Migration of giant planets in planetesimal discs (original) (raw)

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1Dipartimento di Matematica, Università Statale di Bergamo, Piazza Rosate, 2 — I 24129 Bergamo, Italy

2Feza Gürsey Institute, P.O. Box 6 Çengelköy, Istanbul, Turkey

3Bogˇaziçi University, Physics Department, 80815 Bebek, Istanbul, Turkey

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1Dipartimento di Matematica, Università Statale di Bergamo, Piazza Rosate, 2 — I 24129 Bergamo, Italy

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3Bogˇaziçi University, Physics Department, 80815 Bebek, Istanbul, Turkey

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Received:

22 February 2001

Published:

21 August 2001

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Abstract

Planets orbiting a planetesimal circumstellar disc can migrate inward from their initial positions because of dynamical friction between planets and planetesimals. The migration rate depends on the disc mass and on its time evolution. Planets that are embedded in long-lived planetesimal discs, having total mass of 10−4– 0.01 M⊙, can migrate inward a large distance and can survive only if the inner disc is truncated or as a result of tidal interaction with the star. In this case the semimajor axis, a, of the planetary orbit is less than 0.1 au. Orbits with larger a are obtained for smaller values of the disc mass or for a rapid evolution (depletion) of the disc. This model may explain not only several of the orbital features of the giant planets that have been discovered in recent years orbiting nearby stars, but also the metallicity enhancement found in several stars associated with short-period planets.

1 Introduction

According to the most popular theory on the formation of giant planets in the Solar system, such planets were formed by the accumulation of solid cores (Safronov 1969; Wetherill & Stewart 1989; Aarseth et al. 1993), known as planetesimals, in a gaseous disc centred around the sun. When the core mass increases above 10 M⊙, it begins a rapid accretion phase (Mizuno 1980; Bodenheimer & Pollack 1986) in which the protoplanet can capture a gas envelope from the protoplanetary disc leading to the formation of a giant planet (Pollack et al. 1996). Jupiter-mass planets may require most of the lifetime of the disc to accrete (Zuckerman, Forveille & Kastner 1995; Pollack et al. 1996).

Protostellar discs around young stellar objects that have properties similar to those supposed for the Solar nebula are common: between 25 and 75 per cent of young stellar objects in the Orion nebula seem to have discs (Prosser et al. 1994; McCaughrean & Stauffer 1994) with mass and size (Beckwith & Sargent 1996). Moreover, several planetary companions, orbiting extrasolar stars, have recently been discovered. The extrasolar planets census, updated at 2000 October, gives 58 planets. For reasons of space, we report on only a small number of them: the companions orbiting 51 Peg (Mayor & Queloz 1995), τ Boo(Marcy et al. 1997– the San Francisco University Team, hereafter SFSU), v And (SFSU), _ρ_1 Cnc (SFSU), ρ CrB (Noyes et al. 1997 — hereafter the AFOE team), HD 114762, 70 Vir, 16 Cyg and 47 UMa (Butler & Marcy 1996). In the above list, with the exception of 47 Uma, the new planets are all at distances <1 au. In recent years, other extrasolar planets have been discovered, orbiting at distances >1 au from the central star, for example ε Eridani, HD210277, HD 82943, 14 Her, HD 190228, HD 222582, HD10697 and HD 29587: these represent only 15 per cent of all the planets that have been discovered by 2000 October. Three planets (51 Peg, τ Boo, v And) are in extremely tight circular orbits with periods of a few days; two planets (_ρ_1 Cnc and ρ CrB) have circular orbits with periods of the order of tens of days; and three planets with wider orbits (16 Cyg B, 70 Vir and HD 114762) have very large eccentricities. The properties of these planets, most of which are Jupiter-mass objects, are difficult to explain using the quoted standard model for planet formation (Lissauer 1993; Boss 1995). This standard model predicts nearly circular planetary orbits, and giant planets with orbital distances ≥1 au from the central star, at which distance the temperature in the protostellar nebula is low enough for icy materials to condense (Boss 1995, 1996; Wuchterl 1993, 1996). Standard disc models show that at 0.05 au, the temperature is about 2000 K, which is too hot for the existence of any small solid particles. Moreover, the ice condensation radius does not depend strongly on stellar mass, so that it does not move inward rapidly as the stellar mass decreases. For star masses , 0.5, 0.1 M⊙, the ice condensation radius moves inward from ≃6 to ≃4.5 au (Boss 1995). Another problem with the in situ formation of a planetary companion is that even though the present evaporation rate is negligible, this effect would have been of major importance in the past. In fact, during the early history of a planet, its radius may have been a factor of ten larger than the present value, implying that the escape speed was much lower than its present value. Hence evaporation mechanisms and ablation by the stellar wind might prevent the formation of a planet. The question that arises is: if such massive planets cannot form at the actual locations, how did they reach their current positions? Four mechanisms have been proposed to explain this dilemma.

