The Hill stability of the possible moons of extrasolar planets

Transcription

1 Mon. Not. R. Astron. Soc. 406, ) doi:0./j x The Hill stability of the possible moons of extrasolar planets J. R. Donnison Astronomy Unit, School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E 4NS Accepted 00 April 5. Received 00 March ; in original form 00 February 8 ABSTRACT The dynamical Hill stability has been derived for a full three-body system composed of a binary moving on an inclined elliptical orbit relative to a third body where the binary mass is very small compared with the mass of the third body. This physical situation arises in a number of important astronomical contexts including extrasolar planetary systems with a star planet moon configuration. The Hill stability criterion against disruption and component exchange was applied to all the known extrasolar planetary systems and the critical separation of a possible moon from the planet determined for moon/planet mass ratios of 0., 0.0 and 0.00 assuming that the moon moves on a circular orbit. It is clear that in those cases where the planet moves on a circular orbit about the central star, the critical separation of the moon from the planet does not change significantly as the value of the moon/planet mass ratio is reduced. In contrast, for eccentric systems there can be big changes in the critical separation as the mass ratio decreases. The variation in size depends crucially on the size of the eccentricity of the planetary orbit. To determine the effect of an eccentrically orbiting moon, the Hill stability criterion was applied generally to the planet moon binary for a range of moon/planet mass ratios assuming that the planet moved on a circular orbit around the central star. It was found that in all cases the critical distance ratio increased, and hence the regions of Hill stability decreased as the binary eccentricity increased and also as the inclination of the third body to the binary was increased. The stability increased slightly as the moon/planet ratio was decreased. Also as the binary/third body mass ratio decreased the effects of the moon/planet mass ratio became less important and the stability curves tended to merge. These types of changes make exchange or disruption of the component masses more likely. Key words: celestial mechanics planets and satellites: dynamical evolution and stability. INTRODUCTION The first extrasolar planetary companion was discovered orbiting a solar-mass star 5 Pegasi by Mayor & Queloz 995). Currently, there are 44 such planets in 58 systems which have been detected by various techniques. These systems are listed at the web site Since all the giant planets in our Solar system have extensive satellite systems, it would seem likely that the extrasolar planets should also possess satellites. Detection using changes in the radial velocity of the stellar component of the planetary system which has been so important for the detection of extrasolar planets cannot be used in this case as the shifted Doppler spectrum due to the presence of a planet plus satellites would be identical to a single point mass moving with the same total mass in orbit around the host star. Several techniques have, however, been proposed for the detection of extrasolar moons. In particular, the possible detection by transits has been advocated Sartoretti & Schneider 999), with both the planet and the moon or moons producing small dips in the light received by an observer. Also, transit timing effects can produce unique moon signatures Kipping 009; Kipping, Fossey & Campanella 009) and it has been suggested that NASA s Kepler mission should be able to detect large moons around extrasolar planets. Brown et al. 00), based on the Hubble Space Telescope transit light curve of the star HD 09458, was able to put upper limits of. Earth radii and M on any satellite of the transiting planet HD 09458b. The possibility of detecting companions by microlensing has also been discussed Bennett & Rhie 00; Han & Han 00). It has also been proposed that Earth-like satellites of giant planets in the habitable zone could be detected by spectroscopy using a methane gap in the planetary spectrum Williams & Knacke 004). In addition, Lewis, Sackett & Mardling 008) have Downloaded from by guest on 5 September 08 r.donnison@qmul.ac.uk C 00 The Authors. Journal compilation C 00 RAS

