theoretical implications of the psr b1620 …faculty.wcas.northwestern.edu/rasio/papers/56.pdfm...

THE ASTROPHYSICAL JOURNAL, 528 :336È350, 2000 January 12000. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

THEORETICAL IMPLICATIONS OF THE PSR B1620[26 TRIPLE SYSTEM AND ITS PLANET

ERIC B. FORD,1 KRITEN J. JOSHI,2 FREDERIC A. RASIO,3,4 AND BORIS ZBARSKY5Department of Physics, Massachusetts Institute of Technology

Received 1999 May 21; accepted 1999 July 16

ABSTRACTWe present a new theoretical analysis of the PSR B1620[26 triple system in the globular cluster M4,

based on the latest radio pulsar timing data, which now include measurements of Ðve time derivatives ofthe pulse frequency. These data allow us to determine the mass and orbital parameters of the secondcompanion completely (up to the usual unknown orbital inclination angle The current best-Ðti2).parameters correspond to a second companion of planetary mass, , in an orbitm2 sin i2^ 7 ] 10~3 M

_of eccentricity and semimajor axis AU. Using numerical scattering experiments, wee2^ 0.45 a2^ 60study a possible formation scenario for the triple system, which involves a dynamical exchange inter-action between the binary pulsar and a primordial star-planet system. The current orbital parameters ofthe triple are consistent with such a dynamical origin and suggest that the separation of the parent star-planet system was very large, AU. We also examine the possible origin of the anomalously highZ50eccentricity of the inner binary pulsar. While this eccentricity could have been induced during the samedynamical interaction that created the triple, we Ðnd that it could equally well arise from long-termsecular perturbation e†ects in the triple, combining the general relativistic precession of the inner orbitwith the Newtonian gravitational perturbation of the planet. The detection of a planet in this systemmay be taken as evidence that large numbers of extrasolar planetary systems, not unlike those dis-covered recently in the solar neighborhood, also exist in old star clusters.Subject headings : binaries : general È celestial mechanics, stellar dynamics È

globular clusters : individual (M4) È planetary systems È pulsars : general Èpulsars : individual (PSR B1620[26)

1. INTRODUCTION

PSR B1620[26 is a unique millisecond radio pulsar. Thepulsar is a member of a hierarchical triple system located inor near the core of the globular cluster M4. It is the onlyradio pulsar known in a triple system, and the only triplesystem known in any globular cluster. The inner binary ofthe triple contains the ^1.4 neutron star with a ^0.3M

_white dwarf companion in a 191 day orbit (Lyne et al.M_1988 ; McKenna & Lyne 1988) The triple nature of the

system was Ðrst proposed by Backer (1993) in order toexplain the unusually high residual second and third pulsefrequency derivatives left over after subtracting a standardKeplerian model for the pulsar binary.

The pulsar has now been timed for 11 yr since its dis-covery (Thorsett, Arzoumanian, & Taylor 1993 ; Backer,Foster, & Sallmen 1993 ; Backer & Thorsett 1995 ; Arzou-manian et al. 1996 ; Thorsett et al. 1999). These observationshave not only conÐrmed the triple nature of the system, butthey have also provided tight constraints on the mass andorbital parameters of the second companion. Earlier calcu-lations using three pulse frequency derivatives suggestedthat the mass of the second companion could be anywherebetween 10~3 and 1 with corresponding orbitalM

_,

periods in the range D102È103 yr (Michel 1994 ; Rasio1994 ; Sigurdsson 1995). More recent calculations using four

1 6-218M MIT, 77 Massachusetts Avenue, Cambridge, MA 02139 ;eford=mit.edu.

2 6-218M MIT, 77 Massachusetts Avenue, Cambridge, MA 02139 ;kjoshi=mit.edu.

3 6-201 MIT, 77 Massachusetts Avenue, Cambridge, MA 02139 ;rasio=mit.edu.

4 Alfred P. Sloan Research Fellow.5 6-218M MIT, 77 Massachusetts Avenue, Cambridge, MA 02139 ;

bzbarsky=mit.edu.

frequency derivatives and preliminary measurements of theorbital perturbations of the inner binary have further con-strained the mass of the second companion and stronglysuggest that it is a giant planet or a brown dwarf of massD0.01 at a distance D50 AU from the pulsar binaryM

_(Arzoumanian et al. 1996 ; Joshi & Rasio 1997). In thispaper we present a new analysis of the pulsar timing data,including the most recent observations of Thorsett et al.(1999 ; hereafter TACL). The data now include measure-ments of Ðve pulse frequency derivatives, as well asimproved measurements and constraints on various orbitalperturbation e†ects in the triple.

Previous optical observations by Bailyn et al. (1994) andShearer et al. (1996) using ground-based images of M4 hadidentiÐed a possible optical counterpart for the pulsar, con-sistent with a D0.5 main-sequence star, thus contra-M

_dicting the theoretical results, which suggest a much lowermass companion. However, it also seemed possible that theobject could be a blend of unresolved fainter stars, if not achance superpositon. Later Hubble Space Telescope (HST )WFPC2 observations of the same region by C. D. Bailyn etal. (1999, in preparation) have resolved the uncertainty. Themuch higher resolution HST image shows no(D0A.1)optical counterpart at the pulsar position, down to a magni-tude of V ^ 23, therefore eliminating the presence of anymain-sequence star in the system. Thus, the optical obser-vations are now consistent with the theoretical modeling ofthe pulsar timing data.

PSR B1620[26 is not the Ðrst millisecond pulsar systemin which a planet (or brown dwarf) has been detected. TheÐrst one, PSR B1257]12, is an isolated millisecond pulsarwith three clearly detected inner planets (all within 1 AU) ofterrestrial masses in circular orbits around the neutron star(Wolszczan 1994). Preliminary evidence for at least one

336

PSR B1620[26 TRIPLE SYSTEM AND ITS PLANET 337

giant planet orbiting at a much larger distance from theneutron star has also been reported (Wolszczan 1996 ; seethe theoretical analysis in Joshi & Rasio 1997). In PSRB1257]12 it is likely that the planets were formed in orbitaround the neutron star, perhaps out of a disk of debris leftbehind following the complete evaporation of the pulsarÏsstellar binary companion (see, e.g., Podsiadlowski 1995).Such an evaporation process has been observed in severaleclipsing binary millisecond pulsars, such as PSRB1957]20 (Arzoumanian et al. 1994) and PSRJ2051[0827 (Stappers et al. 1996), where the companionmasses have been reduced to D0.01È0.1 by ablation.M

_These companions used to be ordinary white dwarfs, and,although their masses are now quite low, they cannot beproperly called either planets or brown dwarfs.

In PSR B1620[26, the hierarchical triple conÐgurationof the system and its location near the core of a denseglobular cluster suggest that the second companion wasacquired by the pulsar following a dynamical interactionwith another object (Rasio, McMillan, & Hut 1995 ;Sigurdsson 1995). This object could have been a primordialbinary with a low-mass brown dwarf component or a main-sequence star with a planetary system containing at leastone massive giant planet. Indeed the possibility of detecting““ scavenged ÏÏ planets around millisecond pulsars in globu-lar clusters was discussed by Sigurdsson (1992) even beforethe triple nature of PSR B1620[26 was discovered.

Objects with masses D0.001È0.01 have recently beenM_detected around many nearby solar-like stars in Doppler

searches for extrasolar planets (see Marcy & Butler 1998 fora review). In at least one case, t Andromedae, severalobjects have been detected in the same system, clearly estab-lishing that they are members of a planetary system ratherthan a very low-mass stellar (brown dwarf) binary compan-ion (Butler et al. 1999). For the second companion of PSRB1620[26, of mass D0.01 current observations andM

_,

theoretical modeling do not make it possible to determinewhether the object was originally formed as part of a plan-etary system, or as a brown dwarf. In this paper, we willsimply follow our prejudice and henceforth will refer to theobject as ““ the planet.ÏÏ

One aspect of the system that remains unexplained, andcan perhaps provide constraints on its formation anddynamical evolution, is the unusually high eccentricity

of the inner binary. This is much larger thane1\ 0.0253one would expect for a binary millisecond pulsar formedthrough the standard process of pulsar recycling throughaccretion from a red giant companion. During the massaccretion phase, tidal circularization of the orbit throughturbulent viscous dissipation in the red giant envelopeshould have brought the eccentricity down to [10~4(Phinney 1992). At the same time, however, the measuredvalue may appear too small for a dynamically inducedeccentricity. Indeed, for an initially circular binary, theeccentricity induced by a dynamical interaction withanother star is an extremely steep function of the distance ofclosest approach (Rasio & Heggie 1995). Therefore a““ typical ÏÏ interaction would be expected either to leave theeccentricity unchanged or to increase it to a value of orderunity (including the possibility of disrupting the binary). Inaddition, one expects that most ““ exchange ÏÏ interactionswith a star-planet system will lead to the ejection of theplanet while the star is retained in a bound orbit around thepulsar binary (Rasio et al. 1995 ; Heggie, Hut, & McMillan

1996). In this paper we will present the results of newnumerical scattering experiments simulating encountersbetween the binary pulsar and a star-planet system. Thesesimulations allow us to estimate quantitatively the prob-ability of retaining the planet in the triple, while perhaps atthe same time inducing a small but signiÐcant eccentricityin the pulsar binary.

