Secular resonance sweeping of the main asteroid belt during planet migration
Abstract
We calculate the eccentricity excitation of asteroids produced by the sweeping secular resonance during the epoch of planetesimaldriven giant planet migration in the early history of the solar system. We derive analytical expressions for the magnitude of the eccentricity change and its dependence on the sweep rate and on planetary parameters; the sweeping leads to either an increase or a decrease of eccentricity depending on an asteroid’s initial orbit. Based on the slowest rate of sweeping that allows a remnant asteroid belt to survive, we derive a lower limit on Saturn’s migration speed of during the era that the resonance swept through the inner asteroid belt (semimajor axis range –). This rate limit is for Saturn’s current eccentricity, and scales with the square of Saturn’s eccentricity; the limit on Saturn’s migration rate could be lower if Saturn’s eccentricity were lower during its migration. Applied to an ensemble of fictitious asteroids, our calculations show that a prior singlepeaked distribution of asteroid eccentricities would be transformed into a doublepeaked distribution due to the sweeping of the . Examination of the orbital data of main belt asteroids reveals that the proper eccentricities of the known bright () asteroids may be consistent with a doublepeaked distribution. If so, our theoretical analysis then yields two possible solutions for the migration rate of Saturn and for the dynamical states of the premigration asteroid belt: a dynamically cold state (singlepeaked eccentricity distribution with mean of ) linked with Saturn’s migration speed , or a dynamically hot state (singlepeaked eccentricity distribution with mean of ) linked with Saturn’s migration speed .
1 Introduction
The dynamical structure of the Kuiper Belt suggests that the outer solar system experienced a phase of planetesimaldriven migration in its early history (Fernandez & Ip, 1984; Malhotra, 1993, 1995; Hahn & Malhotra, 1999; Levison et al., 2008). Pluto and other Kuiper belt objects that are trapped in mean motion resonances (MMRs) with Neptune are explained by the outward migration of Neptune due to interactions with a more massive primordial planetesimal disk in the outer regions of the solar system (Malhotra, 1993, 1995). In addition, the socalled the scattered disk of the Kuiper belt can also be explained by the outward migration of Neptune (Hahn & Malhotra, 2005), or by the effects of a high eccentricity phase of ice giant planet evolution during the outward migration of Neptune (Levison et al., 2008). The basic premise of planetesimaldriven migration is that the giant planets formed in a more compact configuration than we find them today, and that they were surrounded by a massive () disk of unaccreted icy planetesimals that was the progenitor of the currently observed Kuiper belt (Hahn & Malhotra, 1999). When planetesimals are preferentially scattered either inward (toward the Sun) or outward (away from the Sun), net orbital angular momentum is transferred between the disk and the large body, causing a drift in the large body’s semimajor axis (Fernandez & Ip, 1984; Kirsh et al., 2009). In many simulations of giant planet migration, icy planetesimals are preferentially scattered inward by each of the three outer giant planets (Saturn, Uranus, and Neptune) causing these planets to migrate outward. Due to Jupiter’s large mass, planetesimals that encounter Jupiter are preferentially ejected out of the solar system, leading to a net loss of mass from the solar system and an inward migration of Jupiter.
Planetesimaldriven giant planet migration has been suggested as a cause of the Late Heavy Bombardment (LHB) (Gomes et al., 2005; Strom et al., 2005), however the link between these two events has yet to be definitively established (Chapman et al., 2007; Ćuk et al., 2010; Malhotra & Strom, 2010). Such migration would have enhanced the impact flux of both asteroids and comets onto the terrestrial planets in two ways. First, many of the icy planetesimals scattered by the giant planets would have crossed the orbits of the terrestrial planets. Second, as the giant planets migrated, locations of mean motion and secular resonances would have swept across the asteroid belt, raising the eccentricities of asteroids to planetcrossing values.
Recently, Minton & Malhotra (2009) showed that the patterns of depletion observed in the asteroid belt are consistent with the effects of sweeping of resonances during the migration of the giant planets. The Jupiterfacing sides of some of the Kirkwood gaps (regions of the asteroid belt that are nearly empty due to strong jovian mean motion resonances) are depleted relative to the Sunfacing sides, as would be expected due to the inward migration of Jupiter and the associated inward sweeping of the jovian mean motion resonances. The region within the inner asteroid belt between semimajor axis range – also has excess depletion relative to a model asteroid belt that was uniformly populated and then subsequently sculpted by the gravitational perturbations of the planets over , as would be expected due to the outward migration of Saturn and the associated inward sweeping of a strong secular resonance, the socalled resonance, as explained below. In our 2009 study, we concluded that the semimajor axis distribution of asteroids in the main belt is consistent with the inward migration of Jupiter and outward migration of Saturn by amounts proposed in previous studies based on the Kuiper belt resonance structure(e.g., Malhotra, 1995). However, in that study the migration timescale was not strongly constrained, because only the relative depletion of asteroids in nearby semimajor axis bins could be determined, not their overall level of depletion. In the present paper, we explore in more detail the effect that planet migration would have had on the asteroid belt due to asteroid eccentricity excitation by the sweeping of the secular resonance. From the observed eccentricity distribution of main belt asteroids, we find that it is possible to derive constraints on the secular resonance sweeping timescale, and hence on the migration timescale.