2 Stability of extrasolar moons 99 proposed using pulsar timing to detect the possible moons of pulsar planets. They applied their method to the pulsar planet PSR B60-6b and found that a stable moon could be detected orbiting the planet if the moon planet separation is about /50th of the semimajor axis of the planet relative to the pulsar, the moon being at least /0th of the mass of the planet. So far, no solar planetary moons have yet been detected by any of the methods mentioned. In the Solar system, the satellites of the giant planets fall into two distinct types. There are the regular satellites which are close to the planet and have near planar and circular orbits, while the irregular satellites are smaller bodies moving at large distances from the planet on high eccentricity, high inclination orbits and are often retrograde in motion. While the irregular satellites, because of their orbits, were most likely to have been captured at a much later stage in the evolution of the planet. Whether such formation scenarios are applicable to extrasolar planets, where significantly large numbers of the planets are within 0. au of the parent star and may have been subject to migration, is not clear. An important aspect of these possible extrasolar moons is the stability of their orbits. The tidal stability of the close-in satellites has been investigated by Barnes & O Brien 00). Since the more massive satellites will be removed more quickly than the less massive ones, they derive an upper limit for those satellites that might have survived to the present day. Domingos, Winter & Yokayama 006), using numerical simulations in the restricted three-body problem, examined the border of the stable regions for a planet/star mass ratio of 0. They concluded that extrasolar planets in the habitable zone could harbour Earth-like satellites. The determination of the orbital stability of such hierarchical bound three-body systems has been investigated in general by many authors both analytically and numerically. In these systems two of the bodies form a binary system, while the third body moves on an approximately Keplerian orbit relative to the centre of mass of the binary. In this paper we will consider the situation of the extrasolar moon where the binary mass, comprising the planet and moon, is very small compared with the mass of the third body, the parent star, so that the binary can be viewed as moving around the third body. It was shown by Harrington 968, 969) analytically that in this case, the binary would remain stable against the perturbations of the third body provided that it was at a comparatively large distance from the binary. Determining quantitatively the critical relative separation of the binary from the third body to the separation of the binary components is the major objective of most stability discussions. In general, it is only possible to follow the long-term evolution of such systems and hence their long-term stability by direct numerical integration of the orbits of all three masses. Analytical approaches can, however, provide limits on the range of orbital elements that a system must possess if it is to remain stable and avoid disruption, exchange or collision of the components. One of these approaches is through Hill stability which we will now consider in detail. HILL STABILITY Hill 878) introduced the concept of zero velocity surfaces in the restricted three-body problem. The topology of these surfaces and hence the regions of possible motion are controlled in this case by the Jacobi constant. For full three-body systems, the theory of Hill stability was developed by Gobulev 967), Gobulev 968) and Marchal & Saari 975) using Sundman s inequality, Zare 976, 977) using the Hamiltonian dynamics and reduction, and Bozis 976) using algebraic manipulation of integrals. The important parameter for the full three-body case is c E,whereE is the energy and c the angular momentum of the system of three masses. This parameter controls the topology of the zero velocity surfaces and determines the regions of possible motion, since the bodies may not cross these surfaces. This quantity corresponds to the Jacobi constant in the restricted problem of three bodies. The value of c E for the actual system is compared with the critical value derived for the corresponding three-body Hill surfaces determined by the position of the collinear Lagrangian points. The binary system will then be stable against disruption or exchange of components if the actual value of c E is greater than or equal to the critical value. The system is then considered to be Hill stable. If this condition is not satisfied, then exchange of components, disruption or collision is possible but not inevitable. The condition for stability is a sufficient condition but not a necessary one so that exchange might not occur when the condition is violated but certainly cannot occur if the condition is satisfied. This theory was originally only applied to coplanar systems where all three bodies moved on bound orbits Szebehely & Zare 977; Walker, Emslie & Roy 980; Walker & Roy 98; Marchal & Bozis 98; Donnison & Williams 98). Szebehely & Zare 977) did, however, examine the stability of known bound stellar systems with mutually inclined orbits using this basic theory. More recently, Gladman 99) applied the Marchal and Bozis formalism to the orbital stability of planets on circular orbits, while Veras & Armitage 004) extended this to cover two massive planets having inclined orbits in an extrasolar planetary system. Donnison 984a,b) also extended the theory to include situations where one component of the system moves on an unbound orbit. The systems were considered to be coplanar. Donnison 006) extended the application of the theory relating to an unbound orbit further by considering non-coplanar systems, in particular the situation where the third body moves on a parabolic orbit that is inclined to the binary pair. This was further extended to encompass the situation where the third body is moving on a hyperbolic orbit inclined to the binary pair Donnison 008). Donnison 009) further extended the stability of bound three-body systems to non-planar situations by considering the effect of an inclined third body. The basic theory was applied to extrasolar planetary systems. Here this is further extended by considering the situation where the binary is small in mass compared with the third mass. This is relevant to Kuiper Belt binary systems, very-low-mass binary systems and the Earth Moon Sun system Donnison 00). This was originally discussed by Donnison 988) in the context of small binaries in triple star systems. In this work, we will apply it to the stability of extrasolar planet moon systems in order to obtain the critical separations. Here we extend the stability criteria to inclined systems and obtain a higher order approximation. This is new but the approach is a development of the earlier work of Szebehely & Zare 977). Downloaded from by guest on 5 September 08 C 00 The Authors. Journal compilation C 00 RAS, MNRAS 406, 98 94

