Temporary Capture of Asteroids by an Eccentric Planet
aa r X i v : . [ a s t r o - ph . E P ] F e b Draft version September 14, 2018
Preprint typeset using L A TEX style AASTeX6 v. 1.0
TEMPORARY CAPTURE OF ASTEROIDS BY AN ECCENTRIC PLANET
A. Higuchi
Department of Earth and Planetary Sciences, Faculty of Science, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan
S. Ida
Earth-Life Science Institute, Tokyo Institute of Technology, Meguro, Tokyo 152-8550, Japan
ABSTRACTWe have investigated the probability of temporary capture of asteroids in eccentric orbits by a planetin a circular or an eccentric orbit through analytical and numerical calculations. We found that in thelimit of the circular orbit, the capture probability is ∼ .
1% of encounters to the planet’s Hill sphere,independent of planetary mass and semimajor axis. In general, the temporary capture becomes moredifficult as the planet’s eccentricity ( e p ) increases. We found that the capture probability is almostindependent of e p until a critical value ( e cp ) that is given by ≃ e p > e cp , the probability decreases approximately in proportion to e − .The current orbital eccentricity of Mars is several times larger than e cp . However, since the range ofsecular change in Martian eccentricity overlaps e cp , the capture of minor bodies by the past Mars isnot ruled out. Keywords: planets and satellites: formation INTRODUCTIONIrregular satellites around giant planets, which are small and with elliptical and inclined orbits, are usually thoughtto be captured passing asteroids (e.g., Jewitt & Haghighipour 2007; Nicolson et al. 2008). The objects capturedtemporarily in the Hill sphere of a planet can be permanently captured by some energy loss (e.g., tidal dissipation,drag force from a circumplanetary disk when it existed, or collisions with other solid bodies in the disk). Higuchi & Ida(2016) derived the conditions for the temporary capture by a planet in a circular orbit as functions of the mass andsemimajor axis of the host planet, and clarified the range of semimajor axes of field particles for prograde and retrogradecapture.Higuchi & Ida (2016) commented that the small eccentricity of Jupiter does not affect the capture probability.However, the effect of a high eccentricity like that of Mars has not been investigated. Mars has two satellites: Phobosand Deimos. Two major theories of the origin of these satellites are (1) in situ formation through accretion of animpact-generated debris by a large impact inferred from the Borealis basin (e.g., Citron et al. 2015; Rosenblatt et al.2016) and (2) capture of asteroids (e.g., Burns 1978). While the large impact model may explain the circular, non-inclined orbits of Phobos and Deimos, which is not easily explained by the capture origin, the surface characteristicsof the satellites are similar to those of primitive asteroids. Spectral observations of Phobos and Deimos suggest thatthe material of the satellites is best modeled as a primitive material, which may not be easily explained by the largeimpact origin (Fraeman et al. 2014). Future sample return missions, such as MMX (Mars Moon eXploration), willprovide important clues about the Martian satellite origin. It is important to explore the possibility of the captureorigin model in detail, as well as to investigate the large impact model.In this study, we generalize our previous study to investigate the effects of orbital eccentricity of a planet on thetemporary capture probability through analytical and numerical calculations. We derive the probability of temporarycapture from encounters with the planet’s Hill sphere as a function of planetary eccentricity e p and mass m p . If theencounter frequency is given by other simulations, we can evaluate the probability of temporary capture throughout thehistory of the solar system. Most of our analysis and orbital calculations assume planar orbits, but some calculationsare done with small finite inclinations. Temporary capture is a necessary condition for permanent capture. Therelation between temporary and permanent capture will be investigated in a subsequent paper.We summarize the assumptions, basic formulation, and derivation of the analytical formulae in Section 2. We defineand derive the efficiency of temporary capture in Section 3. The methods and results of numerical calculations arepresented and compared with the analytical prediction in Section 4. In Section 5, we summarize the results andcomment on the origin of Martian satellites. ANALYTICAL DERIVATION OF TEMPORARY CAPTURE EFFICIENCY BY AN ECCENTRIC PLANETWe first derive analytical formulae for temporary capture by an eccentric planet. These formulae give orbital elementsof the asteroids that can be captured, as functions of the mass, eccentricity, and true anomaly of the host planet. As wewill show in section 4, the analytical formulae reproduce the results obtained through numerical orbital integrations.From the analytical derivation, the intrinsic dynamics of temporary capture by an eccentric planet will be revealed.2.1.
