of periastron, T p (MJD) (1) Orbital period, P b (4) days Longitude of periastron,! =76928(2) Advance of periastron, _!... 1=755805(3) yr 1 Gravitational redshift, (7) ms Orbital period derivative, P _ b (10 12 ) (1) Shape of Shapiro delay, s (3) m (8) M Dispersion measure and its derivatives: Dispersion measure, DM (3) cm 3 pc First derivative, DM... _ (9) cm 3 pc yr 1 Second derivative, DM (1) cm 3 pc yr 2 Third derivative, DM ð3þ (1) cm 3 pc yr 3 Fourth derivative, DM ð4þ (7) cm 3 pc yr 4 Fifth derivative, DM ð5þ (1) cm 3 pc yr 5 Sixth derivative, DM ð6þ (4) cm 3 pc yr 6 Seventh derivative, DM ð7þ (7) cm 3 pc yr 7 Note. Figures in parentheses are the formal, derived with TEMPO, 1 uncertainties in the last digits quoted. post-keplerian relativistic parameters of the orbit is presented in Table 1. The best-fit residuals shown in Figure 4 are characterized by an rms value of 5.5 ls. The DM variations shown in Figure 5 are parameterized in terms of the best-fit DM value and its seven time derivatives, also listed in Table 1. The timing model for PSR B derived from the Mark III and the PSPM TOA measurements agrees within errors with the recently published model based on the Mark III and Mark IV data (Stairs et al. 2002), despite the fact that the 430 MHz TOAs included in our analysis are biased by small, apparently orbital phase dependent, systematic errors (e.g., Arzoumanian 1995). The recent convincing detection of a predicted orbital phase dependence of the interstellar scintillation patterns of PSR B (Bogdanov et al. 2002), gives support to the idea that systematic effects seen in the 430 MHz TOAs measured with the filterbank based hardware are probably caused by a combination of an imperfect post-detection dispersion removal and time-variable scintillation characteristics of the pulsar (e.g., Stairs et al. 1998). Fig. 4. Best-fit timing residuals for PSR B obtained from TOA measurements made at Arecibo with the Mark III back end (circles) and the PSPM (triangles). The filled and open symbols are for the 430 and 1400 MHz measurements, respectively. Fig. 5. Dispersion measure variations obtained from the twofrequency timing of PSR B The filled circles represent the values of DM calculated over time intervals indicated by horizontal bars. Errors of DM measurements are smaller than the size of the circles. The solid line denotes the least-squares fit of a polynomial to the DM values (see Table 1).

6 500 KONACKI, WOLSZCZAN, & STAIRS Vol CONSTRAINTS ON THE GEODETIC PRECESSION GEOMETRY 4.1. PSR B The detection of geodetic precession in PSR B represents a considerable challenge because of the long, 700 yr precession period of the pulsar. Although the pulse profile variations attributable to geodetic precession have been detected (Arzoumanian et al. 1999; Stairs et al. 2000c), they are not yet sufficient to set useful limits on the magnitude of the precession signature in the timing data. Moreover, the inclusion of a direct fit for the second rotational frequency derivative ð2þ in the timing model did not produce a statistically significant result. To constrain this parameter, we have performed a search for the 2 minimum in the domain of ð2þ using the fixed best-fit timing model of Table 1. As shown in Figure 6, the 2 minimum at ð2þ ¼ 0: s 3 has the 3 range of ð 8; 9Þ10 30 s 3 which is too wide to yield any significant constraints for geodetic precession. We have performed simulations of the future timing observations of PSR B to predict a minimum time span of the data necessary for a significant ð2þ detection. As shown in Figure 7, the present best-fit value of ð2þ will become 30% accurate after another 25 years of continuing observations. Nevertheless, it is instructive to use this value as an example to demonstrate constraints that can be put on geodetic precession with the aid of the pulse timing and polarization data. The allowed values of the parameters constraining the geometry of the geodetic precession are 2 ½0; Š, 2 ½0; Š, 0 2 ½0; 2Š, i 2 ½0; Š. Timing observations of PSR B give i ¼ 78=4 or i ¼ 101=6 (Stairs et al. 2002), and polarimetric observations yield two plausible orientations of the magnetic moment, ¼ 103 and ¼ 114 (Arzoumanian et al. 1996), with the corresponding impact parameters (at the epoch around 1992) of 2=4 and 19. The remaining two parameters, and 0, can take any value within the intervals [0, ] and[0, 2], respectively. The value of the second derivative of rotational frequency ð2þ can be used to put limits on 0, especially on the misalignment Fig. 7. Time span of the additional years in timing data predicted to be needed to achieve 30% accuracy of ð2þ as a function of ð2þ. angle. Namely, the theoretical values of ð2þ ¼ 1 d 3 2 dt 3 can be computed as a function of and 0. This is shown in Figure 8 for the orbital inclination of i ¼ 78=4, the magnetic inclination of ¼ 114, and the impact parameter of ¼ 19, and in Figure 9 for ¼ 103 and ¼ 2=4: solid white lines give the values of (, ) that predict exactly the simulated measured value of ð2þ, while the black regions predict values of ð2þ that fall within 3 ð2þ of the measured value. Finally, since the impact parameter is also a function of (, ), we can apply the known value of the impact parameter together with its error to select the allowed solutions. For the purpose of this test, we use the value of 19 (Fig. 8) and 2=4 (Fig. 9) and assume the measurement error of 3. The allowed solutions for (, ) fall in the area where the impact parameter is within the 3 interval of 19 or 2=4 (the area between the dashed white lines in Figs. 8 and 9). This finally limits the sought solution for to four areas of the plane (, ) that correspond to the values of 30, 100, 130, or 160. This procedure can be easily repeated for the other values of the orbital and magnetic inclinations as well as for the subsequent rotational frequency derivatives ðnþ ; n ¼ 3;..., if measured. Fig. 6. Change in the 2 of the best-fit timing model for PSR B as a function of ð2þ. The vertical lines are the formal 1, 2, and 3 uncertainty levels. The minimum in 2 corresponds to the value of ð2þ ¼ 0: s 3. Fig. 8. Second derivative of the rotational frequency ð2þ induced by the geodetic precession for ¼ 114, ¼ 19, and i ¼ 78=4 as a function of the misalignment angle and the precessional phase 0 [values of ð2þ outside the interval ð 20; 20Þ10 30 s 3 were blanked out for clarity]. See text for full explanation.