The first mechanism consists of a secular interaction with a distant binary companion (Holman, Touma & Tremaine 1997; Mazeh, Krymolowski & Rosenfeld 1996). While this mechanism can also produce significant eccentricities for the longer period extrasolar planets, it is unable to explain objects like 51 Peg. In fact, 51 Peg has been extensively searched for a binary companion (Marcy et al. 1997), but none has been found. Consequently, in the particular case of 51 Peg, this mechanism is not responsible for the orbital decay.

The second possible mechanism proposed to explain short-period planets is dissipation in the protostellar nebula. Tidal interaction between a massive planet and a circumstellar disc gives rise to an angular momentum transfer between the disc and the planet (Goldreich & Tremaine 1979, 1980; Ward 1986; Lin, Bodenheimer & Richardson 1996; Ward 1997). The motion of the planet in the disc excites density waves both interior and exterior to the planet. A torque originates from the attraction of the protoplanet for these non-axisymmetric density perturbations (Goldreich & Tremaine 1980). Density wave torques repel material on either side of the orbit of the protoplanet and attempt to open a gap in the disc, the size of which depends on the viscosity of the disc and inversely on the mass of the planet (Lin & Papaloizou 1986; Takeuchi, Miyama & Lin 1986). [Note that there exists a minimum mass for gap opening, which is of the order of magnitude of Jovian-mass planets; this prevents the nonsense of an infinitely large gap for a zero-mass planet.] If gap formation is successful (e.g. in the case of a Jupiter-mass planet), the protoplanet becomes locked to the disc and must ultimately share its fate (Ward 1982; Lin & Papaloizou 1986, 1993). This mechanism is called type II drift. The situation is different if the object is not yet large enough to open and sustain a gap. In this case also, the protoplanet migrates inwards, but on a time-scale even smaller than that of type II drift (Ward 1997). This is called type I drift. In both cases, the rate of radial mobility of the planet, with respect to the central star, is indicated by the term ‘drift velocity’ (see Ward 1997). In some cases, see the case of Neptune below, the drift velocity can be directed outwards.

Since, in this model, the time-scale of migration is ≃ (Ward 1997), the migration has to switch off at a critical moment, if the planet is to stop close to the star without falling in. The movement of the planet might be halted by short-range tidal or magnetic effects from the central star (Lin et al. 1996): in any case, as shown by Murray et al. (1998), it is difficult to explain, by means of these stopping mechanisms, planets with semimajor axes .

A resonant interaction with a disc of planetesimals is another possible source of orbital migration. In this model, planet migration begins when the surface density of planetesimals, Σ satisfies the condition , being Σc a critical value for the surface density Σ. The advantage of this mechanism is that the migration is halted naturally at short distances when the majority of perturbed planetesimals collide with the star. Moreover, wide eccentric orbits can also be produced for planets more massive than ≃3_M_J. However the model has some disadvantages, since the protoplanetary disc mass required for the migration of a Jupiter-mass planet to is very large (Ford, Rasio & Sills 1999).

Finally, the fourth and last mechanism deals with dynamical instabilities in a system of giant planets (Rasio & Ford 1996). The orbits of planets could become unstable if the orbital radii evolve secularly at different rates or if the masses increase significantly as the planets accrete their gaseous envelopes (Lissauer 1993). In this model, the gravitational interaction between two planets, during evolution, (Gladman 1993; Chambers, Wetherill & Boss 1996) can give rise to the ejection of one planet, leaving the other in an eccentric orbit. The orbit of this planet can then circularize at an orbital separation of a few stellar radii (Rasio et al. 1996) if the inner planet has a sufficiently small pericentre distance. Simulations of multiple giant-planet systems showed that successive mergers between two or more planets can lead to the formation of a massive (≥10_M_J) object in a wide eccentric orbit (Lin & Ida 1997). While it is almost certain that this mechanism operates in many systems with multiple planets, it is not clear whether they can reproduce the fraction of systems similar to 51 Peg observed.