Assumptions
Following Higuchi & Ida (2016), we split a coplanar three-body problem (Sun-planet-particle) into two independenttwo-body problems (Sun-particle and planet-particle). The particles are candidates that are captured by the planetto become satellites. Hereafter, we refer to particles as ”asteroids,” although the particles do not necessarily originatefrom the asteroid belt. We identify the relative velocity between the asteroid and the planet in heliocentric orbits withthe satellite velocity orbiting around the planet (condition [1]) at the capture point. The capture points are assumedto be the L and L points (condition [2]). The distance of the points from the planet is the Hill radius, r H = r p (cid:18) m p M ⊙ (cid:19) = r p ˆ r H , (1)where m p is the planet mass and M ⊙ is the solar mass. The instantaneous heliocentric distance is given by r p = a p − e e p cos f p , (2)where a p , m p , e p , and f p are semimajor axis, mass, eccentricity, and true anomaly of the planet, respectively. We alsoassume that the geometric condition that the two elliptic orbits are touching at a capture point; the velocity vectorsof the planet and the asteroid are parallel or antiparallel (condition [3]).2.2. Conditions for Temporary Capture
We consider an asteroid and a planet in the Cartesian coordinates ( x , y ) centered on the Sun. The x -axis is towardthe perihelion of the planet’s orbit and the x - y plane lies in the planet’s orbital plane. Let r , a , e , and f be heliocentricdistance, semimajor axis, eccentricity of the asteroid, and true anomaly, respectively.Condition [2] reads as r = a − e e cos( f + θ ) = r p A ∓ , (3)where θ = f p − f and A − = 1 − ˆ r H at L A + = 1 + ˆ r H at L . (4)The heliocentric velocity of the asteroid at capture is v = s GM ⊙ (cid:18) r − a (cid:19) = s GM ⊙ (cid:18) A ∓ r p − a (cid:19) = v p χ, (5)where v p = p GM ⊙ /a p and χ = s p A ∓ − a p a , (6)Φ p = 1 − e e p cos f p . (7)Condition [1] reads as v − v p = v s , (8)where v and v p are heliocentric velocities of the asteroid and the planet and v s is the planetocentric velocity of theasteroid as a satellite at the capture. The velocity of the satellite at the planetocentric distance r s = r H is v s = s Gm p (cid:18) r H − a s (cid:19) = v H p − Φ s , (9)where a s = r H / Φ s is the planetocentric semimajor axis of the satellite,Φ s = 1 − e e s cos f s , (10) v H = r Gm p r H = s p ˆ r H , (11)and e s and f s are the planetocentric eccentricity and true anomaly. Since v H is a circular velocity around the planetat the planetocentric distance r s = r H , ν = v s /v H = p − Φ s (12)is related to the planetocentric orbital eccentricity (which is equivalent to κ appearing in Higuchi & Ida (2016)); ν = 1 corresponds to a circular orbit with the semimajor axis a s = r H and the orbit is hyperbolic for ν > √ α = α p , where α and α p are the angles between the position and velocity vectors ofthe asteroid and those of the planet, which are given bysin α = 1 + e cos( f + θ ) p e + 2 e cos( f + θ ) (13)sin α p = 1 + e p cos f p q e + 2 e p cos f p (14)These angles are given geometrically, applying the law of cosines to a triangle composed of r , the x axis, and thetangent line of the orbit at r . Another way to derive α using the angular momentum is found in Roy (2005).2.3. Equation of Temporary Capture
We combine the equations describing the three conditions above and solve for the orbital elements of temporarilycaptured asteroids. 2.3.1.
