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Implications for Planet Formation in Globular Clusters 3 Has Significant Rotation (See Pringle 1989)

Implications for Planet Formation in Globular Clusters 3 Has Significant Rotation (See Pringle 1989)

arXiv:astro-ph/0409762v1 30 Sep 2004 o.Nt .Ato.Soc. Astron. R. Not. Mon. .E Beer E. M. in formation for clusters implications globular M4: in planet The cetd20 e 5 eevd20 u 8 noiia om2 form original in 18; Aug 2004 Received 15. Sep 2004 Accepted ⋆ keeps re- kick have small only This can formation. binary during kick inner argues small the 5 a in Section ceived . neutron forming the and encou that star disc the passing cir- of a likelihood a tering inner the how the outflows discusses of 4 equatorial suggest evolution Section of binary. of we result phase 3 common-envelope a the Section as during form In may formed. disc mod- it cumbinary proposed how the as for well els as B1620–26 PSR orbiting planet pass- a with interaction a an of in star. result instability ing a that gravitational as propose a disc We through circumbinary 1999). formed al. bi- planet a et inner this with Thorsett the 1993; as (Backer non-coplanar B1620–26) with nary wide (PSR triple a sys- hierarchical in millisecond a planetary binary planet is known This one star Jupiter cluster. any is a globular in a There in than planet. tem lower known far a are with where cluster, would view. system this to low challenge a a in Convinc offer planet 2004). clearly a al. for et core evidence Beer metallic see a ing build (but to further protoplanet accreting a for before need interpretatio the & potential of Valenti terms appealing Fischer, in an 2003; has al. This et 2004). Santos & Marcy 2002; Wallerstein Reid Gonzalez, 1999; 1998; around Saar absent (Gonzalez or metallicity rare low be of to appear planets Extrasolar INTRODUCTION 1

c .Isiueo srnm,MdnlyR,CmrdeC3OHA, CB3 Cambridge Rd, Madingley , Leiceste of of Institute University 2. Group, Astrophysics Theoretical 1. -al [email protected] E-mail: 04RAS 2004 nScin2w umrs h nw rpriso the of properties known the summarise we 2 Section In globular a is challenge a such for arena potential One 1 ⋆ .R King R. A. , 000 – 20)Pitd6Nvme 08(NL (MN 2018 November 6 Printed (2004) 1–8 , neoepaeo h ne iayseouin hsfrainroute words: formation clusters. Key This globular evolution. as such binary’s th systems du inner disc that stellar circumbinary the dense a propose of with We Consequently star phase unlikely. passing applied. envelope a be are of interaction not formation the should through for cores methods rocky env metallic suggested metallicity stars onto very-low main-sequence a gas around is formation of M4 planet as of that model PS propose standard system We planetary M4. the of cluster evolution globular and formation the consider We ABSTRACT ic usr:idvda:PRB1620–26 PSR individual: : - discs 1 n .E Pringle E. J. and lnt n aelts omto lntr ytm:protoplane systems: planetary - formation satellites: and planets ,Lietr E R,UK 7RH, LE1 Leicester, r, n- n - 0 ue9 June 004 UK g n aso h ht wr r 4.8 are The dwarf undermassive. white and the young of is mass dwarf and ind age white (2003) the al. 0.025315 that and et cate d Sigurdsson 191.4 11 pe- by are The Observations an binary respectively. this companion. by of of dwarf eccentricity consists found white and binary been riod a inner with has The pulsar glob- solution (1999). millisecond the al. orbital in et The system Thorsett M4. triple hierarchical cluster a ular is B1620–26 FOR PSR IDEAS PREVIOUS AND SYSTEM THE 2 In it. conclusions to our plane summarising to bound as application well planet its as and general the in result formation this and discuss cluster, we 6 the Section in binary the n 0.34 and hs ri a eido re 0 r(hrete al. et (Thorsett yr 100 order of period a has orbit whose relationshi this disrupt not should for system significantly. model Any the mass. pe- of binary (white-dwarf) origin the core the of hence function and Stel- a model, as this periods. riod in lobe millisecond which Roche radius, to its envelope filling the up transferred constrains was it theory envelope evolution spinning white whose lar al. pulsar, giant the a the et assuming of to core Rappaport from the arise and is mass- dwarf predictions (1987) the Savonije These for (1995). of predictions relation theoretical period the with agreement sue ob .5M 1.35 be to assumed 2 T ORIGIN ITS h hr ebro hstil saJptrms planet Jupiter-mass a is triple this of member third The good in are period binary and mass white-dwarf The ± .4M 0.04 A T E tl l v2.2) file style X ⊙ epciey h aso h usris pulsar the of mass The respectively. ⊙ . hog h accretion the through 12–6i the in B1620–26 R × igtecommon- the ring lntformed planet e 10 sfvue by favoured is h previously the 8 rnetthe ironment ± 1.4 × 10 tary 8 yr is i- p t . 2 M.E. Beer, A.R. King and J.E. Pringle

