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The Solar Nebula, Secular Resonances, Gas Drag, and the Asteroid Belt
ICARUS
129, 134–146 (1997) IS975782
ARTICLE NO.
The Solar Nebula, Secular Resonances, Gas Drag, and the Asteroid Belt Myron Lecar and Fred Franklin Harvard–Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138 E-mail: [email protected] Received November 1, 1996; revised May 29, 1997
We assume that the asteroids were initially distributed uniformly from Mars to Jupiter. Orbits between the 3 : 2 resonance with Jupiter (0.763 a(J), where a(J) is Jupiter’s semimajor axis) and Jupiter were chaotic because of overlapping mean motion resonances and were rapidly removed by close encounters with Jupiter. However, in numerical simulations extending as long as 108 years, we have been unsuccessful in removing asteroids between 0.630 and 0.763 a(J) (the 2 : 1 resonance and the 3 : 2 resonance with Jupiter), when Jupiter and Saturn were the only perturbers. We now suggest that a combination of the secular resonance between the apsidal rotations of Jupiter and the asteroids, moved outward by the gravitational potential of the primitive solar nebula and the gas drag of that nebula, removed those asteroids. A minimum mass solar nebula will move one of the secular resonances outward from its present location at 0.40 a(J) to the 3 : 2 resonance. The sweeping secular resonance pumped up the eccentricities of the asteroids causing many outer belt asteroids to be ejected by close encounters with Jupiter. The high eccentricities also increased the relative motion of the gas and the asteroids, greatly enhancing the efficiency of the drag, which caused other asteroids to spiral into the inner Solar System. That nebula, removed on a time scale of 104 –105 years, cleared the asteroid belt exterior to the 2 : 1 resonance and increased the eccentricities of the remaining asteroids to an average value of about 0.15. We found that a much longer time scale would have removed all the asteroids, while a much shorter time scale would not have increased the initial low eccentricities. 1997 Academic Press
INTRODUCTION
We suggested in 1973 (Lecar and Franklin 1973) that the asteroids were initially distributed uniformly between Mars and Saturn. In numerical simulations, we removed the asteroids between Jupiter and Saturn and the asteroids interior to Jupiter down to the 3 : 2 resonance (Franklin et al. 1989, Soper et al. 1990, Lecar et al. 1992a,b). Wisdom (1980) showed that the orbits of asteroids closer to Jupiter than to the 3 : 2 resonance were chaotic due to overlapping first order mean motion resonances with Jupiter. More 134 0019-1035/97 $25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
recently, we extended our integrations to longer than 108 years (Lecar et al. 1992b) but did not succeed in clearing the asteroid belt between the 2 : 1 resonance and the 3 : 2 resonance. In 1976, Ward et al. suggested that the dispersal of the solar nebula would provide a mechanism for moving the secular resonances. Ward (1980) first proposed that scanning secular resonances (i.e., secular resonances that moved through the asteroid belt as the primitive solar nebular dissipated) could act as a ‘‘cosmogonical broom.’’ Shortly afterward, Heppenheimer (1980) showed that the secular resonances, displaced outward by the primitive solar nebula, could have been responsible for the eccentricities of Mars and the asteroids. He further argued that the dissipation time was a few times 104 years because longer times would have increased the eccentricity of Mars above its present value. Ward (1981) presented a more complete treatment of his earlier calculations and suggested that the dispersal time was between 104 and 105 years in order to avoid pumping up the eccentricity of the Earth in excess of its present value. Our time scale, based on the criterion that we increase the eccentricity of the main belt asteroids but do not remove all the asteroids, is in the same range. A simple, readable treatment of the effects of scanning secular resonances was given by Lemaitre and Dubru (1991). We noticed that the nebula mass required to move the g(6) term in the secular resonance out to the 3 : 2 resonance would be accompanied by significant gas drag. This happens to be the nebula mass proposed by Hayashi (1981) who first argued for the importance of gas drag. Hayashi’s mass estimate was based on the minimum mass to form the planets and was unconcerned with secular resonances. More recently, Ida and Lin (1996) also investigated the effects of gas drag, but without secular resonances. They succeeded in moving asteroids outside the 2 : 1 resonance into the main belt (interior to the 2 : 1 resonance), but their asteroids had much lower eccentricities than the present asteroids, and in addition, the time scales for the drag to operate were correspondingly longer than we derive here.
SECULAR RESONANCE, GAS DRAG, AND THE ASTEROID BELT
Since there are very few asteroids between the 2 : 1 and 3 : 2 resonances, and we intend to use the secular resonance to remove them, we require the secular resonance to be displaced out to the 3 : 2 resonance. This criterion sets the mass scale of the primitive solar nebula, which is almost identical with the minimum mass nebula proposed by Hayashi (1981). We are impressed by this coincidence. It turns out that the secular resonance and gas drag work together to move and to remove asteroids. Without the action of the secular resonance to increase the eccentricity, the asteroids move almost with the gas and the effect of gas drag is minimal. There still is a slight relative motion in the tangential direction because the gas is partially supported by pressure while the asteroids are not, so the gas moves slightly slower than the circular orbital velocity of the asteroids (Whipple 1972). However, the gas has no radial motion, so the relatively high eccentricities induced by the secular resonance provide the relative motion that gives the gas drag something to bite into. In fact, we worried that the gas drag might completely damp the induced eccentricities. However, for a wide range of nebula dispersal times, that did not happen. The drag did tend to reduce the eccentricities but the secular resonance more than offset that effect, and as the nebula evaporated, the asteroids were left with significant eccentricities. Other mechanisms have been studied to both clear the outer asteroid belt and build up the eccentricities of the asteroids that remained. Perhaps the earliest proposal was that collisions by asteroidal objects scattered by close encounters with Jupiter, or collisions and close encounters between larger objects (lunar to martian size) scattered by each other during the process of planet formation, accomplished these aims. The most recent references that we are aware of are Wetherill (1992, 1990) and Ip (1989). It is difficult to evaluate these scenarios without a large scale numerical simulation. We are skeptical that collisions could have cleaned the regions now devoid of asteroids so efficiently that so few were left behind. More recently, Liou and Malhotra (1997) suggested that mean-motion resonances could have scanned the asteroid belt if Jupiter was formed further out and migrated inward to its present position. If the migration occurred in the presence of the gaseous nebula, then gas drag should also be included in that scenario. If the migration occurred after the nebula dissipated, then our mechanism would have preceded theirs. We think their mechanism is plausible and would work if, for some reason, our mechanism failed. We are partial to the ‘‘scanning secular resonances’’ proposed by Ward but we did worry that the effect of gas drag might damp the eccentricities too severely. Our main result is that the gas drag, which was certainly present, changed the way the secular resonance cleared the outer belt (bodies lose energy and spiral in rather than have close encounters with Jupiter) but allows those asteroids
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FIG. 1. Number of minor planets vs semimajor axis. This plot, due to Gareth Williams of the Minor Planet Center, contains over 13,000 objects with well determined orbits.
that survive to remain with substantial eccentricities. We note here that we considered asteroids with the average radius of 10 km. Since the effect of gas drag varies with
FIG. 2. A companion to Fig. 1, giving eccentricity vs semimajor axis. Note that the average eccentricity of the main belt asteroids lies near 0.15.