Derivation of Heliocentric Orbital Elements for Temporary CaptureSemimajor axis. — Using v k v p (condition [3]), v s = νv H (Eq. 9), and v = v p χ (Eq. 12), condition [1] (Eq. 8) becomes | χ − | = νv H . (15)Substituting Equations (3),(4), and (6), into Equation (15), we obtain the heliocentric semimajor axis of the asteroidat temporary capture as aa p ≡ ¯ a tc = Φ p (cid:20) A ∓ − (cid:16)p − Φ p ± √ ν ˆ r H (cid:17) (cid:21) − . (16)Note that Equation (16) has four values corresponding to a combination of prograde or retrograde and L or L . Ifthe sign in front of √ − ” sign represents retrograde capture. Thesign in A ∓ represents L -type or L -type (Eq. (4)). Eccentricity. — The heliocentric orbital angular momentum of the asteroid is h = rv sin α = p GM ⊙ a (1 − e ) . (17)Substituting Equations (17) into condition [2] given by Eq. (3) with α = α p , we obtain the heliocentric eccentricity attemporary capture, e tc = s − sin α p (cid:20) − (cid:18) − Φ p A ∓ ¯ a tc (cid:19)(cid:21) . (18) Angle of perihelion θ . — The perihelion angle at temporary capture is easily obtained from Equation (3), θ tc = f p − acos( g ) , (19) g = ¯ a tc Φ p A ∓ (1 − e ) − e tc = − e − (cid:18) cos α p ∓ sin α p q e − cos α p (cid:19) , (20)where Equation (18) is substituted at the end. Inclination. — If the asteroid has non-zero heliocentric inclination i , the relative velocity is modified. Since the relativevelocity is equal to v s , v = ( v cos i − v p ) + v sin i = v "(cid:18) p A ∓ − a p a (cid:19) + (cid:18) p − (cid:19) − s p A ∓ − a p a s p − i = v (cid:20) χ + (cid:18) p − (cid:19) − χ (cid:18) p − (cid:19) cos i (cid:21) , (21)which is reduced to χ + (cid:18) p − (cid:19) − χ s p − i = ( νv H ) . (22)For this equation to have a solution, the inclination must satisfysin i < s − Φ p ν ˆ r H . (23)The maximum value of i for capture is obtained with f p = 180 ◦ (Φ p = 1 + e p ).2.3.2. Dependence on f p and e p Higuchi & Ida (2016) found that capture is mostly retrograde for asteroids near the planetary orbit and is progradefor those from distant orbits. We found that this property does not change for a planet in an eccentric orbit. Thesolutions to Equation (16) and are plotted against f p with e p = 0 . ν from ν = 0(planetocentric circular orbit case) to ν = √ f p = 0, the plot showsthe following: ¯ a . . , L ]0 . a min ) . ¯ a . . , L ]0 . . ¯ a . .
85 : [prograde , L ] and [retrograde , L ]0 . . ¯ a . . , L , L ]1 . . ¯ a . .