In this paper we propose another model to explain the orbital parameters of extraterrestrial planets. The model is based on dynamical friction of a planet with a planetesimal disc. While the role of dynamical friction on the planetary accumulation process has been studied in several papers (Stewart & Kaula 1980; Horedt 1985; Stewart & Wetherill 1988), very few papers have studied the role of dynamical friction on radial migration of planets or planetesimals in planetary discs (see Melita & Woolfson 1996; Haghighipour 1999). This attitude is a result of the fact that, so far, many people have assumed a priori that radial migration arising from dynamical friction is much slower than the damping of velocity dispersion arising from dynamical friction. Therefore, most studies on dynamical friction were concerned only with damping of velocity dispersion (damping of the eccentricity, e, and inclination, i ), adopting local coordinates. Analytical works by Stewart & Wetherill (1988) and Ida (1990) adopted local coordinates. _N_-body simulation by Ida & Makino (1992) adopted non-local coordinates, but did not investigate radial migration. Moreover, most models of the planetesimal disc assume that, except for the influence of aerodynamic drag, which loses its effectiveness for planetesimals larger than a few kilometers, the primary cause of radial migration is mutual scattering (Hayashi et al. 1977; Wetherill 1990). In order to calculate the migration of protoplanets, we apply the model introduced in Del Popolo, Spedicato & Gambera (1999) to study the migration of Kuiper Belt objects (KBOs) and we suppose that the gas in the disc is dissipated soon after the planet forms so that it has little effect on planet migration. We are particularly interested in studying the role of planetesimals in planet migration and the dependence of migration on the disc mass and on its evolution.

The plan of the paper is as follows in Section 2 we introduce the model used to study radial migration. In Section 3 we show the assumptions used in the simulation. In Section 4 we show the results that can be drawn from our calculations and, finally, Section 5 is devoted to our conclusions.

2 Planetary migration model

In a recent paper by Del Popolo et al. (1999), we studied how dynamical friction, arising from small planetesimals, influences the evolution of KBOs having masses larger than 1022g. We found that the mean eccentricity of large-mass particles is reduced by dynamical friction arising from small-mass particles in time-scales shorter than the age of the Solar system for objects of mass equal to or larger than 1023g. Moreover, the dynamical drag, produced by dynamical friction of objects of masses ≥1024g, is responsible for the loss of angular momentum and the fall through more central regions in a time-scale ≈109yr. Here we use a similar model to study the radial migration of planet. We suppose that a single planet moves in a planetesimal disc under the influence of the gravitational force of the Sun. The equation of motion of the planet can be written as:

(1)

(Melita & Woolfson 1996), where the term _F_⊙ represents the force per unit mass from the Sun, while R is the dissipative force (the dynamical friction term — see Melita & Woolfson 1996). Calculations involving dynamical friction that are used to study planetesimal dynamics often use Chandrasekhar's theory (Stewart & Wetherill 1988; Ida 1990) for homogeneous and isotropic distribution of lighter particles. This choice is not the right one, since dynamical friction in discs differs from that in spherical isotropic three-dimensional systems. This is because disc evolution is influenced by effects different from those producing the evolution of stellar systems:

• (1)
  In a disc, the contribution to the friction coming from close encounters is comparable to that arising from distant encounters (Donner & Sundelius 1993; Palmer, Lin & Aarseth 1993).
• (2)
  Collective effects in a disc are much stronger than those in a three-dimensional system (Thorne 1968).
• (3)
  The peculiar velocities of planetesimals in a disc are small. This means that differential rotation of the disc dominates over the relative velocities of the planetesimals.
• (4)
  The velocity dispersion of particles in a disc potential is anisotropic.

We assume that the matter distribution is disc-shaped and that it has a velocity distribution described by

(2)

(Hornung, Pellat & Borge 1985, Stewart & Wetherill 1988), where _v_∥ and _σ_∥ are the velocity and the velocity dispersion in the direction parallel to the plane while _v_⊥ and _σ_⊥ are those in the perpendicular direction. We suppose that _σ_∥ and _σ_⊥ are constants and that their ratio is simply taken to be 2:1. Then, according to Chandrasekhar (1968) and Binney (1977), we may write the force components as

(3)

(4)

where

(5)

(6)

and

(7)

while _n_¯ is the average spatial density, _m_1 is the mass of the test particle, _m_2 is the mass of a field one, and log Λ is the Coulomb logarithm. The frictional drag on the test particles may be written as

(8)

where _e_∥ and _e_⊥ are two unit vectors parallel and perpendicular to the disc plane.