Secular perturbations in the triple system can also lead toan increase in the eccentricity of the inner binary. A pre-vious analysis assuming nearly coplanar orbits suggestedthat, starting from a circular inner orbit, an eccentricity aslarge as 0.025 could only be induced by the perturbationfrom a stellar-mass second companion (Rasio 1994), whichis now ruled out. For large relative inclinations, however, itis known that the eccentricity perturbations can in principlebe considerably larger (Kozai 1962 ; see Ford, Kozinsky, &Rasio 1999 for a recent treatment). In addition, the Newto-nian secular perturbations due to the tidal Ðeld of thesecond companion can combine nonlinearly with other per-turbation e†ects, such as the general relativistic precessionof the inner orbit, to produce enhanced eccentricity pertur-bations (Ford et al. 1999). In this paper, we reexamine thee†ects of secular perturbations on eccentricity in the PSRB1620[26 triple, taking into account the possibly largerelative inclination of the orbits, as well as the interactionbetween Newtonian perturbations and the general rela-tivistic precession of the pulsar binary.

2. ANALYSIS OF THE PULSAR TIMING DATA

2.1. Pulse Frequency DerivativesThe standard method of Ðtting a Keplerian orbit to

timing residuals cannot be used when the pulsar timing datacover only a small fraction of the orbital period (but seeTACL for an attempt at Ðtting two Keplerian orbits to thePSR B1620[26 timing data). For PSR B1620[26, theduration of the observations, about 11 yr, is very shortcompared to the likely orbital period of the second compan-ion, yr. In this case, pulse frequency derivativesZ100(coefficients in a Taylor expansion of the pulse frequencyaround a reference epoch) can be derived to characterize theshape of the timing residuals (after subtraction of aKeplerian model for the inner binary). It is easy to showthat from Ðve well-measured and dynamically induced fre-quency derivatives one can obtain a complete solution forthe orbital parameters and mass of the companion, up tothe usual inclination factor (Joshi & Rasio 1997 ; hereafterJR97).

In our previous analysis of the PSR B1620[26 system(JR97), we used the Ðrst four time derivatives of the pulsefrequency to solve for a one-parameter family of solutionsfor the orbital parameters and mass of the second compan-ion. The detection of the fourth derivative, which was mar-ginal at the time, has now been conÐrmed (TACL). Inaddition, we now also have a preliminary measurement ofthe Ðfth pulse derivative. This allows us in principle toobtain a unique solution, but the measurement uncertaintyon the Ðfth derivative is very large, giving us correspond-ingly large uncertainties on the theoretically derived param-eters of the system. Equations and details on the method ofsolution were presented in JR97 and will not be repeatedhere.

Our new solution is based on the latest available valuesof the pulse frequency derivatives, obtained by TACL for

338 FORD ET AL. Vol. 528

the epoch MJD 2,448,725.5 :

Spin period P\ 11.0757509142025(18) ms

Spin frequency f\ 90.287332005426(14) s~1f 5\ [5.4693(3)] 10~15 s~2f �\ 1.9283(14)] 10~23 s~3

f (3)\ 6.39(25)] 10~33 s~4f (4)\ [2.1(2)] 10~40 s~5f (5)\ 3(3)] 10~49 s~5 . (1)

Here the number in parentheses is a conservative estimateof the formal 1 p error on the measured best-Ðt value, takinginto account the correlations between parameters (seeTACL for details). It should be noted that the best-Ðt valuefor the fourth derivative quoted earlier by Arzoumanian etal. (1996) and used in JR97, f (4)\ [2.1(6)] 10~40 s~5, hasnot changed, while the estimated 1 p error has decreased bya factor of 3. This gives us conÐdence that the new measure-ment of f (5), although preliminary, will not change signiÐ-cantly over the next few years as more timing data becomeavailable.

Since the orbital period of the second companion is muchlonger than that of the inner binary, we treat the innerbinary as a single object. Keeping the same notation as inJR97, we let be the mass of the inner binarym1\ mNS ] m

cpulsar, with the mass of the neutron star and themNS mcmass of the (inner) companion, and we denote by them2mass of the second companion. The orbital parameters are

the longitude at epoch (measured from pericenter), thej2longitude of pericenter (measured from the ascendingu2node), the eccentricity semimajor axis and inclinatione2, a2,(such that for an orbit seen edge-on). They alli2 sin i2\ 1

refer to the orbit of the second companion with respect tothe center of mass of the system (the entire triple). A sub-script 1 for the orbital elements refers to the orbit of theinner binary. We assume that givingmNS \ 1.35 M

_,

where is the inclination of the pulsarmc

sin i1\ 0.3 M_

, i1binary (Thorsett et al. 1993), and we take for thesin i1\ 1analysis presented in this section since our results dependonly very weakly on the inner companion mass.

The observed value of is in general determined by af 5combination of the intrinsic spin-down of the pulsar and theacceleration due to the second companion. However, in thiscase, the observed value of has changed fromf 5[8.1] 10~15 s~2 to [5.4] 10~15 s~2 over 11 yr (TACL).Since the intrinsic spin-down rate is essentially constant,this large observed rate of change indicates that theobserved is almost entirely acceleration-induced. Similarly,f 5the observed value of is at least an order of magnitudef �larger than the estimate of from intrinsic timing noise,f �which is usually not measurable for old millisecond pulsars(see Arzoumanian et al. 1994, TACL, and JR97). Intrinsiccontributions to the higher derivatives should also be com-pletely negligible for millisecond pulsars. Hence, in ouranalysis, we assume that all observed frequency derivativesare dynamically induced, reÑecting the presence of thesecond companion.

Figure 1 illustrates our new one-parameter family of solu-tions, or ““ standard solution,ÏÏ obtained using the updatedvalues of the Ðrst four pulse frequency derivatives. There areno signiÐcant di†erences compared to the solution obtainedpreviously in JR97. The vertical solid line indicates the

FIG. 1.ÈAllowed values of the semimajor axis mass eccentricitya2, m2,longitude at epoch and longitude of pericenter for the seconde2, j2, u2companion of PSR B1620[26, using the latest available values for fourpulse frequency derivatives. The vertical solid line indicates the completesolution obtained by including the preliminary value of the Ðfth derivative.The two dashed lines indicate the solutions obtained by decreasing thevalue of f (5) by a factor of 1.5 (right), or increasing it by a factor of 1.5 (left).

unique solution obtained by including the Ðfth derivative. Itcorresponds to a second companion mass m2 sin i2\ 7.0] 10~3 eccentricity and semimajor axisM

_, e2\ 0.45,

AU. For a total system massa2\ 57 m1] m2\ 1.65 M_

,this gives an outer orbital period P2\ 308yr.

It is extremely reassuring to see that the new measure-ment of f (5) is consistent with the family of solutionsobtained previously on the basis of the Ðrst four derivatives.The implication is that the signs and magnitudes of theseÐve independently measured quantities are all consistentwith the basic interpretation of the data in terms of a secondcompanion orbiting the inner binary pulsar in a Keplerianorbit.

For comparison, the two vertical dashed lines in Figure 1indicate the change in the solution obtained by decreasingthe value of f (5) by a factor of 1.5 (right) or increasing it by afactor of 1.5 (left). Note that lower values of f (5) give highervalues for If we vary the value of f (5) within its entire 1 pm2.error bar, all solutions are allowed, except for the extremelylow-mass solutions with In particular, them2 [ 0.002.present 1 p error on f (5) does not strictly rule out a hyper-bolic orbit for However, it is still possible to(e2[ 1) m2.derive a strict upper limit on by considering hyperbolicm2solutions and requiring that the relative velocity at inÐnityof the perturber be less than the escape speed from thecluster core. This will be discussed in ° 2.3. A strict lowerlimit on the mass, can be set bym2 Z 2 ] 10~4 M

_,

requiring that the orbital period of the second companionbe longer than the duration of the timing observations(about 10 yr). Note that all solutions then give dynamically

No. 1, 2000 PSR B1620[26 TRIPLE SYSTEM AND ITS PLANET 339

stable triples, even at the low-mass, short period limit (seeJR97 for a more detailed discussion).

