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The Effect of Sweeping Secular Resonances on the Classical Kuiper Belt
CHINESE ASTRONOMY AND ASTROPHYSICS
ELSEVIER
Chinese Astronomy (2008)409–422 409–422 Chinese Astronomyand andAstrophysics Astrophysics 32 32 (2008)
The Effect of Sweeping Secular Resonances on the Classical Kuiper Belt† LI Jian
ZHOU Li-yong
SUN Yi-sui
Department of Astronomy, Nanjing University, Nanjing 210093
Abstract The Kuiper Belt is a disk of small icy objects orbiting the Sun beyond Neptune. The region between 40-48 AU in this disk is supposed to consist of dynamical “ cold” objects on low-inclination orbits and is called the “ Classical Kuiper Belt”. Recent observations reveal that there is a “ hot” population with inclinations being as large as 30◦ residing in this region. Secular resonance sweeping, which took place in the late stage of formation of the planetary system when the residual nebula gas was dispersing, is a possible mechanism that can excite the orbits in this region. In this paper, we investigate in detail the excitation of orbital inclination by this mechanism. It is shown that the excitation depends sensitively on the angle δ between the midplane of the nebula gas and the invariable plane of the solar system. The excitation is very small when δ = 0◦ , but if the gas midplane coincides with the ecliptic, i.e. if δ ≈ 1.6◦ , then objects in the region of classical Kuiper belt can be excited to orbital inclinations as high as 30◦ , provided the nebula gas has the proper initial density and disperses at a proper rate. We also considered the orbital excitation by secular resonance sweeping with Jupiter on an inclined orbit and with migrating Jovian planets, and found the excitation is only slightly affected. Key words: solar system: Kuiper Belt—Solar system: formation—celestial mechanics
1. INTRODUCTION † Supported by Natural Science Foundation, National “973” Project and Program for Ph.D. Training from Ministry of Education Received 2006–07–05; revised version 2006–11–06 A translation of Acta Astron. Sin. Vol. 49, No. 2, pp. 179–191, 2008 [email protected]
0275-1062/08/$-see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.chinastron.2008.10.003
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The existence of a disk consisting of icy celestial bodies beyond Neptune was proposed by Edgeworth[1] and Kuiper[2] as early as the 1940s/50s, but the first object in this now called Kuiper Belt (hereafter KB) was observed only in 1992 by Jewitt and Luu [3] . Since this first Kuiper Belt Object (KBO hereafter), 1992QB1, more than 1000 KBOs have been observed so far 1 . According to their orbital characters, they are divided into three groups (e.g. Ref.[4][5] ): (1) Classical KBOs, located between 42-48AU and constitute about 2/3 of the whole population of KBOs; (2) Resonant KBOs, those trapped in mean-motion resonance with Neptune, mainly the 3:2 resonance (e.g. Pluto with a semimajor axis of 39.4AU), but also the 4:3, 5:3 and 2:1 resonances. About 1/4 of known KBOs are resonant KBOs. (3) Scattered KBOs, consisting of 8% of the population on large eccentric orbits (a > 50 AU and 0.2 < e < 0.85). Generally KBOs are believed to be the residual matter from the early stage of the solar system after the planets have obtained most of their masses, so they are expected to contain hints on the formation and early evolution of the solar system. Malhotra[6−8] proposed a mechanism of orbital migration and resonance capturing to explain the origin of the resonant KBOs. In this model, the Jovian planets experienced orbital migration due to gravitational scatterings with the planetesimals in the residual nebula disk after they obtained most of their masses. Along with the migration of Neptune, its mean-motion resonances swept through the planetesimal disk and captured many objects from the disk, meanwhile exciting the orbital eccentricities of these trapped objects. A number of the so-called “hot” dynamical objects have recently been discovered on highly inclined orbits (with largest inclination of 30◦ ) in the classical KB region. Their origin is a new challenge. Gomes[9] supposed that they were transported from their original orbits at ∼ 25 AU to their present location by gravitational scattering by Neptune during its migration and the scattering also increased their inclinations. Chiang et al.[10] argued that during the period of planet formation, several Neptune-mass oligarchs may have existed in the region 20-40 AU from the Sun and they may have stimulated the KBOs’ orbits before they were finally removed. Li et al.[11] discovered another scenario of exciting the orbits: during the migration of the Jovian planets, the secular resonance ν8 raises the eccentricity of some of the objects initially located beyond 30 AU, decreasing their perihelion distances and leading them to close encounters with Neptune. The inclination was then excited during these encounters. There is another possibility. If the locations of the secular resonances varied in the early history of the solar system, then the secular resonances could sweep through the KB and pump up the orbits there. Indeed, as the residual nebula gas dispersed, changing its gravitational potential, after the formation of the planets, the Secular Resonance Sweeping (SRS hereafter) mechanism began to play a role in exciting those KBOs’ eccentricities and inclinations. Nagasawa et al. found that the SRS could pump up the inclination of orbits outside 40 AU to the observed value, provided the dispersing time-scale of the nebula gas is τdel ∼ 107 yrs and the Jovian planets were on the same orbits as their current ones[12] . However, it is widely believed that Jovian planets had migrated from their original positions after they obtained most of their masses. The original orbits of these planets were more “compact” than nowadays, e.g. in one model the semimajor axes of Jupiter, Saturn, Uranus and Neptune were 5.4, 8.7, 16.3, 23.2 AU respectively[13,14] . The analysis in Ref.[12] shows 1
http://cfa-www.harvard.edu/cfa/ps/lists/TNOs.html
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that a much longer dispersing time-scale τdel ∼ 108 yrs is necessary to excite the KBOs’ orbits to current value if the Jovian planets were in the compact configuration. But the lifetime of the nebula gas around a Sun-like star is proved by observations to be much shorter, only 3 × 106 − 107 yrs[15] . Meanwhile, in the study of the orbital excitation by the SRS under the current configuration of planetary orbits, the excitation was found to depend sensitively on the angle δ (Fig. 1) between the midplane of the nebula gas and the invariable plane of the solar system[16] . If the midplane coincided with the invariant plane (δ = 0), the excitation of KBOs beyond 40 AU would have been very limited.
Fig. 1 The relationship between the midplane of the nebula gas and the invariable plane of the solar system in space. For clarity, their included angle δ is shown exaggerated. If the midplane of the nebula gas coincides with the ecliptic plane, then δ ≈ 1.6◦ .
Considering the uncertainty in the orbital configuration of the solar system before the planetary migration, it is worth investigating the possibility of orbital excitation by the SRS mechanism. Now, the expansion of the perturbation function in the linear theory is not valid for high eccentricities or high inclinations, we shall in this paper study the SRS for the so-called compact planetary configuration and we shall use numerical simulation. We study the orbital excitation of KBOs for the case of δ = 0 and a small δ. And we shall consider the scenario where the dispersion of the gaseous nebula and the orbital migrations of the Jovian planets took place simultaneously.