When , the drag caused by dynamical friction will tend to increase the anisotropy of the velocity distribution of the test particles. In other words, the dynamical drag experienced by an object of mass _m_1 moving through a less-massive non-spherical distribution of objects of mass _m_2 is not exerted in the direction of the relative motion (as in the case of spherically symmetric distribution of matter). Hence the already-flat distribution of more massive objects will be further flattened during the evolution of the system (Binney 1977). As shown by Ida (1990), Ida & Makino (1992) and Del Popolo et al. (1999), damping of eccentricity and inclination is more rapid than radial migration, so in this paper we deal only with radial migration and we assume that the planet has negligible inclination and eccentricity, and that the initial heliocentric distance of the planet is 5.2 au. The objects lying in the plane have no way of knowing that they are moving into a non-spherically symmetric potential. Hence we expect that the dynamical drag is exerted in the direction opposite to the motion of the particle,

(9)

3 Simulation parameters

In order to calculate the effect of dynamical friction on the orbital evolution of the planet, we suppose that and that the dispersion velocities are constant. If the planetesimals attain dynamical equilibrium, their equilibrium velocity dispersion, _σ_m, would be comparable to the surface escape velocity of the dominant bodies (Safronov 1969) such that

(10)

where θ is the Safronov number, and _m_∗ and _r_∗ are the mass and radius of the largest planetesimals (note that the velocity dispersion of the planetesimals _σ_m, now introduced, is the velocity dispersion to be used for calculating the σ which is present in the dynamical friction force). If instead we consider a two-component system, consisting of one protoplanet and many equal-mass planetesimals, the velocity dispersion of the planetesimals in the neighborhood of the protoplanet depends on the mass of the protoplanet. When the mass of the planet, M, is ≤1025g, the value of (_e_m being the eccentricity of the planetesimals) is independent of M, therefore

(11)

(Ida & Makino 1993) where m is the mass of the planetesimals. When the mass of the planet reaches values larger than at 1 au, is proportional to _M_1/3,

(12)

(Ida & Makino 1993). As a consequence also, the dispersion velocity in the disc is characterized by two regimes, it being connected to the eccentricity by the equation

(13)

where _i_m is the inclination of planetesimals and _v_c is the Keplerian circular velocity. The width of the heated region is roughly given by (Ida & Makino 1993) where a is the semimajor axis and is the Hill radius of the protoplanet. The increase in velocity dispersion of planetesimals around the protoplanet decreases the dynamical friction force (see equation 8) and consequently increases the migration time-scale.

In the simulation, we assume that the planetesimals all have equal masses, m, and that , M being the planetary mass. This assumption does not affect the results, since dynamical friction does not depend on the individual masses of these particles but on their overall density. We also assume that the surface density in planetesimals varies as , where Σ⊙, the surface density at 1 au, is a free parameter. The total mass in the planetesimal disc within radius r is then

(14)

We assume that ≃ 1 per cent of the disc mass is in the form of solid particles (Stepinski & Valageas 1996). To be more precise, assuming metal abundance , the disc mass in gas interior to Jupiter's orbit is 0.16 M⊙(Σ⊙/103g cm−2) (Murray et al. 1998). In our model, the gas is almost totally dissipated when the planet begins to migrate and we assume that the value of disc mass reported in the following part of the paper is contained within 40 au (Weidenshilling 1977). We integrated the equations of motion in heliocentric coordinates using the Bulirsch–Stoer method.

4 Results

4.1 Migration in a non-evolving disc

Our model starts with a fully formed gaseous giant planet of 1_M_J at 5.2 au. As mentioned in the previous section, the circumstellar disc is assumed to have a power-law radial density and to be axisymmetric. According to various evidence showing that the disc lifetimes range from 105 to 107yr (Strom, Edwards & Skrutskie 1993; Ruden & Pollack 1991), we assume that the disc has a nominal effective lifetime of 106yr (Zuckerman et al. 1995). This assumption refers to the gas disc. Usually, this decline of gas mass near stars is more rapid than the decline in the mass of orbiting particulate matter (Zuckerman et al. 1995). Moreover, the disc is populated by residual planetesimals for a longer period. We are interested in studying the migration resulting from interaction with planetesimals, and for this reason we suppose that the gas is almost dissipated when the planet starts its migration. Since Jupiter-mass planets may require most of the lifetime of the disc to accrete (106 to 107yr), and meanwhile the disc is subject to evolutionary changes — and since the disc is also subject to evolution after this time interval (Pollack et al. 1996; Zuckerman et al. 1995) — we incorporated this possibility in our model by also running models that allow some part of the disc to dissipate during the migration of the planet. We integrated the model introduced in the previous section for several values of the disc surface density, or equivalently several disc masses: , 0.005, 0.001, 0.0005, 0.0001 M⊙.