2.2. Orbital PerturbationsAdditional constraints and consistency checks on the

model can be obtained by considering the perturbations ofthe orbital elements of the inner binary due to the presenceof the second companion. These include a precession of thepericenter, as well as short-term linear drifts in the inclina-tion and eccentricity. The drift in inclination can bedetected through a change in the projected semimajor axisof the pulsar. The semimajor axis itself is not expected to beperturbed signiÐcantly by a low-mass second companion(Sigurdsson 1995).

The latest measurements, obtained by adding a lineardrift term to each orbital element in the Keplerian Ðt for theinner binary (TACL), give

u5 1\ [5(8)] 10~5 deg yr~1 , (2)

e5 1\ 0.2(1.1)] 10~15 s~1 , (3)

x5p\ [6.7(3)] 10~13 , (4)

where is the projected semimajor axis of thexp\ a

psin i1pulsar. Note that only is clearly detected, while the otherx5

ptwo measurements only provide upper limits.We use these measurements to constrain the system by

requiring that all our solutions be consistent with thesesecular perturbations. To do this, we perform Monte Carlosimulations, constructing a large number of random realiza-tions of the triple system in three dimensions, and acceptingor rejecting them on the basis of compatibility with themeasured orbital perturbations (see JR97 for details). Theeccentricity of the outer orbit is selected randomly assuminga thermal distribution, and the other orbital parameters arethen calculated from the standard solutions obtained in° 2.1 (see below for a modiÐed procedure which includes thepreliminary measurement of f (5)). The unknown inclinationangles and are generated assuming random orienta-i1 i2tions of the orbital planes, and the two position angles ofthe second companion are determined using i1, i2, u2, j2,and an additional undetermined angle a, which (along with

and describes the relative orientation of the twoi1 i2)orbital planes. The perturbations are calculated theoreti-cally for each realization of the system, assuming a Ðxedposition of the second companion. The perturbation equa-tions are given in Rasio (1994) and JR97.

Figure 2 shows the resulting probability distributions forthe mass and the current separation (at epoch) of them2 r12second companion. The Monte Carlo trials were performedusing only our standard solution based on four pulse fre-quency derivatives, since the Ðfth derivative is still onlymarginally detected. The solid line indicates the value givenby the preliminary measurement of f (5). The most probablevalue of is consistent with the range of valuesm2^ 0.01 M

_obtained from the complete solution using the Ðfth deriv-ative. The two dashed lines indicate the values obtainedfrom the complete solution by decreasing the value of f (5)by a factor of 1.5 (right) or increasing it by a factor of 1.5(left).

Figure 3 shows the probability distribution of f (5) pre-dicted by our Monte Carlo simulations, with the verticalsolid line indicating the preliminary measurement, and thedashed lines showing the values of f (5) increased and

FIG. 2.ÈNumber of accepted realizations (N) of the triple for di†erentvalues of and the corresponding distance of the second companionm2 r12from the inner binary in our Monte Carlo simulations. Accepted realiza-tions are those leading to short-term orbital perturbation e†ects consistentwith the current observations. The Monte Carlo trials were performedusing only the standard one-parameter family of solutions obtained fromthe Ðrst four pulse frequency derivatives. The solid line indicates the valuegiven by the preliminary measurement of f (5) and assuming Thesin i2\ 1.two dashed lines indicate the values obtained by decreasing the value off (5) by a factor of 1.5 (right), or increasing it by a factor of 1.5 (left).

decreased by a factor of 1.5, as before. The most probablevalue is clearly consistent with the measured value, wellwithin the formal 1 p error. This result provides anotherindependent self-consistency check on our model, indicat-ing that all the present timing data, including the orbitalperturbations, are consistent with the basic interpretation ofthe system in terms of a triple.

Since the uncertainty in f (5) is still large, it can also beincluded in the Monte Carlo procedure as a new constraint,in addition to the three orbital perturbation parameters.For each realization of the system, we now also calculatethe predicted value of f (5), and we add compatibility withthe measured value (assuming a Gaussian distributionaround the best-Ðt value, with the quoted 1 p standarddeviation) as a condition to accept the system. These MonteCarlo simulations with the additional constraint on f (5)produce results very similar to those of Figure 2, giving inparticular the same value for the most probable secondcompanion mass. This is of course not surprising, given theresults of Figure 3. The only signiÐcant di†erence is that thesmall number of accepted solutions with m2[ 10~3 M

_seen in Figure 2 (small hump seen on the left of the mainpeak) are no longer allowed.

We also conducted Monte Carlo simulations using f (5)alone as a constraint, to determine the a priori probabilitydistributions of the three secular perturbations for solutions


FIG. 3.ÈSame as Fig. 2 but for the predicted value of the Ðfth pulsederivative f (5). It is clear that the measured value of f (5) is in perfectagreement with the theoretical expectations based on the Ðrst four deriv-atives and the preliminary measurements of orbital perturbation e†ects.

that are consistent with all measured pulse frequency deriv-atives. Figure 4 shows the distribution of the absolutevalues of the three secular perturbations (the symmetry ofthe problem allows both positive and negative values withequal probability) that were consistent with the frequencyderivative data. We see that the allowed values of the per-turbations span several orders of magnitude, with the mostprobable values of and being roughly consistentx5

p, e5 1, u5 1with their observed values.

We note from Figure 4 that the wide probability distribu-tion for easily allows (although it does not favor) valuesx5

pmuch smaller than its currently observed value. This sug-gests the possibility that the observed value of may bex5

psigniÐcantly a†ected by external e†ects, such as the propermotion of the pulsar (which, if signiÐcant, can also lead to aslow drift in the inclination of the orbital plane to the line ofsight, through the change in the direction of the line of sightitself). Indeed this possibility was discussed by Arzouma-nian et al. (1996). They pointed out that, if the pulsar propermotion is equal to the published cluster proper motion,k \ 19.5 mas yr~1 (Cudworth & Hansen 1993) and if theproper motion alone determines the measured thisx5

p,

would require the inclination angle to be less than abouti116¡, implying a mass for the (inner) pulsar com-Z1.4 M_panion. While another neutron star or a black hole com-

panion cannot be strictly ruled out, this would require ahighly unusual formation scenario for the system and wouldmake the nearly circular orbit of the binary pulsarextremely difficult to explain.

Note, however, that the published cluster proper motionmay be in error. Indeed, the pulsar timing proper motion

FIG. 4.ÈA priori probability distributions of the three secular pertur-bations, eqs. (2)È(4), based on the one-parameter family of solutions shownin Fig. 1 and using only the preliminary measurement of f (5) as a constraintin the Monte Carlo simulations. The vertical lines show the measuredvalue (for or the 1 p upper limits on the measured values (for andx5

p), e5 1which are all clearly consistent with these theoretical distributions.u5 1),

obtained by TACL, k \ 28.4 mas yr~1, is incompatible withthe published value for the cluster, since it would imply arelative velocity of the pulsar system with respect to thecluster of 78^ 40 km s~1 (TACL), far greater than theescape speed from the cluster. Since the location of thepulsar very near the cluster core makes its association withthe cluster practically certain, it is very likely that either thetiming proper motion or the cluster proper motion is incor-rect (see TACL for further discussion).

If we assumed that the observed is due entirely tox5pproper motion, we can obtain an estimate of the proper

motion. For example, take the inner pulsar companion tobe a white dwarf of mass 0.35 corresponding to anM

_,

inclination We Ðnd that this would require ai1\ 55¡.proper motion k ^ 100 mas yr~1, or about 5 times greaterthan the current value of the optical (cluster) proper motion,and implying a velocity far greater than the escape speed forthe cluster. But this is also about 4 times greater than thepulsar timing proper motion obtained by TACL. Therefore,if the observed were due to proper motion, then both thex5

poptical and timing proper motion would have to be incor-rect, which seems unlikely.

2.3. Hyperbolic SolutionsGiven the uncertainty in the proper motion of the system

and the fact that the measurements of f (5) and the orbitalperturbations would allow it, we have also considered thepossibility that the ““ second companion ÏÏ may in fact be anobject on a hyperbolic orbit, caught in the middle of a close


interaction with the binary pulsar. The probability ofobserving the system during such a transient event is ofcourse very low (D2 ] 10~5 if the binary pulsar is in thecore of M4; see Phinney 1993), but it cannot be ruled out apriori.