1999). This planet was detected through the observed sec- A fundamental problem with both of these models, how- ond period derivative of the pulsar. The published timing ever, is that they require the planet to have already formed has representative solutions for different ellipticities of the in the which is a low metallicity environ- outer orbit. For an eccentricity of 0.20 the period would be ment. M4 has an [Fe/H] of −1.20 (Harris 1996) while the 129 yr and the minimum companion mass would be 3.5 MJ. current lowest known metallicity of a star If the eccentricity were as large as 0.5 the period would be with a planetary companion is HD 6434 which has an [Fe/H] 389 yr and the minimum companion mass would be 6.8 MJ. of −0.52 ± 0.08 (Santos et al. 2003). Fischer et al. (2004) Sigurdsson et al. (2003; 2004) argue that the planet is at have hypothesised that there is a metallicity dependence the lower end of the allowed ranges for mass and semi-major on planet formation. They find that at solar metallicity be- axis. This implies a minimum planet mass of 1.7 MJ, a pe- tween 5–10% of stars host Doppler-detected planets. As stel- riod of 68 yr and an eccentricity of 0.13. lar metallicity drops toward [Fe/H] of −0.5, however, the Phinney (1992) suggested a relationship between eccen- occurrence of detected planets declines to a few percent. tricity and orbital period for binary millisecond pulsars (see also Phinney & Kulkarni 1994). Phinney (1992) argues that this relationship is caused by the non-central (multipole) 3 THE PROPOSED MODEL forces exerted on the due to the fluctuating density of convection cells in the red-giant progenitor of the Figure 1 shows a schematic view of how the system formed . The longer the period and hence the larger the in the proposed model. During the formation of the in- size of the red-giant progenitor the greater the induced ec- ner binary the progenitor of the neutron star underwent centricity. This model fits the observed eccentricity orbital a common-envelope phase (see Bhattacharya & van den period distribution of binary millisecond pulsars well. From Heuvel 1991 for a discussion of likely evolutionary scenar- − this model we would expect an eccentricity of order 10 3 ios). A typical scenario consists of a binary with a high- for the inner binary significantly smaller than the observed mass component (e.g., 15 M⊙) and a low-mass component eccentricity of 0.025. Ford et al. (2000) have proposed that (1 M⊙). At the end of core helium burning of the primary Kozai pumping by the planetary companion could explain its envelope expands until it fills its Roche lobe and enters a the observed eccentricity. For such a large eccentricity Kozai common-envelope phase. During this phase the envelope of (1962) requires that the difference in inclination between the the neutron star progenitor is ejected while the white dwarf ◦ inner binary orbit and the planetary orbit is at least 40 . progenitor spirals in. This leaves the core of the massive Sigurdsson (1992, 1993, 1995 hereafter S93) has pro- star which is a helium star and the low-mass companion. posed a model for the origin of the current configuration. The helium star eventually undergoes a leaving In this model the planet forms around a main sequence star a neutron star possibly in an eccentric orbit (due to the outside the core of M4. The migrates