Secular resonances play an important role in the evolution of the main asteroid belt. The inner edge of the belt nearly coincides with the secular resonance which is defined by , where is the rate of precession of the longitude of pericenter, , of an asteroid and is the sixth eigenfrequency of the solar system planets (approximately the rate of precession of Saturn’s longitude of pericenter). The resonance, is important for the delivery of Near Earth Asteroids (NEAs) to the inner solar system (Scholl & Froeschle, 1991). Williams & Faulkner (1981) showed that the location of the resonance actually forms surfaces in space, and Milani & Knezevic (1990) showed that those surfaces approximately define the “inner edge” of the main asteroid belt. However, as mentioned above, Minton & Malhotra (2009) found that, with the planets in their present configuration, planetary perturbations over the age of the solar system cannot fully account for the detailed orbital distribution of the asteroids in the inner asteroid belt. The pattern of excess depletion in inner asteroid belt noted by Minton & Malhotra (2009) is consistent with the effect of the inward sweeping of the secular resonance. In general, the direction of motion of the is anticorrelated with that of Saturn, so an inward sweeping of the would be produced by an outwardly migrating Saturn.
Sweeping, or scanning, secular resonances have been analyzed in a number of previous works. Sweeping secular resonances due to the changing quadrupole moment of the Sun during solar spindown have been explored as a possible mechanism for explaining the eccentricity and inclination of Mercury (Ward et al., 1976). Secular resonance sweeping due to the effects of the dissipating solar nebula just after planet formation has also been investigated as a possible mechanism for exciting the orbital eccentricities of Mars and of the asteroid belt (Heppenheimer, 1980; Ward, 1981). The dissipating massive gaseous solar nebula would have altered the secular frequencies of the solar system planets in a timedependent way, causing locations of secular resonances to possibly sweep across the inner solar system, thereby exciting asteroids into the eccentric and inclined orbits that are observed today. This mechanism was revisited by Nagasawa et al. (2000), who incorporated a more sophisticated treatment of the nebular dispersal. However, O’Brien et al. (2007) have argued that the excitation (and clearing) of the primordial asteroid belt was unlikely due to secular resonance sweeping due to the dispersion of the solar nebula.
The special case of asteroids on initially circular orbits being swept by the and resonances has been investigated by Gomes (1997). In this paper, we consider the more general case of nonzero initial eccentricities; our analysis yields qualitatively new results and provides new insights into the dynamical history of the asteroid belt. This extends the work of Ward et al. (1976) and Gomes (1997) in developing analytical treatments of the effects of sweeping secular resonances on asteroid orbits. In doing so, we have developed an explicit relationship between the migration rate of the giant planets, the initial eccentricity of the asteroid and its initial longitude of perihelion, and the final eccentricity of the asteroid after the passage of the resonance. We show that for initially nonzero asteroid eccentricity, the sweeping of the resonances can either increase or decrease asteroid eccentricities. Examining the orbits of observed main belt asteroids we find evidence for a doublepeaked eccentricity distribution; this supports the case for a history of sweeping. Quantitative comparison of our analytical theory with the semimajor axis and eccentricity distribution of asteroids yields new constraints on the timescale of planet migration.
We note that although our analysis is carried out in the specific context of the sweeping resonance during the phase of planetesimaldriven migration of Jupiter and Saturn, the techniques developed here may be extended to other similar problems, for example, the sweeping of the inclinationnode resonance in the main asteroid belt, the secular resonance in the Kuiper belt, and farther afield, the sweeping of secular resonances in circumstellar or other astrophysical disks.
2 Analytical theory of a sweeping secular resonance
We adopt a simplified model in which a test particle (asteroid) is perturbed only by a single resonance, the resonance. We use a system of units where the mass is in solar masses, the semimajor axis is in units of AU, and the unit of time is y. With this choice, the gravitational constant, , is unity. An asteroid’s secular perturbations close to a secular resonance can be described by the following Hamiltonian function (Malhotra, 1998):
(1) 
where describes the phase of the pth eigenmode of the linearized eccentricitypericenter secular theory for the Solar system planets (Murray & Dermott, 1999), is the associated eigenfrequency, is the asteroid’s longitude of perihelion, is the canonical generalized momentum which is related to the asteroid’s orbital semimajor axis and eccentricity ; and are the canonically conjugate pair of variables in this 1degreeoffreedom Hamiltonian system. The coefficients and are given by:
(2)  
(3) 
where the subscript refers to a planet, is the amplitude of the mode in the planet’s orbit, , is the ratio of the mass of planet to the Sun, and and are Laplace coefficients; the sum is over all major planets. The summations in equations (2)–(3) are over the 8 major planets, for the greatest accuracy; however, we will adopt the simpler twoplanet model of the SunJupiterSaturn in §3, in which case we sum over only the indices referring to Jupiter and Saturn; then is an eigenfrequency of the secular equations of the two planet system.
With fixed values of the planetary masses and semimajor axes, , and are constant parameters in the Hamiltonian given in equation (1). However, during the epoch of giant planet migration, the planets’ semimajor axes change secularly with time, so that , and become timedependent parameters. In the analysis below, we neglect the timedependence of and , and adopt a simple prescription for the timedependence of (see equation 8 below). This approximation is physically motivated: the fractional variation of and for an individual asteroid is small compared to the effects of the “small divisor” during the resonance sweeping event.