7 No. 1, 2003 RELATIVISTIC BINARY PULSARS 501 Fig. 9. Same as Fig. 8, but for ¼ 103 and ¼ 2=4 Also note that the impact parameter evolves in time. Therefore, for this analysis, it is best to use its value derived from polarimetric observations taken around the epoch t 0 (for which we compute the precessional phase 0 ) PSR B The geometry of geodetic precession for PSR B is parameterized by i ¼ 132=8, ¼ 153, ¼ 22,and ¼ 0 for t ¼ 2128 yr (Kramer 1998). Since the coordinate system used by Kramer is different from the one used in this paper, we express these parameters in the system defined in Figure 1 to get (compare Fig. 4 in Kramer 1998) ¼ ¼ 22, ¼ 180 ¼ 27, i ¼ 180 i ¼ 47=2, and t ¼ t P prec =2, where P prec is the period of precession (300 yr). With the available timing data for PSR B covering the period , choosing t 0 ¼ 1988 (the midpoint of the data), for the above value of t and the precession rate of this pulsar prec ¼ 1=21 yr 1 (Weisberg et al. 1989), it follows that 0 ¼ ðt 0 Þ¼9=7. This gives the following values of the parameters d=dt; d 2 =dt 2 ; and d 3 =dt 3 1 d ¼ 1: s 1 ; 2 dt 1 d 2 2 dt 2 ¼ 4: s 2 ; 1 d 3 2 dt 3 ¼ 1: s 3 : ð15þ The first two parameters contribute to the net rotational frequency ¼ psr =2 and its first derivative _ ¼ _ psr =2, but the third one can in principle be detected. To explore this possibility we computed the timing effects resulting from such precession (see Fig. 10a) as well as from only the third term, d 3 ðt t 0 Þ 3 dt 3 6 (Fig. 10b). Clearly, the timing signature from geodetic precession is substantial even on timescales much shorter than Fig. 10. (a) Predicted timing residuals caused by the geodetic precession of the PSR B (b) Corresponding residuals for the term ½ðd 3 =dt 3 Þðt t 0 Þ 3 Š=6 alone. the precessional period. Figure 10b shows also that the term d 3 ðt t 0 Þ 3 dt 3 6 alone results in timing variations that are measurable with current timing techniques and should in principle lead to determination of a second derivative of the rotational frequency d 2 dt 2 ¼ d3 1 dt 3 2 whose value would verify the geometry of geodetic precession worked out by Kramer (1998). Unfortunately, in recent years the PSR B has been observed mostly in the campaign-style mode: in 2 week sessions every year or two. This type of observing is best suited for determination of the orbital parameters and their evolution and does not produce reliable determination of long-term effects such as the timing signature of geodetic precession (J. Taylor 2002, private communication). It is thus unlikely that the current timing observations will allow us to determine the misalignment angle. 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. M. K. is a Michelson Postdoctoral Fellow and is supported by the Polish Committee for Scientific Research, grant 2P03D A. W. is supported by the NSF under grant AST I. H. S. holds an NSERC UFA and is supported by a Discovery Grant. Arzoumanian, Z. 1995, Ph.D. thesis, Princeton Univ. Arzoumanian, Z., Phillips, J. A., Taylor, J. H., & Wolszczan, A. 1996, ApJ, 470, 1111 Arzoumanian, Z., Taylor, J. H., & Wolszczan, A. 1999, in Pulsar Timing, General Relativity, and the Internal Structure of Neutron Stars, ed. Z. Arzoumanian, F. van der Hooft, & E. P. J. van den Heuvel (Amsterdam: Koninklijke Nederlandse Akademie van Wettenschappen), 85 Barker, B. M., & O Connell, R. F. 1975a, Phys. Rev. D, 12, b, ApJ, 199, L25 Bogdanov, S., Pruszyńska, M., Lewandowski, W., & Wolszczan, A. 2002, ApJ, 581, 495 REFERENCES Cadwell, B. J. 1997, Ph.D. thesis, Pennsylvania State Univ. Cordes, J. M., Wasserman, I., & Blaskiewicz, M. 1990, ApJ, 349, 546 Damour, T., & Deruelle, N. 1986, Ann. Inst. H. Poincaré (Physique Théorique), 44, 263 Damour, T., & Ruffini, R. 1974, CR Acad. Sci. Paris, 279, 971 Damour, T., & Taylor, J. H. 1992, Phys. Rev. D, 45, 1840 Esposito, L. W., & Harrison, E. R. 1975, ApJ, 196, L1 Everett, J. E., & Weisberg, J. M. 2001, ApJ, 553, 341 Hamilton, A. J. S., & Sarazin, G. L. 1982, MNRAS, 198, 59 Hughes, A., & Bailes, M. 1999, ApJ, 522, 504 Jones, D. I., & Andersson, N. 2001, MNRAS, 324, 811

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