2. THE MODEL In our model, the solar system consists of the Sun (with the masses of the terrestrial planets included), four Jovian planets, the gaseous solar nebula and the disk of planetesimals beyond Neptune’s orbit. The planetesimals are massless test particles, gravitationally affected by, but not affecting the other objects. 2.1 The solar nebula We adopt the axial symmetrical minimum mass solar nebula model in which the gas
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density is described by[12,17] ρH = 1.4 × 10−9 (
r −11/4 z/1 AU ) exp[−( )2 ] g cm−3 , 1 AU 0.047(r/1 AU)5/4
(1)
where ρH is the density of the nebula gas and (r, z) are the cylindrical coordinates centered 5/4 at the Sun. We set 0.35 AU< r < 150 AU, |z| < 0.0472rmax. With these assumptions, the total mass of the nebula gas is ∼ 0.028M. Due to the tidal interaction from Jupiter, a gap in the √ nebula gas around Jupiter’s orbit is formed. The half-width of this gap is assumed to be 2 3RH , where RH = (MJ /3M )1/3 aJ is the Hill radius of Jupiter[18,19] . To simplify our simulations, the nebula gas is assumed to deplete uniformly with depletion time-scale τdel [12] : ρ(r, z, t) = ρ(r, z, 0) exp(−t/τdel ),
(2)
Let ρ(r , z , t) be the gas density at moment t at (r , z ). Then the corresponding gravitational potential is V (r, z, t) = r z 0
2π
Gρ(r , z , t)r dr dz dφ . r2 + r2 + (z − z )2 − 2rr cos φ
(3)
The components of gravitational force arising from the nebula gas potential in the r- and zdirections are: Gρ(r , z , t)(r − r cos φ ) ∂V Fr (r, z, t) = − = dV , 3/2 2 2 2 ∂r (r + r + (z − z ) − 2rr cos φ ) Gρ(r , z , t)(z − z ) ∂V = Fz (r, z, t) = − dV . (4) 3/2 ∂z (r2 + r2 + (z − z )2 − 2rr cos φ ) We divide the r − z plane into a grid and using Gauss’s method we integrate Eq. (4) for each grid point, and get the force component Fr and Fz for the grid point at the given moment t0 . Since Fz (r, z, t) t − t0 Fr (r, z, t) = = exp − , (5) Fr (r, z, t0 ) Fz (r, z, t0 ) τdel the forces Fr and Fz at any point (r, z) at any time t can be calculated from a 2D three-point interpolation. 2.2 The planetary system The current orbital elements of the four Jovian planets are taken from the JPL website2 and their masses, listed in Table. 1, are from the DE405 ephemeris[20] . With the other elements fixed, we write their semimajor axes as ai = af − Δa, where af ’s are their current semimajor axes and Δa = −0.2, 0.8, 3.0, 7.0 AU for Jupiter, Saturn, Uranus and Neptune, respectively. These orbits are closer to one another and to the Sun, defining the so-called “compact configuration”. 2 http://ssd.jpl.nasa.gov/elem
planets.html
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Table 1
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The masses of the four giant planets and their orbital elements referred to ecliptic and equinox of J2000 at the J2000 epoch Planet GM /GMP a (AU) Jupiter 1047.35 5.20 Saturn 3497.92 9.54 Uranus 22903.12 19.19 Neptune 19412.36 30.07
e 0.048 0.054 0.047 0.0086
i◦ 1.30 2.48 0.77 1.77
Ω◦ 100.56 113.72 74.23 131.72
ω◦ 274.20 338.72 96.73 273.25
M◦ 19.65 317.51 142.27 259.91
Moreover, the migration of the Jovian planets was simplified to follow the following rule[6,7] : a(t) = af − Δa exp(−t/τmig ) , (6) where a(t) is the semimajor axis at time t and τmig is the orbital migration time-scale. This migration can be mimicked by adding an artificial force on the planet in the direction of the velocity vˆ [7] : t vˆ GM GM − ). (7) exp(− Δ¨ r= τmig af ai τmig
3. SECULAR RESONANCE Secular resonance happens when the precession rate of the perihelion or ascending node of the planetesimal coincides with one of the eigenfrequencies of the solar system. The secular resonance can pump up the eccentricity or the inclination of the planetesimal[12] . In the system consisting of the Sun, Jovian planets and the gaseous solar nebula, the dispersion of the gas induces a change in the gravitational potential of the nebula. This change of potential will induce a change in the locations of the secular resonances, causing the SRS. Expanding Eq. (3) to second order in eccentricity and inclination, and averaging the short-period terms, the perturbation potential of the nebula gas is obtained[21,22] : Rn = C + T (a, t)e2 − S(a, t)I 2 ,
a2 −3 + 2r /a cos φ 3(a − r cos φ )2 T (a, t) = + − 2 4 (a + r2 + z 2 − 2ar cos φ )3/2 (a2 + r2 + z 2 − 2ar cos φ )5/2 GρdV ,
r /a cos φ a2 S(a, t) = − − 2 4 (a + r2 + z 2 − 2ar cos φ )3/2 3z 2 + 2 (8) GρdV , (a + r2 + z 2 − 2ar cos φ )5/2 where C is a constant. Then, for a planetesimal with orbital elements a, n (mean motion), e, I, , Ω, the secular term in the perturbation function reads[23]: ⎡ ⎤ 4 4 1 1 Aj eej cos( − j ) + Bj IIj cos(Ω − Ωj )⎦ , (9) R = na2 ⎣ Ae2 + BI 2 + 2 2 j=1 j=1
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where 1 mj 2 (1) αj b3/2 (αj ) + 2 T, 4 j=1 m na 4
A = +n
1 mj (2) αj b3/2 (αj ) , 4 m 4 2 1 mj (1) αj b3/2 (αj ) − 2 S, B = −n 4 j=1 m na
Aj = −n
Bj = +n
1 mj (1) αj b3/2 (αj ) . 4 m
(10) (1)
(2)
The subscript j refers to the Jovian planets, αj = αj /a and b3/2 (αj ), b3/2 (αj ) are the Laplace coefficients. After introducing the canonic variables: h = e sin ,
k = e cos ,
p = I sin Ω,
q = I cos Ω,
(11)
the equation of motion of the planetesimal can be written as: 1 ∂R 1 ∂R 1 ∂R 1 ∂R ˙ ,k = − 2 , p˙ = + 2 , q˙ = − 2 . h˙ = + 2 na ∂k na ∂h na ∂q na ∂p
(12)
The solution to Eq. (12) is: h = efree sin(At + β) −
8 j=5
k = efree cos(At + β) −
υj sin(gj t + βj ), A − gj
8 j=5
p = Ifree sin(Bt + γ) − q = Ifree cos(Bt + γ) −
υj cos(gj t + βj ), A − gj
18
μj sin(fj t + γj ), B − fj j=15 18 j=15
μj cos(fj t + γj ) , B − fj
(13)
where efree , Ifree , β, βj , γ, γj , νj , μj are all constants of integration, gj (j = 5, 6, 7, 8) and fj (j = 15, 16, 17, 18) are the eigenfrequencies in the solar system in compact configuration. If, for a given planetesimal, the proper frequency of the perihelion precession A = gj , or the proper frequency of the ascending node precession B = fj , then we have the secular resonance. The former, A = gj , stimulate the eccentricity of the planetesimal and are labelled ν5 , ν6 , ν7 , ν8 (g5 > g6 > g7 > g8 > 0), and the latter, B = fj , excite the inclination and are usually labelled ν15 , ν16 , ν17 , ν18 (f16 < f17 < f18 < f15 < 0). According to Eq.(10), the frequencies A and B depend only on the semimajor axis a at time t, thus the locations of the secular resonances can be calculated from A = gj and B = fj [12,22] . See Figs. 2a and 2b. Fig. 2a shows that, as the nebula gas density decreases,
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Fig. 2 Estimated locations of the secular resonances for the compact configuration of the solar system, as functions of the gas density ρ, in units of ρH , the gas density of the minimum mass solar nebula. (a) Locations of the secular resonances ν15 ,ν16 , ν17 , ν18 . (b) Locations of the secular resonances ν5 , ν6 , ν7 ,ν8 .