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the (inverse) first power of the radius, our results could encompass radii up to 50 km, but the larger asteroids with radii in excess of 100 km were probably unaffected by the drag, and for them, Ward’s mechanism worked without modification. The mass of the nebula was determined by requiring that the secular resonance be at (or slightly beyond) the 3 : 2 mean-motion resonance. We assumed that the density profile of the nebula was that used by Hayashi; namely, that the surface density fell off with the 3/2 power of the distance from the Sun. That left the dispersal time as the only free parameter. We set the surface density, s (t) 5 s (0) exp(2t/ t), where t is the ‘‘e-folding time’’ of the dispersal of the nebula. (Henceforth, we will just refer to t as the dispersal time.) If the dispersal time scale was very short, the secular resonance didn’t have a chance to pump up the eccentricities. That limit was less than 104 years. If the dispersal time scale was too long, asteroids with initial distances 0.6 a(J) or greater had their eccentricities built up quickly and were removed by encounters with Jupiter or spiraled into the inner Solar System. That, in itself, might not be a problem as probably a substantial amount of material has been removed from the region of the asteroids (Wetherill 1992). But Ward’s argument that much longer times result in increasing the Earth’s eccentricity above its present value sets an upper limit to the time scale.
The characteristics of the present asteroid belt are displayed in Figs. 1 and 2, which were prepared for this report by Gareth Williams. Figure 1 shows the number of asteroids vs their semimajor axis (in units of a(J)). One of our tasks, in this study, was to explain the scarcity of asteroids between a 5 0.630 (the 2 : 1 resonance) and a 5 0.763 (the 3 : 2 resonance). The scarcity of asteroids exterior to the 3 : 2 resonance has already been accounted for by our previous numerical simulations and by Wisdom’s derivation of the extent of the overlapping mean-motion resonances. Figure 2 shows the eccentricities of the asteroids vs their semimajor axes. We adopt the argument that the high eccentricities were induced by the sweeping secular resonances; i.e., perturbations by Jupiter and Saturn in the present configuration of planets could not increase the eccentricities to such high values. However, we added the effects of gas drag. We were relieved to find that although the gas drag tended to decrease the eccentricities, it did not eradicate the effects of the secular resonance. In addition, we focused on removing asteroids from the region between the 2 : 1 and 3 : 2 resonances. The gas drag accomplished that by removing energy from the orbits and causing the semimajor axes to decrease. It turns out that even without the gas drag, the secular resonance would have removed outer belt asteroids but by another mechanism; they would have had close encounters with Jupiter. However, clearing the outer belt
FIG. 3. Location of the g(5) and g(6) secular resonances as a function of the nebula density normalized to 122 g cm22 (the density that brings g(6) to a 5 0.763). The dashed vertical lines mark density ranges where the eccentricities were driven to at least 0.3.
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FIG. 4. A simulation with a(initial) 5 0.46 a(J), e(initial) 5 0.02, so 5 122 g cm22, and an e-folding dispersal time of 104 T(J). Part (a) shows (1) the presence of one secular resonance [g(5)] between Jupiter (small filled triangles) and the asteroid (crosses) near t 5 20,000 T(J) and (2) entry into a distinct second secular resonance [g(6)], near t 5 50,000 T(J). Part (b) plots eccentricity growth arising from the action of the two secular resonances. Part (c) gives semimajor axis vs time. After t 5 45,000 T(J), when the nebula has largely dissipated, the body lies in the g(6) secular resonance at a 5 0.395.
was not a problem that was addressed by Heppenheimer or Ward. THE SECULAR RESONANCE
We begin with a note on units. Unless otherwise indicated, distances will be in units of Jupiter’s semimajor axis, a(J) 5 5.20280 AU, and times will be in units of Jupiter’s period, T(J) 5 11.86223 years. The perturbations of Saturn on Jupiter’s orbit cause Jupiter’s apse to progress (i.e., to rotate in the direction of its orbital motion). Averaging over the orbital periods of Jupiter and Saturn yields the following expressions for the longitude of Jupiter’s perihelion (g˜ J,S) and Jupiter’s eccentricity (eJ,S) (Franklin et al. 1989) e 2J,S 5 A2 1 B2 1 2AB cos(a 2 b)t
(1)
eJ,S sin(g˜ J,S) 5 A sin(at) 1 B sin (bt)
(2)
and
with A 5 0.0445, B 5 20.0165, 2f/a 5 2.579672 3 105 T(J), and 2f/b 5 3.877938 3 103 T(J). The constants were
chosen to fit the numerical integrations of Cohen et al. (1973). The perturbations of Jupiter on an asteroid cause the longitude of the asteroid’s perihelion to progress at a rate that is, in the present asteroid belt, considerably faster than Jupiter’s (about 14 times faster just interior to the 3 : 2 resonance). The period of the precession of the asteroid’s perihelion caused by Jupiter is given by T(g˜ ast,J) 5 exp(11.0525 2 7.0618 a) in units of T(J), (3) where a is the semimajor axis of the asteroid in units of a(J). Thus when the asteroid’s semimajor axis is 0.395, T(g˜ ast,J) 5 exp(8.2631) 5 3878 T(J), the rotation of the asteroid’s apse, and the short period term in the rotation of Jupiter’s apse, g(6) (in the absence of the nebula), have the same frequency. (We have adopted the notation of calling the short period term in the rotations of Jupiter’s apse g(6) because it has the same period as the rotation of Saturn’s apse, which is usually designated g(6).) At present, this delineates the inner edge of the asteroid belt. Equation (3) was a fit to the results of numerical integrations (in the absence of a nebula) but with Jupiter’s eccentricity and longitude of perihelion given by Eqs. (1) and (2).
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FIG. 4—Continued
For asteroids closer to Jupiter, the asteroid’s apse rotates significantly faster than Jupiter’s apse. To cause the two rates to be the same, i.e., to cause a secular resonance, the nebula must slow down the rotation of the asteroid’s perihelion, speed up the rotation of Jupiter’s perihelion, or both. Ward suggested that Jupiter carved out a gap in the nebula, and in that case, the nebula caused Jupiter’s apsidal line to progress by an amount given by
gJ,neb 5 [2fGs /aJnJ][4Ïj ][o n(2n 1 1)Anj 2n /(4n 1 1)],
(4)
where nJ is Jupiter’s mean motion, the sum goes from n equals 1 to infinity, j 5 1 2 D, where D is the half-width of the gap in units of Jupiter’s distance, and An 5 [(2n)!/ 22n(n!)2]. Note that the gap is symmetric about Jupiter’s semimajor axis. We assumed that the gap extended to the
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FIG. 5. Plots that differ from Fig. 4 only by a 5 times faster dispersal time; i.e., t(dispersal) 5 2000 T(J). Part (a) shows that the decreasing nebula density has moved a secular resonance initially at a 5 0.76, into the vicinity of this body at a 5 0.46 at t 5 5000 T(J). Part (b) indicates that the more rapid removal time has increased the eccentricity only to 0.2. Part (c) shows that the faster dispersal results in a smaller reduction in the body’s semimajor axis.
3 : 2 resonance, whence j 5 0.763. With that choice, gJ,neb 5 [fGs /aJnJ][1.503954].
gast,J 1 gast,neb 5 gJ,S 1 gJ,neb . (5)
The perturbations of Jupiter and the nebula on the asteroid were numerically integrated. The acceleration of the nebula on the asteroid is aneb 5 2fGs F
(6)
in the radial direction, where G is the gravitational constant, s is the surface density of the nebula, and F 5 o [4/4n 1 1][(2n)!/22n(n!)2] with the sum over n going from 0 to y. Numerically, F 5 4.377345. The time-averaged value of the asteroid’s apse (calculated analytically) is gast,neb 5 2fGs F/4an.