3 90 J. R. Donnison Figure. The three-body configuration with masses M and M forming the inner binary system with a semimajor axis and eccentricity e, with barycentre G, which moves on a elliptical orbit relative to the third body M with an inclination i, semimajor axis a and eccentricity e. SYSTEMS WITH THE BINARY MOVING ON AN INCLINED ELLIPTIC ORBIT RELATIVE TO THE CENTRAL BODY Consider three bodies with masses M,M and M, which can have any positive value. The masses M and M form an inner binary system with a semimajor axis and eccentricity e, whose barycentre G moves on a elliptical orbit relative to the third body M with an inclination i, semimajor axis a and eccentricity e see Fig. ). The total energy of the system, using a two-body approximation, is then GM M E = + GM ) μ, ) a where μ = M + M and G is the gravitational constant. The corresponding exact angular momentum squared for the system is then c = G a ) ) ) ) ) M μ a + M μa a M M + M M M cos i μ M, ) where M = M + M + M and 0 i 80.Herei = 0 indicates that M and the binary pair are both moving in the same plane and in thesamesenseoncoplanar prograde orbits, while i = 80 indicates that the motion of the masses is in the opposite sense that is coplanar retrograde motion. The motion is generally prograde if i<90 and retrograde if i 90. The parameter c E controlling the topology of the zero velocity surfaces of the actual system can now, using equations ) and ), be written as S ac where { } c E S ac = = M M M ) μ ) ) a + M G M M μ ) ) μ +M M M cos i M + M M M a a a ) ) + M μ M ) ) ) ) ) μ + M M M cos i M a a ) ). For this system to be Hill stable the system must satisfy the condition S ac S cr 0, where S cr is the critical value of the parameter derived for the corresponding Hill surfaces determined by the position of the collinear Lagrange points. The consequence is that the third body M is then unable to disrupt the binary system or exchange with its components. This condition for stability therefore from equation ) takes on the form M M μ + M M M + M M M cos i + a a μ M M M M μ ) M )) ) ) a ) ) a M M + M μ ) + M μ M a a ) )) ) ) ) Scr 0. 4) ) Downloaded from by guest on 5 September 08 In order to proceed further analytically, we can determine the critical values of e in terms of a / ),e and i for which the binary remains stable by solving this quadratic in ). C 00 The Authors. Journal compilation C 00 RAS, MNRAS 406, 98 94

4 Stability of extrasolar moons 9 From this expression, simple algebra shows that all bound binary orbits are stable for the condition ) a S cr M = M M M μ ) M μ. 5) M M It should be noted that this is independent of the inclination i and depends only on the masses and e. In terms of the pericentric distance q = a ),wehave ) q S cr M = M M M μ + e ) M μ ). 6) M M Donnison 009) showed that although these bounds are useful if we require to solve generally for a / ) in terms of e,e,i for a given S cr, we can rewrite equation 4) and solve it numerically as a fourth-order algebraic equation in the form ) + y β y + βy cos i + ) Ay = 0, 7) where ) a M μ ) M M ) M ) y =,β= 8) a M M M μ 4 and S cr M A = M μ ). This equation uses the same basic parameters as Szebehely & Zare 977) did for a completely bound coplanar three-body system. To proceed further, we need to determine the critical value of c E expressed in the form S cr. 4 THE CRITICAL VALUE OF c E The critical value for the surfaces is found by evaluating c E at the position of a collinear Lagrangian point for a central configuration M :M :M. The critical value depends only on the three masses present. If the ratio of the distances M M : M M is taken to be :x, then x, the position of the appropriate collinear equilibrium point, is the real solution to the usual Lagrange quintic equation see Roy 005 for derivation): M + M ) x 5 + M + M ) x 4 + M + M ) x M + M ) x M + M ) x M + M ) = 0. 9) It was shown by Zare 977) that the critical value of c E here can be computed in terms of x in the convenient form { } c E S cr M,M,M,x) = = f x) g x) G cr M, 0) where f x) = M M + M M + x + M M x and g x) = M M + M M + x) + M M x, ) and as before M = M + M + M. Only in certain special cases is it possible to express x and hence S cr in a closed form. In general, equation 9) has to be solved numerically. We will now consider a particularly important combination of masses which gives closed solution systems. Donnison 009) discussed the three special cases where the masses of all three components are equal M = M = M, where the components M = M and the situation where M M and M. Here we concentrate on the situation where the binary is very much smaller than the third component M,thatisM M + M. This physical situation as mentioned previously arises in a number of important astronomical contexts. Donnison 988) also previously discussed the particular case where a binary stellar system is orbiting a large star in terms of variations in e for planar systems. Here we will derive a new better approximation and apply this to star planet moon configurations in extrasolar planetary systems. Downloaded from by guest on 5 September 08 5 APPROXIMATE SOLUTIONS FOR SYSTEMS WITH M LARGE Donnison 009) found, using exact solutions of equation 9), solutions in an easily manageable form. We can also find similar solutions using approximations to the solution of equation 9). This occurs when the third component of the system very much exceeds the mass of the binary component. This type of orbit is often classified as a satellite orbit Szebehely 980). C 00 The Authors. Journal compilation C 00 RAS, MNRAS 406, 98 94