the secular resonances νj (j > 10) migrate outwards with, however, the ν16 , ν17 , ν18 always located inside 30 AU, so the classical KBOs’ orbits can not be excited by them. On the other hand, the ν15 reaches 42.73438 AU when the nebula gas density is 10−3 ρH , so it must have swept through the region of the present-day “classical KB” as the gas density continued to decrease. As we focus on the orbital excitation of the classical KBOs in this paper, we will set the moment when the nebula gas density reaches one thousandth of its initial density as the “time zero” in our numerical simulations. Now, Fig. 2b shows that the secular resonances ν6 , ν7 , ν8 passed through the classical KB before the gas density was decreased 10−2 ρH , while the ν5 was always inside the inner edge of the KB. So, in our numerical simulations beginning from ρ = 10−3 ρH , the eccentricities of the KBOs will not be pumped up by the secular resonances ν5 , ν6 , ν7 and ν8 . It should be pointed out that the existence and variation of the nebula gas density not only change the locations of the secular resonances, but also introduce new secular resonance: in the case of no nebula gas[24] we have ν15 = 0 and there are only seven secular resonances.
4. THE NUMERICAL RESULTS For the compact planetary configuration, we numerically simulated the orbital excitation of classical KBOs by the SRS mechanism for the following three cases: (1) When the angle δ is taken to be zero. (2) When the angle δ is non-zero, but small. (3) When the orbital migration of the Jovian planets happens at the same time as the SRS. In each simulation, 20 massless test particles are distributed in 40 − 50 AU, with uniformly spaced semimajor axes, and the initial eccentricity e and inclination i both set to be 0.001 and the other angles of ascending node Ω, perihelion argument ω and mean anomaly M randomly selected in [0, 2π]. We used the second-order symplectic integrator[25] as improved by Mikkola et al.[26,27] that can deal with the non-canonical perturbations, e.g. the gas drag Fr , Fz and the artificial
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friction force Δ¨ r . We used a step size of 200 days, about one-twentieth of Jupiter’s orbital period. 4.1 Case I: δ = 0◦ We consider first the case of δ = 0◦ , that is, when the nebula gas midplane coincides with the invariant plane of the solar system. We adopt a nebula gas depletion time-scale τdel = 1.7 × 107 yrs ( unless otherwise specified, this value τdel will be understood hereafter. This is the lower limit for exciting the KBOs’ inclination to 30◦ when δ = 0◦ . See details in Section 4.2.). With these set values, the system was numerically followed to 108 yr. The results are partly illustrated in Fig. 3.
Fig. 3 The midplane of nebula gas coincides with the invariable plane of solar system. The density of initial nebula gas is 10−3 · ρH and the depletion timescale τdel = 1.7 × 107 yr. The system is integrated for 108 yr. The final orbital elements of the test particles are values averaged over the last 3 × 106 yr. (Left panel) Plot of final semi-major axes vs. final inclinations of the test particles. (Right panel) Plot of final semi-major axes vs. final eccentricities of the test particles.
When the gas midplane coincides with the invariant plane in the compact planetary configuration, the SRS can not efficiently pump up the inclination (the maximum inclination is 0.13◦ as shown in Fig. 3), although the secular resonance ν15 slowly sweeps through the classical KB. A similar conclusion was obtained by Hahn et al.[16] , who studied the excitation of the inclination of KBOs by the SRS mechanism with the planets in their current orbital configuration. To find out the influence of the gas depletion rate on the orbital excitation, we also calculated for τdel = 108 yrs as did Ref. [12], and found that the maximum inclination is still quite small, imax = 0.3◦ . In summary, the orbital inclination can not be efficiently excited by the SRS mechanism when δ = 0◦ . Meanwhile, the eccentricity of KBOs can not be excited by the SRS either, as we have analyzed in Section 3. Therefore we will not describe the eccentricities of the KBOs from now on but only show the a vs. e distribution of the test particles in our numerical simulations. The excitation of the test particle’s inclination Δi depends linearly on the inclinations of the Jovian planets[12] . On the other hand, the secular resonance ν15 mainly originated from Jupiter’s inclination with respect to the invariant plane, which is only iJ = 0.31◦ . Jupiter’s small inclination limits the orbital excitation of test particles. In order to clarify the dependence of inclination excitation on Jupiter’s inclination, we artificially set iJ = 1.21◦
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Fig. 4 As Figure 3, but with Jupiter’s orbital inclination (with respect to the invariable plane of the solar system) increased from the observed 0.31◦ to 1.21◦
and numerically simulated the system again to 108 yrs. As shown in Fig. 4, the maximum inclination of the test particles in this situation is now 0.4◦ , which is about 3 times larger than the 0.13◦ in the former case with iJ = 0.31◦ . However, compared to the observed value in the classical KB region (30◦ ), the inclination excitation is still very limited. Although a higher Jupiter inclination is helpful to pump up the KBOs’ inclinations in our model, it would not be reasonable to expect that Jupiter’s inclination had been much higher than nowadays. Therefore, there must have been some other mechanism at work, for example density wave in the nebula gas[28] , to damp a highly inclined orbit to the current one. But in our model the excitation happens just before the nebula gas depletion (when 99.9% of the gas had been dispersed), that is, there is very little probability for Jupiter to adjust its orbit and attain the current one. 4.2 Case II: δ = 0◦ Generally, the solar nebula gas disk should coincide with the dust disk, that is, the angle δ between midplane of the gas and the invariant plane should be zero. Nevertheless, the orbital configuration of the Jovian planets before the final orbital migration took place must be complex and full of uncertainties; hence we can not exclude the possibility that there is a small angle, δ = 0◦ , between the two planes. In this subsection we assume that the gas midplane is the same as the ecliptic plane, as was done in Ref. [12]. In this case, δ = 1.6◦ . As pointed out by Nagasawa et al.[22] , the inclination excitation of the test particles √ is proportional to τdel . We performed test runs with different τdel values and found the lower limit τdel ∼ 1.7 × 107 yrs for the inclination of the test particle beyond 40 AU to be excited by the SRS to the observed value (∼ 30◦ ). We show in Fig. 5 the orbital elements of the test particles in a run with δ = 1.6◦ , τdel = 1.7 × 107 yrs and a total integrating time of 108 yrs. The results in Fig. 5 can be summarized as follows: (1) The inclinations of the test particles are efficiently pumped up, with the maximum inclination larger than 30◦ . (2) The most excited test particles are located around 41 AU and the excitation decreases with increasing semimajor axis. According to the linear analysis in Section 3, the initial location of
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Fig. 5 As Figure 3, but the angle between the midplane of solar system and the invariable plane of nebula gas is taken to be about 1.6◦ .