45 : [retrograde , L ] and [prograde , L ]1 . . ¯ a . . a max ) : [prograde , L ]2 . . ¯ a : [no capture , L ]The asteroids from these regions to the planet’s Hill sphere have orbital eccentricities given by Eq. (18). As seen inFigure 1a and b, the boundaries of individual regions depend on f p . The planet can capture asteroids from furtherregions near perihelion ( f p =0/360 ◦ ) than near aphelion. During a planet’s orbital period, the instantaneous Hill radius r H and v H change. At its perihelion, v H has the largest value, so that the planet captures asteroids from distant regionsthat have large relative velocity. Equation (16) suggests that the range of encounters, ¯ a max − ¯ a min , increases with m p and e p , because ˆ r H ∝ m / and Φ s ∝ e p (for e ≪ a max and ¯ a min are plotted inFigure 2.Figure 1c and d show the solutions to Equation (20) with ν = √ e p . For e p ∼ θ tc coversall the range (0 ◦ -360 ◦ ) as f p changes from 0 ◦ -360 ◦ . The whole range is covered for small values of θ tc with slightmodulation. However, for e p larger than a threshold value ( e cp ), the coverage of θ tc is only a part of 0 ◦ to 360 ◦ . We willshow that capture probability decreases with the increase in e p when e p > e cp . Since we found that e cp is the largestfor ν = √
2, we define the value for ν = √ m p as e cp for m p .2.3.3. The dependence of e cp on the planetary mass The values of e cp are obtained numerically, by finding if the point satisfying d θ tc / d f p = 0. Figure 3 shows e cp for fourtypes of temporary capture for ν = √ m p . The dependence of e cp on m p is approximately given by e c ≃ r H ∝ m / .In the figure, the current values of the eccentricities of the eight planets of the solar system are also plotted. Thebars attached to the points show the maximum variation ranges over past 10 Myr, calculated by following the methoddeveloped by Ito et al. (1995) which is based on the secular perturbation theory of Laskar (1988). As we will showlater, the analytically derived values of e p , beyond which the temporary capture probability drops, agree with theresults obtained by numerical orbital integration. Jupiter, Saturn, and Neptune always have e p < e cp . This meansthat their rates of temporary capture have remained relatively high. The maximum e p values for Venus, Earth, andUranus are slightly higher than e cp but the current values and most of the error-bar ranges of e p are below e cp .Mars, which has relatively high e p , apparently has less chance to capture asteroids with its current orbit. However,the bar of e p for Mars shows that the Martian e p can have the values of e p much smaller than e cp during orbitalvariations. Mercury never has e p < e cp . THE EFFICIENCY OF TEMPORARY CAPTURE BY AN ECCENTRIC PLANETNow we estimate the dependences of the probability of temporary capture on e p and m p of the host planet. Wedefine the probability as K tc /K enc , where K enc and K tc are the phase space volume that satisfies the conditions forencounters with the planet’s Hill sphere, and that for temporary capture, respectively. Encounters wth the Hill sphereare defined as those with minimum distance to the planet less than their instantaneous Hill radius r H . For simplicity,we here set r p = a p (1 + e / a <