(7)
The sense of the effect of the nebula is to cause the asteroid’s apse to regress. Thus the nebula, with a gap centered on a(J), speeds up rotation of Jupiter’s apse and slows down the rotation of the asteroid’s apse, both effects contributing to bringing the asteroid into a secular resonance. The condition for a secular resonance at a semimajor axis, a, is that
(8)
We remind the reader that the term gJ,S is the time derivative of the longitude of Jupiter’s perihelion, which, to a high degree of accuracy, contains two dominant terms, one of which is the same as the frequency of Saturn’s perihelion. Saturn makes itself felt only through this term; it is not included in the numerical integrations. Using Hayashi’s formulation,
s (a) 5 soa23/2,
(9)
we write the condition for secular resonance as
so 5 [(Mo 1 MJ)/fa 2(J)][exp(28.26306) 3 P(a)] (10a) with P(a) 5 hexp[2.78936(a/0.395) 2 1] 2 1j/h1.50395 1 0.72764/aj,
(10b) where Mo is the mass of the Sun. The value of so required to displace the g(6) term in the secular resonance to a 5 0.763 (i.e., the 3 : 2 resonance) is 122 g/cm2, which is 85% of Hayashi’s value. (Note that
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FIG. 5—Continued
Hayashi’s constant of 1700 has to be multiplied by (5.2028)23/2 to put the distance in units of a(J) instead of AU, which reduces it to 143). The numerator of Eq. (10b) results from the difference between the perihelion rotations of the asteroid (due to Jupiter) and Jupiter (due to Saturn) and dominates the
location of the secular resonance. The denominator is provided by the effects of nebula on the rotation of Jupiter’s and the asteroid’s perihelion. If, for example, we neglect the effect of the nebula on Jupiter completely, the constant 1.50395 is removed from the denominator; that would increase so by a factor of about 2 (for 0.395 , a , 0.763).
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FIG. 6. A case near the center of the main belt, differing from Fig. 4 only because a(initial) 5 0.56 and the dispersal time is 2500 T(J). Although entry into or passage through secular resonances dominates the dynamical behavior of all bodies, passage through the 3 : 1 mean motion resonance at a 5 0.481 generates an enhanced eccentricity which, as can be seen in part (c), the drag can exploit.
In fact, Ward admits to some uncertainty in the shape of the gap which Jupiter carved out of the nebula. This would change the numerical value of the constant (1.503954) which would, in turn, change the numerical value of so . The values of so required to place the short period term in the secular resonance [g(6)] and the long period term [g(5)] at distance a, as given by Eqs. (10), are displayed in Fig. 3. In Fig. 3, the vertical lines represent numerical verifications; values of so in the indicated ranges are capable of increasing asteroid eccentricities to at least 0.3. THE GAS DRAG
The deceleration caused by the gas drag is given by adrag 5 2(3/8)( rgas / rast rast)VrelVrel ,
(11a)
where Vrel 5 [Vr]r 1 [Vu 2 Vc(1 2 h)]u
(11b)
and r and u are unit vectors in the radial and tangential direction, and boldfaced characters represent vectors. In Eq. (11b), V 2c 5 GMo /r and h 5 0.0040 r 1/2 (h represents the pressure support of the gas which results in its circular velocity to be somewhat less than that of the asteroid). See Ida and Lin (1996), but note that our numerical value reflects our convention that r is units of a(J). We adopt
the nebula proposed by Hayashi (1981) which gives, in our units, s 5 143.25 r 23/2 g/cm2. The gas density is given by rgas 5 s /2h with 2h 5 9.825 3 1012 r 1.25 cm. (We depart from our convention and give 2h in cm so that s divided by 2h will give r in g/cm3, but note that r is in units of a(J).) We maintain this relation between the surface density and the space density even as the nebula evaporates. This amounts to keeping the temperature of the nebula constant. We set d 5 (8/3)( rast rast / rgas) 5 7128 r 2.75, for asteroids with radii of 10 km and densities of 3 g/cm3. The numerical simulations used Eqs. (11). The main effect of the drag can be estimated by letting adrag > d/dt Vr , and Vrel > Vr . The evolution of Vr > eVc is given by Vr(t)/Vr(o) 5 1/[1 1 Vr(o)t/d].
(12)
Equation (12) predicts that the eccentricity decreases by a factor of 2 in a time t1/2 5 d/e(0)Vc 5 2160 T(J) (using e(0) 5 0.1 r 5 0.6 a(J) so Vc 5 16.9 km/sec), a time in the range of acceptable nebula dispersal times. This remains valid for asteroids a few times larger than 10 km, but for asteroids with radii in excess of 100 km, the gas drag was largely ineffective. THE NUMERICAL SIMULATIONS
The numerical integrations were done with a fourth order Taylor series, similar to what we have used previously,
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FIG. 6—Continued
but of lower order. The numerical integrations were confined to a plane and so we did not investigate the effects of the secular resonance and gas drag on the inclinations. This would be a worthwhile extension. Figures 4a–4c illustrate an interesting example in which an asteroid, thanks to the drag, passes through one secular resonance and finally becomes lodged in the g(6) secular resonance. For
times in the range 18,000 to 30,000 T(J), Fig. 4a shows that the precession of Jupiter’s apse as driven almost completely by the nebula and the asteroid’s apse as forced by the nebula and Jupiter have, on average, the same rate. When lying in the first secular resonance, the body therefore experiences a surge in eccentricity, shown in Fig. 4b, from e 5 0 to e . 0.3. Note that the increase in the eccentricity,
SECULAR RESONANCE, GAS DRAG, AND THE ASTEROID BELT
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FIG. 7. A fairly unusual case in which a body in the outer region (a(initial) 5 0.68) is captured into the main belt. More frequently, such objects experience close jovian encounters and may collide with that planet. This one, as shown in part (c), made a close approach (its distance to Jupiter was 0.103 5 0.535 AU) before being captured into a stable orbit with a 5 0.585 and e 5 0.28.
in the presence of the nebula, proceeds more rapidly than if the nebula were absent (cf. Scholl and Froeschle 1991). This is primarily because the nebula speeds up the rotations of both Jupiter’s apse and the asteroid’s apse (by a factor of three or more). In addition, in the presence of the nebula, the asteroid is somewhat closer to Jupiter when it is caught in the secular resonance. As a further check on the rapid increase in eccentricity, we performed a number of integrations in which we artificially speeded up the rotation of Jupiter’s apse to 2000 and 1000 T(J), but simplified the simulation by removing the nebula. In both cases, the eccentricity of an asteroid in secular resonance increased in 1–2 rotations of Jupiter’s apse. In units of T(J), the increase was twice as fast when the period of Jupiter’s apse was shortened by one-half. Figure 4a plots the longitudes of the perihelia of the asteroid (x’s) and Jupiter (filled triangles). The vertical axis is in degrees, and the horizontal axis is time. Figure 4b shows the eccentricity versus time and demonstrates that once either secular resonance takes hold, the eccentricity climbs rapidly. Figure 4c shows the semimajor axis versus time. Once the eccentricity climbs, the drag extracts energy from the orbit and the semimajor axis decreases from 0.46 to less than 0.40. Thus this asteroid spirals into the inner Solar System, interior to the main asteroid belt. In contrast, Figs. 5a–5c start with the same orbit but the dispersal time of the nebula was five times shorter. In this
case the maximum eccentricity was only slightly less but the asteroid remained in the secular resonance for a shorter time and its semimajor axis decreased only slightly (from 0.460 to 0.450). This asteroid remained in the main belt with a substantially enhanced eccentricity so we accept this dispersal time. Figures 6a–6c illustrate an example where an asteroid started with a 5 0.56 which decayed to a 5 0.45, with a final eccentricity of 0.3. Figures 7a–7b illustrate a rare case of an outer belt asteroid (a(initial) 5 0.68) that ends up being captured into the main belt. In general, our results can be summarized as follows. For short dispersal times, on the order of 1000 T(J) or less, nothing much happens. The eccentricities remain below 0.1 and the drag has little effect on the semimajor axes. For longer dispersal times, but less than about 5000 T(J), asteroids remain in the main belt (0.40 , a , 0.63) or spiral into the main belt from further out. The essential point is that they remain with substantial eccentricities. For even longer dispersal times, all the asteroids are removed or spiral into a narrow region (a , 0.45) near the inner edge of the main belt. A sample of final semimajor axes, a(final), and final eccentricities, e(final), for three dispersal times is given in Table I. In Table I and in Table II, the symbols .... denote collision with Jupiter and ,,,, denote spiraling into the inner Solar System belt (a , 0.395). Finally, we were curious to see how the original scenario envisioned by Ward, which neglected gas drag, would play
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FIG. 7—Continued
out. In Table II, we present final eccentricities for three values of the dispersal time, without gas drag. In this case, the only role of the dispersal time is to set the rate of sweeping of the secular resonance; e.g., short dispersal times do not have a chance to substantially increase the eccentricity. Since there is no drag, the semimajor axes are
unaffected. As predicted, the eccentricities of the main belt asteroids are elevated to approximately the observed values. Furthermore, the secular resonance does clear the outer belt by producing encounters with Jupiter rather than by spiraling bodies into the inner Solar System and it does increase the eccentricities of the main belt asteroids.