the ν15 was at 42.73438 AU, where the maximum excitation should happen. This discrepancy in the location of maximum excitation is probably due to the truncation error arising from the ignoring of higher order terms in the perturbation function. Due to the continuous decrease of the gas density, the secular resonance ν15 migrates outwards and excites test particles on farther orbits. The sweeping speed keeps on increasing during the migration and the excitation effect decreases. (3) The SRS pumps up the orbital inclinations of all the test particles between 40 AU and 50 AU to over 15◦ , leaving no “cold” orbits (i < 4◦ ) in this region. (4) The eccentricity is not excited. These excited test particles eventually run on highly-inclined, near-circular orbits. To check the dependence of the inclination excitation on the angle δ, we also calculated for the case of δ = 0.88◦, and the results show that the excitation is considerably reduced with the maximum inclination now imax < 12◦ . This confirms that the orbital excitation by the SRS mechanism depends sensitively on the angle between the gas midplane and the invariant plane. This dependence deserves a thorough investigation in future. 4.3 Case III: SRS accompanied by orbital migration Since the time-scales for the orbital migration τmig and for the nebula gas depletion τdel are both of the order of 107 yrs, it is reasonable to assume that they could happen simultaneously. We numerically simulate this process of SRS accompanied by the orbital migration, for τmig = 2 × 107 yrs[11] . The system is integrated with the initial gas density of ρ = 10−3 ρH to 3 × 107 yrs. At the end of our simulation, the final gas density is ρ ≈ 1.7 × 10−4ρH and the secular resonance ν15 had swept through the classical KB (Fig. 2), and the excitation by the SRS had ended. First we assume that the midplane of the nebula gas coincides with the invariant plane of solar system (δ = 0◦ ). We compare in Fig. 6 the numerical results of the case of SRS + planetary migration (filled circles) with the case of planetary orbital migration only (open circles). It is clear that the inclination excitation is nearly the same in the two models. In fact, the orbits are excited mainly by the planetary migration and mean-motion resonance capturing, and the contribution from the SRS mechanism is negligibly small.
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Fig. 6 The midplane of nebula gas coincides with the invariable plane of the solar system. The density of initial nebula gas is 10−3 ·ρH and the depletion timescale τdel = 1.7 × 107 yr. The system is integrated for 3 × 107 yr, plotted are the actual final orbital elements of the test particles. Left panel: plot of final semi-major axis vs. final inclination. Right panel: plot of final semi-major axis vs. final eccentricity. The filled circles stand for the case of SRS along with the migration of the giant planets, the open circles for the case of migration of giant planets only.
Fig. 7 shows the results for the same two models (SRS + planetary migrations, filled circles) and (SRS only, open circles) when the nebula gas midplane coincides with the ecliptic (δ = 1.6◦ ). In the first model (filled circles), the SRS can pump up the inclinations of the test particles to above 28◦ , but in the second model (open circles) the excitation is apparently smaller. This is due to the fact that in the latter case the secular resonance ν15 moves outwards faster because of the migration of planets, thus the time during which ν15 affects the test particles is shorter than in the former case. Indeed, starting from the same position, the ν15 arrives at 43.52 AU and 43.83 AU at the moment t = 1.7 × 106 yrs in the former and latter cases, respectively. In a word, although the planetary orbital migration reduces the inclination excitation, the SRS mechanism still can pump up in a reasonable time the inclination of KBOs to the observed value, provided the midplane of the nebula gas is on the ecliptic.
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Fig. 7 As Figure 6, but the angle between the midplane of nebula gas and the invariable plane of solar system is now set at about 1.6◦ . The filled circles stand for the case of SRS along with the migration of giant planets, the open circles, that of SRS only.
5. CONCLUSIONS AND DISCUSSION Some high-inclusion (the maximum inclination imax > 30◦ ) objects have been observed in the region of the classical Kuiper belt in recent years, which poses a new challenge to the study of the Kuiper belt. Secular resonances are one of the most plausible mechanisms that can pump up the KBOs’ inclination. Secular resonance sweeping (SRS)[12] , resulting from the varying gravitational potential of the residual nebula gas during its depletion, can pump up the inclination of the KBOs. But the excitation depends sensitively on the angle δ between the midplane of the nebula gas and the invariant plane of the solar system[16] . Based on these considerations, we investigated in this paper the inclination excitation of the classical KBOs by the SRS for the compact planetary orbital configuration. Several different initial conditions were discussed and our results can be summarized as below: (1) When the nebula gas midplane coincides with the invariant plane of the solar system, i.e. δ = 0◦ , SRS can not efficiently pump up the inclinations of orbits in the region of the classical KB. Even if the relative inclination of the Jovian planets had been larger than at present, the variation Δi of the test particles is still very limited. (2) When δ = 1.6◦ , that is, when the gas midplane coincides with the ecliptic, the inclination of test particles can be excited by the SRS to values over 30◦ , provided the initial density of the nebula gas is 0.1% of the gas density in the minimum mass solar nebula model and the gas depletion time-scale is τdel = 1.7 × 107 yrs. This time-scale is consistent with the observed lifespan of the nebula gas disk around the T-Tauri stars (3×106 −107 yrs). Meanwhile, the eccentricities of test particles are not much affected by the SRS and retain small values (e < 0.01), therefore this orbital evolution scenario can explain the origin of the high-inclination, low-eccentricity orbits in the classical KB. (3) When the SRS and planetary orbital migration happen simultaneously, if δ = 0◦ , the contribution to the orbital excitation of the test bodies by the SRS is very small: the main contribution comes for the planeatary migration. If δ = 1.6◦ , the SRS can effectively excite the orbital inclinations, but because the changes in the orbits of the Jovian planets cause the secular resonance ν15 to move outward faster, the excitation of the orbital inclination of
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the test bodies is slightly less than in the case of no planetary migration. The linear analysis by Nagasawa et al.[12] indicates that the nebula gas depletion timescale should be τdel > 108 yrs if the inclination of orbits beyond 40 AU had been pumped up to i > 30◦ by the SRS in the compact planetary configuration. But our numerical simulations in this paper show that the necessary time-scale is one order of magnitude smaller, at τdel ∼ 1.7 × 107 . The reason for this difference lies in at least two facts. First, the expansion of the perturbation function is not valid for large inclinations. Second, and more importantly different values of the angle δ between the gas midplane and the invariant plane were adopted in their and our papers. For example, the results in Section 4.2 show that the inclination can be pumped up by the SRS to 12◦ when δ = 0.88◦ and τdel = 1.7 × 107 yrs. √ According to the linear estimation, Δi ∝ τdel , so to excite the inclination to i = 30◦ needs a depletion time-scale τdel ≈ 1.1 × 108 yrs. The initial gas density ρ0 adopted in this paper is 0.1% of the density in the minimum mass solar nebula model. With this value of ρ0 , the secular resonance ν15 is beyond 40 AU while the ν6 , ν7 , ν8 have swept through the Kuiper belt so that low-eccentricity planetesimals are preserved inside 40 AU (as shown in Fig. 8). Afterwards, during the outward migration of Neptune, the 2:1 mean-motion resonance can capture these planetesimals, bring them into the Kuiper belt and form the population of classical KBOs of low inclinations. In fact the initial density ρ0 is a very subtle parameter. The secular resonances ν6 , ν7 , ν8 will sweep through the Kuiper belt at the moments when ρ0 attains the values 0.231ρH , 0.141ρH, and 0.067ρH respectively. If a larger ρ0 was adopted, these secular resonances would successively excite the eccentricities of KBOs, and then the 2:1 mean-motion resonance could hardly trap those highly-eccentricity objects and bring them into the Kuiper belt[7] . On the other hand, the secular resonance ν15 sweeps outwards through the planetesimal disk beyond Neptune outside 30 AU from the moment of ρ0 ≈ 0.01ρH and pumps up the inclination of the planetesimals, then it would be very difficult for the currently observed configuration of the Kuiper blet to form.