5 9 J. R. Donnison In this case where M M + M, a closed solution is possible. A reasonable approximation of the quintic equation 9) is then given by Walker et al. 980; Walker 98) ) M + M x =. ) M Since x is very small, the critical value S cr can be expanded in terms of increasing x to give see Donnison 988 for an earlier approximation) { S cr = β + λ) + 9λ λ + ) x + λ λ ) + λ) ) x where + λ + 5λ) + λ + λ) + 9λ ) ) + λ x 4 + O } x 5) +, + λ) λ = M and β = M M. M Similarly, the actual value S ac can be expanded in terms of x to give { S ac = β + λ) ) + ) ) a ) λ + λ) ) cos i + λ + λ) a +O } x 4.5) +. ) ) λ + + λ) a a Subtracting ) from 4) gives { S ac S cr = β + λ) ) + λ + λ) + λ + λ) a a ) ) λ + + λ) x ) ) ) ) + λ) x a a a ) ) ) cos i + λ) + 9λ λ + ) x + λ λ ) + λ) ) x + λ + 5λ) + λ + λ) + 9λ ) ) )} + λ x λ) For stability to fourth order in x, we have + λ) ) + ) ) a ) λ + λ) ) cos i + λ + λ) a a ) ) λ + + λ) a + λ) + 9λ λ + ) x λ λ ) + λ) ) x λ + 5λ) + λ + λ) + 9λ ) ) + λ x λ) Setting X = ), we can write this as a quadratic in X : a a ) x ) ) ) ) + λ) x x ) ) ) ) + λ) x ) 4) 5) 6) Downloaded from by guest on 5 September 08 X + + λ) X λ + + λ) λ a ) a ) + λ) ) x cos i )) + λ + λ) a + λ) 9λ λ + ) x λ λ ) + λ) ) x λ + 5λ) + λ + λ) + 9λ ) ) + λ x 4 + λ) ) ) ) ) + λ) x = 0 7) C 00 The Authors. Journal compilation C 00 RAS, MNRAS 406, 98 94

6 Stability of extrasolar moons 9 From equation 7) or by expanding equation 7), we find that the Hill stability for this system for all bound orbits for the binary occurs when ) a = + λ) )) λx + λ) ) + ) + x λ ) ) λ + 5λ) + λ λ λ ) + λ) + λ + λ) + λ) + 9λ ) ) + λ x + λ) ) + λ + λ) 8) ) and that this result is totally independent of the inclination. In terms of the pericentre distance q = a ), we have the critical condition ) q = + λ) )) + λ) + + λx + e ) + e ) x λ + e ) ) λ + 5λ) + λ λ λ ) + λ) + λ) + 9λ ) ) + λ x + λ) 9) + +. λ + λ) + e ) λ + λ) + e ) In general, the critical value of q / ) lies in the range + ) λ + 5λ) + λ λ) λ λ ) + λ) + λ) + 9λ ) ) + λ x + λ) ) q + λ) + + +, 0) x λ 6λ + λ) 6λ + λ) 6λx with the lower limit corresponding to circular orbits and the upper to parabolic orbits for λ. Similarly if we consider the variations in e we find by setting X = ), we can write equation 6) as a quadratic in X : { + λ) + λ) x ) ) } a + λ x X { + λ + λ) + λ + λ) a a a a ) ) ) } ) ) x cos i X + λ) 9λ λ + ) x λ λ ) + λ) ) x λ + 5λ) + λ + λ) + 9λ ) ) + λ x 4 + λ) = 0. We will now consider the application of these equations. 6 PREVIOUS WORK ON PLANETS WITH CIRCULAR ORBITS AROUND THE CENTRAL STAR Before discussing the new results in detail, we will consider previous work relevant to the stability of systems where both the binary components move on circular orbits. In particular, in Donnison 988) the critical Hill stability distances were approximated for circular orbits for different mass ratios by the expressions ) ) a 8M = ) M + M for systems with prograde orbits and ) ) a 8M =.06 ) M + M for retrograde orbits. Donnison & Mikulskis 99) found numerically that for prograde three-body systems moving on circular orbits, there was a good correspondence between the results obtained for the critical orbital or Lagrange stability and the c E Hill stability method we have employed here. Donnison & Mikulskis 994) and Eggleton & Kiselev995) have also shown numerically that retrograde circular orbits tend to be more stable than their prograde counterparts. The result for retrograde orbits in ) is not sufficiently restrictive and therefore of a more limited value. If we take the Hill radius of the binary as see Donnison & Williams 975) ) Downloaded from by guest on 5 September 08 ), M + M R H = a M then we can write, using equation ), the critical size of the binary if it moves on a prograde orbit in terms of the Hill radius as = R H = ) 5) C 00 The Authors. Journal compilation C 00 RAS, MNRAS 406, 98 94