SECULAR RESONANCE, GAS DRAG, AND THE ASTEROID BELT
TABLE I Final a’s and e’s for a Few Dispersal Times (Td)
a(initial) 0.750 0.740 0.730 0.720 0.710 0.700 0.690 0.680 0.670 0.660 0.640 0.620 0.600 0.580 0.560 0.540 0.520 0.500 0.480 0.460 0.440 0.420
Td 5 5000 T(J)
Td 5 2500 T(J)
Td 5 2000 T(J)
a(final) e(final)
a(final) e(final)
a(final) e(final)
,,,, 0.430 .... ,,,, ,,,, ,,,, ,,,, ,,,, ,,,, 0.425 ,,,, 0.429 0.437 ,,,, ,,,, ,,,, 0.424 ,,,, 0.432 0.436 0.416 ,,,,
0.423 ,,,, .... 0.670 ,,,, .... .... 0.404 ,,,, 0.447 0.422 ,,,, ,,,, 0.500 0.444 0.454 0.455 0.428 0.456 0.451 0.433 0.413
0.422 ,,,, .... 0.680 ,,,, 0.575 ,,,, ,,,, ,,,, 0.437 0.528 0.440 ,,,, 0.548 0.545 0.524 0.503 0.476 0.466 0.450 0.432 0.414
0.11
0.14 0.09 0.10
0.30 0.12 0.08 0.16
0.14
0.12
0.10 0.18 0.39
0.19 0.26 0.30 0.27 0.36 0.28 0.13 0.06 0.17
0.16
0.21 0.13
0.18 0.09 0.06 0.15 0.11 0.16 0.18 0.23 0.21 0.20 0.21 0.30
Note. .... encounter with Jupiter, ,,,, a(final) , 0.395.
CONCLUSIONS
We suggest that the secular resonance between the rotation of the longitude of perihelia of Jupiter and asteroids
TABLE II Final Eccentricities, e(final) for Three Dispersal Times with No Drag Td 5 5000 T(J)
Td 5 2000 T(J)
Td 5 400 T(J)
a(initial)
e(final)
e(final)
e(final)
0.74 0.72 0.70 0.68 0.66 0.64 0.62 0.60 0.58 0.56 0.54 0.52 0.50 0.48 0.46 0.44 0.42
.... .... .... .... .... 0.14 .... .... 0.33 .... 0.46 0.29 0.27 .... 0.30 0.12 0.38
.... .... .... .... .... 0.13 .... .... 0.24 0.14 0.21 0.20 0.23 0.19 0.22 0.24 0.21
.... 0.10 .... .... .... 0.14 0.16 0.11 0.10 0.09 0.09 0.09 0.10 0.09 0.07 0.08 0.11
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pumped up the eccentricities of the asteroids from initial values near zero to values even in excess of 0.3. The asteroids initially closer to Jupiter than the 2 : 1 resonance (0.630 a(J)) were removed by close encounters with Jupiter, or because gas drag removed so much energy from their orbits that they spiraled into the inner Solar System. For asteroids with radii in excess of 100 km the gas drag was ineffective and they were removed by close encounters with Jupiter. The removal beyond 0.630 is so efficient that we suggest this may have been responsible for removing as much as an Earth mass of rocky material from that region. We pointed out in 1973, as did Weidenschilling (1977), that, compared to the smooth r 23/2 decay in the surface density obtained by smearing out the rocky material of the planets (Hayashi 1981), there is a drop of more than a factor of 1000 in the vicinity of the asteroid belt. This material could have been removed by the mechanism proposed here: a pumping of the eccentricity by the secular resonance, followed by either a close encounter with Jupiter or a spiraling into the inner Solar System. Since the present mass of the asteroid belt is on the order of 1024 Earth masses, the mechanism would had to have been very efficient. We have not yet made enough runs to verify that the statistics work out, but the results so far are suggestive that the mechanism would operate successfully. In addition to removing asteroids from 0.630 , a , 0.763, there has been a problem of producing the rather high eccentricities of the remaining asteroids. The secular resonance seems to also account for them. We were encouraged by the coincidence that the value of so needed to displace the secular resonance from its present position at 0.40 to 0.76 was so close to the value of so derived by Hayashi as the minimum nebula required to form the planets. We were also encouraged by the coincidence between the dispersal times which we derived so as not to remove all the asteroids and the value that Ward derived to keep the eccentricity of the Earth from becoming unacceptably large. Thus this mechanism, to remove most of the asteroids between 0.630 and 0.763 a(J) and to enhance the eccentricities of the remaining asteroids, seems to follow naturally from a primitive solar nebula just massive enough to form the planets. As far as this study is concerned, the nebula could have lasted for an indefinite time in a stationary state. Once it starts to disperse, and the secular resonance migrates inward, we limit the dispersal time to less than about 100,000 years. ACKNOWLEDGMENT We dedicate this paper to Fred L. Whipple on the occasion of his 90th birthday. Professor Whipple established the Smithsonian Astrophysical Observatory in Cambridge more than four decades ago. He has been our mentor and friend, as well as a friend to the other small bodies in the Solar System.
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REFERENCES Cohen, C. J., E. C. Hubbard, and C. Oesterwinter 1973. Elements of the outer planets for one million years. Astron. Pap. Am. Eph. Naut Alman. 22, 3–92. Franklin, F., M. Lecar, and P. Soper 1989. On the original distribution of the asteroids II. Do stable orbits exist between Jupiter and Saturn? Icarus 79, 223–227. Hayashi, C. 1981. Formation of the planets. In Fundamental Problems in the Theory of Stellar Evolution, IAU Symposium No. 93 (D. Sugimoto, D. Q. Lamb, and D. N. Schramm, Eds.), pp. 113–128. D. Reidel, Dordrecht. Heppenheimer, T. A. 1980. Secular resonances and the origin of the eccentricities of Mars and the asteroids. Icarus 41, 76–88. Ida, S., and D. N. C. Lin 1996. Long term gas drag effect on the structure of the asteroid belt and its implications for the solar nebula. Astron. J. 112, 1239–1246. Ip, W.-H. 1989. Dynamical injection of Mars-sized planetoids into the asteroidal belt from the terrestrial planetary accretion zone. Icarus 78, 270–279. Lecar, M., and F. Franklin 1973. On the original distribution of asteroids I. Icarus 20, 422–436. Lecar, M., F. Franklin, and M. Murison 1992a. On predicting long-term orbital instability: A relation between the Lyapunov time and sudden orbital transitions. Astron. J. 104, 1230–1236. Lecar, M., F. Franklin, and P. Soper 1992b. On the original distribution of the asteroids IV. Numerical experiments in the outer asteroid belt. Icarus 96, 234–250.