The results of this first set of calculations (assuming that the disc does not evolve) are shown in Fig. 1. The curves show the evolution of a 1-M J planet in a disc with planetesimals having a surface density . The simulation is started with the planet at 5.2 au and . In any case, similarly to what was shown by Murray et al. (1998), planets with masses during their migration can increase the value of _e_p. The curves correspond, from bottom to top (short-dashed long-dashed line, dotted line, long-dashed line, dot-short-dashed line, solid line) to the following values of _M_D: 0.01, 0.005, 0.001, 0.0005, 0.0001 M⊙. As expected, the most massive disc (0.01 M⊙) produces a rapid radial migration of the planet. Discs having masses lower than 0.01 M⊙ produce a smaller radial migration of the planet.In particular, we found that for , the planet moves to 0.05 au in ≃. The migration halts for the following reason (by the term ‘halt’ we mean that the planet has not had time to migrate any further, even if it is still migrating). If , 0.001, 0.0005, 0.0001 M⊙ we have, respectively, for the time needed to reach 0.05 au: ≃, ≃, ≃, ≃. In other words, disc mass is one of the parameters that controls radial migration. An interesting feature of the model is that migration naturally halts without needing any peculiar mechanisms that stop the planet from plunging into the central star. In fact, as shown in Fig. 1, the migration time to reach ≃0.05 au increases with decreasing disc mass. If the disc mass is ≤0.000 08 M⊙, the time needed to reach the quoted position is larger than the age of the stellar system and the planet does not fall into the star. Then, the planet can halt its migration without falling in the star if the initial disc mass is ≤0.000 08 M⊙. Even if the disc density does not fall below the critical value, the planet must halt at several _R_∗ from the surface of the star (_R_∗ is the stellar radius). In fact solid bodies cannot condense at distances ≤7_R_∗, and planetesimals cannot survive long at distances ≤2_R_∗. When the planet arrives at this distance, the dynamical friction force switches off and its migration stops. This means that the minimum value of the semimajor axis that a planet can reach is ≃ 0.03 au.

Figure 1.

The evolution of the a(t) of a Jupiter-mass planet, in a disc with for several values of the planetesimal disc mass, (short-dashed long-dashed line), 0.005 M⊙ (dotted line), 0.001 M⊙ (long dashed line), 0.0005 M⊙ (dot-short dashed line) and 0.0001 M⊙ (solid line). Σ is supposed to remain constant in time.

4.2 Migration in an evolving disc

In the previous calculations, we supposed that the disc mass did not undergo time evolution, but in reality the disc evolves as the planet moves inward and tends to dissipate.

In our model, we are fundamentally interested in the evolution of solid matter and planetesimals in discs. To this end, it is very important to note that the distribution of solid particles follows a global time evolution, which accompanies the time evolution of the gaseous component of the disc. As a result of viscous torques, the gaseous disc spreads and its mass diminishes. If initially the solid particles are small and coupled to the gas, they decouple from it when they gain mass because of coagulation. (Stepinski & Valageas 1996). Particles having radius can be considered perfectly coupled to the gas, while those having can be considered completely decoupled from it and their mean velocities remain practically unchanged with time (Stepinski & Valageas 1996). Moreover, as shown in a recent study by Ida et al. (2000), the radial migration of a planet of Jupiter-mass produces a very rapid capture of planetesimals in the 2:1 and 3:2 resonances: the resonance capture occurs if the migration time, _τ_mig, of the planet is for 2:1 resonance and if for 3:2 resonance. If the result is correct, this means that the disc should be rapidly depleted with a consequent rapid stopping of migration. As may be understood from our results above disc evolution depends on disc and system characteristics.

In this paper we have tried to take account of disc evolution by supposing that the total mass in solids decays with time from its original value to the present value as shown in Fig. 2 (see Stepinski & Valageas 1996). The results of this calculation are shown in Figs 2–5.