It is useful to make a canonical transformation to new variables defined by the following generating function,
(4) 
Thus, and . The new Hamiltonian function is ,
(5) 
where we have retained to denote the canonical momentum, since . It is useful to make a second canonical transformation to canonical eccentric variables,
(6) 
where is the canonical coordinate and is the canonically conjugate momentum. The Hamiltonian expressed in these variables is
(7) 
As discussed above, during planetary migration, the secular frequency is a slowly varying function of time. We approximate its rate of change, , as a constant, so that
(8) 
We define as the epoch of exact resonance crossing, so that (cf. Ward et al., 1976). Then, , and the equations of motion from the Hamiltonian of equation (7) can be written as:
(9)  
(10) 
These equations of motion form a system of linear, nonhomogenous differential equations, whose solution is a linear combination of a homogeneous and a particular solution. The homogeneous solution can be found by inspection, giving:
(11)  
(12) 
where and are constant coefficients. We use the method of variation of parameters to find the particular solution. Accordingly, we replace the constants and in the homogeneous solution with functions and , to seek the particular solution of the form
(13)  
(14) 
Substituting this into the equations of motion we now have:
(15)  
(16) 
therefore
(17)  
(18) 
Equations (17) and (18) do not have a simple closedform solution, but their solution can be expressed in terms of Fresnel integrals (Zwillinger, 1996). The Fresnel integrals are defined as follows:
(19)  
(20) 
and have the following properties:
(21)  
(22)  
(23) 
Therefore
(24)  
(25) 
We denote initial conditions with a subscript , and write the solution to equations (9) and (10) as
(26)  
(27) 
Because the asteroid is swept over by the secular resonance at time , we can calculate the changes in by letting and evaluating the coefficients far from resonance passage, i.e., for , by use of equation (23). Thus we find
(28)  
(29) 
The final value of long after resonance passage is therefore given by
(30) 
With a judicious choice of the initial time, , and without loss of generality, the cosine in the last term becomes , and therefore
(31) 
The asteroid’s semimajor axis is unchanged by the secular perturbations; thus, the changes in reflect changes in the asteroid’s eccentricity . For asteroids with nonzero initial eccentricity, the phase dependence in equation (31) means that secular resonance sweeping can potentially both excite and damp orbital eccentricities. We also note that the magnitude of eccentricity change is inversely related to the speed of planet migration. In linear secular theory (equation (1)), eccentricity and inclination are decoupled, and therefore the effect of the sweeping does not depend on the inclination. However, as Williams & Faulkner (1981) showed, the location of the does depend on inclination, but the dependence is weak for typical inclinations of main belt objects. Nevertheless, there are populations of main belt asteroids at high inclination (such as the Hungaria and Phocaea families), and an analysis of secular resonance sweeping that incorporates coupling between eccentricity and inclination would be valuable for understanding the effects of planet migration on these populations; we leave this to a future investigation.
For small , we can use the approximation . Considering all possible values of , an asteroid with initial eccentricity that is swept by the resonance will have a final eccentricity in the range to , where
(32) 
and
(33) 
Equations (31)–(33) have the following implications:

Initially circular orbits become eccentric, with a final eccentricity .

An ensemble of orbits with the same and initial nonzero but uniform random orientations of pericenter are transformed into an ensemble that has eccentricities in the range to ; this range is not uniformly distributed because of the dependence in equation (31), rather the distribution peaks at the extreme values (see Fig. 1 below).

An ensemble of asteroids having an initial distribution of eccentricities which is a singlepeaked Gaussian (and random orientations of pericenter) would be transformed into one with a doublepeaked eccentricity distribution.
3 Application to the Main Asteroid Belt
In light of the above calculations, it is possible to conclude that if the asteroid belt were initially dynamically cold, that is asteroids were on nearly circular orbits prior to secular resonance sweeping, then the asteroids would be nearly uniformly excited to a narrow range of final eccentricities, the value of which would be determined by the rate of resonance sweeping. Because asteroids having eccentricities above planetcrossing values would be unlikely to survive to the present day, it follows that an initially cold asteroid belt which is uniformly excited by the sweeping will either lose all its asteroids or none. On the other hand, an initially excited asteroid belt, that is a belt with asteroids that had nonzero eccentricities prior to the sweeping, would have asteroids’ final eccentricities bounded by equation (32), allowing for partial depletion and also broadening of its eccentricity distribution. In this section, we apply our theoretical analysis to the problem of the resonance sweeping through the asteroid belt, and compare the theoretical predictions with the observed eccentricity distribution of asteroids.
3.1 Parameters
In order to apply the theory, we must find the location of the resonance as a function of the semimajor axes of the giant planets orbits, and also obtain values for the parameter (equation 3), for asteroids with semimajor axis values in the main asteroid belt. The location of the resonance is defined as the semimajor axis, , where the rate, (equation 2), of pericenter precession of a massless particle (or asteroid) is equal to the eigenfrequency of the solar system. In the current solar system, the frequency is associated with the secular mode with the most power in Saturn’s eccentricity–pericenter variations. During the epoch of planetesimaldriven planet migration, Jupiter migrated by only a small amount but Saturn likely migrated significantly more (Fernandez & Ip, 1984; Malhotra, 1995; Tsiganis et al., 2005), so we expect that the variation in location of the secular resonance is most sensitive to Saturn’s semimajor axis. We therefore adopt a simple model of planet migration in which Jupiter is fixed at and only Saturn migrates. We neglect the effects of the ice giants Uranus and Neptune, as well as secular effects due to the previously more massive Kuiper belt and asteroid belt. With these simplifications, the frequency varies with time as Saturn migrates, so we parametrize as a function of Saturn’s semimajor axis. In contrast with the variation of , there is little variation of the asteroid’s apsidal precession rate, , as Saturn migrates. Thus, finding is reduced to calculating the dependence of on Saturn’s semimajor axis.