Lemaitre, A., and P. Dubru 1991. Secular resonances in the primitive solar nebula. Celestial Mech. Dynam. Astron. 52, 57–78. Liou, J-C., and R. Malhotra 1997. Depletion of the outer asteroid belt. Science 275, 375–377. Scholl, H., and Ch. Froeschle 1991. The n6 secular resonance near 2 AU: A possible source of meteorites. Astron. Astrophys. 245, 316–321. Soper, P., F. Franklin, and M. Lecar 1990. On the original distribution of the asteroids III. Orbits between Jupiter and Saturn. Icarus 87, 265–284. Ward, W. R. 1980. Scanning secular resonances: A cosmological broom? Abstracts of Papers Submitted to the Eleventh Lunar and Planetary Science Conference, Lunar and Planetary Institute, Houston, Texas. Ward, W. R. 1981. Solar nebula dispersal and the stability of the planetary system I. Scanning secular resonance theory. Icarus 47, 234–264. Ward, W. R., G. Colombo, and F. Franklin 1976. Secular resonance, solar spin down and the orbit of Mercury. Icarus 28, 441–452. Weidenschilling, S. 1977. The distribution of mass in the planetary system and the solar nebula. Astrophs. Space Sci. 51, 153–158. Wetherill, G. 1989. Origin of the asteroid belt. In Asteroids II (R. P. Binzel, T. Gehrels, and M. S. Mathews, Eds.), Univ. of Arizona Press, Tucson, AZ. Wetherill, G. 1992. An alternative model for the formation of the asteroids. Icarus 100, 307–325. Whipple, F. L. 1972. On certain aerodynamic processes for asteroids and comets. In From Plasma to Planets: Proc. Nobel Symposium No. 21 (A. Elvius, Ed.), pp. 211–232. Wiley, New York. Wisdom, J. 1980. The resonance overlap criterion and the onset of stochastic behavior in the restricted three-body problem. Astron. J. 85, 1122– 1133.
129, 134–146 (1997) IS975782
ARTICLE NO.
The Solar Nebula, Secular Resonances, Gas Drag, and the Asteroid Belt Myron Lecar and Fred Franklin Harvard–Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138 E-mail: [email protected] Received November 1, 1996; revised May 29, 1997
We assume that the asteroids were initially distributed uniformly from Mars to Jupiter. Orbits between the 3 : 2 resonance with Jupiter (0.763 a(J), where a(J) is Jupiter’s semimajor axis) and Jupiter were chaotic because of overlapping mean motion resonances and were rapidly removed by close encounters with Jupiter. However, in numerical simulations extending as long as 108 years, we have been unsuccessful in removing asteroids between 0.630 and 0.763 a(J) (the 2 : 1 resonance and the 3 : 2 resonance with Jupiter), when Jupiter and Saturn were the only perturbers. We now suggest that a combination of the secular resonance between the apsidal rotations of Jupiter and the asteroids, moved outward by the gravitational potential of the primitive solar nebula and the gas drag of that nebula, removed those asteroids. A minimum mass solar nebula will move one of the secular resonances outward from its present location at 0.40 a(J) to the 3 : 2 resonance. The sweeping secular resonance pumped up the eccentricities of the asteroids causing many outer belt asteroids to be ejected by close encounters with Jupiter. The high eccentricities also increased the relative motion of the gas and the asteroids, greatly enhancing the efficiency of the drag, which caused other asteroids to spiral into the inner Solar System. That nebula, removed on a time scale of 104 –105 years, cleared the asteroid belt exterior to the 2 : 1 resonance and increased the eccentricities of the remaining asteroids to an average value of about 0.15. We found that a much longer time scale would have removed all the asteroids, while a much shorter time scale would not have increased the initial low eccentricities. 1997 Academic Press
INTRODUCTION
We suggested in 1973 (Lecar and Franklin 1973) that the asteroids were initially distributed uniformly between Mars and Saturn. In numerical simulations, we removed the asteroids between Jupiter and Saturn and the asteroids interior to Jupiter down to the 3 : 2 resonance (Franklin et al. 1989, Soper et al. 1990, Lecar et al. 1992a,b). Wisdom (1980) showed that the orbits of asteroids closer to Jupiter than to the 3 : 2 resonance were chaotic due to overlapping first order mean motion resonances with Jupiter. More 134 0019-1035/97 $25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
recently, we extended our integrations to longer than 108 years (Lecar et al. 1992b) but did not succeed in clearing the asteroid belt between the 2 : 1 resonance and the 3 : 2 resonance. In 1976, Ward et al. suggested that the dispersal of the solar nebula would provide a mechanism for moving the secular resonances. Ward (1980) first proposed that scanning secular resonances (i.e., secular resonances that moved through the asteroid belt as the primitive solar nebular dissipated) could act as a ‘‘cosmogonical broom.’’ Shortly afterward, Heppenheimer (1980) showed that the secular resonances, displaced outward by the primitive solar nebula, could have been responsible for the eccentricities of Mars and the asteroids. He further argued that the dissipation time was a few times 104 years because longer times would have increased the eccentricity of Mars above its present value. Ward (1981) presented a more complete treatment of his earlier calculations and suggested that the dispersal time was between 104 and 105 years in order to avoid pumping up the eccentricity of the Earth in excess of its present value. Our time scale, based on the criterion that we increase the eccentricity of the main belt asteroids but do not remove all the asteroids, is in the same range. A simple, readable treatment of the effects of scanning secular resonances was given by Lemaitre and Dubru (1991). We noticed that the nebula mass required to move the g(6) term in the secular resonance out to the 3 : 2 resonance would be accompanied by significant gas drag. This happens to be the nebula mass proposed by Hayashi (1981) who first argued for the importance of gas drag. Hayashi’s mass estimate was based on the minimum mass to form the planets and was unconcerned with secular resonances. More recently, Ida and Lin (1996) also investigated the effects of gas drag, but without secular resonances. They succeeded in moving asteroids outside the 2 : 1 resonance into the main belt (interior to the 2 : 1 resonance), but their asteroids had much lower eccentricities than the present asteroids, and in addition, the time scales for the drag to operate were correspondingly longer than we derive here.
SECULAR RESONANCE, GAS DRAG, AND THE ASTEROID BELT
Since there are very few asteroids between the 2 : 1 and 3 : 2 resonances, and we intend to use the secular resonance to remove them, we require the secular resonance to be displaced out to the 3 : 2 resonance. This criterion sets the mass scale of the primitive solar nebula, which is almost identical with the minimum mass nebula proposed by Hayashi (1981). We are impressed by this coincidence. It turns out that the secular resonance and gas drag work together to move and to remove asteroids. Without the action of the secular resonance to increase the eccentricity, the asteroids move almost with the gas and the effect of gas drag is minimal. There still is a slight relative motion in the tangential direction because the gas is partially supported by pressure while the asteroids are not, so the gas moves slightly slower than the circular orbital velocity of the asteroids (Whipple 1972). However, the gas has no radial motion, so the relatively high eccentricities induced by the secular resonance provide the relative motion that gives the gas drag something to bite into. In fact, we worried that the gas drag might completely damp the induced eccentricities. However, for a wide range of nebula dispersal times, that did not happen. The drag did tend to reduce the eccentricities but the secular resonance more than offset that effect, and as the nebula evaporated, the asteroids were left with significant eccentricities. Other mechanisms have been studied to both clear the outer asteroid belt and build up the eccentricities of the asteroids that remained. Perhaps the earliest proposal was that collisions by asteroidal objects scattered by close encounters with Jupiter, or collisions and close encounters between larger objects (lunar to martian size) scattered by each other during the process of planet formation, accomplished these aims. The most recent references that we are aware of are Wetherill (1992, 1990) and Ip (1989). It is difficult to evaluate these scenarios without a large scale numerical simulation. We are skeptical that collisions could have cleaned the regions now devoid of asteroids so efficiently that so few were left behind. More recently, Liou and Malhotra (1997) suggested that mean-motion resonances could have scanned the asteroid belt if Jupiter was formed further out and migrated inward to its present position. If the migration occurred in the presence of the gaseous nebula, then gas drag should also be included in that scenario. If the migration occurred after the nebula dissipated, then our mechanism would have preceded theirs. We think their mechanism is plausible and would work if, for some reason, our mechanism failed. We are partial to the ‘‘scanning secular resonances’’ proposed by Ward but we did worry that the effect of gas drag might damp the eccentricities too severely. Our main result is that the gas drag, which was certainly present, changed the way the secular resonance cleared the outer belt (bodies lose energy and spiral in rather than have close encounters with Jupiter) but allows those asteroids
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FIG. 1. Number of minor planets vs semimajor axis. This plot, due to Gareth Williams of the Minor Planet Center, contains over 13,000 objects with well determined orbits.
that survive to remain with substantial eccentricities. We note here that we considered asteroids with the average radius of 10 km. Since the effect of gas drag varies with
FIG. 2. A companion to Fig. 1, giving eccentricity vs semimajor axis. Note that the average eccentricity of the main belt asteroids lies near 0.15.