to- supernova kick). This orbit circularizes due to tides and if wards the core of M4 where it encounters a neutron-star the companion fills its Roche lobe on the branch binary. The neutron star captures the star and planet and then mass transfer occurs. The red-giant envelope is trans- ejects its original companion. The system is now a hierar- ferred onto the neutron star spinning it up to millisecond chical triple with an inner neutron-star plus main-sequence periods while the degenerate helium core of the red-giant binary and an outer planet. The main sequence star evolves remains as the white-dwarf companion. The exact details into a red giant and transfers mass to the neutron star. This of the common-envelope phase and associated mass loss are spins it up to millisecond periods, eventually leaving a mil- not well understood although it is generally agreed that this lisecond pulsar plus white dwarf binary. In this model, dur- phase is short. In the scenario outlined above the mass loss 3 4 ing the encounter which forms the hierarchical triple, the is of order 10 M⊙ and occurs on a timescale of 10 –10 yr system has a recoil which displaces the binary from the core (van den Heuvel 1976). of the cluster. Sigurdsson et al. (2003) state that a binary Possible evidence for the post-common-envelope struc- sinks into the cluster core on the relaxation timescale (of or- ture comes from observations of planetary nebulae and pre- 1 der a Gyr). This is correct but does not give the full picture. planetary nebulae . In order to explain the shape of many As (in the proposed model) the system was initially in the planetary nebulae a slow dense wind confined to a plane core, and has undergone no other exchanges, its orbit takes it is required prior to a final fast spherically symmetric wind back into the core on the crossing timescale (around a Myr) which interacts with it (see the review of Balick & Frank which is much shorter than both the relaxation timescale 2002). This would form the observed bipolar outflows ob- and the age of the white dwarf. For this model to be correct served in many planetary nebulae (Gieseking, Becker & Solf the system must have formed very recently in order to avoid 1985; Miranda & Solf 1992; L´opez, Steffen & Meaburn 1997). encounters which unbind the planet (see section 4). This slow, dense wind has been directly observed in some Alternatively, Joshi & Rasio (1997) proposed that a pre-planetary nebulae through HCN emission e.g., in the planetary system containing a main sequence star encoun- ‘Egg ’ (Bieging & Nguyen-Quang-Rieu 1988) and in ters a preexisting binary . During this en- the ‘Westbrook nebula’ (S´anchez Contreras & Sahai 2004). counter the main sequence star is ejected, although as the In the ‘Egg nebula’ there is evidence that this slow wind authors note, one would expect the the main sequence star (as the more massive object) to be retained rather than the 1 Pre-planetary nebulae are also referred to as proto-planetary planet. This model can successfully explain the observed in- nebulae, but we do not use the phrase here to avoid confusion ner binary eccentricity without the requirement for Kozai with the use of proto-planetary nebulae as being centrifugally pumping which requires a high relative inclination of the supported discs of accreting matter around stars in which planets two . may form.