Figure 2.

The assumed evolution of the mass of the planetesimals. The mass, from the initial value M o, reduces to 10−3_M_o, in ≃107yr (solid line), ≃108yr (dotted line), and ≃ (dashed line).

Figure 3.

Same as Fig.1 but now we suppose that the disc mass decreases exponentially in (see Fig. 2).

Figure 4.

Same as Fig. 2 but with mass of the planetesimals evolving in 108yr (see Fig. 2).

Figure 5.

Same as Fig. 4 but with mass of the planetesimals evolving in 107yr (see Fig. 2).

Fig. 2 shows the evolution of the disc mass used in the calculations of radial migration (see Stepinski & Valageas 1996, fig. 6). We suppose that the mass in the disc decreases exponentially with time from its original value M o to 10−3_M_ o in (dashed line), 108yr (dotted line) and 107yr (solid line). Fig. 3 is the same as Fig. 1 but now we suppose that the mass in the disc decreases as described.

If , the planet stops its migration at , while if , 0.001, 0.0005, 0.0001 M⊙, we have , 3.5, 4.3,5 au, respectively. Fig. 4 is obtained by supposing that the mass decreases, as previously quoted, in 108yr. As can be shown, the planet embedded in a disc having , 0.005, 0.001, 0.0005, 0.0001 M⊙ respectively migrates to 1.47, 2.7, 4.58, 4.88, 5.1 au. Finally, in Fig. 5 the time-scale for disc evolution is 107yr. The planet embedded in a disc with , 0.005, 0.001, 0.0005, 0.0001 M⊙ migrates respectively to 4.58, 4.88, 5.13, 5.17, 5.2 au.

Our distribution of final masses and heliocentric distance predicts that massive planets can be present at any heliocentric distances between their formation locations and extremely small orbits, depending on the initial mass of the disc and its evolution. As we shall show in the following, the model can explain the locations not only of the close companions (at ≤0.1 au), but it can also reproduce other observed planets, including Jupiter.

4.3 Comparison of the model results with observations

Configurations of planets like τ Bootis b, 51 Peg b, having very small semimajor axes, can be reproduced by models with no evolution (see Fig. 1) or with high values of the initial disc mass and low time evolution (e.g., , (Boss 1996): having a disc mass a bit lower can explain the configuration of planets like 55 Cnc b and ρ CrB b . The parameters of 47 UMa b can be explained, for example, by supposing a low-mass disc and evolution (e.g. , . Planets like 70 Vir b , and HD 114762 b have high eccentricities and masses larger than that of Jupiter. The low semimajor axis can be explained by radial migration, as shown, while the high value of eccentricities can be explained in a model like ours of interaction between planets and planetesimals, in a way similar to that shown by Murray et al. (1998). If a planet having mass , which is the case of 70 Vir b and HD 114762, moves in a planetesimal disc during interactions, planetesimals scattered from their Hill sphere can be ejected with (where Δ_E_ and Δ_L_ are respectively the energy and angular momentum removed from a planet by the ejection of a planetesimal), and the eccentricity _e_p tends to increase. For sake of completeness we must say that there are also some systems that are a bit puzzling. For example, v And has and a high eccentricity, HD210277 has and . Such high eccentricities could be explained by supposing an encounter with an object having mass ≃ _M_⊕, or alternatively by supposing that the systems are seen at small _i_p. In the case of the companion to 16 Cyg B , the high eccentricity may be the result of interactions with the stellar companion (Murray et al. 1998).

Our Solar system could have been subject to giant-planet migration. For example, a shrinkage of the orbit of Jupiter of 0.1–−0.2 au could naturally explain the depletion of the outer asteroid belt (Fernandez & Ip 1984; Liou & Malhotra 1997). Some of our model runs produce a 1-M J planet that moves from 5.2 au inwards for a fraction of 1 au. The planetesimal disc enabling this small migration has a lifetime ≃107yr, so that the disc gas must have disappeared soon after Jupiter fully formed (Boss 1996).