Fig. 8 As Figure 5, but with the initial semi-major axes of test particles distributed uniformly in the range 30—50 AU.
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ELSEVIER
Chinese Astronomy (2008)409–422 409–422 Chinese Astronomyand andAstrophysics Astrophysics 32 32 (2008)
The Effect of Sweeping Secular Resonances on the Classical Kuiper Belt† LI Jian
ZHOU Li-yong
SUN Yi-sui
Department of Astronomy, Nanjing University, Nanjing 210093
Abstract The Kuiper Belt is a disk of small icy objects orbiting the Sun beyond Neptune. The region between 40-48 AU in this disk is supposed to consist of dynamical “ cold” objects on low-inclination orbits and is called the “ Classical Kuiper Belt”. Recent observations reveal that there is a “ hot” population with inclinations being as large as 30◦ residing in this region. Secular resonance sweeping, which took place in the late stage of formation of the planetary system when the residual nebula gas was dispersing, is a possible mechanism that can excite the orbits in this region. In this paper, we investigate in detail the excitation of orbital inclination by this mechanism. It is shown that the excitation depends sensitively on the angle δ between the midplane of the nebula gas and the invariable plane of the solar system. The excitation is very small when δ = 0◦ , but if the gas midplane coincides with the ecliptic, i.e. if δ ≈ 1.6◦ , then objects in the region of classical Kuiper belt can be excited to orbital inclinations as high as 30◦ , provided the nebula gas has the proper initial density and disperses at a proper rate. We also considered the orbital excitation by secular resonance sweeping with Jupiter on an inclined orbit and with migrating Jovian planets, and found the excitation is only slightly affected. Key words: solar system: Kuiper Belt—Solar system: formation—celestial mechanics
1. INTRODUCTION † Supported by Natural Science Foundation, National “973” Project and Program for Ph.D. Training from Ministry of Education Received 2006–07–05; revised version 2006–11–06 A translation of Acta Astron. Sin. Vol. 49, No. 2, pp. 179–191, 2008 [email protected]
0275-1062/08/$-see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.chinastron.2008.10.003
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The existence of a disk consisting of icy celestial bodies beyond Neptune was proposed by Edgeworth[1] and Kuiper[2] as early as the 1940s/50s, but the first object in this now called Kuiper Belt (hereafter KB) was observed only in 1992 by Jewitt and Luu [3] . Since this first Kuiper Belt Object (KBO hereafter), 1992QB1, more than 1000 KBOs have been observed so far 1 . According to their orbital characters, they are divided into three groups (e.g. Ref.[4][5] ): (1) Classical KBOs, located between 42-48AU and constitute about 2/3 of the whole population of KBOs; (2) Resonant KBOs, those trapped in mean-motion resonance with Neptune, mainly the 3:2 resonance (e.g. Pluto with a semimajor axis of 39.4AU), but also the 4:3, 5:3 and 2:1 resonances. About 1/4 of known KBOs are resonant KBOs. (3) Scattered KBOs, consisting of 8% of the population on large eccentric orbits (a > 50 AU and 0.2 < e < 0.85). Generally KBOs are believed to be the residual matter from the early stage of the solar system after the planets have obtained most of their masses, so they are expected to contain hints on the formation and early evolution of the solar system. Malhotra[6−8] proposed a mechanism of orbital migration and resonance capturing to explain the origin of the resonant KBOs. In this model, the Jovian planets experienced orbital migration due to gravitational scatterings with the planetesimals in the residual nebula disk after they obtained most of their masses. Along with the migration of Neptune, its mean-motion resonances swept through the planetesimal disk and captured many objects from the disk, meanwhile exciting the orbital eccentricities of these trapped objects. A number of the so-called “hot” dynamical objects have recently been discovered on highly inclined orbits (with largest inclination of 30◦ ) in the classical KB region. Their origin is a new challenge. Gomes[9] supposed that they were transported from their original orbits at ∼ 25 AU to their present location by gravitational scattering by Neptune during its migration and the scattering also increased their inclinations. Chiang et al.[10] argued that during the period of planet formation, several Neptune-mass oligarchs may have existed in the region 20-40 AU from the Sun and they may have stimulated the KBOs’ orbits before they were finally removed. Li et al.[11] discovered another scenario of exciting the orbits: during the migration of the Jovian planets, the secular resonance ν8 raises the eccentricity of some of the objects initially located beyond 30 AU, decreasing their perihelion distances and leading them to close encounters with Neptune. The inclination was then excited during these encounters. There is another possibility. If the locations of the secular resonances varied in the early history of the solar system, then the secular resonances could sweep through the KB and pump up the orbits there. Indeed, as the residual nebula gas dispersed, changing its gravitational potential, after the formation of the planets, the Secular Resonance Sweeping (SRS hereafter) mechanism began to play a role in exciting those KBOs’ eccentricities and inclinations. Nagasawa et al. found that the SRS could pump up the inclination of orbits outside 40 AU to the observed value, provided the dispersing time-scale of the nebula gas is τdel ∼ 107 yrs and the Jovian planets were on the same orbits as their current ones[12] . However, it is widely believed that Jovian planets had migrated from their original positions after they obtained most of their masses. The original orbits of these planets were more “compact” than nowadays, e.g. in one model the semimajor axes of Jupiter, Saturn, Uranus and Neptune were 5.4, 8.7, 16.3, 23.2 AU respectively[13,14] . The analysis in Ref.[12] shows 1
http://cfa-www.harvard.edu/cfa/ps/lists/TNOs.html
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that a much longer dispersing time-scale τdel ∼ 108 yrs is necessary to excite the KBOs’ orbits to current value if the Jovian planets were in the compact configuration. But the lifetime of the nebula gas around a Sun-like star is proved by observations to be much shorter, only 3 × 106 − 107 yrs[15] . Meanwhile, in the study of the orbital excitation by the SRS under the current configuration of planetary orbits, the excitation was found to depend sensitively on the angle δ (Fig. 1) between the midplane of the nebula gas and the invariable plane of the solar system[16] . If the midplane coincided with the invariant plane (δ = 0), the excitation of KBOs beyond 40 AU would have been very limited.
Fig. 1 The relationship between the midplane of the nebula gas and the invariable plane of the solar system in space. For clarity, their included angle δ is shown exaggerated. If the midplane of the nebula gas coincides with the ecliptic plane, then δ ≈ 1.6◦ .
Considering the uncertainty in the orbital configuration of the solar system before the planetary migration, it is worth investigating the possibility of orbital excitation by the SRS mechanism. Now, the expansion of the perturbation function in the linear theory is not valid for high eccentricities or high inclinations, we shall in this paper study the SRS for the so-called compact planetary configuration and we shall use numerical simulation. We study the orbital excitation of KBOs for the case of δ = 0 and a small δ. And we shall consider the scenario where the dispersion of the gaseous nebula and the orbital migrations of the Jovian planets took place simultaneously.