In Figure 5, we show the predicted value of f (5) as afunction of the mass of the second companion (m2 sin i2),obtained by extending the solutions of Figure 1 into thehyperbolic domain. Values of corre-m2 sin i2[ 0.012 M

_spond to a hyperbolic orbit for the second companionThe solid horizontal line indicates the measured(e2 [ 1).

value of f (5), and the dashed lines indicate the current 1 perror bar. We see that the current uncertainty in f (5) allowsthe entire range of hyperbolic orbits for the second compan-ion.

However, a strict upper limit on can be derived fromm2the requirement that the pulsar remain bound to the cluster.For we Ðnd that the relative velocitym2 sin i2 [ 0.055 M

_,

(““ at inÐnity ÏÏ) would exceed the escape speed from thecluster core (^12 km s~1 ; see Peterson, Rees, & Cudworth1995). To reach a stellar mass, say wouldm2\ 0.5 M

_,

require a relative velocity at inÐnity ^45 km s~1, whichwould be well above the escape speed from the cluster.Thus, even in the hyperbolic regime, the mass of the secondcompanion is constrained to remain substellar. In addition,

FIG. 5.ÈTheoretically predicted value of f (5) as a function of the massof the second companion for the one-parameter family of solutions(m2)based on the Ðrst four derivatives. Here we have extended these solutionsinto the hyperbolic regime. Values of correspond tom2 sin i2[ 0.012 M

_a hyperbolic orbit for the second companion The solid horizontal(e2[ 1).line indicates the measured value of f (5), and the dashed lines correspond tothe current 1 p error. Thus the current measurement uncertainty allows allhyperbolic solutions for the second companion and hence does not directlyconstrain However, for the relative velocity (““ atm2. m2[ 0.055 M

_,

inÐnity ÏÏ) of the hyperbolic orbit would exceed the escape speed from thecluster core (^12 km s~1).

we Ðnd that the hyperbolic solutions require the presentposition of the second companion to be very close to peri-center This is not surprising, since the hyper-( o j2 o\ 10¡).bolic solutions are a smooth extension of the standardbound solutions shown in Figure 1 (in which is alreadyj2small for solutions with but it makes the hyperbolice2^ 1),solutions even more unlikely.

3. TRIPLE FORMATION PROCESS

3.1. Recycling in the TripleOne plausible formation scenario for PSR B1620[26

begins with an old neutron star in a binary system, whichhas a dynamical interaction with a star-planet system(Sigurdsson 1993, 1995). The original companion of theneutron star is ejected in the interaction, while the main-sequence star and its orbiting giant planet are retained inorbit around the neutron star. The planet is typicallyretained in a wider orbit around the new inner binarysystem. The main-sequence star later evolves to become ared giant, transferring mass onto the neutron star and recy-cling it into a millisecond pulsar, while at the same time theinner orbit is circularized through tidal dissipation. Thus, inthis scenario, the triple system is formed before the recyclingof the neutron star takes place.

Sigurdsson (1993) Ðnds through numerical scatteringexperiments, that in approximately 15% of such exchangeinteractions (in which the main-sequence star remainsbound to the neutron star), the planet is left in a bound,wide orbit around the system. In a majority of such cases,the planet has a Ðnal semimajor axis 10È100 times largerthan the initial semimajor axis of the neutron star binaryand an eccentricity in the range 0.3È0.7. Sigurdsson (1993)also Ðnds that the probability of retaining a planet is pro-portionately higher if the main-sequence star has more thanone planet orbiting it. Thus it seems reasonable to considersuch an exchange interaction as a possible formation sce-nario for PSR B1620[26.

However, there are two main difficulties with this sce-nario. First, it requires that the age of the millisecond pulsarbe at most comparable to the age of the triple system. Ourmodeling of the timing data (° 2) indicates that the semi-major axis of the planetÏs orbit is AU. This makes theZ50outer orbit of the triple ““ soft ÏÏ in the cluster, implying thatany close interaction with a passing star will disrupt thetriple (ejecting the planet). Since gravitational focusing isnegligible for such a large object, the lifetime of the triplecan then be written

qd^ 8.5] 107 yr g

d2A SmT0.8 M

_

BA104 M_

pc~3o

B

]A5 km s~1

pBA50 AU

a2

B2. (5)

Here SmT is the average stellar mass in the cluster, o is thestellar density, p is the three-dimensional velocity disper-sion, is the semimajor axis of the outer orbit, anda2 g

d4

where is the average distance of closest approacha2/rd, rdfor an encounter that will disrupt the outer orbit. Based on

an extensive set of scattering experiments (see ° 3.2 below),we Ðnd that and therefore the factorr

d/a2^ 1.4 g

d2^ 0.5.

Hence, in the core of M4, the expected lifetime of the tripleis yr. This is much shorter than the estimatedq

dD 3 ] 107

age of the binary pulsar, yr (Thorsett et al. 1993).qpZ 109


However, outside the core, the expected lifetime of the triplecan be up to 2 orders of magnitude longer owing to themuch lower number density of stars in the halo. Forexample, at the half-mass radius of M4, the stellar density isdown to D102 pc~3, and the lifetime becomesM

_qhm D

109 yr. Hence, this scenario would require that the triplesystem, currently observed (in projection) near the edge ofthe cluster core, is in fact on an orbit that extends faroutside the core, allowing it to spend most of its lifetime inthe less dense cluster halo.

Indeed, Sigurdsson (1995) showed through numericalsimulations that a triple system with a low-mass secondcompanion, such as PSR B1620[26, is more likely to beobserved outside the cluster core than inside the core.Sigurdsson also Ðnds that such triples can easily survive forup to 2 ] 109 yr by spending most of their time outside thecore, but that once they enter the core, they are quicklydisrupted. In contrast, triples with stellar-mass second com-panions sink to the core much more quickly owing todynamical friction (on a timescale yr) and have a[109lifetime of order 108 yr in the core (Sigurdsson 1995 ; Rasioet al. 1995). Thus the fact that PSR B1620[26 is observedjust outside the core suggests that it probably contains alow-mass second companion (which is consistent with ourtiming solution) or that it is a very young system formed lessthan 5 ] 108 yr ago (which is difficult to reconcile with theestimated age of the binary pulsar in this scenario).

The other difficulty concerns the eccentricity of the pulsarbinary. Since the circularization of the inner binary takesplace after the triple is formed, this scenario leaves thehigher observed eccentricity of the inner binary unex-plained. To produce the higher eccentricity, a seconddynamical interaction of the triple with a passing star canbe invoked (Sigurdsson 1995). However, a very close inter-action is needed to induce a signiÐcant eccentricity in aninitially circular binary (Rasio et al. 1995 ; see Heggie &Rasio 1996 for a general treatment of this problem). There-fore, such an encounter is likely to disrupt the outer orbit inthe triple system. An encounter with a distance of closestapproach of ^2.5 AU to the inner binary could induce theobserved eccentricity. But this is much smaller than thesemimajor axis of the outer orbit, and roughly half of suchencounters will disrupt the outer orbit (Sigurdsson 1995).Such a close interaction occurs on average once inD4 ] 108 yr. Therefore, for each interaction that couldhave produced the eccentricity of the inner binary, weexpect D10 encounters that could have disrupted the outerorbit, leaving the probability of surviving at [0.01.

Alternatively, it is possible that the eccentricity of theinner orbit could have been induced later through secularperturbations in the triple. Rasio (1994) argued that this isunlikely for a low-mass second companion with low relativeinclination between the two orbits. However, we Ðnd thatunder certain conditions, the inner eccentricity can beexplained as arising from the combined e†ects of the tidalperturbation by the second companion and the generalrelativistic precession of the inner orbit. We discuss this inmore detail in ° 4.

3.2. Preexisting Binary PulsarWe now propose an alternative formation scenario,

which involves a dynamical exchange interaction between apreexisting binary millisecond pulsar and another widersystem containing a giant planet in orbit around a main-

sequence star. This interaction could lead to the ejection ofthe main-sequence star, leaving the planet in a wide orbitaround the binary pulsar. An exchange interaction of thistype might also simultaneously induce the observed eccen-tricity of the binary pulsar if either the main-sequence staror the planet passes sufficiently close to the binary pulsarduring the interaction. This scenario is a natural extensionof the mechanism studied by Rasio et al. (1995) for theformation of a stellar triple containing a millisecond pulsar.Here, however, an added difficulty is that one expects theplanet, and not the main-sequence star, to be preferentiallyejected during the interaction.