c 2004 RAS, MNRAS 000, 1–8 The planet in M4: implications for planet formation in globular clusters 3 has significant rotation (see Pringle 1989). The ‘Westbrook High mass binary system Timescale nebula’ (also known as CRL 618) has been interferometri- cally mapped by S´anchez Contreras & Sahai (2004). They inferred a dense, equatorial torus expanding at 17.5 kms−1. These observations could be of the excited part of a much MS MS slower (or even non-moving) torus. This torus extends out 15 Myr to 5.5 arcsec which at their adopted distance of 900 pc corre- phase sponds to an outer radius of 3300 AU. They estimate a torus mass of 0.25 M⊙ (260 MJ). The most plausible methods for the formation of this He dense, confined gas are either the expulsion of the wind of a MS red giant by a binary companion in the orbital plane (Mor- 3 4 10 − 10 yr ris 1981) or the emergence from a common-envelope phase (Soker & Livio 1991; Reyes-Ruiz & L´opez 1999). There is a strong resemblance between these winds and proto- Slow dense equatorial wind planetary discs (Pringle 1989; Kastner & Weintraub 1995). There is no reason why a dense, slow, equatorial wind in the progenitor of a binary millisecond pulsar would not be produced during/after the common-envelope phase. Two- and three-dimensional simulations of common-envelope evo- Encounter with passing star lution show that the envelope mass is lost preferentially in directions near the orbital plane (Yorke, Bodenheimer & Taam 1995; Sandquist et al. 1998). MS Mastrodemos & Morris (1998; 1999) performed three- Planet forms through dimensional hydrodynamical modelling of gas flows in bipo- gravitational instability lar nebulae including the formation of accretion discs. We NS use the term accretion discs loosely here as the matter in MS them is not centrifugally supported. In fact the discs con- tain significant rotation but are not strongly unbound. The 10 accretion discs in their simulations extended out to between 10 yr 100 and 1000 AU and had terminal velocities in the range − Planet 10–27 kms 1. If the general wind loss rate in these systems − − is 10 4 M⊙yr 1 (a typical rate for planetary nebulae) then −1 a 10 kms wind would have 5 MJ within 100 AU. Simula- tions by Sandquist et al. (1998) found similar results with a NS disc extending out to 100 AU and a mass > 1 M⊙ although RG not all the mass in their simulations was unbound. Such Inner binary evolves − recycling discs have enough mass at large enough radii to form the neutron star to a millisecond pulsar planet around PSR B1620–26. This planet has a mass of a few Jupiter and a periastron distance of ∼30 AU 8 5 x 10 yr (Thorsett et al. 1999). Planet formation in a post-common-envelope disc could proceed by the standard model for the formation of the giant planets (Wetherill 1980; Mizuno 1980; Steven- son 1982) if the disc is metal rich. This standard model Planet assumes that planets form initially through the agglomera- tion of dust into grains, pebbles, rocks and thence planetes- Figure 1. A figure showing the stages during the formation of imals which form the planetary cores. This formation route, the hierarchical triple pulsar planetary system and their associ- however, would not provide the large relative inclination of ated timescales. Initially a high mass X-ray binary system con- the two orbits necessary for the to pro- sisting of e.g. a 15 M⊙ and a 1 M⊙ star evolves on the timescale duce the high eccentricity of the inner binary. M4 is a low- of the most massive component. Once the high-mass component metallicity cluster but the environment around the primary evolves off the main sequence mass transfer becomes unstable and a common-envelope phase ensues. The common-envelope phase which loses its envelope during the asymptotic-giant-branch lasts 103–104 yr. During the common envelope phase a slow, dense primary could be dust rich (Livio & Pringle 2003). If the equatorial wind is formed. A passing star encounters this wind planet is to form bound from the initially unbound matter and the planet forms through a dynamical instability of the disc. in this disc/wind phase an interaction with a passing star is The system now consists of the progenitor of the binary mil- required (see below). lisecond pulsar and the planet. The system undergoes no fur- Gravitational instability in the disc (Boss 2001; Rice et ther significant encounters on a timescale of the age of the clus- 10 al. 2003; Mayer et al. 2004) provides another possible for- ter 10 yrs. Eventually the inner binary evolves spinning up the mation mechanism. This instability requires a change in disc neutron star to a millisecond pulsar and forming the young white properties on a timescale comparable to or less than the dy- dwarf observed today. namical timescale of the disc. Interaction with a passing star