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the (inverse) first power of the radius, our results could encompass radii up to 50 km, but the larger asteroids with radii in excess of 100 km were probably unaffected by the drag, and for them, Ward’s mechanism worked without modification. The mass of the nebula was determined by requiring that the secular resonance be at (or slightly beyond) the 3 : 2 mean-motion resonance. We assumed that the density profile of the nebula was that used by Hayashi; namely, that the surface density fell off with the 3/2 power of the distance from the Sun. That left the dispersal time as the only free parameter. We set the surface density, s (t) 5 s (0) exp(2t/ t), where t is the ‘‘e-folding time’’ of the dispersal of the nebula. (Henceforth, we will just refer to t as the dispersal time.) If the dispersal time scale was very short, the secular resonance didn’t have a chance to pump up the eccentricities. That limit was less than 104 years. If the dispersal time scale was too long, asteroids with initial distances 0.6 a(J) or greater had their eccentricities built up quickly and were removed by encounters with Jupiter or spiraled into the inner Solar System. That, in itself, might not be a problem as probably a substantial amount of material has been removed from the region of the asteroids (Wetherill 1992). But Ward’s argument that much longer times result in increasing the Earth’s eccentricity above its present value sets an upper limit to the time scale.
The characteristics of the present asteroid belt are displayed in Figs. 1 and 2, which were prepared for this report by Gareth Williams. Figure 1 shows the number of asteroids vs their semimajor axis (in units of a(J)). One of our tasks, in this study, was to explain the scarcity of asteroids between a 5 0.630 (the 2 : 1 resonance) and a 5 0.763 (the 3 : 2 resonance). The scarcity of asteroids exterior to the 3 : 2 resonance has already been accounted for by our previous numerical simulations and by Wisdom’s derivation of the extent of the overlapping mean-motion resonances. Figure 2 shows the eccentricities of the asteroids vs their semimajor axes. We adopt the argument that the high eccentricities were induced by the sweeping secular resonances; i.e., perturbations by Jupiter and Saturn in the present configuration of planets could not increase the eccentricities to such high values. However, we added the effects of gas drag. We were relieved to find that although the gas drag tended to decrease the eccentricities, it did not eradicate the effects of the secular resonance. In addition, we focused on removing asteroids from the region between the 2 : 1 and 3 : 2 resonances. The gas drag accomplished that by removing energy from the orbits and causing the semimajor axes to decrease. It turns out that even without the gas drag, the secular resonance would have removed outer belt asteroids but by another mechanism; they would have had close encounters with Jupiter. However, clearing the outer belt
FIG. 3. Location of the g(5) and g(6) secular resonances as a function of the nebula density normalized to 122 g cm22 (the density that brings g(6) to a 5 0.763). The dashed vertical lines mark density ranges where the eccentricities were driven to at least 0.3.
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FIG. 4. A simulation with a(initial) 5 0.46 a(J), e(initial) 5 0.02, so 5 122 g cm22, and an e-folding dispersal time of 104 T(J). Part (a) shows (1) the presence of one secular resonance [g(5)] between Jupiter (small filled triangles) and the asteroid (crosses) near t 5 20,000 T(J) and (2) entry into a distinct second secular resonance [g(6)], near t 5 50,000 T(J). Part (b) plots eccentricity growth arising from the action of the two secular resonances. Part (c) gives semimajor axis vs time. After t 5 45,000 T(J), when the nebula has largely dissipated, the body lies in the g(6) secular resonance at a 5 0.395.
was not a problem that was addressed by Heppenheimer or Ward. THE SECULAR RESONANCE
We begin with a note on units. Unless otherwise indicated, distances will be in units of Jupiter’s semimajor axis, a(J) 5 5.20280 AU, and times will be in units of Jupiter’s period, T(J) 5 11.86223 years. The perturbations of Saturn on Jupiter’s orbit cause Jupiter’s apse to progress (i.e., to rotate in the direction of its orbital motion). Averaging over the orbital periods of Jupiter and Saturn yields the following expressions for the longitude of Jupiter’s perihelion (g˜ J,S) and Jupiter’s eccentricity (eJ,S) (Franklin et al. 1989) e 2J,S 5 A2 1 B2 1 2AB cos(a 2 b)t
(1)
eJ,S sin(g˜ J,S) 5 A sin(at) 1 B sin (bt)
(2)
and
with A 5 0.0445, B 5 20.0165, 2f/a 5 2.579672 3 105 T(J), and 2f/b 5 3.877938 3 103 T(J). The constants were
chosen to fit the numerical integrations of Cohen et al. (1973). The perturbations of Jupiter on an asteroid cause the longitude of the asteroid’s perihelion to progress at a rate that is, in the present asteroid belt, considerably faster than Jupiter’s (about 14 times faster just interior to the 3 : 2 resonance). The period of the precession of the asteroid’s perihelion caused by Jupiter is given by T(g˜ ast,J) 5 exp(11.0525 2 7.0618 a) in units of T(J), (3) where a is the semimajor axis of the asteroid in units of a(J). Thus when the asteroid’s semimajor axis is 0.395, T(g˜ ast,J) 5 exp(8.2631) 5 3878 T(J), the rotation of the asteroid’s apse, and the short period term in the rotation of Jupiter’s apse, g(6) (in the absence of the nebula), have the same frequency. (We have adopted the notation of calling the short period term in the rotations of Jupiter’s apse g(6) because it has the same period as the rotation of Saturn’s apse, which is usually designated g(6).) At present, this delineates the inner edge of the asteroid belt. Equation (3) was a fit to the results of numerical integrations (in the absence of a nebula) but with Jupiter’s eccentricity and longitude of perihelion given by Eqs. (1) and (2).
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FIG. 4—Continued
For asteroids closer to Jupiter, the asteroid’s apse rotates significantly faster than Jupiter’s apse. To cause the two rates to be the same, i.e., to cause a secular resonance, the nebula must slow down the rotation of the asteroid’s perihelion, speed up the rotation of Jupiter’s perihelion, or both. Ward suggested that Jupiter carved out a gap in the nebula, and in that case, the nebula caused Jupiter’s apsidal line to progress by an amount given by
gJ,neb 5 [2fGs /aJnJ][4Ïj ][o n(2n 1 1)Anj 2n /(4n 1 1)],
(4)
where nJ is Jupiter’s mean motion, the sum goes from n equals 1 to infinity, j 5 1 2 D, where D is the half-width of the gap in units of Jupiter’s distance, and An 5 [(2n)!/ 22n(n!)2]. Note that the gap is symmetric about Jupiter’s semimajor axis. We assumed that the gap extended to the
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FIG. 5. Plots that differ from Fig. 4 only by a 5 times faster dispersal time; i.e., t(dispersal) 5 2000 T(J). Part (a) shows that the decreasing nebula density has moved a secular resonance initially at a 5 0.76, into the vicinity of this body at a 5 0.46 at t 5 5000 T(J). Part (b) indicates that the more rapid removal time has increased the eccentricity only to 0.2. Part (c) shows that the faster dispersal results in a smaller reduction in the body’s semimajor axis.