To study this binary-binary interaction scenario quanti-tatively, we have performed numerical scattering experi-ments similar to those done by Rasio et al. (1995) for stellartriples. The binary pulsar was treated as a single body ofmass 1.65 scattering o† several di†erent binary systemsM

_,

containing a main-sequence star with a very low-mass com-panion (which we refer to henceforth as the ““ star-planetsystem ÏÏ). The assumption that one can treat the innerpulsar binary as a single mass is justiÐed because its semi-major axis (0.77 AU) is considerably smaller than theorbital radius of the planet in all relevant cases. In addition,close approaches to the binary pulsar by either the planet orthe main-sequence star are forbidden, since they wouldinduce an eccentricity larger than the one observed. Forexample, an approach by a 0.01 object to within aboutM

_1.2 AU (1.5 semimajor axes) of the pulsar binary would beenough to induce an eccentricity greater than 0.025 (cf.Heggie & Rasio 1996). Hence in all relevant interactions,the pulsar binary could be treated as a single point mass.

Four di†erent outcomes are possible for each simulatedinteraction : (1) ionization ; (2) no exchange (as in a Ñyby) ; (3)exchange in which the planet is retained around the binarypulsar ; or (4) exchange in which the star remains bound tothe binary pulsar. Exchanges can be further divided into““ resonant ÏÏ and ““ nonresonant ÏÏ or ““ direct.ÏÏ In a resonantexchange interaction, all stars involved remain together fora time long compared to the initial orbital periods (typicallyD102È103 dynamical times), leading eventually to the ejec-tion of one object while (in this case) the others remain in abound hierarchical triple conÐguration. In direct exchangeinteractions, either the planet or the main-sequence star isejected promptly, following a close approach by the binarypulsar. In general, the type of outcome depends on theimpact parameter, the mass of the planet, and the initialplanet-star separation.

Heggie et al. (1996) have computed cross sections forexchange interactions in binaryÈsingle-star encounters,both numerically using scattering experiments and analyti-cally in various limiting regimes. They include results forbinary mass ratios as extreme as D0.01. However, theirresults are all obtained for hard binaries, whereas the star-planet system in our scenario represents a very soft binary.Therefore, we do not expect the results of Heggie et al.(1996) to be directly applicable here, although they doprovide some qualitative predictions. For example, for abinary mass ratio and the ratio ofm2/m1^ 0.01 m3/m1\ 1,cross sections for ejection of the star to ejection of the(m1)planet is about 0.12 for direct exchanges and 0.04 for(m2)resonant exchanges, with the total cross section for all reso-nant exchanges being about 2.5 times larger than that of alldirect exchanges. For soft binaries we Ðnd that directexchanges are dominant and that the relative probability of


ejecting the more massive binary member is higher (seebelow).

Our scattering experiments were performed using thescatter3 module of the STARLAB stellar dynamics package(McMillan & Hut 1996 ; Portegies Zwart et al. 1997). Themass of the main-sequence star was held constant at 0.8

close to the main-sequence turno† point of the cluster.M_

,The planet was placed on an initially circular orbit. Therelative velocity at inÐnity for all encounters was taken tobe 4 km s~1, slightly lower than the three-dimensionalvelocity dispersion in the cluster core (^6 km s~1 ; seePeterson et al. 1995). The impact parameter b was randomlyselected between zero and with probability p(b) P 2nb.bmax,The upper limit was selected such that all interactions with

led to a Ñyby with no exchange. In units of theb º bmaxinitial semimajor axis of the star-planet system, variedbmax

from about 5 to 35 depending on the mass of the planet.Three di†erent masses were selected for the ““ planet ÏÏ : 0.001

0.01 and 0.1 In each case, the semimajor axisM_

, M_

, M_

.of the star-planet system was initially chosen to be 5 AU,since this is the typical distance at which giant planets areexpected to form. In addition, in an e†ort to match betterthe orbital parameters of triple systems such PSRB1620[26, we also considered initial star-planet semimajoraxes of 30 and 50 AU.

We Ðrst address the possibility of capturing the planetinto a wide orbit around the pulsar binary, while ejectingthe star during an exchange interaction. Figure 6 shows thedistributions of the semimajor axis and eccentricity of theplanet in the triple for those interactions in which the planetwas captured and the binary pulsar was not disrupted. Wesee that the Ðnal semimajor axis of the planet in the triple is

FIG. 6.ÈDistributions of the Ðnal eccentricity and semimajor axis following an exchange interaction in which a planet of mass is captured by thee2 a2 mpbinary pulsar. The semimajor axis is given in units of the original star-planet separation Values of and are (a) 0.001 5 AU; (b) 0.01 5 AU;a

p. m

pap

M_

, M_

,(c) 0.1 5 AU; (d) 0.001 30 AU; (e) 0.01 50 AU.M

_, M

_, M

_,


likely to be within a factor of D2 of the initial star-planetseparation (note that, in this section, an index p refers toa

pthe orbit of the star-planet system, and not to the pulsar, asin ° 2). The eccentricity distribution is very broad, andapproaches a ““ thermal ÏÏ distribution [p(e)P e] for verywide initial separations.

Figures 7aÈ7e show the corresponding probabilities ofthe three possible outcomes (branching ratios) for di†erentplanet masses and initial separations of the star-planetsystem. The distance of closest approach is given in unitsr

pof the initial star-planet semimajor axis Since the planetap.

was the lowest mass object participating in the interaction,it was expected to have a much higher probability of beingejected. However, we Ðnd that the branching ratio for inter-actions resulting in the capture of the planet remains signiÐ-

cant for all interactions close enough to result in anexchange. For example, from Figure 7e, we see that for aninteraction between the binary pulsar and a star-planetsystem with separation 50 AU and planet mass m

p\ 0.01

the branching ratio for planet capture (ejection of theM_

,star) is in the range ^10%È30% for distances of closestapproach in the range D10È50 AU. For this case, there areno resonant interactions, implying that retention of the starhas negligible probability, while the branching ratio for ion-ization is in the range ^20%È60%. The remainder of theinteraction cross section (10%È70%) corresponds to““ clean ÏÏ Ñybys (without capture of either star or planet, orionization).

To focus our study on interactions in which the Ðnaltriple conÐguration is similar to that of PSR B1620[26, we

FIG. 7.ÈBranching ratios for various possible outcomes as a function of the distance of closest approach between the binary pulsar and therpmain-sequence star in the parent star-planet system. The stars represent ionization, the triangles, the ejection of the star, the squares, the ejection of the

planet, and the pentagons, a simple Ñyby with no exchange. Here the values of and are (a, f ) 0.001 5 AU; (b, g) 0.01 5 AU; (c, h) 0.1 5 AU;mp

ap

M_

, M_

, M_

,(d, i) 0.001 30 AU; (e, j) 0.01 50 AU. In panels (a)È(e) the only constraint imposed is that the binary pulsar was not disrupted during the encounterM

_, M

_,

(as in Fig. 6) ; in panels ( f )È( j) a stronger constraint was imposed, namely that the eccentricity of the binary pulsar was not perturbed to a value exceeding thepresently observed value of 0.025. In all cases, we note that the branching ratio for capture of the planet remains signiÐcant, at about 10%È30%.


now eliminate cases in which the eccentricity induced in thebinary pulsarÏs orbit would have been larger that thepresent value of 0.025. The induced eccentricity is estimatedbased on the results of Heggie & Rasio (1996). In makingthese estimates, all unknown angles, including the relativeinclination of the two orbits, are chosen randomly(assuming random orientations). Figures 7fÈ7j show thebranching ratios for those interactions in which the eccen-tricity induced in the binary pulsar during the interaction isbelow 0.025. The main di†erence with the general results(Figs. 7aÈ7e) is that the branching ratio for ejection of thestar now goes to zero for very close interactions, since thoseinteractions in which the star passes very close to the binarypulsar would have induced an eccentricity larger thanallowed. Figure 8 shows the corresponding distributions ofsemimajor axis and eccentricity of the planet in the triple(like Fig. 6 but with the additional constraint on induced

eccentricity). Here the only signiÐcant di†erence is in thelower probability of retaining the planet on a highly eccen-tric orbit, especially for more massive planets (compare, e.g.,Fig. 8c to Fig. 6c).