c 2004 RAS, MNRAS 000, 1–8 4 M.E. Beer, A.R. King and J.E. Pringle would provide this and is likely in a globular cluster envi- ronment. Any encounter would probably be non-coplanar and so we expect the resulting planets to be non-coplanar as well. Detailed numerical simulations are required, how- ever, to verify if this formation route could indeed produce the planet around PSR B1620–26. Gravitational instabilities typically generate many planetary mass objects (Rice et al. 2003). Some of these may escape from the system as a result of the encounter while others in the case of PSR B1620–26 may not yet be detectable. Hall, Clarke & Pringle (1996) have investigated the re- sponse of a circumstellar accretion disc to the fly-by of a perturbing star on a parabolic orbit. They considered a per- turbing star of equal mass to the central star and a disc of non-interacting particles i.e. they did not include hy- drodynamical forces in the disc. Their analysis investigated which regions of the disc remain bound after an encounter in addition to whether energy and is gained/lost from the perturbing star. They found that for prograde fly-bys matter outside of periastron is lost from the system (i.e. made unbound) while matter within 60 per cent of the periastron distance is not only retained but because of a corotation resonance loses energy and angular momentum Figure 2. A figure showing a King model for the density profile to the binary orbit.2 In retrograde fly-bys there is no coro- of the globular cluster M4. The long-dashed line shows the mean tation resonance and the binary orbit loses energy to the density inside the half-mass radius and the short-dashed lines r disc principally in the sense of matter outside of periastron the positions and densities at the core radius ( c) and half-mass r becoming unbound from the system. radius ( h). Boffin et al. (1998) have simulated star-disc encounters using a smoothed particle hydrodynamical approach. They find that the formation of spiral arms and fragmentation oc- 4 DYNAMICAL CONSIDERATIONS cur in the circumstellar disc. The spiral arms have density contrasts of greater than two orders of compared In this section we discuss the possible dynamical history of to the initial disc and it is in these that fragmentation oc- PSR B1620–26. The aim of this section is to show that the curs. Pfalzner (2003) has also simulated star-disc encounters system could be outside the core of M4 and that it would and finds formation of spiral arms in the disc. Both of these not have sunk into the core of the core within a Hubble time. simulations had a limited number of particles (11,300 and We start by considering the possible position of the system 50,000 particles respectively) and higher resolution simula- in the cluster with respect to its observed position which tions of star-disc encounters need to be carried out to in- is within the core. We then discuss the relaxation time for vestigate the possible fragmentation of the disc due to an an object in M4 to sink into the core using a King profile encounter further. for the radial number density. We also discuss the encounter In non-coplanar encounters the passing star imparts a timescale for the system and the effects encounters have on to the accretion disc in its vertical direction the planetary orbit which may initially have been tighter (Heller 1993). This tilts the accretion disc compared to its than currently observed. unperturbed orientation. If the disc were to fragment due to this encounter then the planets formed would have inclined orbits relative to the initial disc. 4.1 Relaxation into cluster core In the model it is easy to envisage a prograde encounter The observed offset of PSR B1620–26 from the core of M4 with a periastron greater than the the disc radius in which (0.767 arcmin) is less than the observed core radius (0.83 ar- the planet forms (of order 100 AU) shocking material inside cmin) and considerably less than the half mass radius (3.65 of this radius and extracting energy and angular momentum arcmin). This could simply be a projection effect however, from this material making it bound. It is in this region that with the system actually far outside the core. This is re- the observed planet can form. Detailed simulations are re- quired in both the model described here and that in S93. quired to verify this scenario but those already performed This would not be unprecedented for globular cluster pul- demonstrate the overdensities which may be induced in addi- sars e.g. pulsars A and C in NGC 6752, PSR J1748–2444 in tion to the extraction of energy from the disc matter. Indeed Terzan 5 and PSR B1718–19 in NGC 6342 all lie outside the the initial binary is so massive that the primary evolves be- half mass radius (see Freire 2004 for a list of the observed fore the cluster is fully formed. The encounter which forms properties of globular cluster pulsars). The chance of the the planet could occur during this phase. system having a projected distance (rapp) less than the core radius (rc) when it is actually at the half-mass radius (rh) is given by 2 Here binary orbit refers to the orbit of the two stars which is initially parabolic i.e. has an eccentricity of 1. f(rapp

