system, and Ðnally, in 7, we perform numerical tests which show how it can be used in practice.

THE ASTROPHYSICAL JOURNAL 544:921È926 2000 December 1 ( 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A. IMPROVED TIMING FORMULA FOR THE PSR B1257]12 PLANETARY SYSTEM MACIEJ KONACKI1 AND ANDRZEJ J. MACIEJEWSKI2 Torun Centre for Astronomy Nicolaus Copernicus University Gagarina 11 87-100 Torun Poland AND ALEX WOLSZCZAN3 Department of Astronomy and Astrophysics Pennsylvania State University University Park PA 16802; and Torun Centre for Astronomy Nicolaus Copernicus University ul. Gagarina 11 87-100 Torun Poland Received 2000 February 8; accepted 2000 June 27 ABSTRACT We present a new analysis of the dynamics of the planetary system around the pulsar B1257]12. A semianalytical theory of perturbation between terrestrial-mass planets B and C is developed and applied to improve the multiorbit timing formula for this obect. We use numerical simulations of the pulse arrival times for PSR B1257]12 to demonstrate that our new timing model can serve as a tool to determine the masses of the two planets. Subect headings: celestial mechanics stellar dynamics È planetary systems È pulsars: individual (PSR B1257]12) 1. INTRODUCTION The Ðrst extrasolar planetary system has been discovered around the millisecond pulsar B1257]12 (Wolszczan & Frail 1992). The system consists of three planets named A B and C with planets B and C having the orbits close to a 3: 2 commensurability. This circumstance allows us to analyze the dynamics of the system beyond the classical Keplerian approximation. Namely in such a conðguration the gravitational interactions of planets B and C give rise to observable time variations of the B and C orbital elements. It was thoroughly discussed by Rasio et al. (1992) and Malhotra (1993a) (see also Malhotra et al. 1992; Malhotra 1993b; Rasio et al. 1993; Peale 1993). Subsequently these studies were used to conðrm the existence of the PSR B1257]12 planetary system through detecting in the timing observations of the pulsar the presence of those non- Keplerian variations (Wolszczan 1994). However the application of non-keplerian dynamics goes further than the conðrmation of the discovery. It can be used to derive some interesting information about the system which is not otherwise accessible. The aim of this paper is to apply the theory of perturbed planetary motion and derive an improved model for the timing observations of PSR B1257]12. Such a model as we show on simulations should lead to a determination of the masses of planets B and C as well as the inclinations of their orbits. To this end in 2 we analyze the equations of motion in the barycentric and Jacobi coordinate systems which we use in the paper. In 3 we show how these two slightly di erent representations are related to the commonly used timing model for pulsars with companions. In 4 we demonstrate how to express such a timing formula in terms of the osculating orbital elements. In 5 we show how to obtain the osculating elements of B and C. In 6 we present an improved timing model describing the motion of this 1 kmc=astri.uni.torun.pl. 2 macieka=astri.uni.torun.pl. 3 alex=astro.psu.edu. system and Ðnally in 7 we perform numerical tests which show how it can be used in practice. 2. EQUATIONS OF MOTION Let us consider a system consisting of a neutron star P 0 with mass m and N planets P with masses m. In an arbi- 0 i i trary inertial reference frame equations of motion of this system have the form m i R i \[G ; N m m i R3 (R i [ R ) (1) /0 i Ei R i \ p R i [ R p i \ 0... N and G is the gravitational constant. The Hamiltonian function for equation (1) has the form H \ 1 2 ; N 1 Gm m P2[ ; i (2) m i R i/0 i 0yi:yN i and equation (1) can be written as HamiltonÏs equations d dt R i \ LH d LP dt P i \[LH i \ 0... N. (3) LR i i Analytical perturbation theory for a planetary system is usually formulated in the so-called Jacobi coordinates r i \ 0... N which are deðned in the following way: i r \ R [ 1 k~1 ; mi R for k \ 1... N (4) k k k i k~1 i/0 r \ 1 N k ; mi R k \ ; mi. (5) 0 k i k N i/0 i/0 In other words r is the radius vector from the center of mass of bodies P i... P to body P and r is the center 0 i~1 i 0 of mass of the system. The above formulae deðne a one-to- 921