In Table 1, we list the cross sections for the various out-comes, obtained from our scattering experiments for each ofthe Ðve cases. The cross sections are given in units of theinitial geometrical cross section of the star-planet binary(na

p2).We now turn to the question of whether the eccentricity

of the binary pulsar could have been increased signiÐcantlyduring the interaction. Figure 9 shows the overall distribu-tion of the Ðnal eccentricity induced in the binary pulsarfollowing an exchange interaction in which the planet wasretained. In every case, we see that the induced eccentricityvaries over a large range, including the currently observedeccentricity of the binary pulsar. Thus, a scenario in which

FIG. 8.ÈSame as Fig. 6 but with the additional constraint that the eccentricity of the binary pulsar was not perturbed to a value exceeding the presentlyobserved value of 0.025.


TABLE 1

INTERACTION CROSS SECTIONS

NORMAL einduced \ 0.025

CASE Star Ejection Planet Ejection Ionization Star Ejection Planet Ejection Ionization

(a) 5 AU, 0.001 M_

. . . . . . . 1.13^0.01 0.00^0.01 1.66^0.01 0.13^0.07 0.00^0.01 0.68^0.03(b) 5 AU, 0.01 M

_. . . . . . . . 1.19^0.01 0.13^0.05 1.52^0.02 0.12^0.07 0.00^0.01 0.63^0.03

(c) 5 AU, 0.1 M_

. . . . . . . . . . 1.05^0.02 2.45^0.01 0.00^0.01 0.14^0.07 0.79^0.03 0.00^0.01(d) 30 AU, 0.001 M

_. . . . . . 0.007^0.004 0.000^0.001 0.008^0.004 0.001^0.001 0.000^0.001 0.003^0.001

(e) 50 AU, 0.01 M_

. . . . . . . 0.001^0.001 0.000^0.001 0.003^0.002 0.0002^0.0001 0.0000^0.0001 0.0007^0.0002

NOTE.ÈCross sections for the various outcomes for the cases described in ° 3.2. In each case, the velocity of the incoming star at inÐnity was 4km s~1.

an exchange interaction successfully forms the triple andsimultaneously induces the inner binary eccentricity we seetoday is certainly possible, although the present value of theeccentricity is not likely in this scenario. Alternatively, thetriple could have formed with an initial inner binary eccen-tricity lower than the present value. The eccentricity couldthen be raised to the present value by long-term secularperturbation e†ects (° 4). Forming the triple with an initialinner binary eccentricity higher than currently observed andlater reducing it through secular perturbations is also pos-sible.

Our analysis of the PSR B1620[26 timing data (° 2)indicates that the second companion is in a wide orbitaround the pulsar binary, with a semimajor axis AU.Z50The results of our scattering experiments indicate that theÐnal semimajor axis of the planet in the triple should becomparable to the initial star-planet separation (see Figs. 6

FIG. 9.ÈDistribution of the eccentricity induced in the binary pulsarduring a dynamical exchange interaction in which a planet was captured.The panels are labeled as in Fig. 6.

and 8). Hence, the interactions that could have produced asystem like PSR B1620[26 are those involving a widesystem with a planet mass and an initialm

pD 0.01 M

_separation AU, as shown in Figures 7j and 8e. JustapD 50

like the triple itself, such a wide star-planet system is ““ soft ÏÏ(binding energy smaller than the typical center-of-massenergy of an encounter) and has a large interaction crosssection, implying a short lifetime in the cluster core, q

dD 3

] 107 yr (cf. eq. [5]). Well outside of the cluster core, itslifetime could be much longer, but since an exchange inter-action with another binary is likely to take place only in thecore, it requires the star-planet system to drift Ðrst into thecore through dynamical friction. The interaction thatdestroyed the star-planet system once it entered the corecould be the one that produced the PSR B1620[26 triple.

4. SECULAR ECCENTRICITY PERTURBATIONS

We now examine in detail the possibility that the innerbinary eccentricity was induced by the secular gravitationalperturbation of the second companion. In the hierarchicalthree-body problem, analytic expressions for the maximuminduced eccentricity and the period of long-term eccentric-ity oscillations are available in certain regimes, dependingon the eccentricities and relative inclination of the orbits.

4.1. Planetary RegimeFor orbits with small eccentricities and a small relative

inclination the classical solution for the long-term(i [ 40¡),secular evolution of eccentricities and longitudes of peri-centers can be written in terms of an eigenvalue and eigen-vector formulation (Brouwer & Clemence 1961 ; for anexcellent pedagogical summary, see Dermott & Nicholson1986). This classical solution is valid to all orders in theratio of semimajor axes. In this regime the eccentricitiesoscillate as angular momentum is transferred between thetwo orbits. The precession of the orbits (libration orcirculation) is coupled to the eccentricity oscillations, butthe relative inclination remains approximately constant. Inthis regime it can be shown that a stellar mass second com-panion would be necessary to induce the observed eccen-tricity in the inner binary (Rasio 1994, 1995). Such a largemass for the third body has been ruled out by recent timingdata (see ° 2), implying that secular perturbations from athird body in a nearly coplanar and circular orbit does notexplain the observed inner eccentricity. The lack of a stellar-mass companion is also supported by the absence of anoptical counterpart for the pulsar in the HST observationsby C. D. Bailyn et al. (1999, in preparation).

Sigurdsson (1995) has suggested that it may be possiblefor the secular perturbations to grow further because of


random distant interactions of the triple with other clusterstars at distances D100 AU, which would perturb the long-term phase relation between the inner and outer orbits,allowing the inner eccentricity to ““ random walk ÏÏ up to amaximum value up to 2 orders of magnitude higher thanthat predicted for an isolated triple. However, the currentmost probable solution from the timing data requires thesecond companion to be in a very wide orbit (separation

AU), giving it a very short lifetime in the core, andZ40leaving it extremely vulnerable to disruption by repeatedweak encounters (see JR97 for a more detailed discussion).

4.2. High-Inclination RegimeFor a triple system formed through a dynamical inter-

action, there is no reason to assume that the relative inclina-tion of the two orbits should be small. When the relativeinclination of the two orbital planes is a di†erentZ40¡,regime of secular perturbations is encountered. This regimehas been studied in the past using the quadrupole approx-imation (Kozai 1962 ; see Holman, Touma, & Tremaine1987 for a recent discussion). Here the relative inclination iof the two orbits and the inner eccentricity are couplede1by the integral of motion (KozaiÏs integral)

# \ (1[ e12) cos2 i . (6)

Thus, the amplitude of the eccentricity oscillations is deter-mined by the relative inclination. It can be shown thatlarge-amplitude eccentricity oscillations are possible onlywhen (Holman et al. 1997). For an initial eccentricity# \ 35and initial inclination this impliese1^ 0 i0 i0[cos~1 (3/5)1@2^ 40¡, and the maximum eccentricity is thengiven by

e1max \ (1[ 53 cos2 i0)1@2 , (7)

which approaches unity for relative inclinations approach-ing 90¡. For a sufficiently large relative inclination, this sug-gests that it should always be possible to induce anarbitrarily large eccentricity in the inner binary and thatthis could provide an explanation for the anomalously higheccentricity of the binary pulsar in the PSR B1620[26system (Rasio, Ford, & Kozinsky 1997). However, there aretwo additional conditions that must be satisÐed for thisexplanation to hold.

First, the timescale for reaching a high eccentricity mustbe shorter than the lifetime of the system. Although themasses, initial eccentricities, and ratio of semimajor axes donot a†ect the maximum inner eccentricity, they do a†ect theperiod of the eccentricity oscillations (see Holman et al.1997 ; Mazeh, Krymolowski, & Rosenfeld 1996). The innerlongitude of periastron precesses with this period, whichcan be quite long, sometimes exceeding the lifetime of thesystem in the cluster. The masses also a†ect the period, butthey decrease the amplitude of the eccentricity oscillationsonly when the mass ratio of the inner binary approachesunity (Ford et al. 1999). In Figure 10 we compare the periodof the eccentricity oscillations to the lifetime of the triple inM4 (see ° 4.3 for details on how the various estimates wereobtained). It is clear that for most solutions the timescale toreach a large eccentricity exceeds the lifetime of the triple.The only possible exceptions are for very low-mass planets

and with the triple residing far outside the(m2[ 0.002 M_

)cluster core. These cannot be excluded but are certainly notfavored by the observations (see ° 2).

FIG. 10.ÈComparison of the various secular precession timescales inthe PSR B1620[26 triple, as a function of the mass of the second compan-ion in the standard solution of ° 2. See text for details.

The second problem is that other sources of pertur-bations may become signiÐcant over these long timescales.In particular, for an inner binary containing compactobjects, general relativistic e†ects can become important.This turns out to play a crucial role for the PSR B1620[26system and will be discussed in detail in the next section.