2. THE MODEL In our model, the solar system consists of the Sun (with the masses of the terrestrial planets included), four Jovian planets, the gaseous solar nebula and the disk of planetesimals beyond Neptune’s orbit. The planetesimals are massless test particles, gravitationally affected by, but not affecting the other objects. 2.1 The solar nebula We adopt the axial symmetrical minimum mass solar nebula model in which the gas
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density is described by[12,17] ρH = 1.4 × 10−9 (
r −11/4 z/1 AU ) exp[−( )2 ] g cm−3 , 1 AU 0.047(r/1 AU)5/4
(1)
where ρH is the density of the nebula gas and (r, z) are the cylindrical coordinates centered 5/4 at the Sun. We set 0.35 AU< r < 150 AU, |z| < 0.0472rmax. With these assumptions, the total mass of the nebula gas is ∼ 0.028M. Due to the tidal interaction from Jupiter, a gap in the √ nebula gas around Jupiter’s orbit is formed. The half-width of this gap is assumed to be 2 3RH , where RH = (MJ /3M )1/3 aJ is the Hill radius of Jupiter[18,19] . To simplify our simulations, the nebula gas is assumed to deplete uniformly with depletion time-scale τdel [12] : ρ(r, z, t) = ρ(r, z, 0) exp(−t/τdel ),
(2)
Let ρ(r , z , t) be the gas density at moment t at (r , z ). Then the corresponding gravitational potential is V (r, z, t) = r z 0
2π
Gρ(r , z , t)r dr dz dφ . r2 + r2 + (z − z )2 − 2rr cos φ
(3)
The components of gravitational force arising from the nebula gas potential in the r- and zdirections are: Gρ(r , z , t)(r − r cos φ ) ∂V Fr (r, z, t) = − = dV , 3/2 2 2 2 ∂r (r + r + (z − z ) − 2rr cos φ ) Gρ(r , z , t)(z − z ) ∂V = Fz (r, z, t) = − dV . (4) 3/2 ∂z (r2 + r2 + (z − z )2 − 2rr cos φ ) We divide the r − z plane into a grid and using Gauss’s method we integrate Eq. (4) for each grid point, and get the force component Fr and Fz for the grid point at the given moment t0 . Since Fz (r, z, t) t − t0 Fr (r, z, t) = = exp − , (5) Fr (r, z, t0 ) Fz (r, z, t0 ) τdel the forces Fr and Fz at any point (r, z) at any time t can be calculated from a 2D three-point interpolation. 2.2 The planetary system The current orbital elements of the four Jovian planets are taken from the JPL website2 and their masses, listed in Table. 1, are from the DE405 ephemeris[20] . With the other elements fixed, we write their semimajor axes as ai = af − Δa, where af ’s are their current semimajor axes and Δa = −0.2, 0.8, 3.0, 7.0 AU for Jupiter, Saturn, Uranus and Neptune, respectively. These orbits are closer to one another and to the Sun, defining the so-called “compact configuration”. 2 http://ssd.jpl.nasa.gov/elem
planets.html
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Table 1
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The masses of the four giant planets and their orbital elements referred to ecliptic and equinox of J2000 at the J2000 epoch Planet GM /GMP a (AU) Jupiter 1047.35 5.20 Saturn 3497.92 9.54 Uranus 22903.12 19.19 Neptune 19412.36 30.07
e 0.048 0.054 0.047 0.0086
i◦ 1.30 2.48 0.77 1.77
Ω◦ 100.56 113.72 74.23 131.72
ω◦ 274.20 338.72 96.73 273.25
M◦ 19.65 317.51 142.27 259.91
Moreover, the migration of the Jovian planets was simplified to follow the following rule[6,7] : a(t) = af − Δa exp(−t/τmig ) , (6) where a(t) is the semimajor axis at time t and τmig is the orbital migration time-scale. This migration can be mimicked by adding an artificial force on the planet in the direction of the velocity vˆ [7] : t vˆ GM GM − ). (7) exp(− Δ¨ r= τmig af ai τmig
3. SECULAR RESONANCE Secular resonance happens when the precession rate of the perihelion or ascending node of the planetesimal coincides with one of the eigenfrequencies of the solar system. The secular resonance can pump up the eccentricity or the inclination of the planetesimal[12] . In the system consisting of the Sun, Jovian planets and the gaseous solar nebula, the dispersion of the gas induces a change in the gravitational potential of the nebula. This change of potential will induce a change in the locations of the secular resonances, causing the SRS. Expanding Eq. (3) to second order in eccentricity and inclination, and averaging the short-period terms, the perturbation potential of the nebula gas is obtained[21,22] : Rn = C + T (a, t)e2 − S(a, t)I 2 ,
a2 −3 + 2r /a cos φ 3(a − r cos φ )2 T (a, t) = + − 2 4 (a + r2 + z 2 − 2ar cos φ )3/2 (a2 + r2 + z 2 − 2ar cos φ )5/2 GρdV ,
r /a cos φ a2 S(a, t) = − − 2 4 (a + r2 + z 2 − 2ar cos φ )3/2 3z 2 + 2 (8) GρdV , (a + r2 + z 2 − 2ar cos φ )5/2 where C is a constant. Then, for a planetesimal with orbital elements a, n (mean motion), e, I, , Ω, the secular term in the perturbation function reads[23]: ⎡ ⎤ 4 4 1 1 Aj eej cos( − j ) + Bj IIj cos(Ω − Ωj )⎦ , (9) R = na2 ⎣ Ae2 + BI 2 + 2 2 j=1 j=1
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where 1 mj 2 (1) αj b3/2 (αj ) + 2 T, 4 j=1 m na 4
A = +n
1 mj (2) αj b3/2 (αj ) , 4 m 4 2 1 mj (1) αj b3/2 (αj ) − 2 S, B = −n 4 j=1 m na
Aj = −n
Bj = +n
1 mj (1) αj b3/2 (αj ) . 4 m
(10) (1)
(2)
The subscript j refers to the Jovian planets, αj = αj /a and b3/2 (αj ), b3/2 (αj ) are the Laplace coefficients. After introducing the canonic variables: h = e sin ,
k = e cos ,
p = I sin Ω,
q = I cos Ω,
(11)
the equation of motion of the planetesimal can be written as: 1 ∂R 1 ∂R 1 ∂R 1 ∂R ˙ ,k = − 2 , p˙ = + 2 , q˙ = − 2 . h˙ = + 2 na ∂k na ∂h na ∂q na ∂p
(12)
The solution to Eq. (12) is: h = efree sin(At + β) −
8 j=5
k = efree cos(At + β) −
υj sin(gj t + βj ), A − gj
8 j=5
p = Ifree sin(Bt + γ) − q = Ifree cos(Bt + γ) −
υj cos(gj t + βj ), A − gj
18
μj sin(fj t + γj ), B − fj j=15 18 j=15
μj cos(fj t + γj ) , B − fj
(13)
where efree , Ifree , β, βj , γ, γj , νj , μj are all constants of integration, gj (j = 5, 6, 7, 8) and fj (j = 15, 16, 17, 18) are the eigenfrequencies in the solar system in compact configuration. If, for a given planetesimal, the proper frequency of the perihelion precession A = gj , or the proper frequency of the ascending node precession B = fj , then we have the secular resonance. The former, A = gj , stimulate the eccentricity of the planetesimal and are labelled ν5 , ν6 , ν7 , ν8 (g5 > g6 > g7 > g8 > 0), and the latter, B = fj , excite the inclination and are usually labelled ν15 , ν16 , ν17 , ν18 (f16 < f17 < f18 < f15 < 0). According to Eq.(10), the frequencies A and B depend only on the semimajor axis a at time t, thus the locations of the secular resonances can be calculated from A = gj and B = fj [12,22] . See Figs. 2a and 2b. Fig. 2a shows that, as the nebula gas density decreases,
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Fig. 2 Estimated locations of the secular resonances for the compact configuration of the solar system, as functions of the gas density ρ, in units of ρH , the gas density of the minimum mass solar nebula. (a) Locations of the secular resonances ν15 ,ν16 , ν17 , ν18 . (b) Locations of the secular resonances ν5 , ν6 , ν7 ,ν8 .