To calculate as a function of Saturn’s semimajor axis, we proceed as follows. For fixed planetary semimajor axes, the LaplaceLagrange secular theory provides the secular frequencies and orbital element variations of the planets. This is a linear perturbation theory, in which the disturbing function is truncated to secular terms of second order in eccentricity and first order in mass (Murray & Dermott, 1999). In the planar twoplanet case, the secular perturbations of planet , where is Jupiter and is Saturn, are described by the following disturbing function:
(34) 
where is the mean motion, and is a matrix with elements
(35)  
(36) 
for , , and ; , and
(37) 
The secular motion of the planets is then described by a set of linear differential equations for the eccentricity vectors, ,
(38) 
For fixed planetary semimajor axes, the coefficients are constants, and the solution is given by a linear superposition of eigenmodes:
(39) 
where are the eigenfrequencies of the matrix and are the corresponding eigenvectors; the amplitudes of the eigenvectors and the phases are determined by initial conditions. In our 2planet model, the secular frequencies and depend on the masses of Jupiter, Saturn, and the Sun and on the semimajor axes of Jupiter and Saturn.
For the current semimajor axes and eccentricities of Jupiter and Saturn the LaplaceLagrance theory gives frequency values and , which are lower than the more accurate values given by Brouwer & van Woerkom (1950) by and , respectively (Laskar, 1988). Brouwer & van Woerkom (1950) achieved their more accurate solution by incorporating higher order terms in the disturbing function involving , which arise due to Jupiter and Saturn’s proximity to the 5:2 resonance (the socalled “Great Inequality”). By doing an accurate numerical analysis (described below), we found that the effect of the 5:2 resonance is only important over a very narrow range in Saturn’s semimajor axis. More significant is the perturbation owing to the 2:1 nearresonance of Jupiter and Saturn. Malhotra et al. (1989) developed corrections to the LaplaceLagrange theory to account for the perturbations from resonances in the context of the Uranian satellite system. Applying that approach to our problem, we find that the 2:1 nearresonance between Jupiter and Saturn leads to zeroth order corrections to the elements of the matrix^{1}^{1}1Corrections due to nearresonances are of order , where is eccentricity and is the order of the resonance. The 5:2 is a third order resonance, so its effect is . The 2:1 is a first order resonance, so that its effect does not depend on . Therefore the discrepancy between linear theory and numerical analysis (or the higher order theory of Brouwer & van Woerkom) arising from the Great Inequality would be much less if Jupiter and Saturn were on more circular orbits, but the effect due to the 2:1 resonance would remain.. Including these corrections, we determined the secular frequencies for a range of values of Saturn’s semimajor axis; the result for is shown in Fig. 2a (dashed line).
We also calculated values for the eccentricitypericenter eigenfrequencies by direct numerical integration of the full equations of motion for the twoplanet, planar solar system. In these integrations, Jupiter’s initial semimajor axis was , Saturn’s semimajor axis, , was one of 233 values in the range –, initial eccentricities of Jupiter and Saturn were , and initial inclinations were zero. The initial longitude of pericenter and mean anomalies of Jupiter were and , and Saturn were and . In each case, the planets’ orbits were integrated for 100 myr, and a Fourier transform of the time series of the yields their spectrum of secular frequencies. For regular (nonchaotic) orbits, the spectral frequencies are well defined and are readily identified with the frequencies of the secular solution. The frequency as a function of Saturn’s semimajor axis was obtained by this numerical analysis; the result is shown by the solid line in Fig. 2a.
The comparison between the numerical analysis and the LaplaceLagrange secular theory indicates that the linear secular theory, including the corrections due to the 2:1 nearresonance, is an adequate model for the variation in as a function of . We adopted the latter for its convenience in the needed computations. The value of as a function of Saturn’s semimajor axis was thus found by solving for the value of asteroid semimajor axis where ; was calculated using equation (2) and is the eigenfrequency associated with the eigenmode (at each value of Saturn’s semimajor axis). The result is shown in Fig. 2b.
We also used the analytical secular theory to calculate the eigenvector components in the secular solution of the 2planet system, for each value of Saturn’s semimajor axis. We adopted the same values for the initial conditions of Jupiter and Saturn as in the direct numerical integrations discussed above. Finally, we computed the values of the parameter at each location of the secular resonance. The result is plotted in Fig. 3. Despite the complexity of the computation, the result shown in Fig. 3 is approximated well by a simple exponential curve, , in the semimajor axis range .