922 KONACKI MACIEJEWSKI & WOLSZCZAN Vol. 544 one relationship between Cartesian inertial coordinates and Jacobi coordinates. The inverse relationship has the form R \ r ] k N m k~1 r [ ; i r k \ 0... N (6) k 0 k k k i k i/k`1 i we assume k \ 0. In terms of canonical Jacobi coordinates the Hamiltonian ~1 (eq. [2]) reads H \ 1 N k N m m p2] ; i p2[g ; 0 i k 0 k m i r N i/1 i~1 i i/1 i ]H3 1 (r 1... r N ) (7) H3 is its perturbative part. If we 1 \ H3 1 (r 1... r N ) assume that masses of planets are small and all are of the same order v then H3 is of order v2 i.e. H3 1 1 \ H 1 ] O(v3) A 1 B H \[G ; m m [ r i Æ r. (8) 1 i r r 1yi:yN i i From the form of the Hamiltonian (eq. [7]) it follows that p is a Ðrst integral and that the equations for variables 0 Mr... r p... p N do not depend on its particular 1 N 1 N value. Thus we can assume that p \ 0 implying that r is 0 0 constant and can be set to r \ 0. This is equivalent to the 0 assumption that our inertial frame is a certain barycentric reference frame. The dynamics of the system are governed by HamiltonÏs equations with the Hamiltonian of the form H \ H ] H ] ÉÉÉ (9) 0 1 N k N m m H \ ; i p2[g ; 0 i (10) 0 k m i r i/1 i~1 i i/1 i is the unperturbed part of the Hamiltonian describing a system of N independent planets. It follows from the form of equation (10) that each planet moves in a Keplerian orbit in the same way as a body with the mass m k /k around a i i~1 i Ðxed gravitational center m m. Each of these Keplerian 0 i motions can be parameterized by Keplerian elements MT a e i u )NÈthe time of pericenter semimaor axis p eccentricity inclination argument of pericenter and longitude of ascending node respectively. 3. TIMING MODEL AND COORDINATE SYSTEMS Timing observations of pulsars represent measurements of the times of arrival (TOAs) of pulsars pulses. An extraordinary precision of timing measurements allows a detection of very low level e ects in timing residuals (see for review Lyne & Graham-Smith 1998). In the case of a binary pulsar the observed TOAs exhibit periodic variations resulting from the motion of the pulsar around the center of mass. Such variations are modeled with the formula *t \ x[(cos E [ e) sin u ] J1 [ e2 sin E cos u] x \ a sin i/c E [ e sin E \ n(t [ T p ) n \ 2n P (11) and *t is the additional delay/advance in TOAs E is the eccentric anomaly a e u sin i P T are Keplerian elements of the orbit and c is the speed of p light. When a pulsar has N planets the TOA variations become N *t \ ; x C (cos E [ e ) sin u /1 D ]J1 [ e2 (sin E cos u ). (12) Note that in practice the parameters of the model which we determine by means of the least-squares Ðt to the data are x e u P T i.e. the proection of semimaor axis of the pulsarïs orbit p eccentricity argument of pericenter orbital period and time of pericenter. In order to precisely understand and interpret these parameters we describe the pulsarïs motion in the barycentric reference frame with the z-axis of the system directed toward the barycenter of the solar system and x-y plane of the reference frame in the plane of the sky. This way we have *t \[ 1 c R 0 Æ ZŒ R 0 \[ 1 N ; mi R (13) m i 0 i/1 ZŒ is the unit vector along the z-axis of the pulsar system barycentric frame and R are barycentric positions of planets. Assuming that m a sin i \ x c G (1 ] m / )2 \ n 2 a 3 (14) with planetsï