c 2004 RAS, MNRAS 000, 1–8 The planet in M4: implications for planet formation in globular clusters 5

Parameter Value Reference

Distance 2.2 kpc Harris (1996) ′ Core radius (rc) 0.83 Trager, Dvorgovski & King (1993) ′ Half-mass radius (rh) 3.65 Harris (1996) ′ Tidal radius (rt) 32.49 Harris (1996) (MV) −7.20 Harris (1996) Velocity dispersion (σobs) 3.88 ±0.64 Peterson & Latham (1986) 4.44 ±0.71 Rastorguev & Samus (1991) 3.50 ±0.2 Peterson, Rees & Cudworth (1995) Concentration (c) 1.59 Trager, Dvorgovski & King (1993) Metallicity [Fe/H] −1.20 Harris (1996)

Table 1. A table listing the properties of the globular cluster M4.

which for M4 is 8 per cent. clusters it is possible that more have undergone a similar for- The relaxation time for an object to enter the core of the mation route and have planetary mass companions in wide globular cluster is inversely proportional to the stellar den- non-coplanar orbits. In low density clusters like M4 these sity (see Djorgovski 1993). The half-mass relaxation time for planets may survive without being disrupted by stellar en- M4 is of order 1 Gyr (Harris 1996). However, this assumes a counters (Davies & Sigurdsson 2001). stellar density which is equivalent to the mean stellar density We now consider the timescale for the system (including inside the half-mass radius. King (1962) finds the density the planet) to encounter another star and what the outcome profile of a globular cluster to be of this encounter will be. Davies & Sigurdsson (2001) give 1 arccos z the timescale for an encounter between a binary and a single ρ(r) ∝ − 1 − z2 , (2) star as z2 z h p i 1 where τenc = , (4) nσv 2 1+(r/rc) where n is the number density of stars, v is the velocity of z = 2 , (3) r 1+(rt/rc) the binary (which we take to be the velocity dispersion of M4) and σ is the cross-section for two stars having a relative and rt is the tidal radius of the globular cluster. Figure 2 shows the density profile of the globular cluster M4 out to 4 velocity at infinity of v∞ to pass within a distance bmin of arcmin. Also shown on figure 2 are the mean number den- one another sity inside of the half-mass radius (calculated below) and 2 2 G M σ = πbmin 1+ , (5) the densities at the core and half mass radius respectively.  bmin v∞  Outside the half-mass radius the stellar density is much less where M is the total mass of all the components and the sec- than the mean density inside the half-mass radius (at least ond term on the right is due to gravitational focusing and an order of magnitude) and so the relaxation time is much dominates in the low dispersion environment of a globular greater. Indeed a system outside the half-mass radius would cluster. We assume that v∞ is similar to the velocity disper- not necessarily enter inside the half-mass radius within a sion (see Section 5). The encounter timescale may now be Hubble time. We conclude that if the system was formed written outside the half-mass radius of the cluster then the system 5 −3 M ∞ would still be outside this radius and would not have relaxed × 8 10 pc AU ⊙ v τenc = 7 10 −1 yr . (6) into the cluster core. n bmin M 10 kms We are interested in encounters within 100 AU. These can affect the planet without perturbing the inner binary. Fol- 4.2 Stellar encounters lowing Djorgovski (1993) and Harris (1996) the mass of the cluster (Mcl) is given by During the initial formation stages of a globular cluster there occurs a phase of residual gas expulsion which occurs on a log (Mcl/2) = 0.4 (4.79 − MV) (7) timescale of 107 yrs (Meylan & Heggie 1997). Irregular mass loss may cause a phase of dynamical mixing - the violent where MV is the cluster absolute magnitude and a mass to relaxation stage (Lynden-Bell 1962; 1967). This violent re- light ratio of 2 has been assumed. Table 1 contains a list of laxation occurs on a timescale of a few crossing times for the properties of the cluster M4. The absolute magnitude is 5 the globular cluster. The crossing time of the cluster is of −7.2 which corresponds to a cluster mass of 1.25 × 10 M⊙. order 106 yrs so the violent relaxation timescale is compa- The mean number density (N) inside the half-mass radius rable to the 15 Myr timescale on which the primary evolves is given by and the planet formed in the model described here. There- 3 Mcl fore the encounter required to form the planet in the model N = (8) 4π mr3 may have occured during this violent relaxation stage (an h encounter within ∼100 AU is all that is required). Indeed where m is the mean (taken to be 1/3 M⊙ fol- given the number of binary millisecond pulsars in globular lowing Harris 1996) and rh is the half mass radius which is