3.2 Four test cases
We checked the results of our analytical model against four full numerical simulations of the restricted fourbody problem (the SunJupiterSaturn system with test particle asteroids) in which the test particles in the asteroid belt are subjected to the effects of a migrating Saturn. The numerical integration was performed with an implementation of a symplectic mapping (Wisdom & Holman, 1991; Saha & Tremaine, 1992), and the integration stepsize was . Jupiter and Saturn were the only massive planets that were integrated, and their mutual gravitational influence was included. The asteroids were approximated as massless test particles. The current solar system values of the eccentricity of Jupiter and Saturn were adopted and all inclinations (planets and test particles) were set to zero. An external acceleration was applied to Saturn to cause it to migrate outward linearly and smoothly starting at at the desired rate. As shown in Fig. 2b, the resonance was located at at the beginning of the simulation, and swept inward past the current location of the inner asteroid belt in all simulations. In each of the four simulations, 30 test particles were placed at and given different initial longitudes of pericenter spaced apart. The semimajor axis value of was chosen because it is far away from the complications arising due to strong mean motion resonances. The only parameters varied between each of the four simulations were the initial osculating eccentricities of the test particles, , and the migration speed of Saturn, . The parameters explored were:

, ;

, ;

, ;

, .
These values were chosen to illustrate the most relevant qualitative features. The migration rates of and are slow enough so that the change in eccentricity is substantial, but not so slow that the nonlinear effects at high eccentricity swamp the results. These test cases illustrate both how well the analytical model matches the numerical results, and where the analytical model breaks down.
Two aspects of the analytical model were checked. First, the perturbative equations of motion, equations (9) and (10), were numerically integrated, and their numerical solution compared with the numerical solution from the direct numerical integration of the full equations of motion. For the perturbative solution, we adopted values for that were approximately equivalent to the values of in the full numerical integrations. Second, the eccentricity bounds predicted by the analytical theory, equation (32), were compared with both numerical solutions. The results of these comparisons for the four test cases are shown in Fig. 4. We find that the analytically predicted values of the maximum and minimum final eccentricities (shown as horizontal dashed lines) are in excellent agreement with the final values of the eccentricities found in the numerical solution of the perturbative equations, and in fairly good agreement with those found in the full numerical solution. Not surprisingly, we find that the test particles in the full numerical integrations exhibit somewhat more complicated behavior than the perturbative approximation, and equation (32) somewhat underpredicts the maximum final eccentricity: this may be due to higher order terms in the disturbing function that have been neglected in the perturbative analysis and which become more important at high eccentricity; effects due to close encounters with Jupiter also become important at the high eccentricities.
3.3 Comparison with observed asteroid eccentricities
Does the eccentricity distribution of main belt asteroids retain features corresponding to the effects of the resonance sweeping? To answer this question, we need to know the main belt eccentricity distribution free of observational bias, and also relatively free of the effects of of collisional evolution subsequent to the effects of planetary migration. We therefore obtained the proper elements of the observationally complete sample of asteroids with absolute magnitude from the AstDys online data service (Knežević & Milani, 2003); we excluded from this set the members of collisional families as identified by Nesvorný et al. (2006). These same criteria were adopted in Minton & Malhotra (2010) in a study of the long term dynamical evolution of large asteroids. This sample of 931 main belt asteroids is a good approximation to a complete set of large asteroids that have been least perturbed by either dynamical evolution or collisional evolution since the epoch of the last major dynamical event that occurred in this region of the solar system; therefore this sample likely preserves best the postmigration orbital distribution of the asteroid belt. The proper eccentricity distribution of these asteroids is shown in Fig. 5. This distribution has usually been described in the literature by simply quoting its mean value (and sometimes a dispersion) (Murray & Dermott, 1999; O’Brien et al., 2007). Our best fit single Gaussian distribution to this data has a mean, and standard deviation, , given by and , and is plotted in Fig. 5. However, we also note (by eye) a possible indication of a doublepeak feature in the observed population. Our best fit double Gaussian distribution (modelled as two symmetrical Gaussians with the same standard deviation, but different mean values) to the same data has the following parameters:
More details of how these fits were obtained are described in Appendix A, where we also discuss goodnessoffit of the single and double gaussian distributions. A KolmogorovSinai test shows that there is only a 4.5% probability that the observed eccentricities actually are consistent with the bestfit single Gaussian distribution, and a 73% probability that they are consistent with the bestfit double Gaussian distribution; while the double gaussian is apparently a better fit to the data compared to the single gaussian, a KS probability of only 73% is quite far from a statistically significant level of confidence. A dip test for multimodality in the observed eccentricity distribution (Hartigan & Hartigan, 1985) is also inconclusive; the likely reason for this is that our data sample is not very large, and the separation between the two putative peaks is too small in relation to the dispersion. Nevertheless, we bravely proceed in the next section with the implications of a doubleGaussian eccentricity distribution, with the caveat that some of the conclusions we reach are based on this statistically marginal result.
4 A constraint on Saturn’s migration rate
By relating the secular frequency to the semimajor axis of Saturn, can be related to the migration rate of Saturn, . In this section only the effects of the sweeping secular resonance will be considered, and effects due to sweeping jovian mean motion resonances will be ignored. Because Jupiter is thought to have migrated inward a much smaller distance than Saturn migrated outward during planetesimaldriven migration, the effects due to migrating jovian MMRs were likely confined to narrow regions near strong resonances (Minton & Malhotra, 2009). In the inner asteroid belt between –, these would include the 3:1 and 5:2 resonances, currently located at approximately and , respectively. As shown in Fig. 2b, plausible parameters for the outward migration of Saturn would have allowed the to sweep across the entire inner asteroid belt. Therefore the resonance would have been the major excitation mechanism across the – region of the main belt (and possibly across the entire main asteroid belt, depending on Saturn’s premigration semimajor axis) during giant planet migration.