orbital parameters a n e u T we can p most naturally interpret the motion of the pulsar as a superposition of the elliptic motions of its planets around the barycenter of the system. However as it was mentioned in the previous section the analytical perturbation theory is usually formulated in Jacobi coordinates in which the TOA variations become *t \[ 1 c R 0 Æ ZŒ R 0 \[; N i r i \ m (15) k /1 r are positions of planets. Furthermore we have the following relations: Gm k i a sin i \ x c 0 \ G \ n2 a3. (16) k 1 [ i ~1 Thus from the timing formula for TOA variations of a pulsar with planets it is possible to obtain two somewhat di erent descriptions of the pulsar motion. Although they both represent a sum of a certain number of elliptic motions the interpretation of some of their parameters is slightly di erent. Throughout the rest of this paper we will use Jacobi coordinates as they are more convenient in the formulation of the theory of perturbed motion. 4. OSCULATING ORBITAL ELEMENTS The non-keplerian motion of the PSR B1257]12 system can be described by means of the osculating ellipses (i.e. by means of ellipses whose parameters change with time). The time evolution of orbital elements can be determined by solving the classical LagrangeÏs perturbation

No. 2 2000 PSR B1257]12 PLANETARY SYSTEM 923 equations (see for example Brouwer & Clemence 1961) with the perturbation Hamiltonian given by equation (8). Such approach leads to the solution in the form x \ x0]*x(t [ t 0 ) (17) x stands for a speciðc orbital element x0 for its initial value and *x for its time-dependent part of small magnitude *x(t [ t )/x0 > 1. In the case of the PSR B1257]12 planetary systems 0 the most signiðcant part of the perturbations comes from planets B and C. Therefore we can assume that the orbital elements of planet A are approximately constant while the elements of planets B and C change with time. Thus using the formulae for *t from the previous section (eq. [12]) and assuming the time evolution of orbital elements in the form of equation (17) the additional TOA variations dt \ MB CN due to the inter- actions between planets B and C can be expressed as follows: cdt i sin i0 \[*h C3 0)D 2 a 0]1 2 a 0 cos (2 ] 1 2 *k a 0 sin (2 0) ] *a sin 0]* a 0 cos 0 (18) h \ e sin u k \ e cos u \ n (t [ T p ) ] u (19) and equation (18) is given to Ðrst order in *a * *h *k and the lowest order in e (or h and k since the eccentric- ities of planets B and C are very small). The above equations can be obtained from the following well-known expansions: cos E \[ e 2 ] 2 ; 1 k J (ke) cos (km) k~1 k Z0 sin E \ 2 e ; 1 k J (ke) sin (km) M \ [ u (20) k~1 k Z0 J (x) is a Bessel function of the Ðrst kind of order k and argument k x Z denotes the set of all positive and negative integers excluding 0 zero and an approximate relation for Bessel functions of small arguments the case of PSR B1257]12 periodic variations can be related to conunctions ( close encounters ÏÏ) of planets B and C with the frequency n \ n [ n and to the 3:2 near commensurability with the ce frequency B C n \ 3n [ 2n. The dynamics of this system was studied by r Rasio C et al. B (1992) and Malhotra (1993a) (see also Malhotra 1993b; Rasio et al. 1993; Peale 1993). Malhotra (1993a) solved the LagrangeÏs perturbation equations for the orbital elements assuming a nonresonant system with coplanar orbits. This Ðrst-order perturbation theory is valid for (sin i)~1 Z 10 (orbital inclinations larger than about 6 ) which corresponds to mass ratios m /m Z 6 ] 10~5 (and the nonresonant case). When planets B and 0 C are locked in the exact resonance it is necessary to develop a di erent theory of motion (Malhotra et al. 1992; Rasio et al. 1993). It turns out that the Ðrst-order perturbation theory for this system developed by Malhotra (1993a) is not accurate