7 94 J. R. Donnison The corresponding value for binaries on retrograde orbits is, from equation ), = R H.06 = ) Using the Jacobi constant in the restricted three-body problem for circular orbits, Hamilton & Krivov 997) find corresponding values of 0.5 and 0.69, respectively, for the /R H ratio. Domingos et al. 006) found numerically, using a planet/star mass ratio of 0,valuesof and for circular orbits. The prograde values in all cases show reasonable agreement. These expressions are therefore good first approximations to the critical separation values for circular orbits. In the next section, we will consider the new approximation which has been applied to currently known extrasolar planetary systems and which covers the full range of eccentricities. 7 APPLICATION TO EXTRASOLAR PLANETARY SYSTEMS The results for Hill stability in the previous sections can be applied to extrasolar planetary systems. The critical distances and hence regions of stability that possible moons of known extrasolar planets can possess can be determined using equation 6). Here the mass of the moon and planet together is M + M = M p sin i p + λ), wherem p is the mass of the planet, i p is the inclination angle of the orbit plane to the line of sight and λ is the moon/planet mass ratio. This expression will be used in our calculations. Clearly, our calculations can only be made for systems where the mass of the central star, M = M star, is known. The errors due to the approximation we are using are of the order of x 4.5,wherex = [ ] M p sin i p + λ) /M star, and are clearly negligible. Since the mass of any moon is usually small compared to its parent planet that is λ, then we have that M + M M p sini p. In all cases where we apply the theory to known systems, we will consider the possible moon to be moving on a circular orbit around the planet. A list of the known extrasolar planetary systems is shown in Table. The data are taken from the web site In this catalogue, the planetary masses included satisfy M p sin i p < Jupiter masses M J ), which is below the critical lithium burning mass and additional companions with masses M p sin i p < 0 M J. Currently as of 00 January, 44 planets have been discovered by the various observational techniques discussed earlier. Of these 4 planetary systems with 94 planets, 4 of which are multiple systems have been detected by radial velocity measurements of the stellar component due to the reflex action caused by the motion of the planetary companion. This technique only allows the measurement of the quantity M p sin i p so that the masses determined for these systems may only be considered as the lower limits to the actual masses. Some 69 planetary systems with 69 planets, three of which are multiple planet systems, have also been detected by direct transit observations and subsequently confirmed by radial velocity observations. Nine planetary systems with 0 planets, one of which is a multiple system, have been detected by microlensing. Nine planetary systems with planets, one of which is a multiple, have been detected by direct imaging. The remaining six planetary systems with nine planets, two of which are multiple, have been detected by timing methods. In Table, we show the observed data for these extrasolar systems with the system name, planet mass M p sini p in terms of Jupiter masses, semimajor axis a p in au, eccentricity e p and the mass of the parent star M star in solar masses, in Columns ) 5). In Columns 6) 8) the derived critical distance, in units of 0 au, is shown for moon planet ratios λ of 0., 0.0 and 0.00 for all the known extrasolar planetary systems. These ratios span the range from a large mass moon, of the order of twice Neptune s mass, to an Earth-mass moon for a Jupiter-mass planet. These results will be considered in detail in the next section. The masses of the planets lie in the range M J M p 5.0M J with a mean value of. M J. The semimajor axes of these systems lie in the range 0.04au a p 670 au, with a mean of 5.7 au, though more importantly the median is au, indicating the small separation of most of the planets from their parent star, while their eccentricities lie in the range 0.0 e p 0.97, with a mean of 0.. Only 66 have known inclinations, i p, and lie in the range 0. 0 i p 90. 0; the mean is 8. 0 and the median The vast majority of these have been obtained by transit determinations and are all very close to 90 as would be expected using this method, so that only a handful of systems have inclinations below The corresponding stellar components with a mass M star lie in the range 0.05 M M star 4.5M with a mean of.08 M, showing clearly the predominance of solar mass stars in the various surveys. 7. Variation in orbital eccentricity e In order to determine the boundary of the possible regions of stability around a known extrasolar planet in which a moon could move, we used the extrasolar planet data from Table that are M p sini p,a p,e p and M star and determine the critical distance ratio by solving equation 6) and hence the critical separation distance. If the eccentricity of the extrasolar planet is not known, we have assumed a circular orbit, that is e p = e = 0. From the numerically determined critical ratio of a / ) cr, the critical separation was determined. The results, as mentioned earlier, for the different values of λ that is 0., 0.0 and 0.00 are shown in Columns 6) 8) in units of 0 au. It is clear that in those cases where the planet moves on a circular orbit about the central star that is e p = 0.0, the critical separation of the moon from the planet,, does not change significantly as the value of the mass ratio λ is reduced. In contrast, for eccentric systems where e p 0, there can be big changes in the critical separation as λ decreases. The variation in critical separation depends crucially on the size of the eccentricity of the planetary orbit. In many cases for larger e p, as the value of λ is reduced from λ = 0.toλ = 0.0 and finally to λ = 0.00, the critical distance mirrors this change and is reduced by a factor of /0 each time λ is reduced by /0. However for small eccentricities, that is e p 0., there is not much change in the critical distance between λ = 0.0 and λ = These general results are to be expected as systems with circular orbits are far more stable than those with eccentric orbits. We find that for the 7 planets for which we have the required data, the critical separation lies in the range au 9.97 au with a mean of 0.08 au while the median is au, for λ = 0.. Here Downloaded from by guest on 5 September 08 C 00 The Authors. Journal compilation C 00 RAS, MNRAS 406, 98 94