We used the results of our analytical model to set limits on the rate of migration of Saturn assuming a linear migration profile, with the caveat that many important effects are ignored, such as asteroidJupiter mean motion resonances, and JupiterSaturn mean motion resonances (with the exception of the 2:1 resonance). We have confined our analysis to only the region of the main belt spanning –. Beyond strong jovian mean motion resonance become more numerous. Due to the high probability that the icy planetesimals driving planet migration would be ejected from the solar system by Jupiter, Jupiter likely migrated inward. The migration of Jupiter would have caused strong jovian mean motion resonances to sweep the asteroid belt, causing additional depletion beyond that of the sweeping resonance (Minton & Malhotra, 2009). A further complication is that sweeping jovian mean motion resonances may have also trapped icy planetesimals that entered the asteroid belt region from their source region beyond Neptune (Levison et al., 2009). The effects of these complications are reduced when we consider only the inner asteroid belt. From Fig. 2b, we find that the would have swept the inner asteroid belt region between – when Saturn was between –. Therefore the limits on that we set using the inner asteroid belt as a constraint are only applicable for this particular portion of Saturn’s migration history.
Our theoretically estimated final eccentricity as a function of initial asteroid semimajor axis and eccentricity is shown in Fig. 6 for three different adopted migration rates of Saturn. The larger the initial asteroid eccentricities, the wider the bounds in their final eccentricities. If we adopt the reasonable criterion that an asteroid is lost from the main belt when it achieves a planetcrossing orbit (that is, crossing the orbits of either Jupiter or Mars) and that initial asteroid eccentricities were therefore confined to , then from Fig. 6 Saturn’s migration rate must have been when the resonance was sweeping through the inner asteroid belt. Our results indicate that if Saturn’s migration rate had been slower than when it was migrating across –, then the inner asteroid belt would have been completely swept clear of asteroids by the resonance.
In light of our analysis and the observed dispersion of eccentricities in the asteroid belt (Fig. 5), we can also immediately conclude that the premigration asteroid belt between and had significantly nonzero eccentricities. This is because, as discussed at the top of §3, an initially cold asteroid belt swept by the resonance would either lose all its asteroids or none, and very low initial eccentricities would result in final asteroid eccentricities in a very narrow range of values (cf. equation 32), in contradiction with the fairly wide eccentricity dispersion that is observed. This conclusion supports recent results from studies of planetesimal accretion and asteroid and planet formation that the asteroids were modestly excited at the end of their formation (e.g., Petit et al., 2002).
In Appendix A we show that the doublegaussian distribution is a slightly better fit to the main belt asteroid eccentricity distribution, but the statistical tests do not rule out a singlepeaked distribution. We boldly proceed with considering the implications of the doublepeaked eccentricity distribution to further constrain the migration rate of Saturn, with the caveat that these results can only be said to be consistent with the observations, rather than uniquely constrained by them.
If the presweeping asteroid belt had a Gaussian eccentricity distribution, then the lower peak of the postsweeping asteroid belt should be equal to the lower bound of equation (32). We use the analytical theory to make a rough estimate of the parameter (and hence ) that would yield a final distribution with lower peak near and upper peak near (which is similar to the bestfit double Gaussian in Fig. 5).
Applying equation (32), we see that there are two possible solutions: , and ,. A corresponding migration rate of Saturn can be estimated from the value of using equation (33), and the parameter relationships plotted in Figs. 2 and 3. The former solution () requires a migration rate for Saturn of . We mention this implausible solution here for completeness, but we will not discuss it any further. The latter solution () requires a migration rate for Saturn of . We dub this solution the “cold belt” solution. This rate is comparable to the rates of planet migration found in the “Jumping Jupiter” scenario proposed by Brasser et al. (2009). A third solution exists if we consider that eccentricities in the main belt are restricted by the orbits of Mars and Jupiter on either side, such that stable asteroid orbits do not cross the planetary orbits. This limits asteroid eccentricities to values such that neither the aphelion of the asteroid crosses the perihelion distance of Jupiter, nor the perihelion of the asteroid crosses the aphelion distance of Mars. Maximum asteroid eccentricity is therefore a function of semimajor axis, where , where and are planet aphelion and perihelion, respectively, and is the semimajor axis of the asteroid. In this case, an initial single Gaussian eccentricity distribution with a mean greater than would be severely truncated, therefore we need only fit the lower peak of the double Gaussian distribution at . Applying equation (32), we find that provides a good fit. The corresponding migration rate of Saturn is . We dub this solution the “hot belt” solution.
We illustrate the two possible solutions for an ensemble of hypothetical asteroids having semimajor axes uniformly distributed randomly in the range to . In Fig. 7a the initial eccentricity distribution is modeled as a Gaussian distribution with a mean and a standard deviation of . This initial standard deviation was chosen so that the final standard deviation would be the same as that of the observed main belt. Fig. 7b shows the eccentricity distribution after resonance sweeping has occurred due to the migration of Saturn at a rate of . The final distribution was calculated with equation (31); we used values of shown in Fig. 3, and the value of was calculated with the aid of Fig. 2 which relates the value of to the semimajor axis of Saturn. As expected, when an ensemble of asteroids with a singlepeaked eccentricity distribution is subjected to the sweeping secular resonance, the result is a doublepeaked eccentricity distribution. Because of the slight bias towards the upper limit of the eccentricity excitation band, proportionally more asteroids are found in the upper peak.