c 2004 RAS, MNRAS 000, 1–8 6 M.E. Beer, A.R. King and J.E. Pringle

2.34 pc using the distance to M4 of 2.2 kpc. This gives a mean accretion onto the neutron star to spin it up to millisecond − number density inside the half-mass radius of 8×103 pc 3. periods is transient i.e. long quiescent periods during which If the number density in the region of M4 where the no accretion occurs and short outburst periods during which − system currently is of order 103 pc 3 then equation (6) gives accretion occurs but the accretion rate is super-Eddington the encounter timescale as 150 Myr. The exact distance of (see Taam, King & Ritter 2000). During these outbursts of the system from the core of M4 is unknown so the timescale super-Eddington accretion most of the transferred matter is may be even larger than this. Indeed the initial orbit of the expelled from the system while a small amount is accreted planet once formed may have been tighter than that cur- by the neutron star spinning it up to millisecond periods. rently observed (because of the softening effects of encoun- The ejection of matter from the inner binary results in the ters - see below). Consequently the encounter timescale (for planet moving further out. The change in semi-major axis an encounter of closest approach comparable to the plane- of the planet due to mass lost is given by tary orbit) may have been larger as well. If the system were Mi at the core radius then the timescale for an encounter would a = ai , (10) M be much less (4.5 Myr) as the number density is higher. This demonstrates that the planet is far more likely to survive if where a and M are the semi-major axis of the planet and it is outside of the core of M4. total mass of the triple and the subscript i refers to their During encounters there is a tendency to equipartition respective values prior to mass transfer and spin-up of the energy between the total kinetic energy of the objects in- neutron star. If half a was ejected from the system volved and the internal energy of the system (Heggie & Hut to give the pulsar (assumed mass 1.35 M⊙) and the 0.34 M⊙ 2003). Hard systems have greater internal energy so encoun- white dwarf then the semi-major axis increased by 30 %. ters result in a tighter system and greater kinetic energy. Soft Although this only represents a small factor in encounter systems have greater kinetic energy which results in a widen- probability it means that the encounter timescale was longer ing of the system. So during encounters, hard systems get until recently (as the white dwarf is young) which may help harder and soft systems get softer. The boundary between explain its survival. these two regimes is known as the hard/soft boundary (Rhs) and is given by (Bonnell et al. 2001) 4.3 How radial is the orbit?