enough for the purpose of determination of the planetsï masses. Therefore in this paper we present a new approach to this problem which precisely addresses the issue. Namely Ðrst we assume that the orbits of B and C are not coplanar however their relative inclination I is small. The geometry of the system can be represented as in Figure 1 and the perturbative part of Hamiltonian expanded to the Ðrst order in I and the product of the mass of planets B and C m m has the following form: 1 2 H 1 \[ Gm 1 m 2 r 2 GC 1 [ 2 r 1 r2 cos t ] Ar 1 r2 B2D [ r 1 r2 cos t H Π 1= ω1 τ1 Π = ω τ 2 2 2 z Π 1 Π2 n 12 (23) t \ f ] u [ f [ u [ q q \ q [ q 1 1 2 2 1 2 r \ a (1 [ e 2) \ 12 1 ] e cos f and f is the true anomaly. Note that in the above expan- sion H does not depend on I. The Ðrst term in which I 1 J k (x) B xk 2kk! x > 1. (21) Now our next step is to Ðnd the explicit form of the functions O Ω 2 I τ 1 τ 2 i 2 y a (t) \ a 0]*a (t) (t) \ 0]* (t) h (t) \ h 0]*h (t) k (t) \ k 0]*k (t). (22) x Ω 1 n 1 i 1 n2 5. SEMIANALYTICAL PERTURBATION THEORY Mutual gravitational interactions between planets cause periodic and secular changes of their orbital elements. In FIG. 1.ÈGeometry of the system. The angles u u are the arguments of pericenter ) ) longitudes of ascending node 1 and 2 i i inclinations of the orbit of the 1 planets 2 B and C; q and q are the angles 1 2 n On and n On respectively. The angles I 1 q q can 2 be found by solving 1 12 the spherical 2 12 triangle n n n. 1 2 1 12 2

924 KONACKI MACIEJEWSKI & WOLSZCZAN Vol. 544 appears is of the second order; precisely it is sin2 (I/2). Thus as long as sin2 (I/2) is negligible the above expansion is valid. Next from the classical form of LagrangeÏs perturbation equations (see for example Brouwer & Clemence 1961) we can obtain the following set of the Ðrst-order di erential equations for the elements a e u : a5 \ [2 k n a LH 1 Lp e5 \ 1 k n a 2 e C [ A 1 [ e 2 B LH 1 Lp ] J1 [ e 2 LH 1 Lu D u5 \ [J1 [ e 2 k n a 2 e LH 1 Le 5 \ n ] 2 LH e J1 [ e2 1 [ k n a La k n a2(1 ] J1 [ e2) (24) n is the mean motion and p is related to time of pericenter through p \[n T so the Kepler equation p reads E [ e sin E \ n t ] p \ [ u (25) and the true anomaly f is related to the eccentric anomaly E through tan f 2 \S1 ] e 1 [ e tan E 2. (26) And it should be noted that the elements ) i remain approximately constant in the case of small relative inclinations. In principle such a set of equations could be solved for a e u. In practice however it is extremely compli- cated. On the other hand in fact we do not need analytical formulae such as those presented in Malhotra (1993a). Thus the most suitable approach is to solve this problem numerically. In order to do so we ust need the explicit form of the right-hand-side functions of equation (24) which can be easily obtained using the perturbing Hamiltonian H. Subsequently the equations can be solved numerically 1 for a e u. As we show in 7 this approach gives very accurate results much more accurate than any reasonable analytical treatment. From the form of equation (24) it follows that the change of orbital parameters *x (strictly speaking here *x stands for * *h *k and *a/a ) is proportional to m /m and m /m. Precisely corrections 0 *x are proportional 1 to 0 m /m 2 and 0 *x are proportional to 1 m /m. Let us Ðnally note 2 0 that in this 2 model there is one additional 1 0 parameter q which in general is not known a priori. However it can be determined through a leastsquares analysis of the data. 6. TIMING FORMULA We now have all of the components necessary to derive a useful timing formula which will describe the motion of the PSR B1257]12 system. First of all let us note that due to the small mass of planet A we will have i A \ i 1 \ m A m i \ i \ B B m B B 2 m ] m ] m m ] A B 0 B m m i \ i \ C B