In Fig. 7c the initial eccentricity distribution is modeled as a truncated Gaussian: a Gaussian with mean and standard deviation , but truncated at the semimajor axis–dependent Marscrossing value. We used equation (31) to calculate the eccentricity distribution of this hypothetical ensemble after resonance sweeping with . Again, allowing that only those asteroids whose final eccentricities are below the Marscrossing value will remain, the resulting postmigration eccentricity distribution is shown in Fig. 7d. In this case, we find the lower peak at the same eccentricity value as the lower peak in the observed main belt distribution (see Fig. 5).
In both cases of possible solutions (initially cold main belt with and ; and initially hot main belt with and ), the theoretical models yield an excess of asteroids with eccentricities greater than than in the observed main belt. However, as shown by Minton & Malhotra (2010), on gigayear timescales, the population of the asteroid belt is dynamically more unstable than the population. Thus, both solutions may be consistent with the observations, as postsweeping dynamical erosion could result in a final eccentricity distribution resembling more closely the observed distribution.
The estimates of Saturn’s migration rate quoted above depend strongly on the eccentricities of the giant planets during their migration. In deriving the above estimates, we adopted the present values of the giant planets’ orbital eccentricities. The resonance strength coefficient (equation 3) is proportional to the amplitude of the mode, which is related to the eccentricities of the giant planets (namely Saturn and Jupiter). From equation (39), and the definition , the value of is a linear combination of the eccentricities of the giant planets. Because Saturn is the planet with the largest amplitude of the mode, from equation (31) the relationship between the sweep rate and the value of Saturn’s eccentricity is approximately . Therefore, to increase the limiting timescale by a factor of ten would only require that the giant planets’ eccentricities were their current value (i.e., ).
5 Conclusion and Discussion
Based on the existence of the inner asteroid belt, we conclude that Saturn’s migration rate must have been as Saturn migrated from to (as the resonance migrated from 2.8 AU to 2.1 AU). Migration rates lower than would be inconsistent with the survival of any asteroids in the inner main belt, as the secular resonance would have excited asteroid eccentricities to planetcrossing values. This lower limit for the migration rate of Saturn assumes that Jupiter and Saturn had their current orbital eccentricities; the migration rate limit is inversely proportional to the square of the amplitude of the secular mode; if Jupiter and Saturn’s eccentricities were their current value (i.e., ) during the planet migration epoch, the limit on the migration rate decreases by a factor of . (This caveat also applies to the migration rate limits quoted below.)
Our analysis of secular resonance sweeping predicts that a singlepeaked eccentricity distribution will be transformed into a doublepeaked eccentricity by secular resonance sweeping. We find that the observed eccentricity distribution of asteroids may be consistent with a doublepeaked distribution function, although we acknowledge that the statistics are poor. We used a doubleGaussian function to model the observed asteroid eccentricity distribution to set even tighter, albeit modeldependent, constraints on the migration rate of Saturn.
We identified two possible migration rates that depend on the premigration dynamical state of the main asteroid belt. The first, the “cold primordial belt” solution, has an asteroid belt with an initial eccentricity distribution modeled as a Gaussian with ; Saturn’s migration rate of yields a final eccentricity distribution consistent with the observed asteroid belt. The second, the “hot primordial belt” solution, has an asteroid belt with an initial eccentricity distribution modeled as a Gaussian with , but truncated above the Marscrossing value of eccentricity; in this case, Saturn’s migration rate of is generally consistent with the observed asteroid belt.
Each of these solutions has very different implications for the primordial excitation and depletion of the main asteroid belt. The cold belt solution, with , would lead to little depletion of the asteroid belt during giant planet migration, as the resonance would be unable to raise eccentricities to Marscrossing values. This implies an initial dynamically quite cold asteroid belt with not more than times the mass of the current main belt; the latter estimate comes from accounting for dynamical erosion over the age of the solar system (Minton & Malhotra, 2010).
The hot belt solution, with , would lead to loss of asteroids directly due to excitation of eccentricities above planetcrossing values, by about a factor of . In this case, the main asteroid belt was more dynamically excited prior to resonance sweeping than we find it today. This implies much greater loss of asteroids prior to sweeping, as the peak of the eccentricity distribution would be near the Marscrossing value, and subject to strong dynamical erosion (Minton & Malhotra, 2010).
Each of these solutions has different implications for the model that the Late Heavy Bombardment of the inner solar system is linked to the epoch of planetesimaldriven giant planet migration (Gomes et al., 2005; Strom et al., 2005). These implications will be explored in a future paper.
We remind the reader that in order to elucidate the effects of resonance sweeping, we have made a number of simplifying assumptions to arrive at an analytically tractable model. These simplifications include neglecting the effects of planets other than Jupiter and Saturn, the effects of sweeping jovian mean motion resonances on asteroids, the effects of a presumed massive Kuiper belt during the epoch of planet migration, and the selfgravity and collisional interactions of a previously more massive asteroid belt. In addition, our analysis was carried out in the planar approximation, thereby neglecting any eccentricityinclination coupling effects. These neglected effects can be expected to reduce somewhat the lower limit on Saturn’s migration speed that we have derived, because in general they would reduce the effectiveness of the in exciting asteroid eccentricities. Perhaps more importantly, giant planet migration would also lead to the sweeping of the main asteroid belt by the inclination secular resonance (Williams & Faulkner, 1981) whose effects could be used to infer additional constraints.