GM1M2(M1 + M2 + M3) Rhs = 2 , (9) In the model described above the system formed outside the M3(M1 + M2)venc half-mass radius of the cluster and has an orbit which does where M1, M2 and M3 are the masses of the inner binary, not take it into the core i.e. the orbit does not have a strong planet and encountered star respectively and venc is the en- radial component. This is a prediction of the model and may counter velocity which may be taken to be the same as the be testable in future. velocity dispersion for the cluster. This gives the hard/soft In the model described in S93 the system has been boundary for the planet as a separation of under 1 AU. The ejected from the core of the cluster due to an exchange en- planet in this system is consequently always softly bound counter. From here the planet describes an orbit which takes and encounters/flybys on average lead to it becoming less it out of the core and back in again. The period of this orbit strongly bound. The outcomes of stellar encounters has been is similar to a crossing time (of order a Myr). It could be deeply investigated in the case of equal mass components argued that although the system’s orbit takes it back into (Hut 1984; Heggie & Hut 1993), but only a few investiga- the core on a Myr timescale the system does not spend much tions have been carried out for planetary systems (Bonnell time in the core as it passes through quickly. Consequently et al. 2001; Davies & Sigurdsson 2001; Woolfson 2004). it is both more likely to be found outside the core and less In their calculations Bonnell et al. (2001) assumed that likely to have undergone an encounter which unbinds the if a planet was soft it became unbound during an encounter. planet from its orbit. This is not necessarily the case as Woolfson (2004) has shown If the system had a radial orbit, although it is more when he considered the survivability of planets in wide or- likely to be found outside the core, we should still expect bits. Woolfson (2004) considered an (venc ∼ it to have undergone a number of encounters in the core by − 1kms 1) and integrated for 107 yrs. For planets with a of now. This can be seen by the following simple argument. order 100 AU he found a large survival probability even for Assuming the orbit is entirely radial, in a medium with a encounters with periastron distances of a similar size. This density equal to the mean density inside the half-mass ra- is because the outcome of an encounter depends on the po- dius, and that it orbits between the core and the half-mass sition of the planet in its orbit and the relative orientation radius, the equation of motion is of the planetary orbit and the path of the encountered star. 2 d r GMclm Woolfson (2004) found that encounters tendered to soften = − r , (11) dt2 2 r3 rather than harden orbits as expected but with little change h in energy. Consequently it is possible for the planet to un- and the system is a simple harmonic oscillator. It is simple to dergo a number of encounters without unbinding it from the show that the system spends a fraction arcsin(rc/rh) equal system. More numerical simulations are required, however, to 13 per cent of its orbit inside the core radius. Using the to verify this scenario before we can authoritatively say what age of the white dwarf as a lower limit to the interaction the timescale for disruption of the hierarchical triple is. timescale this corresponds to 62.4 Myr. As we show above it In the scenario described above after the planet has is probable that the system would have undergone a number formed the inner binary evolves to form the binary millisec- of encounters in the core of the cluster within this timescale. ond pulsar observed today. As this binary is wide (191 d) the The main motivation for this paper, however, remains

c 2004 RAS, MNRAS 000, 1–8 The planet in M4: implications for planet formation in globular clusters 7 how the planet could have formed in a low metallicity envi- orbit as that in PSR B1620–26 (which has a periastron dis- ronment and not whether the model proposed by S93 would tance of 30 AU), or as heavy a mass (all the planets orbiting have been disrupted in the core. PSR B1257+12 have Earth masses). To conclude, the globular cluster M4 is a very-low metallicity environment and so the standard model of planet formation should not be applied to the observed planetary 5 A LOW-KICK VELOCITY FOR THE system in it. Consequently the previously suggested meth- NEUTRON STAR ods for formation are unlikely. The Jupiter mass planet ob- served in M4 is a member of a hierarchical triple. When Before accepting the picture sketched above, we should the inner binary underwent a common-envelope phase a cir- check that the system would not be unbound by the na- cumbinary disc formed as a slow, dense, equatorial wind. tal supernova kick felt by the neutron star. Gnedin et al. This disc extended beyond 100 AU and we propose it un- (2002) find that the relation between escape velocity (vesc), derwent an encounter with a passing star. This encounter velocity dispersion (σobs) and concentration (c) of a globular caused a gravitational instability in the disc and consequent cluster is given by planet formation. This planet was formed in a wide, eccen- vesc tric, non-coplanar orbit around the inner binary. No other = 3.7 + 0.9(c − 1.4) . (12) σobs encounters/exchanges are required to explain the formation of this system. The data shown in Table 1 then give the mean velocity dis- − − persion of 4 kms 1 and an escape velocity of 15.25 kms 1 from the core of M4. Any kick the neutron star received must have been less than this or the binary would have es- ACKNOWLEDGEMENTS caped from the cluster. Consequently the neutron star kick must have been small. Pfahl et al. (2002) have argued that We thank the anonymous referee for comments which have there is a population of neutron-star binaries which receive improved this paper. Theoretical astrophysics research at low kicks. The neutron star in PSR B1620–26 may either Leicester is supported by a PPARC rolling grant. ARK have received one of these low kicks or could be at the low- gratefully acknowledges a Royal Society Wolfson Research velocity end of the high-kick velocity distribution. Merit Award. MEB acknowledges the support of a UKAFF The planet is not strongly bound at formation, so we fellowship. might expect that a neutron-star kick would unbind it. The neutron-star kick, however, may be towards the planet rather than away from it, making the planet more strongly REFERENCES bound. 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