8 Stability of extrasolar moons 95 Table. We show the data for the known extrasolar systems with the name, M p sin i p, semimajor axis a p, eccentricity e p and the mass of the parent star M star, in solar masses listed in Columns ) 5). In Columns 6) 8), the derived critical separation distance, in units of 0 au, is shown for the moon/planet ratios λ of 0., 0.0 and 0.00 for all these known extrasolar planetary systems. Planet M p sin i p a p e p M star M J ) M ) 0 au) 0 au) 0 au) λ = 0. λ = 0.0 λ = 0.00 OGLE MOA5b M07 b GQLup b AB Pic b.5 75 SCR 845b >8.5 >4.5 UScoCTIO 08b WASP-b OGLE-TR-56b TrES HD 4004 Bb WASP-4b OGLE-TR-b CoRoT-Exo-b WASP-5b OGLE-TR-b CoRoT-Exo-b SWEEPS WASP-b Gliese 876d Gliese 876c Gliese 876b WASP-9b. 0.0 HD 8608b WASP-b HAT-P-7b HD 897b WASP-4b HD 0b TrES WASP-b HD 756b XO-b GJ 46b HAT-P-5b Cnc e Cnc b Cnc c Cnc f Cnc d HD 6454b HD 4906b HAT-P-b HD 844b HD 4675b TrES HAT-P-4b HD 79949b HD 87b HD 87c OGLE-TR-0b XO-b Tau Boo b WASP-6b HD 0075b HD 88b HD 68b BD-0 66b HD 7589b Downloaded from by guest on 5 September 08 C 00 The Authors. Journal compilation C 00 RAS, MNRAS 406, 98 94

9 96 J. R. Donnison Table continued Planet M p sin i p a p e p M star M J ) M ) 0 au) 0 au) 0 au) λ = 0. λ = 0.0 λ = 0.00 HD 09458b TrES TW Hya b OGLE-TR-b WASP-b WASP-5b HD 988b HAT-P-6b Lupus-TR-b XO-b HD 76700b OGLE-TR-8b OGLE-TR-b HD 494b HD 095b SWEEPS-04 < Peg b WASP-b HAT-P-b Ups And b Ups And c Ups And d GJ 674b HD 49674b WASP-7b HD 09749b GI 58 b GI 58 c GI 58 d WASP-0b HAT-P-b HD 80b HD 68988b HD 68746b HIP 480b HIP 480c HD 8569b HD 707b HD 707c WASP-8b HD 600b HD 6509b HD 6980b HD 6980c HD 6980d HD 6069d HD 6069e HD 6069b HD 6069c HD 85968b HD HD 0847b HD 859b HD 859c HD 408b GI 86b HD 9949b HD 9060c HD 9060b HD 7894b HD 8b HD 9509b Downloaded from by guest on 5 September 08 C 00 The Authors. Journal compilation C 00 RAS, MNRAS 406, 98 94

10 Stability of extrasolar moons 97 Table continued Planet M p sin i p a p e p M star M J ) M ) 0 au) 0 au) 0 au) λ = 0. λ = 0.0 λ = 0.00 HD 07b HD 756b HD 644b HD 96b PSR 57+b 7e PSR 57+c PSR 57+d HD 469b HD 469b HD 964b HD 964c rho CrB b HD 0748b HD 7456b HD 7456d HD 7456c HD 768b HD 7605b HD 6844b HD 6844c HD 65b HD 504b HD 090b HD 789B b HD 64b HD 476b HD 80606b Vir b HD 6770b HD 565b HD 08487b GJ 0b ksi Aql b HD 70b HD 908b HD 74b HD 74d HD 74c HD 9449b.9 0. HD 8688b HD 756b HD 756c HD 5558b HD 5558c HD 04985b HD 75898b HD 894c HD 894b HD 6980b HD 6980c HD 8574b HD 006b HD 006c HD 89744b HD 4987b HD 66b HD 66c HD HD 40979b Uma b HD 7554b HD 59686b Downloaded from by guest on 5 September 08 C 00 The Authors. Journal compilation C 00 RAS, MNRAS 406, 98 94