Recently, Morbidelli et al. (2010) found through numerical simulations with slow rates of planet migration (efolding timescales exceeding 5 Myr) that the surviving asteroids in the main belt tend to be clumped around mean motion resonances. The semimajor axis distribution of survivors is found very different from the observed distribution for asteroids, and also that a large proportion of survivors have inclinations above 20, both inconsistent with the current main asteroid belt; comparisons made with the unbiased main belt asteroid distributions as described in Minton & Malhotra (2009). This is likely due to the effects of the sweeping inclinationlongitude of ascending node secular resonance, analogous to the sweeping eccentricitypericenter secular resonance that we analyzed in the present study. While the effects of the sweeping resonance are analogous to the , only affecting inclinations instead of eccentricities, a full analysis of the asteroid belt inclinations is beyond the scope of the present work, but will be explored in a future study.
A number of other studies have derived limits on the speed of planetesimaldriven giant planet migration. MurrayClay & Chiang (2005) exclude an folding migration timescale to confidence based on the lack of a large observed asymmetry in the population of Kuiper belt objects in the two libration centers of the 2:1 Neptune mean motion resonance. Boué et al. (2009) exclude based on the observed obliquity of Saturn. The latter lower limit on the migration timescale is slightly incompatible with the lower limit on the rate of Saturn’s migration of we derive based on the existence of the inner asteroid belt. One way these can be reconciled is if Saturn’s orbital eccentricity were a factor smaller than its present value as it migrated from 8.5 AU to 9.2 AU; then, some mechanism would need to have increased Saturn’s eccentricity up to its present value by the time Saturn reached its present semimajor axis of .
Appendix A The Main belt eccentricity distribution
The binned eccentricity distribution may be modeled as a Gaussian probability distribution function, given by:
(A1) 
where is the standard deviation, is the mean, and is the random variable; in our case is the eccentricity. With an appropriate scaling factor, equation (A1) can be used to model the number of asteroids per eccentricity bin. However, rather than fit the binned distribution, we instead perform a least squares fit of the unbinned sample to the Gaussian cumulative distribution function given by:
(A2) 
For the eccentricities of our sample of 931 main belt asteroids, the best fit parameters are:
We also fit the data to a doubleGaussian distribution,
(A3) 
The cumulative distribution function for equation (A3) is
(A4) 
For the eccentricities of our sample of 931 main belt asteroids, we performed a least squares fit to equation (A4) and obtained the following bestfit parameters:
We evaluated the goodness of fit using the KolmogorovSmirnov (KS) test. The KS test determines the probability that two distributions are the same, or in our case how well our model distributions fit the observed data (Press et al., 1992). The KS test compares the cumulative distribution of the data against the model cumulative distribution function. We found that our asteroid sample has a probability of that it comes from the best fit single Gaussian (equation (A2)), but a probability of that it comes from the doubleGaussian (equation (A4)). Therefore, the KS tests indicate that the doubleGaussian is a better fit to the data than the singleGaussian.
We performed Hartigan’s dip test (Hartigan & Hartigan, 1985) to test whether the observational data is consistent with a multipeaked distribution. Hartigan’s dip test calculates the probability that the distribution being tested has a single peak. Applying Hartigan’s dip test to a given distribution yields in a test statistic; together with the sample size, the test statistic is matched to a pvalue range in a precomputed table provided by Hartigan & Hartigan (1985). The pvalue is a measure of the probability that the distribution actually has only one peak (the nullhypothesis, for this problem). The smaller the calculated pvalue, the less likely is the distribution to have a single peak and the more likely it is to have at least two significant peaks. A pvalue of indicates that the nullhypothesis is very unlikely, and that the given distribution has more than one peak. We applied the dip test to the eccentricity distribution of our sample of 931 main belt asteroids; we determined that the test statistic is . From Hartigan & Hartigan (1985), this corresponds to a pvalue range of (based on a sample size of 1000). This indicates that the Hartigan dip test does not rule out the null hypothesis, i.e., a singlepeaked distribution cannot be ruled out.
To further aid the interpretation of this test, we compare this result to that obtained by applying Hartigan’s dip test to synthetic distributions that were explicitly doublepeaked by construction. Ten model distributions (of 1000 eccentricity values) were generated from the doublegaussian function of equation (A4), with parameter values , , and (same parameter values as the bestfit for our sample of asteroids). Only the random seed was varied between each model distribution. Applying the Hartigan dip test, we find that the test statistic ranged between and , corresponding to pvalues between and . This means that, according to the dip test, the null hypothesis – i.e., a singlepeaked distribution – could not be ruled out for any of the model distributions (since none had ), despite the fact that they were each drawn from an explicitly doublepeaked distribution by construction. This illustrates that even with a sample size of nearly 1000, the dispersion in eccentricity around the two peaks is too large in comparison to the distance between the peaks, that the dip test is not sufficiently sensitive to detect the doublepeaked underlying distribution. We interpret this to mean that the results of both the KS test and Hartigan’s dip test indicate that the main asteroid belt eccentricity distribution is consistent with being drawn from a doublepeaked distribution, but that this cannot be definitely shown.
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