C. (27) C 3 m ] m ] m ] m m ] m ] A B C 0 B C The problem of Ðnding the masses and inclinations of planets B and C can now be formulated as a least-squares problem in which we Ðt to the data the following function: *t \ *t K (() ] dt B (( c B c C ) ] dt C (( c B c C ) (28) ( \ Mx0 n0 e0 u0 T 0 x0 n0 e0 u0 T 0 N B B B B pb C C C C pc c B \ m B c C \ m C (29) *t (() describes the Keplerian part of the motion given by K equation (12) and dt B \ x B 0 G [*h B C3 2 ] 1 2 cos (2 B 0)D ] 1 2 *k B sin (2 B 0) ] *a B a B 0 sin B 0]* B cos B 0H dt C \ x C 0 G [*h C C3 2 ] 1 2 cos (2 C 0)D ] 1 2 *k C sin (2 C 0) ] *a C a C 0 sin C 0]* C cos C 0H (30) describe its non-keplerian part. In this model the initial values of osculating orbital elements replace the Keplerian elements as parameters of the model and we additionally have the parameters c and c related to the masses of B B C and C. We also need the derivatives of *t with respect to the parameters M( c c N. With sufficient accuracy the derivatives with respect B C to ( can be computed from the Keplerian part *t of the model only and the derivatives with respect to c K and c can be easily obtained as dt and dt are proportional B to them: C B C L*t Lc B \ dt C c B L*t Lc C \ dt B c C. (31) Now determination of the masses of B and C and inclinations i and i can be carried out in the following way. First we B assume C that the mass of the pulsar m is canonical (i.e. 1.4 M ) and derive m and m directly fro c and c. Subsequently _ inclinations B can be C computed from B the following C equations: \ MB CN. cx 0\i a 0 sin i 0 G 1 [ i \ n 02 a 03 (32)

No. 2 2000 PSR B1257]12 PLANETARY SYSTEM 925 7. NUMERICAL TESTS AND CONCLUSIONS We start the tests with the comparison of our approach for computing osculating elements with that of Malhotra (1993a). It turns out that the most signiðcant component of the non-keplerian motion comes from the changes of mean longitude. Therefore in Figure 2 we present * for the case of coplanar edge-on orbits. As we can see the approach developed in this paper is in fact as good as the integration of full equations of motion. For other elements we obtain very similar results. It should be mentioned that because the model of Malhotra (1993a) is not accurate enough in predicting the secular change of * and because * are proportional to c and c using such a model would result in a signiðcant B error in C determination of the planetsï masses. Next we compared the TOA residuals due to the non-keplerian part of the motion calculated from our model with those obtained by means of numerical integration of full equations of motion. In Figure 3 as one can see for small relative inclinations (Figs. 3a and 3b) our model is very accurate. For larger I (Figs. 3c and 3d) one can see small di erences but this is consistent with the assumptions made to obtain the model. Finally we performed the tests which show how the derived timing model can be used to obtain the masses of planets. To this end we used the TEMPO software package4 modiðed to include our model of the non- Keplerian motion of PSR B1257]12. We simulated two di erent sets of artiðcial TOAs assuming the parameters of PSR B1257]12 (Wolszczan 1994). The Ðrst set of TOAs sampled every day covered a period of 10 years. We also added Gaussian noise with p \ 0.1 ks. The second set was prepared to resemble the real timing observations of PSR B1257]12. Under such conditions we simulated TOAs for the cases described in Figure 3. Subsequently we applied the modiðed TEMPO to obtain the masses of planets B and C. The results are presented in Table 1. The tests performed on Set 1 were used to establish a limit on applicability of our model. As one can see for the relative inclinations I of about 10 the model becomes inaccurate and the relative error of determination of masses is at the level of 20%. Because the relative inclination weakens the interactions of planets using the model which assumes a small I results in such situations in the determined masses that are smaller than the real ones. From the tests performed on set 2 we also learn that this error is bigger than uncertainty originating in TOA measurement errors. On the other hand when I is relatively small we obtain a very 4 See http://pulsar.princeton.edu/tempo. FIG. 2.