4.3. General Relativistic E†ectsAdditional perturbations that alter the longitude of

periastron can indirectly a†ect the evolution of the eccen-tricity of the inner binary in a hierarchical triple. Forexample, tidally or rotationally induced quadrupolar dis-tortions, as well as general relativity, can cause a signiÐcantprecession of the inner orbit for a sufficiently compactbinary. If this additional precession is much slower than theprecession owing to the secular perturbations, then theeccentricity oscillations are not signiÐcantly a†ected.However, if the additional precession is faster than thesecular perturbations, then eccentricity oscillations areseverely damped (Holman et al. 1997 ; Lin et al. 1999). Inaddition, if the two precession periods are comparable, thena type of resonance can occur that leads to a signiÐcantincrease in the eccentricity perturbation (Ford et al. 1999).

Figure 10 compares the various precession periods forPSR B1620[26 as a function of the second companionmass for the entire one-parameter family of standardm2solutions constructed in ° 2. Also shown for comparison isthe lifetime of the triple, both in the cluster core and at thehalf-mass radius. We assumed that the stellar densities inthe core and at the half-mass radius are 104 pc~3 andM

_102 pc~3, with average stellar masses SmT \ 0.8M_

M_and 0.3 respectively, and set the three-dimensionalM

_,

velocity dispersions to 6 km s~1 in the core and 3 km s~1 atthe half-mass radius. Disruption of the triple is assumed tooccur for any encounter with pericenter distance smallerthan (corresponding to setting in eq. [5]). Thea2 g

d\ 1


timescale between such encounters was then estimated fromsimple kinetic theory (see, e.g., eq. [8-123] of Binney & Tre-maine 1987), taking into account gravitational focusing(which is signiÐcant at the low-mass, short-period end of thesequence of solutions ; instead, our eq. [5] is valid in thelimit at which gravitational focusing is negligible). The pre-cession period for high inclinations was calculatedPHigh iusing the approximate analytic expression given byHolman et al. (1997, eq. [3]), while is from BrouwerPLow i& Clemence (1961 ; see Rasio 1995 for simpliÐed expressionsin various limits). The general relativistic precession period,

is derived, e.g., in Misner, Thorne, & Wheeler (1973,PGR° 40.5). Note that, although we have labeled the plotassuming only has an explicit dependencesin i2\ 1, PLow ion (and it is calculated here form2 sin i2\ 1).

For nearly all solutions we Ðnd that the general rela-tivistic precession is faster than the precession owing to theNewtonian secular perturbations. The only exceptions arefor low-mass second companions in low-(m2[ 0.005 M

_)

inclination orbits. However, we have already mentionedabove (° 4.1) that in this case the maximum induced eccen-tricity cannot reach the present observed value (for detailedcalculations, see Rasio 1994, 1995, and ° 4.4).

Most remarkably, we see immediately in Figure 10 thatfor the most probable solution (based on the current mea-sured value of f (5) and indicated by the vertical solid line inthe Ðgure), the two precession periods are nearly equal for alow-inclination system. This suggests that resonant e†ectsmay play an important role in this system, a possibility thatwe have explored in detail using numerical integrations. Wedescribe the numerical results in the next section.

4.4. Numerical Integrations of the SecularEvolution Equations

If the quadrupole approximation used for the high-inclination regime is extended to octupole order, theresulting secular perturbation equations approximate verywell the long-term dynamical evolution of hierarchicaltriple systems for a wide range of masses, eccentricities, andinclinations (Ford et al. 1999). We have used the octupole-level secular perturbation equations derived by Ford et al.(1999) to study the long-term eccentricity evolution of thePSR B1620[26 triple. We integrate the equations using thevariables ande1 sin u1, e1 cos u1, e2 sin u2, e2 cos u2,where and are the longitudes of pericenters. Thisu1 u2avoids numerical problems for near-circular orbits andallows us to incorporate easily the Ðrst-order post-Newtonian correction into our integrator, which is basedon the Burlish-Stoer integrator of Press et al. (1992).

We assume that the present inner eccentricity is dueentirely to the secular perturbations and that the initialeccentricity of the binary pulsar was much smaller than itspresent value. In addition, we restrict our attention to thestandard one-parameter family of solutions constructed in° 2. From the numerical integrations we can then determinethe maximum induced eccentricity, which depends only onthe relative inclination.

In Figure 11 we show this maximum induced eccentricityin the inner orbit as a function of the second companionmass for several inclinations. For most inclinations andmasses, we see that the maximum induced eccentricity issigniÐcantly smaller than that the observed value, asexpected from the discussion of ° 4.3. However, for a smallbut signiÐcant range of masses near the most probable

FIG. 11.ÈMaximum eccentricity of the binary pulsar induced bysecular perturbations in the triple, as a function of the mass of the secondcompanion in the standard solution of ° 2. See text for details.

value (approximately 0.0055 theM_

\ m2 \ 0.0065 M_

),induced eccentricity for low-inclination systems can reachvalues Z0.02.

If the initial eccentricity of the outer orbit had been largerthan presently observed (as calculated in Fig. 1) but laterdecreased through secular perturbations, the evolution ofthe inner orbit may have been di†erent. We have performednumerical integrations for this case as well and obtainedresults very similar to those of Figure 11. The ““ resonancepeaks ÏÏ are slightly broader to the right side and themaximum induced eccentricity is somewhat larger when thesecond companion mass is large. In particular, we Ðnd thatan inner eccentricity can then be achieved for massesZ0.02as high as However, from the results of ° 2m2^ 0.012 M

_.

we see that the corresponding orbital separation in the stan-dard solution is extremely large, AU, and thea2 Z 500outer eccentricity implying a very short lifetimee2 Z 0.95,for the triple in M4, yr.q

d\ 106

For relative inclinations we do not Ðnd a50¡ [ i [ 70¡peak in the maximum induced eccentricity as a function of

consistent with the timing data. For inclinationsm2 Z75¡we again Ðnd a peak in the maximum induced eccentricitythat becomes smaller and moves toward larger masses asthe relative inclination of the two orbits is increased. AsigniÐcant induced eccentricity in the inner orbit is alsopossible for if the two orbits are very nearlym2[ 0.004 M

_orthogonal (cf. ° 4.3). However, the results of our MonteCarlo simulations incorporating the measured value of f (5)do not support this scenario.

As already pointed out in ° 4.3, the maximum inducedeccentricity may also be limited by the lifetime of the triplesystem. From Figure 10 we see that near resonance weexpect yr, which is comparable to thePLowi

BPGRD 107lifetime of the triple in the cluster core, and much shorterthan the lifetime outside of the core. For solutions near aresonance the inner eccentricity initially grows linearly ate1approximately the same rate as it would without the general


relativistic perturbation. However, the period of the eccen-tricity oscillation can be many times the period of the clas-sical eccentricity oscillations. Although this allows theeccentricity to grow to a larger value, the timescale for thisgrowth can then be longer than the expected lifetime of thetriple in the cluster core. For example, with m2\ 0.006 M

_we Ðnd that the inner binary reaches an eccentricity of 0.025after about 1.5 times the expected lifetime of the triple in thecore of M4. Thus, even if the system is near resonance, itmust still be residing somewhat outside the core for thesecular eccentricity perturbation to have enough time togrow to the currently observed value.

5. SUMMARY

Our theoretical analysis of the latest timing data for PSRB1620[26 clearly conÐrms the triple nature of the system.Indeed, the values of all Ðve measured pulse frequencyderivatives are consistent with our basic interpretation of abinary pulsar perturbed by the gravitational inÑuence of amore distant object on a bound Keplerian orbit. The resultsof our Monte Carlo simulations based on the four well-measured frequency derivatives and preliminary measure-ments of short-term orbital perturbation e†ects in the tripleare consistent with the complete solution obtained when weinclude the preliminary measurement of the Ðfth frequencyderivative. This complete solution corresponds to a secondcompanion of mass in an orbit ofm2 sin i2^ 7 ] 10~3 M

_eccentricity and semimajor axis AUe2^ 0.45 a2^ 60(orbital period yr). Although the present formal 1P2 ^ 300p error on f (5) is large, we do not expect this solution tochange signiÐcantly as more timing data become available.