11 98 J. R. Donnison Table continued Planet M p sin i p a p e p M star M J ) M ) 0 au) 0 au) 0 au) λ = 0. λ = 0.0 λ = 0.00 HR 80b HD 4b HD 070b HD 40b HD 9699b HD 56846b HD 709b HD 9788b HD 885b HD 00777b HD 445b HD 564b HD 7780b HD 08874b HD 08874c HD 40b HD 54857b HD 6704b HD 744b HD 4756b HD 4756c HD 9909b HD 077b HD 8b HD 8c HD 9994b HD 87b HD 8805b HD 89b HD 067b HD HD 56b HIP 75458b HD 4b HD 708b HD 475b HD 58b HD 078b HD eps Tau b HD 656b HD 86b HD 497b HD 4004 A b HD 59b NGC 449 No 7b GJ7b HD 977b NGC 4 b HD 079b Cyg B b HD 408b HD 7057b HD 675b HD 486b HD 4550b Gamma Cephei b HD 40b HD 406b HD 87085b HD 59868b Del b Downloaded from by guest on 5 September 08 C 00 The Authors. Journal compilation C 00 RAS, MNRAS 406, 98 94

12 Stability of extrasolar moons 99 Table continued Planet M p sin i p a p e p M star M J ) M ) 0 au) 0 au) 0 au) λ = 0. λ = 0.0 λ = 0.00 HD 8040b HD 90647b HD 0647b HD 0697b Uma b Uma c HD 908b HD HD b HD 70469b HD 649b V9 Peg b Kappa CrB b HD 09 b HD 68b HD 7b HD HD 506b HD 96050b HD 647b HD 96885b HD 645b HD 065b HD 596b ChaHa8 b Her b OGLE-06-09L b OGLE-06-09L c Gj849b HD 40A b HD 6648b HD 909b HD 7064b Epsilon Eridani b HD 50499b HD 707b HD 077b HD 8907b HD 7659b OGLE-05-69L b HD 5445b OGLE-05-90L b OGLE-05-07L b PSR B60-6b MOA-008-BLG-0-L b CT Cha b MOA-007-BLG-9-L b MOA-007-BLG-400-L b CoRoT HD HAT-P- b HAT-P- c HAT-P-8 b WASP-6 b Gl 58 e WASP-7 b HAT-P-9 b CoRoT-5 b HD 4786 b HD 4786 c XO-4 b Downloaded from by guest on 5 September 08 C 00 The Authors. Journal compilation C 00 RAS, MNRAS 406, 98 94

13 90 J. R. Donnison Table continued Planet M p sin i p a p e p M star M J ) M ) 0 au) 0 au) 0 au) λ = 0. λ = 0.0 λ = 0.00 XO-5 b CoRoT- b HD 4007 b HD 4007 c HD 4007 d HAT-P- b HD 794 d HIP 480 d CoRoT-6 b. CoRoT-4 b HD 84 b HD 84 c HD 84 d HD 647 b HD 4565 b HD 4708 b HD 4708 c HD 4577 b HD 07 b HD 07 c HD 5467 b HD 506 c And b HD 605 b HD 605 c HD 4564 b HD 4564 c BD b VB 0 b HD 0579 b HD 746 b BD0 457 b BD0 457 c HD 0868 b HD 6760 b Dra b HD 9667 b HD 400 b HD 86 c BD-7 6 b HD 4865 b HD 004 b Lyn b Cet b HD 46 b HD 957 b HD 767 b HD 78 b HD 04 b HD 664 b HD 4848 b HW Vir c 9. HW Virb 8.5 GJ 8 b beta Pic d HR 8799 d HR 8799 c HR 8799 b Fomalhaut b HD b HD 58 b HD 4847 b Downloaded from by guest on 5 September 08 C 00 The Authors. Journal compilation C 00 RAS, MNRAS 406, 98 94

14 Stability of extrasolar moons 9 Table continued Planet M p sin i p a p e p M star M J ) M ) 0 au) 0 au) 0 au) λ = 0. λ = 0.0 λ = 0.00 UMi b HD 056 b HD 8664 b HD 754 b HD WASP-9 b CoRoT-7 c WASP-8 b GJ 4 b < HD HAT-P-b Kepler-4 b Kepler-6 b Kepler-8 b Kepler-5 b HD 5497 b HD 5497 c HD 56 c HD 56 d Vir b Vir c Vir d HD b Kepler-7 b BD-088 b BD-088 c HD46 b GJ 4 b 0.09 GJ 667C b 0.08 BD b HD 5595 b HD 9446 b HD 9446 c HD 097 b HD 9056 b HD b HIP 96 b 0.47 HIP HD 4987 c HD 407 b Ari B b HIP 558 b HD 6765 b gamm Leo b HD 449 b 0.58 HD GI 649 b HD 588 b.96. HD 8590 b HD 564 b HD 870 b GJ 676A b 4 HD HD 854 b.6.06 HD 855 b HD 678 b QS Vir b HIP >5 0.6 HD b. 0.9 DP Leo b Downloaded from by guest on 5 September 08 C 00 The Authors. Journal compilation C 00 RAS, MNRAS 406, 98 94