This last result is obviously strictly valid only for a single planet orbiting around the Sun because in the presence of several planets, migration becomes more complex. A close example is that of the Solar system. In this case two planets, Uranus and Neptune, were subject to outward migration, which is the opposite of what would be expected. Several models have been proposed to explain this outward migration. A first model is connected to gravitational scattering between planet and residual planetesimals (Malhotra 1993; Ida et al. 2000). A second model allowing Neptune to have outward migration is connected to the dissipation in the protostellar nebula. In this case both inward and outward planetary migration are allowed. In fact in a viscous disc, gas inside a particular radius, known as the radius of maximum viscous stress, _r_mvs, drifts inwards as it loses angular momentum, while gas outside _r_mvs expands outwards as it receives angular momentum (Lynden-Bell & Pringle 1974). The outward migration of Neptune is a result of the fact that the gas in the Neptune-forming region has a tendency to migrate outwards (Ruden & Lin 1986). In Fig. 6 we show the drift velocity, , as a function of mass, M. As shown, objects having masses < _M_⊕ have velocity drift increasing as M, while after a threshold mass, any further mass increase begins to slow down the drift. As the threshold is exceeded, the motion fairly abruptly converts to a slower mode in which the drift velocity is independent of mass. As previously explained, this behaviour is a result of the transition from a stage in which the dispersion velocity is independent of M to a stage in which it increases with _M_1/3 (Ida & Makino 1993). This last stage is known as the protoplanet-dominated stage. The phenomenon is equivalent to that predicted in the density-wave approach (Goldreich & Tremaine 1980; Ward 1997). In this approach, the density wave torques repel material on either side of the orbit of a protoplanet and attempt to open a gap in the disc. Only very large objects are able to open and sustain such a gap. After gap formation, the drift rate of the planet is set by disc viscosity and is generally smaller than in the absence of the gap. We stress that the decaying portion of the curve corresponding to the transition from the first to the second stage does not correspond to any particular model because, following Ida & Makino (1993), we do not have information on the evolution of σ in the transition regime.

Figure 6.

Drift velocity, , as a function of mass. Velocities are normalized to where M_⊕ is an Earth mass, Σ the surface density, Ω is the angular velocity and σ the dispersion velocity. The assumed conditions are those considered appropriate for the Jovian region and assuming that . The line ∝_M corresponds to the model described by Ward (1997) and its behaviour is valid until ≃0.1_M_⊕, but beyond this value there is a transition to a behaviour ∝_M_0.

4.4 Enhancements of metallicity

Another important point is that the dynamical processes leading to planetary migration can also affect the evolution of the central star. Gonzales (1997, 1998a,b) showed that several stars with short-period planets have high metallicities, . Gonzales (1998b) proposes that their metallicities have been enhanced by the accretion of high-Z material which leads to the speculation that there may be a relationship between stars with higher metallicities and stars with planets. Alternatively, the correlation could arise, if metal-rich stars have metal-rich discs which are more likely to form planets.

Several mechanisms have been proposed to explain the high metallicity of stars having extrasolar planets. One of these mechanisms is related to the Lin et al. (1996) migration model, but this model has two severe drawbacks. Models in which gas disc material accretes on to the star are not able to significantly alter the observed metallicity, because the disc has a metallicity slightly larger than the star. Good results are obtained in models in which asteroids or planetesimals accrete on to the star (Murray et al. 1998). As previously quoted, Murray et al. (1998) suggest that a giant planet can induce eccentricity growth among residual planetesimals through resonant interactions. Subsequent close encounters cause most of the affected planetesimals to be ejected outwards while the planet migrates inward. A substantial population of planetesimals could induce a Jupiter-mass planet to migrate a large distance inward. Neglecting any planetesimals that are scattered into the star until the planet reaches its final orbit, and assuming that the fraction of planetesimals scattered on to the star consists of only those that become planet-crossing before colliding with the star, the mass accreted on to the star is given by

(15)

where

(16)

(Ford et al. 1999), where ƒ(a) is the fraction of planetesimals scattered on to the star by the planet at distance a and α is a parameter . As shown by equation 16, the fraction of planetesimals scattered on to the star increases with decreasing distance from it. At if . For example, the mass of planetesimals which would be scattered into 51 Peg, the final orbit of which has , is 130_M_⊕ (Ford et al. 1999). By starting from a star with Solar metallicity and adding 130_M_⊕ of asteroids to 51 Peg, the observed [Fe/H] increases to 0.48 (Ford et al. 1999). The value found is an inferior limit because it takes account only of planetesimals scattered from the planet located in its final orbit. Moreover, it is calculated for a value of less than the maximum disc mass used in Murray et al. (1998). Using the largest values of Σ⊙ used by Murray et al. (1998), one would expect a value of _M_acc double that previously quoted.