At least two formation scenarios are possible for thetriple system, both involving dynamical exchange inter-actions between binaries in the core of M4. In the one sce-nario that we have studied in detail, a preexisting binarymillisecond pulsar has a dynamical interaction with a widestar-planet system that leaves the planet bound to thebinary pulsar while the star is ejected. From numerical scat-tering experiments we Ðnd that the probability of retainingthe planet, although smaller than the probability ofretaining the star, is always signiÐcant, with a branchingratio ^10%È30% for encounters with pericenter distances

in the range where AU is the initialrp

0.2È1ap, a

pD 50

star-planet separation. All the observed parameters of thetriple system are consistent with such a formation scenario,which also allows the age of the millisecond pulsar (mostlikely to be much larger than the lifetime of theZ109yr)triple (as short as D107 yr if it resides in the core of thecluster).

It is also possible that the dynamical interaction thatformed the triple also perturbed the eccentricity of thebinary pulsar to the anomalously large value of 0.025

observed today. However, we have shown that through asubtle interaction between the general relativistic correc-tions to the binary pulsarÏs orbit and the Newtonian gravi-tational perturbation of the planet, this eccentricity couldalso have been induced by long-term secular perturbationsin the triple after its formation. The interaction arises fromthe near equality between the general relativistic precessionperiod of the inner orbit and the period of the Newtoniansecular perturbations for a low-inclination system. It allowsthe eccentricity to slowly build up to the presently observedvalue, on a timescale that can be comparable to the lifetimeof the triple.

All dynamical formation scenarios have to confront theproblem that the lifetimes of both the current triple and itsparent star-planet system are quite short, typically D107È108yr as they approach the cluster core, where the inter-action is most likely to occur. Therefore, the detection of aplanet in orbit around the PSR B1620[26 binary clearlysuggests that large numbers of these star-planet systemsmust exist in globular clusters, since most of them will bedestroyed before (or soon after) entering the core, and mostplanets will not be able to survive long in a wide orbitaround any millisecond pulsar system (where they maybecome detectable through high-precision pulsar timing).Although a star-planet separation AU may seema

pD 50

quite large when compared to the orbital radii of all recent-ly detected extrasolar planets (which are all smaller than afew AU; see Marcy & Butler 1998), one must rememberthat the current Doppler searches are most sensitive toplanets in short-period orbits and that they could neverhave detected a low-mass companion with an orbital period?10 yr. In addition, it remains of course possible that theparent system was a primordial binary star with a low-mass, brown dwarf component, and not a main-sequencestar with planets.

We are very grateful to Steve Thorsett for many usefuldiscussions and for communicating results of observationsin progress. We thank Piet Hut and Steve McMillan forproviding us with the latest version of the STARLAB soft-ware. We are also grateful to C. Bailyn and S. Sigurdssonfor helpful comments, and to J. Bostick for help with thecalculations of ° 3.2. This work was supported in part byNSF grant AST 96-18116 and a NASA ATP grant at MIT.F. A. R. was supported in part by an Alfred P. SloanResearch Fellowship. E. B. F. was supported in part bythe Orlo† UROP Fund and the UROP program at MIT.F. A. R. also thanks the Theory Division of the Harvard-Smithsonian Center for Astrophysics for hospitality. Thiswork was also supported by the National ComputationalScience Alliance under grant AST 97-0022N and utilized theSGI/Cray Origin2000 supercomputer at Boston University.

REFERENCES

Arzoumanian, Z., Joshi, K. J., Rasio, F. A., & Thorsett, S. E. 1996, in IAUColloq. 160, Pulsars : Problems and Progress, ed. S. Johnston et al. (ASPConf. Ser. 105 ; San Francisco : ASP), 525

Arzoumanian, Z., Nice, D. J., Taylor, J. H., & Thorsett, S. E. 1994, ApJ,422, 621

Backer, D. C. 1993, in ASP Conf. Ser., 36, Planets around Pulsars, ed. J. A.Phillips et al. (San Francisco : ASP), 11

Backer, D. C., Foster, R. S., & Sallmen, S. 1993, Nature, 365, 817Backer, D. C., & Thorsett, S. E. 1995, in ASP Conf. Ser. 72, Millisecond

Pulsars : A Decade of Surprise, ed. A. S. Fruchter, M. Tavani, D. C.Backer (San Francisco : ASP), 387

Bailyn, C. D., Rubinstein, E. P., Girard, T. M., Dinescu, D. I., Rasio, F. A.,& Yanny, B. 1994, ApJ, 433, L89

Binney, J., & Tremaine, S. 1987, Galactic Dynamics (Princeton : PrincetonUniv. Press)

Brouwer, D. & Clemence, G. M. 1961, Methods of Celestial Mechanics(New York : Academic)

Butler, R. P., Marcy, G. W., Fisher, D. A., Brown, T. W., Contos, A. R.,Korzennik, S. G., Nisenson, P., & Noyes, R. W. 1999, ApJ, 526, 916

Cudworth, K. M., & Hansen, R. B. 1993, AJ, 105, 168Dermott, S. F., & Nicholson, P. D. 1986, Nature, 319, 115Ford, E. B., Kozinsky, B., & Rasio, F. A. 1999, ApJ, submittedHeggie, D. C., Hut, P., & McMillan, S. L. 1996, ApJ, 467, 359Heggie, D. C., & Rasio, F. A. 1996, MNRAS, 282, 1064Holman, M., Touma, J., & Tremaine, S. 1997, Nature, 386, 254Joshi, K. J., & Rasio, F. A. 1997, ApJ, 479, 948 (erratum 488, 901)

350 FORD ET AL.

Kozai, Y. 1962, AJ, 67, 591Lin, D. N. C., Papaloizou, J. C. B., Bryden, G., Ida, S., & Terquem, C. 1999,

in Protostars and Planets IV, ed. A. Boss, V. Mannings, & S. Russell, inpress

Lyne, A. G., Biggs, J. D., Brinklow, A., Ashworth, M., & McKenna, J. 1988,Nature, 332, 45

Marcy, G. W., & Butler, R. P. 1998, ARA&A, 36, 57Mazeh, T., Krymolowski, Y., & Rosenfeld, G. 1996, ApJ, 466, 415McKenna, J., & Lyne, A. G. 1988, Nature, 336, 226 (erratum 336, 698)ÈÈÈ. 1996, ApJ, 467, 348Michel, F. C. 1994, ApJ, 432, 239Misner, C. W., Thorne, K. S., & Wheeler, J. A. 1973, Gravitation (New

York : Freeman)Peterson, R. C., Rees, R. F., & Cudworth, K. M. 1995, ApJ, 443, 124Phinney, E. S. 1992, Philos. Trans. R. Soc. London A, 341, 39ÈÈÈ. 1993, in ASP Conf. Ser. 50, Structure and Dynamics of Globular

Clusters, ed. S. G. Djorgovski & G. Meylan (San Francisco : ASP), 141Podsiadlowski, P. 1995, in ASP Conf. Ser. 72, Millisecond Pulsars : A

Decade of Surprise, ed. A. S. Fruchter, M. Tavani, & D. C. Backer (SanFrancisco : ASP), 411

Portegies Zwart, S. F., Hut, P., McMillan, S. L. W., & Verbunt, F. 1997,A&A, 328, 143

Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992,Numerical Recipes in C: The Art of ScientiÐc Computing (2d ed. ; NewYork : Cambridge Univ. Press)

Rasio, F. A. 1994, ApJ, 427, L107ÈÈÈ. 1995, in ASP Conf. Ser. 72, Millisecond Pulsars : A Decade of

Surprise, ed. A. S. Fruchter, M. Tavani, & D. C. Backer (San Francisco :ASP), 424

Rasio, F. A., Ford, E. B., & Kozinsky, B. 1997, AAS Meeting 191, 44.16Rasio, F. A., & Heggie, D. C. 1995, ApJ, 445, L133Rasio, F. A., McMillan, S., & Hut, P. 1995, ApJ, 438, L33Shearer, A., Butler, R., Redfern, R. M., Cullum, M., & Danks, A. C. 1996,

ApJ, 473, L115Sigurdsson, S. 1992, ApJ, 399, L95ÈÈÈ. 1993, ApJ, 415, L43ÈÈÈ. 1995, ApJ, 452, 323Stappers, B. W., et al. 1996, ApJ, 465, L119Thorsett, S. E., Arzoumanian, Z., Camilo, F., & Lyne, A. G. 1999, ApJ, 523,

763Thorsett, S. E., Arzoumanian, Z., & Taylor, J. H. 1993, ApJ, 412, L33Wolszczan, A. 1994, Science, 264, 538ÈÈÈ. 1996, in IAU Colloq. 160, Pulsars : Problems and Progress, ed.

S. Johnston et al. (ASP Conf. Ser. 105 ; San Francisco : ASP), 91

theoretical implications of the psr b1620 …faculty.wcas.northwestern.edu/rasio/papers/56.pdfm...

Documents