the secular resonances νj (j > 10) migrate outwards with, however, the ν16 , ν17 , ν18 always located inside 30 AU, so the classical KBOs’ orbits can not be excited by them. On the other hand, the ν15 reaches 42.73438 AU when the nebula gas density is 10−3 ρH , so it must have swept through the region of the present-day “classical KB” as the gas density continued to decrease. As we focus on the orbital excitation of the classical KBOs in this paper, we will set the moment when the nebula gas density reaches one thousandth of its initial density as the “time zero” in our numerical simulations. Now, Fig. 2b shows that the secular resonances ν6 , ν7 , ν8 passed through the classical KB before the gas density was decreased 10−2 ρH , while the ν5 was always inside the inner edge of the KB. So, in our numerical simulations beginning from ρ = 10−3 ρH , the eccentricities of the KBOs will not be pumped up by the secular resonances ν5 , ν6 , ν7 and ν8 . It should be pointed out that the existence and variation of the nebula gas density not only change the locations of the secular resonances, but also introduce new secular resonance: in the case of no nebula gas[24] we have ν15 = 0 and there are only seven secular resonances.
4. THE NUMERICAL RESULTS For the compact planetary configuration, we numerically simulated the orbital excitation of classical KBOs by the SRS mechanism for the following three cases: (1) When the angle δ is taken to be zero. (2) When the angle δ is non-zero, but small. (3) When the orbital migration of the Jovian planets happens at the same time as the SRS. In each simulation, 20 massless test particles are distributed in 40 − 50 AU, with uniformly spaced semimajor axes, and the initial eccentricity e and inclination i both set to be 0.001 and the other angles of ascending node Ω, perihelion argument ω and mean anomaly M randomly selected in [0, 2π]. We used the second-order symplectic integrator[25] as improved by Mikkola et al.[26,27] that can deal with the non-canonical perturbations, e.g. the gas drag Fr , Fz and the artificial
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friction force Δ¨ r . We used a step size of 200 days, about one-twentieth of Jupiter’s orbital period. 4.1 Case I: δ = 0◦ We consider first the case of δ = 0◦ , that is, when the nebula gas midplane coincides with the invariant plane of the solar system. We adopt a nebula gas depletion time-scale τdel = 1.7 × 107 yrs ( unless otherwise specified, this value τdel will be understood hereafter. This is the lower limit for exciting the KBOs’ inclination to 30◦ when δ = 0◦ . See details in Section 4.2.). With these set values, the system was numerically followed to 108 yr. The results are partly illustrated in Fig. 3.
Fig. 3 The midplane of nebula gas coincides with the invariable plane of solar system. The density of initial nebula gas is 10−3 · ρH and the depletion timescale τdel = 1.7 × 107 yr. The system is integrated for 108 yr. The final orbital elements of the test particles are values averaged over the last 3 × 106 yr. (Left panel) Plot of final semi-major axes vs. final inclinations of the test particles. (Right panel) Plot of final semi-major axes vs. final eccentricities of the test particles.
When the gas midplane coincides with the invariant plane in the compact planetary configuration, the SRS can not efficiently pump up the inclination (the maximum inclination is 0.13◦ as shown in Fig. 3), although the secular resonance ν15 slowly sweeps through the classical KB. A similar conclusion was obtained by Hahn et al.[16] , who studied the excitation of the inclination of KBOs by the SRS mechanism with the planets in their current orbital configuration. To find out the influence of the gas depletion rate on the orbital excitation, we also calculated for τdel = 108 yrs as did Ref. [12], and found that the maximum inclination is still quite small, imax = 0.3◦ . In summary, the orbital inclination can not be efficiently excited by the SRS mechanism when δ = 0◦ . Meanwhile, the eccentricity of KBOs can not be excited by the SRS either, as we have analyzed in Section 3. Therefore we will not describe the eccentricities of the KBOs from now on but only show the a vs. e distribution of the test particles in our numerical simulations. The excitation of the test particle’s inclination Δi depends linearly on the inclinations of the Jovian planets[12] . On the other hand, the secular resonance ν15 mainly originated from Jupiter’s inclination with respect to the invariant plane, which is only iJ = 0.31◦ . Jupiter’s small inclination limits the orbital excitation of test particles. In order to clarify the dependence of inclination excitation on Jupiter’s inclination, we artificially set iJ = 1.21◦
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Fig. 4 As Figure 3, but with Jupiter’s orbital inclination (with respect to the invariable plane of the solar system) increased from the observed 0.31◦ to 1.21◦
and numerically simulated the system again to 108 yrs. As shown in Fig. 4, the maximum inclination of the test particles in this situation is now 0.4◦ , which is about 3 times larger than the 0.13◦ in the former case with iJ = 0.31◦ . However, compared to the observed value in the classical KB region (30◦ ), the inclination excitation is still very limited. Although a higher Jupiter inclination is helpful to pump up the KBOs’ inclinations in our model, it would not be reasonable to expect that Jupiter’s inclination had been much higher than nowadays. Therefore, there must have been some other mechanism at work, for example density wave in the nebula gas[28] , to damp a highly inclined orbit to the current one. But in our model the excitation happens just before the nebula gas depletion (when 99.9% of the gas had been dispersed), that is, there is very little probability for Jupiter to adjust its orbit and attain the current one. 4.2 Case II: δ = 0◦ Generally, the solar nebula gas disk should coincide with the dust disk, that is, the angle δ between midplane of the gas and the invariant plane should be zero. Nevertheless, the orbital configuration of the Jovian planets before the final orbital migration took place must be complex and full of uncertainties; hence we can not exclude the possibility that there is a small angle, δ = 0◦ , between the two planes. In this subsection we assume that the gas midplane is the same as the ecliptic plane, as was done in Ref. [12]. In this case, δ = 1.6◦ . As pointed out by Nagasawa et al.[22] , the inclination excitation of the test particles √ is proportional to τdel . We performed test runs with different τdel values and found the lower limit τdel ∼ 1.7 × 107 yrs for the inclination of the test particle beyond 40 AU to be excited by the SRS to the observed value (∼ 30◦ ). We show in Fig. 5 the orbital elements of the test particles in a run with δ = 1.6◦ , τdel = 1.7 × 107 yrs and a total integrating time of 108 yrs. The results in Fig. 5 can be summarized as follows: (1) The inclinations of the test particles are efficiently pumped up, with the maximum inclination larger than 30◦ . (2) The most excited test particles are located around 41 AU and the excitation decreases with increasing semimajor axis. According to the linear analysis in Section 3, the initial location of
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Fig. 5 As Figure 3, but the angle between the midplane of solar system and the invariable plane of nebula gas is taken to be about 1.6◦ .