Even if we assume , this value is larger than that observed in 51 Peg . Following the Ford et al. (1999) model for 51 Peg, the prediction for metallicity abundance is larger than that observed. This means that unless most of the acquired heavy elements are able to diffuse in the radiative interior, the planetesimal-scattering scenario for orbital migration would require more than 90 per cent of the close encounters to result in the outward ejection of planetesimals (Sandquist et al. 1998).

Our model predicts a smaller value for metallicity close to the observed value for 51 Peg. In fact, our model, similarly to that of Murray et al. (1998), explains the migration of the planets by planet–planetesimal interaction, but unlike that of Murray et al. (1998), our model needs less planetesimal mass for radial migration. When the planet reaches its final location, equation 15÷16 (giving the quantity of planetesimals scattered in the star) can be applied after scaling the Murray et al. (1998) result to reflect our disc mass. In the case of the more massive disc , we find and .

An important point to stress is that the plausibility of such an explanation depends, among other things, on the size of the stellar convective envelope at the time of accretion. In order to be efficient, the accretion must take place sufficiently late in the stellar evolution, when the outer convective envelope is shallow. Accretion taking place while the star was still on the pre-main-sequence, and consequently having a large convective envelope, would have little effect on the observed metallicities of stars The time it takes for a planet to migrate to a <0.1 au orbit, in our model, is >107yr. Once the planet stops its migration, planetesimals inside its orbit are quickly cleared out. As shown by Ford et al. (1999), the right time to produce the observed metallicities is at . This value can therefore be regarded as an important constraint for disc models.

5 Conclusions

The discovery (Mayor & Queloz 1995; Marcy & Butler 1996; Butler & Marcy 1996; Butler et al. 1997; Cochran et al. 1997; Noyes et al. 1997) of extrasolar planets has revitalized the discussion on the theory of planetary system formation and evolution. Although close giant-planet formation may be theoretically possible (Wuchterl 1993, 1996), it requires the initial formation of a solid core of at least , which may be difficult to achieve very close to the parent star. It is therefore more likely that Jupiter-mass extrasolar planets cannot form at small heliocentric distances (Boss 1995; Guillot et al. 1996). After these discoveries, the idea that planets can migrate radially (Goldreich & Tremaine 1980; Ward & Hourigan 1989; Lin & Papaloizou 1993; Lin et al. 1996) for long distances has been taken more seriously of late than it has in the past.

In this paper, we have shown that dynamical friction between the planet and a planetesimals disc is an important mechanism for planet migration. We showed that migration of a 1-M J planet to small heliocentric distances (0.05 au) is possible for a disc with a total mass of (we remember that, according to Stepinski & Valageas 1996, and Murray et. al 1998, only ≃1 per cent of the disc mass is in the form of solid particles) if the planetesimal disc does not dissipate during the planet migration or if the disc has and the planetesimals are dissipated in ∼108yr. The model predicts that massive planets can be present at any heliocentric distances for the right value of disc mass and time evolution.

We also showed that the drift velocity of planets and the migration time are very similar to the predictions of the density wave approach (Ward 1997): the drift velocity increases as M for masses smaller than 0.1_M_⊕ and is constant for larger masses. Finally, we showed that the metallicity enhancement observed in several stars having extrasolar planets can also be explained, in much the same way as proposed by Murray et al. (1998) and Ford et al. (1999), by means of the scattering of planetesimals on to the parent star after the planet reached its final configuration. Comparing our model with other models that attempt to explain planetary migration, we think our model has some advantages.

• (1)
  unlike models based on the density wave theory (Goldreich & Tremaine 1980; Ward 1986, 1997), our model does not require a peculiar mechanism to stop the inward migration (Lin et al. 1996). Planet halt is naturally provided by the model. It can explain planets found at heliocentric distances of >0.1 au or planets having larger values of eccentricity. It can explain metallicity enhancements observed in stars having planets in short-period orbits.
• (2)
  unlike the Murray et al. (1998) model, our model shows that radial migration is possible with a not-too-massive planetesimals disc (which is one of the drawbacks of the Murray et al. model) and predicts the right metallicity enhancement.

Acknowledgments

We thank the anonymous referees whose comments and suggestions helped us to improve the quality of this work. We are grateful to E. Ford and E. Spedicato for stimulating discussions during the period in which this work was performed.

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