15 9 J. R. Donnison.0 x=m +M )/M ) / =0. Dot-dash λ=0.0 Solid lines λ=0. Dashed lines λ= e a /a Figure. The plot shows for x = [M + M ) /M ] = 0. the critical a / ) values against binary eccentricity e for the inclinations i = 0, 0, 45, 60, 75 and 90 with e = 0.0. The dashed curves show the plot of the critical a / ) values for the mass ratio λ = 0.5, while the solid curves correspond to λ = 0. and the dot dashed curves are for λ = 0.0. The regions of Hill stability lie to the right-hand side of the curves in each case. The critical a / ) value increases and hence the Hill surfaces decrease as the value of e increases and it also decreases as the inclination i increases. The regions of stability increase in this case as λ decreases. All other values of λ,where0.0 <λ<0.5, lie between these curves and are not shown to avoid confusion. This value of x is in the range of the expected values of extrasolar planetary systems where a solar mass star is being orbited by a few Jupiter-mass planet with an accompanying moon. we have left out the exceptionally large system UScoCTO 08 which has a semimajor axis of 670 au, in order not to distort the statistics. As λ decreases the range of increases as the minimum value is decreased, the mean value of is however only decreased slightly, but the median value is more significantly altered. 7. Variation in binary orbital eccentricity e Donnison 009) applied the general theory to stellar binaries when the binary orbit about the parent star is a circular orbit. In this way, the variation with binary eccentricity and inclination was investigated. It was found that the stability varied with both e and i but that the changes were small compared to the variations in e which we have investigated. However, these secondary changes can be important. We will now apply the new approximation theory generally to the moons in extrasolar planetary systems. Here we consider variation in the binary orbit composed of a planet and its accompanying moon which is assumed to move on a circular orbit relative to the planet. Hence, equation 6) hastobesolvedtogivethecriticala / ) values for a range of e and i values for given x,λ and e.infig.,theplotofthecriticala / ) values against binary eccentricity e is shown for the inclinations i = 0, 0, 45, 60, 75 and 90 with e = 0.0, with masses satisfying x = [M + M ) /M ] = 0., where M + M is the mass of the planet and moon and M is the mass of the central star. The dashed curves show the situation where the moon is fairly massive with a binary ratio λ = 0.5. The curves for λ = 0. areshownassolidcurves for the same values of inclination. The dot dashed curves give the stability curves for λ = 0.0. The intermediate values of λ in the range 0.0 <λ<0.5 lie between the sets of curves and are not shown to avoid cluttering the figure. There is obviously a degree of overlap for the different λ values. The systems are Hill stable against exchange if they lie to the right of the curves. To the left of the curves, the surfaces open out and exchange or collision of component masses is possible but not inevitable. In general, the critical distances increase and the regions of stability decrease as e and inclination i increase in value. This is in line with the previous results of Donnison 009). As the value of λ was decreased, the stability curves moved slightly to the left so that systems with smaller λ were found to be more stable. This value of x is in the range of the expected values of extrasolar planetary systems where a solar-mass star is being orbited by a few Jupiter-mass planet with an accompanying moon. In Fig., the value of the mass parameter x is decreased so that x = [M + M ) /M ] = 0.0 and again the curves show the plot of the critical a / ) values against binary eccentricity e for the inclinations i = 0, 0, 45, 60, 75 and 90 with e = 0.0, for λ = 0.5thedashedcurves),λ = 0. the solid curves) and λ = 0.0 the dot dashed curves). In this case, the critical distance ratio a / ) has increased by a factor of about 0 and the stability curves have moved by this factor to the right indicating the decrease in stability of this configuration compared to the case where x = 0.. In this case the difference between the three sets of curves has become smaller and they are beginning to merge, indicating that the importance of the mass ratio λ diminishes with decreasing x. This value of x is applicable to an extrasolar planetary system with an Earth-mass planet with an orbiting moon. Further reductions in x not shown) further support these conclusions. Potentially for these mass ranges, the stability curves can give an estimate of the mass of the binary system. This is particularly useful where the masses of the system are unknown and obviously can also be used as an independent check on systems with known masses. Recently, Szenkovits & Makó 008), using an elliptic restricted three-body criterion applied to extrasolar planets in stellar binary systems, found similar stability variations with inclination Downloaded from by guest on 5 September 08 C 00 The Authors. Journal compilation C 00 RAS, MNRAS 406, 98 94