ÈTime evolution of the mean longitude *(t) \ (t) [ 0(t) for the planets B and C in the case of coplanar edge-on orbits. The solid line indicates the solution by means of the numerical integration of full equations of motion the open circles indicate the solution obtained with the approach presented in this paper and the dash-dotted line with the model by Malhotra (1993a). accurate determination of masses thus proving that our model can be successfully applied as long as the assumptions made to derive it are satisðed. To sum up our model can be applied in all the cases when the observational ÏÏ uncertainties of the planetsï masses are bigger than the uncertainties resulting from the assumptions made to derive the timing model. This in general depends on masses of planets and the relative inclination of the orbits I as well as the characteristic of the observations but one can estimate that in the case of the PSR B1257]12 data I should be smaller than 10. We should also mention that for non-coplanar orbits the angle q will be in general di erent from zero therefore we have to Ðnd it in order to get the proper values of the planetsï masses. It can be done by computing the least-squares value of s2 for a range of q and then choosing the q that corresponds to the minimum of s2. And eventually one has to remember that the PSR B1257]12 planetary system could be in such conðguration that a more signiðcant alteration must be done to our model. First of all the relative inclination of the orbits could be large. In such case one would have to use the full form of the perturbative Hamiltonian H which would lead to a much more complicated model with 1 four additional parameters ) ) i i (instead of one q). Second the presence of a massive B C distant B C planet if signiðcant for the theory of motion would have to be taken into account. TABLE 1 ASSUMED AND DERIVED PARAMETERS OF SIMULATIONS ASSUMED SET 1 SET 2 q I m B m C m B m C m B m C SIMULATION (deg) (deg) (M^) (M^) (M^) (M^) (M^) (M^) a... 0.0 0.0 3.41 2.83 3.42(1) 2.82(1) 3.53(51) 2.63(51) b... 0.0 2.0 4.99 4.32 4.98(1) 4.26(1) 5.07(51) 4.06(48) c... 0.0 10.0 4.82 4.94 4.06(3) 4.11(3) 4.06(51) 3.79(48) d... 9.7 10.3 9.96 16.33 8.42(12) 13.39(12) 8.25(45) 13.16(42) NOTE.ÈValues in parentheses are 3 p uncertainties in the last digits quoted.

926 KONACKI MACIEJEWSKI & WOLSZCZAN FIG. 3.ÈTOA residuals due to the non-keplerian part of motion of the PSR B1257]12 in four di erent conðgurations. The solution obtained by means of the numerical integration of full equations of motion is indicated with the dots and the one obtained using our model with the solid line. Brouwer D. & Clemence G. M. 1961 Methods of Celestial Mechanics (New York: Academic Press) Lyne A. G. & Graham-Smith F. 1998 Pulsar Astronomy (New York: Cambridge Univ. Press) Malhotra R. 1993a ApJ 407 266 ÈÈÈ. 1993b in ASP Conf. Ser. 36 Planets around Pulsars ed. J. A. Phillips S. E. Thorsett & S. R. Kulkarni (San Francisco: ASP) 89 Malhotra R. Black D. Eck A. & Jackson A. 1992 Nature 356 583 REFERENCES Peale S. J. 1993 AJ 105 1562 Rasio F. A. Nicholson P. D. Shapiro S. L. & Teukolsky S. A. 1992 Nature 355 325 ÈÈÈ. 1993 in ASP Conf. Ser. 36 Planets around Pulsars ed. J. A. Phillips S. E. Thorsett & S. R. Kulkarni (San Francisco: ASP) 107 Wolszczan A. 1994 Science 264 538 Wolszczan A. & Frail D. A. 1992 Nature 355 145