1. The maximum eccentricity e is required for an orbit with its aphelion at the L point;¯ a (1 + e ) = (cid:18) e (cid:19) (1 + ˆ r H ) → e = (cid:18) e (cid:19) (cid:18) r H ¯ a (cid:19) . (24)In a similar way, the minimum eccentricity e satisfies¯ a (1 + e ) = (cid:18) e (cid:19) (1 − ˆ r H ) → e = (cid:18) e (cid:19) (cid:18) − ˆ r H ¯ a (cid:19) . (25)Then, the range of eccentricity for close encounters is given by∆ e = e − e = (cid:18) e (cid:19) r H ¯ a . (26)The range of eccentricity for close encounters with ¯ a > θ/ π , where we can set ∆ θ = 2ˆ r H . Then we obtain K enc as the phase space volume by integrating∆ e · ∆ θ/ π over ¯ a with the time weight ( ∝ ¯ a − / ), K enc = ˆ r H π Z ¯ a max ¯ a min ∆ e (¯ a ) ¯ a − d¯ a = 2ˆ r π (cid:18) e (cid:19) (cid:16) ¯ a − min − ¯ a − max (cid:17) , (27)where we assumed a uniform a -distribution of asteroids. We use ¯ a tc , min ,L and ¯ a tc , max ,L for ¯ a min and ¯ a max , which areobtained from Equation (16). We set the upper limit of ¯ a max = 3 to avoid the divergence in the calculation of K enc .This is used only in cases of Jovian mass planets. Assuming e p ≪ r H ≪
1, one can find that K enc ∝ ˆ r .The phase space volume for temporary capture is much more restricted than for the encounters. In a similar way aswe defined K enc , the phase volume of temporary capture is given by K tc = 1 T p Z T p Z ¯ a max ¯ a min ∆ e tc ∆ θ tc π ¯ a − tc d¯ a tc d t. (28)Because a tc , e tc , and θ tc are correlated, it is useful to rewrite ∆ e tc , ∆ θ tc , and d a tc as ∆ e tc = (d e tc / d ν tc )∆ ν tc ,∆ θ tc = (d θ tc / d ν tc )ˆ r H ∆ ν tc , and d¯ a tc = (d¯ a tc / d ν tc ) dν tc . Using these relations, we change the integral of K tc by d a tc tothat by ∆ ν tc . For e p = 0, we set ∆ θ tc = ∆ γ · ˆ r H , where ∆ γ ≪
1. Because the integrands depend on f p , we also addedtime averaging over an orbital period of the planet ( T p = 1).Thereby, the temporary capture rate is given from Equations (16) as K tc = (∆ ν ) ˆ r H πT p Z T p Z ν max ν min d e tc d ν tc d θ tc d ν tc d¯ a tc d ν tc ¯ a − tc d ν tc d t, (29) d¯ a tc d ν = ± a Φ p √ r H (cid:16)p − Φ p ± √ r H ν (cid:17) (30)d e tc d ν = d e tc d¯ a tc d¯ a tc d ν (31)d θ tc d ν = d g d ν p − g (32)d e tc d¯ a tc = Φ p A ∓ sin α p (cid:18) − Φ p A ∓ ¯ a tc (cid:19) ¯ a − e − (33)d g d ν = d g d e tc d e tc d ν ; d g d e tc = − ge − ± sin α p p e − cos α p . (34)For e p = 0, K tc = ∆ ν ∆ γ ˆ r H π Z ν max ν min d e tc d ν tc d¯ a tc d ν tc ¯ a − tc d ν tc (35)Assuming e p ≪ r H ≪
1, one can find (d a tc / d ν tc ) ∝ ˆ r H , and (d e tc / d¯ a tc ), (d g/ d e tc ), and g are independent ofˆ r H . This leads to K tc ∝ ˆ r , which is the same as K enc , implying that K tc /K enc is independent of m p for e p ≪ ν min < ν tc < ν max , can be simply estimated in the framework of the two-body problem(planet-particle) as follows. The physical radius of the planet may give the value of ν min . A planetocentric temporarilycaptured orbit has its apocenter distance at a s (1 + e s ) ≃ r H . The pericenter distance, a s (1 − e s ), must be larger thanthe physical radius of the planet, R p , to avoid a collision. From these two equations, e s < − ( R p /r H )1 + ( R p /r H ) (36)Since ν = √ κ = √ − e s for f s = 0, ν min = s R p /r H R p /r H ) ≃ q R p /r H . (37)The simplest assumption for the maximum value in the framework of the two-body problem is ν max = √
2, which isthe upper limit for an elliptic orbit around the planet. However, the effect of the third body (Sun) changes them.We found that it is more appropriate to assume ν min = 0 .
25 and ν max = 2 for a prograde trap and ν min = 0 . ν max = √ K tc /K enc as a function of e p for planets with Martian, Jovian, Earth, and Neptunian mass.Each plot has four curves for the temporary capture types, and the sum of the four types (the black line). The totalratio (black) is almost constant or rather gradually increases with e p . until e p exceeds e cp . The asymptotic values of K tc /K enc at e p → m p ) and semimajor axis ( a p ), as we predicted. As shown inFigure 4, e cp ≃ r H where ˆ r H = 4 . × − for Mars, ˆ r H = 0 .
068 for Jupiter, ˆ r H = 0 .