3 : 2 resonance, whence j 5 0.763. With that choice, gJ,neb 5 [fGs /aJnJ][1.503954].
gast,J 1 gast,neb 5 gJ,S 1 gJ,neb . (5)
The perturbations of Jupiter and the nebula on the asteroid were numerically integrated. The acceleration of the nebula on the asteroid is aneb 5 2fGs F
(6)
in the radial direction, where G is the gravitational constant, s is the surface density of the nebula, and F 5 o [4/4n 1 1][(2n)!/22n(n!)2] with the sum over n going from 0 to y. Numerically, F 5 4.377345. The time-averaged value of the asteroid’s apse (calculated analytically) is gast,neb 5 2fGs F/4an.
(7)
The sense of the effect of the nebula is to cause the asteroid’s apse to regress. Thus the nebula, with a gap centered on a(J), speeds up rotation of Jupiter’s apse and slows down the rotation of the asteroid’s apse, both effects contributing to bringing the asteroid into a secular resonance. The condition for a secular resonance at a semimajor axis, a, is that
(8)
We remind the reader that the term gJ,S is the time derivative of the longitude of Jupiter’s perihelion, which, to a high degree of accuracy, contains two dominant terms, one of which is the same as the frequency of Saturn’s perihelion. Saturn makes itself felt only through this term; it is not included in the numerical integrations. Using Hayashi’s formulation,
s (a) 5 soa23/2,
(9)
we write the condition for secular resonance as
so 5 [(Mo 1 MJ)/fa 2(J)][exp(28.26306) 3 P(a)] (10a) with P(a) 5 hexp[2.78936(a/0.395) 2 1] 2 1j/h1.50395 1 0.72764/aj,
(10b) where Mo is the mass of the Sun. The value of so required to displace the g(6) term in the secular resonance to a 5 0.763 (i.e., the 3 : 2 resonance) is 122 g/cm2, which is 85% of Hayashi’s value. (Note that
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FIG. 5—Continued
Hayashi’s constant of 1700 has to be multiplied by (5.2028)23/2 to put the distance in units of a(J) instead of AU, which reduces it to 143). The numerator of Eq. (10b) results from the difference between the perihelion rotations of the asteroid (due to Jupiter) and Jupiter (due to Saturn) and dominates the
location of the secular resonance. The denominator is provided by the effects of nebula on the rotation of Jupiter’s and the asteroid’s perihelion. If, for example, we neglect the effect of the nebula on Jupiter completely, the constant 1.50395 is removed from the denominator; that would increase so by a factor of about 2 (for 0.395 , a , 0.763).
SECULAR RESONANCE, GAS DRAG, AND THE ASTEROID BELT
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FIG. 6. A case near the center of the main belt, differing from Fig. 4 only because a(initial) 5 0.56 and the dispersal time is 2500 T(J). Although entry into or passage through secular resonances dominates the dynamical behavior of all bodies, passage through the 3 : 1 mean motion resonance at a 5 0.481 generates an enhanced eccentricity which, as can be seen in part (c), the drag can exploit.
In fact, Ward admits to some uncertainty in the shape of the gap which Jupiter carved out of the nebula. This would change the numerical value of the constant (1.503954) which would, in turn, change the numerical value of so . The values of so required to place the short period term in the secular resonance [g(6)] and the long period term [g(5)] at distance a, as given by Eqs. (10), are displayed in Fig. 3. In Fig. 3, the vertical lines represent numerical verifications; values of so in the indicated ranges are capable of increasing asteroid eccentricities to at least 0.3. THE GAS DRAG
The deceleration caused by the gas drag is given by adrag 5 2(3/8)( rgas / rast rast)VrelVrel ,
(11a)
where Vrel 5 [Vr]r 1 [Vu 2 Vc(1 2 h)]u
(11b)
and r and u are unit vectors in the radial and tangential direction, and boldfaced characters represent vectors. In Eq. (11b), V 2c 5 GMo /r and h 5 0.0040 r 1/2 (h represents the pressure support of the gas which results in its circular velocity to be somewhat less than that of the asteroid). See Ida and Lin (1996), but note that our numerical value reflects our convention that r is units of a(J). We adopt
the nebula proposed by Hayashi (1981) which gives, in our units, s 5 143.25 r 23/2 g/cm2. The gas density is given by rgas 5 s /2h with 2h 5 9.825 3 1012 r 1.25 cm. (We depart from our convention and give 2h in cm so that s divided by 2h will give r in g/cm3, but note that r is in units of a(J).) We maintain this relation between the surface density and the space density even as the nebula evaporates. This amounts to keeping the temperature of the nebula constant. We set d 5 (8/3)( rast rast / rgas) 5 7128 r 2.75, for asteroids with radii of 10 km and densities of 3 g/cm3. The numerical simulations used Eqs. (11). The main effect of the drag can be estimated by letting adrag > d/dt Vr , and Vrel > Vr . The evolution of Vr > eVc is given by Vr(t)/Vr(o) 5 1/[1 1 Vr(o)t/d].
(12)
Equation (12) predicts that the eccentricity decreases by a factor of 2 in a time t1/2 5 d/e(0)Vc 5 2160 T(J) (using e(0) 5 0.1 r 5 0.6 a(J) so Vc 5 16.9 km/sec), a time in the range of acceptable nebula dispersal times. This remains valid for asteroids a few times larger than 10 km, but for asteroids with radii in excess of 100 km, the gas drag was largely ineffective. THE NUMERICAL SIMULATIONS
The numerical integrations were done with a fourth order Taylor series, similar to what we have used previously,
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FIG. 6—Continued
but of lower order. The numerical integrations were confined to a plane and so we did not investigate the effects of the secular resonance and gas drag on the inclinations. This would be a worthwhile extension. Figures 4a–4c illustrate an interesting example in which an asteroid, thanks to the drag, passes through one secular resonance and finally becomes lodged in the g(6) secular resonance. For
times in the range 18,000 to 30,000 T(J), Fig. 4a shows that the precession of Jupiter’s apse as driven almost completely by the nebula and the asteroid’s apse as forced by the nebula and Jupiter have, on average, the same rate. When lying in the first secular resonance, the body therefore experiences a surge in eccentricity, shown in Fig. 4b, from e 5 0 to e . 0.3. Note that the increase in the eccentricity,
SECULAR RESONANCE, GAS DRAG, AND THE ASTEROID BELT
143
FIG. 7. A fairly unusual case in which a body in the outer region (a(initial) 5 0.68) is captured into the main belt. More frequently, such objects experience close jovian encounters and may collide with that planet. This one, as shown in part (c), made a close approach (its distance to Jupiter was 0.103 5 0.535 AU) before being captured into a stable orbit with a 5 0.585 and e 5 0.28.