the ν15 was at 42.73438 AU, where the maximum excitation should happen. This discrepancy in the location of maximum excitation is probably due to the truncation error arising from the ignoring of higher order terms in the perturbation function. Due to the continuous decrease of the gas density, the secular resonance ν15 migrates outwards and excites test particles on farther orbits. The sweeping speed keeps on increasing during the migration and the excitation effect decreases. (3) The SRS pumps up the orbital inclinations of all the test particles between 40 AU and 50 AU to over 15◦ , leaving no “cold” orbits (i < 4◦ ) in this region. (4) The eccentricity is not excited. These excited test particles eventually run on highly-inclined, near-circular orbits. To check the dependence of the inclination excitation on the angle δ, we also calculated for the case of δ = 0.88◦, and the results show that the excitation is considerably reduced with the maximum inclination now imax < 12◦ . This confirms that the orbital excitation by the SRS mechanism depends sensitively on the angle between the gas midplane and the invariant plane. This dependence deserves a thorough investigation in future. 4.3 Case III: SRS accompanied by orbital migration Since the time-scales for the orbital migration τmig and for the nebula gas depletion τdel are both of the order of 107 yrs, it is reasonable to assume that they could happen simultaneously. We numerically simulate this process of SRS accompanied by the orbital migration, for τmig = 2 × 107 yrs[11] . The system is integrated with the initial gas density of ρ = 10−3 ρH to 3 × 107 yrs. At the end of our simulation, the final gas density is ρ ≈ 1.7 × 10−4ρH and the secular resonance ν15 had swept through the classical KB (Fig. 2), and the excitation by the SRS had ended. First we assume that the midplane of the nebula gas coincides with the invariant plane of solar system (δ = 0◦ ). We compare in Fig. 6 the numerical results of the case of SRS + planetary migration (filled circles) with the case of planetary orbital migration only (open circles). It is clear that the inclination excitation is nearly the same in the two models. In fact, the orbits are excited mainly by the planetary migration and mean-motion resonance capturing, and the contribution from the SRS mechanism is negligibly small.
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Fig. 6 The midplane of nebula gas coincides with the invariable plane of the solar system. The density of initial nebula gas is 10−3 ·ρH and the depletion timescale τdel = 1.7 × 107 yr. The system is integrated for 3 × 107 yr, plotted are the actual final orbital elements of the test particles. Left panel: plot of final semi-major axis vs. final inclination. Right panel: plot of final semi-major axis vs. final eccentricity. The filled circles stand for the case of SRS along with the migration of the giant planets, the open circles for the case of migration of giant planets only.
Fig. 7 shows the results for the same two models (SRS + planetary migrations, filled circles) and (SRS only, open circles) when the nebula gas midplane coincides with the ecliptic (δ = 1.6◦ ). In the first model (filled circles), the SRS can pump up the inclinations of the test particles to above 28◦ , but in the second model (open circles) the excitation is apparently smaller. This is due to the fact that in the latter case the secular resonance ν15 moves outwards faster because of the migration of planets, thus the time during which ν15 affects the test particles is shorter than in the former case. Indeed, starting from the same position, the ν15 arrives at 43.52 AU and 43.83 AU at the moment t = 1.7 × 106 yrs in the former and latter cases, respectively. In a word, although the planetary orbital migration reduces the inclination excitation, the SRS mechanism still can pump up in a reasonable time the inclination of KBOs to the observed value, provided the midplane of the nebula gas is on the ecliptic.
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Fig. 7 As Figure 6, but the angle between the midplane of nebula gas and the invariable plane of solar system is now set at about 1.6◦ . The filled circles stand for the case of SRS along with the migration of giant planets, the open circles, that of SRS only.
5. CONCLUSIONS AND DISCUSSION Some high-inclusion (the maximum inclination imax > 30◦ ) objects have been observed in the region of the classical Kuiper belt in recent years, which poses a new challenge to the study of the Kuiper belt. Secular resonances are one of the most plausible mechanisms that can pump up the KBOs’ inclination. Secular resonance sweeping (SRS)[12] , resulting from the varying gravitational potential of the residual nebula gas during its depletion, can pump up the inclination of the KBOs. But the excitation depends sensitively on the angle δ between the midplane of the nebula gas and the invariant plane of the solar system[16] . Based on these considerations, we investigated in this paper the inclination excitation of the classical KBOs by the SRS for the compact planetary orbital configuration. Several different initial conditions were discussed and our results can be summarized as below: (1) When the nebula gas midplane coincides with the invariant plane of the solar system, i.e. δ = 0◦ , SRS can not efficiently pump up the inclinations of orbits in the region of the classical KB. Even if the relative inclination of the Jovian planets had been larger than at present, the variation Δi of the test particles is still very limited. (2) When δ = 1.6◦ , that is, when the gas midplane coincides with the ecliptic, the inclination of test particles can be excited by the SRS to values over 30◦ , provided the initial density of the nebula gas is 0.1% of the gas density in the minimum mass solar nebula model and the gas depletion time-scale is τdel = 1.7 × 107 yrs. This time-scale is consistent with the observed lifespan of the nebula gas disk around the T-Tauri stars (3×106 −107 yrs). Meanwhile, the eccentricities of test particles are not much affected by the SRS and retain small values (e < 0.01), therefore this orbital evolution scenario can explain the origin of the high-inclination, low-eccentricity orbits in the classical KB. (3) When the SRS and planetary orbital migration happen simultaneously, if δ = 0◦ , the contribution to the orbital excitation of the test bodies by the SRS is very small: the main contribution comes for the planeatary migration. If δ = 1.6◦ , the SRS can effectively excite the orbital inclinations, but because the changes in the orbits of the Jovian planets cause the secular resonance ν15 to move outward faster, the excitation of the orbital inclination of
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the test bodies is slightly less than in the case of no planetary migration. The linear analysis by Nagasawa et al.[12] indicates that the nebula gas depletion timescale should be τdel > 108 yrs if the inclination of orbits beyond 40 AU had been pumped up to i > 30◦ by the SRS in the compact planetary configuration. But our numerical simulations in this paper show that the necessary time-scale is one order of magnitude smaller, at τdel ∼ 1.7 × 107 . The reason for this difference lies in at least two facts. First, the expansion of the perturbation function is not valid for large inclinations. Second, and more importantly different values of the angle δ between the gas midplane and the invariant plane were adopted in their and our papers. For example, the results in Section 4.2 show that the inclination can be pumped up by the SRS to 12◦ when δ = 0.88◦ and τdel = 1.7 × 107 yrs. √ According to the linear estimation, Δi ∝ τdel , so to excite the inclination to i = 30◦ needs a depletion time-scale τdel ≈ 1.1 × 108 yrs. The initial gas density ρ0 adopted in this paper is 0.1% of the density in the minimum mass solar nebula model. With this value of ρ0 , the secular resonance ν15 is beyond 40 AU while the ν6 , ν7 , ν8 have swept through the Kuiper belt so that low-eccentricity planetesimals are preserved inside 40 AU (as shown in Fig. 8). Afterwards, during the outward migration of Neptune, the 2:1 mean-motion resonance can capture these planetesimals, bring them into the Kuiper belt and form the population of classical KBOs of low inclinations. In fact the initial density ρ0 is a very subtle parameter. The secular resonances ν6 , ν7 , ν8 will sweep through the Kuiper belt at the moments when ρ0 attains the values 0.231ρH , 0.141ρH, and 0.067ρH respectively. If a larger ρ0 was adopted, these secular resonances would successively excite the eccentricities of KBOs, and then the 2:1 mean-motion resonance could hardly trap those highly-eccentricity objects and bring them into the Kuiper belt[7] . On the other hand, the secular resonance ν15 sweeps outwards through the planetesimal disk beyond Neptune outside 30 AU from the moment of ρ0 ≈ 0.01ρH and pumps up the inclination of the planetesimals, then it would be very difficult for the currently observed configuration of the Kuiper blet to form.
Fig. 8 As Figure 5, but with the initial semi-major axes of test particles distributed uniformly in the range 30—50 AU.
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