01 for Earth, and ˆ r H = 0 . e p > e cp s, K tc decays with e p approximately as ∝ e − . As will be shown in the next section, thefunctional form of the predicted K tc /K enc agrees very well with the results of numerical orbital integrations, while theallowance for temporary capture ∆ ν and ∆ γ cannot be estimated by analytical arguments here. Because ν expressesthe satellite orbital energy at the Hill radius, it is expected that the allowance ∆ ν is independent of m p and a p aswell. Also the independence of ∆ γ is expected since the angle ∆ θ tc would be a function only of ˆ r H . From comparisonwith the numerical simulations, we empirically set ∆ ν ∼ .
025 and ∆ γ ∼ . COMPARISON WITH NUMERICAL RESULTSWe perform numerical calculations for the temporary capture of bodies by planets with Mars, Jupiter, Earth, andNeptune masses to evaluate the relevance of our analytical formulae.4.1.
Methods and Initial Conditions
We compute the orbital evolution of massless bodies, which correspond to asteroids, perturbed by a planet in acircular or eccentric orbit, using a 4th-order Hermite integration scheme. The parameters are summarized in Table 1.The number of the massless bodies in each run is 5 × . Asteroids are initially uniformly distributed on the a, e -planebetween ¯ a tc , min ,L < ¯ a < ¯ a tc , max ,L , e min < e < e max , which are derived analytically and numerically in Section 2.3.2and summarized in Figure 2. The parameter θ is also uniformly distributed between 0 and 2 π . We set the upper limitof ¯ a tc , max ,L = 3. In most runs we assume i = 0 for the asteroids. In several additional runs, we give i with a uniformdistribution for 0 < i < i tc , max where i tc , max is given by Equation (23) for f p = 180 ◦ and ν = √
2. We regard asteroidsas temporarily captured bodies if they stay within r H from the planet longer than one orbital period of the planet T p .Using the planetocentric location and the relative velocity vector to the planet, at the moment when an asteroidenters the r H region around the planet for the first time, we define the type of temporary capture: [prograde- L ],[retrograde- L ], [retrograde- L ], and [prograde- L ].In this paper, we focus on the equilibrium state where the ratio of temporary capture and encounter rates becomesconstant with time. To obtain this state, we first perform several long-time calculations with 10 particles for 10 T p and choose the time range where the ratio is constant with time. Note that Higuchi & Ida (2016) presented thecumulative number of captured bodies over 10 years, which is not directly compared with the results presented here.4.2. Results
Figure 5 shows θ of the temporarily captured bodies against f p for a Martian mass planet with various e p . Theanalytical prediction (eq. (19)) for 0 ≤ ν ≤ √ e cp ), agrees well with the numerical results.Figure 6 shows the ratio ( n tc /n enc ) of the temporary capture and encounter rates as a function of e p for planets withMartian, Jovian, Earth, and Neptunian mass, respectively. The ratio drops beyond the predicted values of e cp ≃ r H ,which are 0.02, 0.27, 0.04, and 0.1 for Martian, Jovian, Earth, and Neptunian mass. This drop of n tc /n enc is wellreproduced by the analytical prediction in Figure 4.The value of n tc /n enc for e p < e cp is ∼ − , which is almost independent of the planetary mass, as predicted. Weperformed additional numerical calculations using particles with i < i max for ν = √ n tc /n enc for the 3D calculations are similar to those for the 2D calculations (within afactor of 2). SUMMARY AND DISCUSSIONIn order to explore the origins of irregular or minor satellites around the planets in the solar system, we haveinvestigated the probability of temporary capture through semi-analytical arguments and numerical integration. Weextended the analysis of temporary capture around a planet in a circular orbit developed by Higuchi & Ida (2016)to that around a planet in an eccentric orbit, allowing us to discuss the origins of the Martian satellites. We derivedthe capture probability as a function of planetary mass ( m p ) and eccentricity ( e p ). Analytical formulae reproduce thenumerical integrations very well.We found that the temporary capture occurs at ∼ .