in the presence of the nebula, proceeds more rapidly than if the nebula were absent (cf. Scholl and Froeschle 1991). This is primarily because the nebula speeds up the rotations of both Jupiter’s apse and the asteroid’s apse (by a factor of three or more). In addition, in the presence of the nebula, the asteroid is somewhat closer to Jupiter when it is caught in the secular resonance. As a further check on the rapid increase in eccentricity, we performed a number of integrations in which we artificially speeded up the rotation of Jupiter’s apse to 2000 and 1000 T(J), but simplified the simulation by removing the nebula. In both cases, the eccentricity of an asteroid in secular resonance increased in 1–2 rotations of Jupiter’s apse. In units of T(J), the increase was twice as fast when the period of Jupiter’s apse was shortened by one-half. Figure 4a plots the longitudes of the perihelia of the asteroid (x’s) and Jupiter (filled triangles). The vertical axis is in degrees, and the horizontal axis is time. Figure 4b shows the eccentricity versus time and demonstrates that once either secular resonance takes hold, the eccentricity climbs rapidly. Figure 4c shows the semimajor axis versus time. Once the eccentricity climbs, the drag extracts energy from the orbit and the semimajor axis decreases from 0.46 to less than 0.40. Thus this asteroid spirals into the inner Solar System, interior to the main asteroid belt. In contrast, Figs. 5a–5c start with the same orbit but the dispersal time of the nebula was five times shorter. In this
case the maximum eccentricity was only slightly less but the asteroid remained in the secular resonance for a shorter time and its semimajor axis decreased only slightly (from 0.460 to 0.450). This asteroid remained in the main belt with a substantially enhanced eccentricity so we accept this dispersal time. Figures 6a–6c illustrate an example where an asteroid started with a 5 0.56 which decayed to a 5 0.45, with a final eccentricity of 0.3. Figures 7a–7b illustrate a rare case of an outer belt asteroid (a(initial) 5 0.68) that ends up being captured into the main belt. In general, our results can be summarized as follows. For short dispersal times, on the order of 1000 T(J) or less, nothing much happens. The eccentricities remain below 0.1 and the drag has little effect on the semimajor axes. For longer dispersal times, but less than about 5000 T(J), asteroids remain in the main belt (0.40 , a , 0.63) or spiral into the main belt from further out. The essential point is that they remain with substantial eccentricities. For even longer dispersal times, all the asteroids are removed or spiral into a narrow region (a , 0.45) near the inner edge of the main belt. A sample of final semimajor axes, a(final), and final eccentricities, e(final), for three dispersal times is given in Table I. In Table I and in Table II, the symbols .... denote collision with Jupiter and ,,,, denote spiraling into the inner Solar System belt (a , 0.395). Finally, we were curious to see how the original scenario envisioned by Ward, which neglected gas drag, would play
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FIG. 7—Continued
out. In Table II, we present final eccentricities for three values of the dispersal time, without gas drag. In this case, the only role of the dispersal time is to set the rate of sweeping of the secular resonance; e.g., short dispersal times do not have a chance to substantially increase the eccentricity. Since there is no drag, the semimajor axes are
unaffected. As predicted, the eccentricities of the main belt asteroids are elevated to approximately the observed values. Furthermore, the secular resonance does clear the outer belt by producing encounters with Jupiter rather than by spiraling bodies into the inner Solar System and it does increase the eccentricities of the main belt asteroids.
SECULAR RESONANCE, GAS DRAG, AND THE ASTEROID BELT
TABLE I Final a’s and e’s for a Few Dispersal Times (Td)
a(initial) 0.750 0.740 0.730 0.720 0.710 0.700 0.690 0.680 0.670 0.660 0.640 0.620 0.600 0.580 0.560 0.540 0.520 0.500 0.480 0.460 0.440 0.420
Td 5 5000 T(J)
Td 5 2500 T(J)
Td 5 2000 T(J)
a(final) e(final)
a(final) e(final)
a(final) e(final)
,,,, 0.430 .... ,,,, ,,,, ,,,, ,,,, ,,,, ,,,, 0.425 ,,,, 0.429 0.437 ,,,, ,,,, ,,,, 0.424 ,,,, 0.432 0.436 0.416 ,,,,
0.423 ,,,, .... 0.670 ,,,, .... .... 0.404 ,,,, 0.447 0.422 ,,,, ,,,, 0.500 0.444 0.454 0.455 0.428 0.456 0.451 0.433 0.413
0.422 ,,,, .... 0.680 ,,,, 0.575 ,,,, ,,,, ,,,, 0.437 0.528 0.440 ,,,, 0.548 0.545 0.524 0.503 0.476 0.466 0.450 0.432 0.414
0.11
0.14 0.09 0.10
0.30 0.12 0.08 0.16
0.14
0.12
0.10 0.18 0.39
0.19 0.26 0.30 0.27 0.36 0.28 0.13 0.06 0.17
0.16
0.21 0.13
0.18 0.09 0.06 0.15 0.11 0.16 0.18 0.23 0.21 0.20 0.21 0.30
Note. .... encounter with Jupiter, ,,,, a(final) , 0.395.
CONCLUSIONS
We suggest that the secular resonance between the rotation of the longitude of perihelia of Jupiter and asteroids
TABLE II Final Eccentricities, e(final) for Three Dispersal Times with No Drag Td 5 5000 T(J)
Td 5 2000 T(J)
Td 5 400 T(J)
a(initial)
e(final)
e(final)
e(final)
0.74 0.72 0.70 0.68 0.66 0.64 0.62 0.60 0.58 0.56 0.54 0.52 0.50 0.48 0.46 0.44 0.42
.... .... .... .... .... 0.14 .... .... 0.33 .... 0.46 0.29 0.27 .... 0.30 0.12 0.38
.... .... .... .... .... 0.13 .... .... 0.24 0.14 0.21 0.20 0.23 0.19 0.22 0.24 0.21
.... 0.10 .... .... .... 0.14 0.16 0.11 0.10 0.09 0.09 0.09 0.10 0.09 0.07 0.08 0.11
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pumped up the eccentricities of the asteroids from initial values near zero to values even in excess of 0.3. The asteroids initially closer to Jupiter than the 2 : 1 resonance (0.630 a(J)) were removed by close encounters with Jupiter, or because gas drag removed so much energy from their orbits that they spiraled into the inner Solar System. For asteroids with radii in excess of 100 km the gas drag was ineffective and they were removed by close encounters with Jupiter. The removal beyond 0.630 is so efficient that we suggest this may have been responsible for removing as much as an Earth mass of rocky material from that region. We pointed out in 1973, as did Weidenschilling (1977), that, compared to the smooth r 23/2 decay in the surface density obtained by smearing out the rocky material of the planets (Hayashi 1981), there is a drop of more than a factor of 1000 in the vicinity of the asteroid belt. This material could have been removed by the mechanism proposed here: a pumping of the eccentricity by the secular resonance, followed by either a close encounter with Jupiter or a spiraling into the inner Solar System. Since the present mass of the asteroid belt is on the order of 1024 Earth masses, the mechanism would had to have been very efficient. We have not yet made enough runs to verify that the statistics work out, but the results so far are suggestive that the mechanism would operate successfully. In addition to removing asteroids from 0.630 , a , 0.763, there has been a problem of producing the rather high eccentricities of the remaining asteroids. The secular resonance seems to also account for them. We were encouraged by the coincidence that the value of so needed to displace the secular resonance from its present position at 0.40 to 0.76 was so close to the value of so derived by Hayashi as the minimum nebula required to form the planets. We were also encouraged by the coincidence between the dispersal times which we derived so as not to remove all the asteroids and the value that Ward derived to keep the eccentricity of the Earth from becoming unacceptably large. Thus this mechanism, to remove most of the asteroids between 0.630 and 0.763 a(J) and to enhance the eccentricities of the remaining asteroids, seems to follow naturally from a primitive solar nebula just massive enough to form the planets. As far as this study is concerned, the nebula could have lasted for an indefinite time in a stationary state. Once it starts to disperse, and the secular resonance migrates inward, we limit the dispersal time to less than about 100,000 years. ACKNOWLEDGMENT We dedicate this paper to Fred L. Whipple on the occasion of his 90th birthday. Professor Whipple established the Smithsonian Astrophysical Observatory in Cambridge more than four decades ago. He has been our mentor and friend, as well as a friend to the other small bodies in the Solar System.
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