1% of encounters that enter Hill sphere of a planet, independentof m p , a p (semimajor axis) and e p up to a critical value e cp ≃ m p / M ⊙ ) / . For e p > e cp , the probability decays withincreasing e p as ∝ e − (1 − .The current eccentricity of Mars is several times larger than e cp , so that the capture origin of Phobos and Deimos looksunfavored. However, as shown in Figure 3, the Martian eccentricity changes with time and can be lower than e cp forsome fraction of time, and temporary capture may have been available in the past. Note again that temporary captureis a necessary condition for permanent capture and their respective probabilities are not necessarily proportional toeach other. As will be discussed in a separate paper, tight capture could be found in the cases where e p > e cp . Ina subsequent paper, we will discuss the probability of permanent capture and the possibility of the capture origin ofPhobos and Deimos.We thank an anonymous referee for his/her useful comments that helped to improve the paper. This work wassupported by JSPS KAKENHI grant Number 23740335 and 15H02065. Data analyses were in part carried out on thePC cluster at the Center for Computational Astrophysics, National Astronomical Observatory of Japan.REFERENCES Burns, J. A. (1978) Symposium on the Satellites of Mars,Washington, D.C., Aug. 11, 1977. Vistas in Astronomy, vol.22, pt. 2,Citron, R.I., Genda, H., & Ida, S. (2015) Icarus, 252, 334 Fraeman, A. A., Murchie, S.L., Arvidson,R.E., Clark, R.N.,Morris, R.V., Rivkin, A.S., & Vilas, F. (2014) Icarus, 229, 196Higuchi, A. & Ida, S. 2016, AJ, 151, 16Ito, T., Masuda, K., Hamano, Y., & Matsui, T. (1995)) Journalof Geophysical Research, 100, 15147
Jewitt, D & Haghighipour, N. 2007, ARA&A, 45, 261Laskar, J. 1988, A&A, 198, 341Murray, C. D.& Dermott, S. F. 1999, Solar System Dynamics,Cambridge: Cambridge University Press Nicolson, P. D., ´Cuk, M., Sheppard, S. S., Nesvorn´y, D., &Johonson, T. V. 2008, in The Solar System Beyond Neptune,ed. M. A. Barucci et al. (Tucson, AZ: Univ. Arizona Press),411Rosenblatt, P., Charnoz, S., Dunseath, K. M., Terao-Dunseath,M., Trinh, A., Hyodo, R., Genda, H., & Toupin, S. 2016Nature Geoscience Nature Geoscience, 9, 581Roy, A. E. 2005, Orbital Motion, 4th edition: CRC Press
Planet a p (au) m p ( M ⊙ ) e p RangeEarth 1 3.00e −
06 0.004-0.36Mars 1.52 3.72e −
07 0.002-0.18Jupiter 5.2 9.55e −
04 0.01-0.9Neptune 30.1 5.15e −
05 0.005-0.5
Table 1 . Parameters of planets Used in Numerical Calculations. a / a p f p [deg] ν =00.10.20.51 √
2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 0 45 90 135 180 225 270 315 360(b) a / a p f p [deg] ν =00.10.20.51 √ θ [ deg ] f p [deg] e p =00.050.10.20.5 -180-135-90-45 0 45 90 135 180 0 45 90 135 180 225 270 315 360(d) θ [ deg ] f p [deg] e p =00.050.10.20.5 Figure 1 . The solutions to Equations (16) are plotted against f p with e p = 0 . ν = 0(black),0.1 (orange), 0.2 (light blue), 0.5 (green), 1 (yellow), and √ L -type and (b) L -type captures. The solutions toEquations (19) with ν = √ e p = 0, 0.01, 0.02, 0.05, and 0.09: (c) L -type and (d) L -type captures. The solid and dashed curves are for prograde and retrograde captures, respectively. a min a max e min e max a m i n / a p , a m a x / a p e m i n , e m a x e p a min e min e max a m i n / a p , a m a x / a p e m i n , e m a x e p a min a max e min e max a m i n / a p , a m a x / a p e m i n , e m a x e p a min a max e min e max a m i n / a p , a m a x / a p e m i n , e m a x e p Figure 2 . Ranges of initial orbital elements are summarized for each planetary mass against e p (top-left: Martian mass, top-right: Jovian mass, bottom-left: Earth mass, bottom-right: Neptunian mass.) Black curves show ¯ a max (solid) and ¯ a min (dashed)on the left y − axis and orange curves show e max (solid) and e min (dashed) on the right y − axis. -7 -6 -5 -4 -3 MercuryMars Venus Earth UranusNeptune SaturnJupiter e cc en t r i c i t y m p [M Sun ]critical eccentricity: pro, L retro, L retro, L pro, L Figure 3 . Critical eccentricity e cp for ν = √ m p . The curve types indicate the capture type (solid:[prograde, L ], long-short dashed: [retrograde, L ], dashed: [retrograde, L ], short dashed: [prograde, L ].) The orange curveshows e p = 5ˆ r H . Current eccentricities of eight planets of the solar system are also plotted against their mass. The error barsshow the variations over 10 Myr calculated following Laskar (1988). -3 -2 -1 K t c / K en c e p allpro, L retro, L retro, L pro, L -3 -2 -1 K t c / K en c e p allpro, L retro, L retro, L pro, L -3 -2 -1 K t c / K en c e p allpro, L retro, L retro, L pro, L -3 -2 -1 K t c / K en c e p allpro, L retro, L retro, L pro, L Figure 4 . Efficiency of temporary capture K tc /K enc plotted against e p for planets with Martian (top-left), Jovian (top-right),Earth (bottom-left), and Neptunian (bottom-right) mass using ν min = 0 .
25 and ν max = 2 for prograde and ν min = 0 . ν max = √ ν ∼ .
025 and ∆ γ ∼ .
05. The K tc for each temporary capture type is plotted in color;[prograde, L ] (green), [retrograde, L ] (blue), [retrograde, L ] (orange), and [prograde, L ] (pink). The black curve shows thesum of the four types. We adopt Equation (35) for K tc if K tc with Equation (29) for e p < e cp is less than that with Equation(35). -180-135-90-45 0 45 90 135 180 0 90 180 270 360 e p =0.005 θ [ deg ] f p [deg] -180-135-90-45 0 45 90 135 180 0 90 180 270 360 e p =0.01 θ [ deg ] f p [deg] -180-135-90-45 0 45 90 135 180 0 90 180 270 360 e p =0.014 θ [ deg ] f p [deg]-180-135-90-45 0 45 90 135 180 0 90 180 270 360 e p =0.02 θ [ deg ] f p [deg] -180-135-90-45 0 45 90 135 180 0 90 180 270 360 e p =0.03 θ [ deg ] f p [deg] -180-135-90-45 0 45 90 135 180 0 90 180 270 360 e p =0.05 θ [ deg ] f p [deg]-180-135-90-45 0 45 90 135 180 0 90 180 270 360 e p =0.09 θ [ deg ] f p [deg] -180-135-90-45 0 45 90 135 180 0 90 180 270 360 e p =0.12 θ [ deg ] f p [deg] -180-135-90-45 0 45 90 135 180 0 90 180 270 360 e p =0.18 θ [ deg ] f p [deg] Figure 5 . Argument of perihelion of temporarily captured bodies by a Martian mass planet with various e p at the moment ofentering the Hill sphere for the first time are plotted against f p . The solution to Equation (19) for each temporary capture typefor ν = 0 (black), 0.1 (orange), 0.2 (light blue), 0.5 (green), 1 (yellow), and √ -5 -4 -3 n t c / n en c e p -5 -4 -3 n t c / n en c e p -5 -4 -3 n t c / n en c e p -5 -4 -3 n t c / n en c e p Figure 6 . The ratio of the number of temporary captures n tc to that of encounters n enc is plotted against e p for planets withMartian (top-left), Jovian (top-right), Earth (bottom-left), and Neptunian (bottom-right) mass. The colors indicate the typesof temporary capture; [prograde, L ] (green), [retrograde, L ] (blue), [retrograde, L ] (orange), [prograde, L2