Temporary Capture of Asteroids by an Eccentric Planet

TL;DR

This study analyzes the likelihood of asteroids being temporarily captured by planets, revealing that capture probability decreases with planetary eccentricity and is influenced by the planet's orbital parameters.

Contribution

It provides an analytical and numerical assessment of how planetary eccentricity affects asteroid capture probability, including a critical eccentricity threshold.

Findings

Capture probability is about 0.1% for circular orbits.

Capture probability decreases as planetary eccentricity exceeds a critical value.

Mars's current eccentricity suggests past asteroid captures are possible.

Abstract

We have investigated the probability of temporary capture of asteroids in eccentric orbits by a planet in a circular or an eccentric orbit through analytical and numerical calculations. We found that in the limit of the circular orbit, the capture probability is $\sim 0.1\%$ of encounters to the planet's Hill sphere, independent of planetary mass and semimajor axis. In general, the temporary capture becomes more difficult as the planet's eccentricity ($e_{\rm p}$) increases. We found that the capture probability is almost independent of $e_{\rm p}$ until a critical value ($e_{\rm p}^{\rm c}$) that is given by $\simeq$ 5 times the Hill radius scaled by the planet's semimajor axis. For $e_{\rm p} > e_{\rm p}^{\rm c}$, the probability decreases approximately in proportion to $e_{\rm p}^{-1}$. The current orbital eccentricity of Mars is several times larger than $e_{\rm p}^{\rm c}$. However, since the range of secular change in Martian eccentricity overlaps $e_{\rm p}^{\rm c}$, the capture of minor bodies by the past Mars is not ruled out.

Tables

Table 1. Table 1: Parameters of planets Used in Numerical Calculations.

Planet	$a_p$ (au)	$m_p$ ($M_{\odot }$)	$e_p$ Range
Earth	1	3.00e$-$06	0.004-0.36
Mars	1.52	3.72e$-$07	0.002-0.18
Jupiter	5.2	9.55e$-$04	0.01-0.9
Neptune	30.1	5.15e$-$05	0.005-0.5

Equations

$$r_{\mathrm{H}}=r_{\mathrm{p}}{\left(\frac{m_{\mathrm{p}}}{3M_{\odot }}\right)}^{\frac{1}{3}}=r_{\mathrm{p}}{\hat{r}}_{\mathrm{H}},$$

$$r_{\mathrm{p}}$$

$$r=a\frac{1-e^2}{1+e\cos (f+\theta )}=r_{\mathrm{p}}{A}_{\mp },$$

$$\displaystyle\left\{\begin{array}[]{l}A_{-}=1-\hat{r}_{\rm H}\;\;{\rm at}\;\;L_{1}\\ A_{+}=1+\hat{r}_{\rm H}\;\;{\rm at}\;\;L_{2}.\end{array}\right.$$

$$v=\sqrt{G{M}_{\odot }\left(\frac{2}{r}-\frac{1}{a}\right)}=\sqrt{G{M}_{\odot }\left(\frac{2}{A_{\mp }{r}_{\mathrm{p}}}-\frac{1}{a}\right)}=v_{\mathrm{p}}\chi ,$$

$$\chi =\sqrt{\frac{2}{{\Phi }_{\mathrm{p}}{A}_{\mp }}-\frac{a_{\mathrm{p}}}{a}},$$

$${\Phi }_{\mathrm{p}}=\frac{1-e_{\mathrm{p}}^2}{1+e_{\mathrm{p}}\cos f_{\mathrm{p}}}.$$

$$\mathbf{v}-{\mathbf{v}}_{\mathrm{p}}={\mathbf{v}}_{\mathrm{s}},$$

$$v_{\mathrm{s}}=\sqrt{G{m}_{\mathrm{p}}\left(\frac{2}{r_{\mathrm{H}}}-\frac{1}{a_{\mathrm{s}}}\right)}=v_{\mathrm{H}}\sqrt{2-{\Phi }_{\mathrm{s}}},$$

$${\Phi }_{\mathrm{s}}$$

$$v_{\mathrm{H}}$$

$$\nu =v_{\mathrm{s}}/v_{\mathrm{H}}=\sqrt{2-{\Phi }_{\mathrm{s}}}$$

$$\sin \alpha$$

$$\sin {\alpha }_{\mathrm{p}}$$

$$|\chi -1|=\nu v_{\mathrm{H}}.$$

$$\frac{a}{a_{\mathrm{p}}}\equiv {\bar{a}}_{\mathrm{tc}}={\Phi }_{\mathrm{p}}{\left[\frac{2}{A_{\mp }}-{\left(\sqrt{2-{\Phi }_{\mathrm{p}}}\pm \sqrt{3}\nu {\hat{r}}_{\mathrm{H}}\right)}^2\right]}^{-1}.$$

$$h=rv\sin \alpha =\sqrt{G{M}_{\odot }a(1-e^2)}.$$

$$e_{\mathrm{tc}}=\sqrt{1-\sin {\alpha }_{\mathrm{p}}\left[1-\left(1-\frac{{\Phi }_{\mathrm{p}}{A}_{\mp }}{{\bar{a}}_{\mathrm{tc}}}\right)\right]}.$$

$${\theta }_{\mathrm{tc}}$$

$$g$$

$$v_{\mathrm{s}}^2$$

$${\chi }^2+\left(\frac{2}{{\Phi }_{\mathrm{p}}}-1\right)-2\chi \sqrt{\frac{2}{{\Phi }_{\mathrm{p}}}-1}\cos i=(\nu v_{\mathrm{H}}{)}^2.$$

$$\sin i<\sqrt{\frac{3}{2-{\Phi }_{\mathrm{p}}}}\nu {\hat{r}}_{\mathrm{H}}.$$

$$\displaystyle\begin{array}[]{rclcl}&\bar{a}&\lesssim 0.6&:&\;\;[\mbox{no capture},\;L_{1}]\\ 0.6\;(=\bar{a}_{\rm min})\;\lesssim&\bar{a}&\lesssim 0.8&:&\;\;[\mbox{prograde},\;L_{1}]\\ 0.8\lesssim&\bar{a}&\lesssim 0.85&:&\;\;[\mbox{prograde},\;L_{1}]\;\mbox{and}\;[\mbox{retrograde},\;L_{2}]\\ 0.85\lesssim&\bar{a}&\lesssim 1.2&:&\;\;[\mbox{retrograde},\;L_{1},L_{2}]\\ 1.2\lesssim&\bar{a}&\lesssim 1.45&:&\;\;[\mbox{retrograde},\;L_{1}]\;\mbox{and}\;[\mbox{prograde},\;L_{2}]\\ 1.45\lesssim&\bar{a}&\lesssim 2.9\;(=\bar{a}_{\rm max})&:&\;\;[\mbox{prograde},\;L_{2}]\\ 2.9\lesssim&\bar{a}&&:&\;\;[\mbox{no capture},\;L_{1}]\end{array}$$

$$\bar{a}(1+e_1)=\left(1+\frac{1}{2}{e}_{\mathrm{p}}^2\right)\left(1+{\hat{r}}_{\mathrm{H}}\right)$$

$$\bar{a}(1+e_2)=\left(1+\frac{1}{2}{e}_{\mathrm{p}}^2\right)\left(1-{\hat{r}}_{\mathrm{H}}\right)$$

$$\Delta e=e_1-e_2=\left(1+\frac{1}{2}{e}_{\mathrm{p}}^2\right)\frac{2{\hat{r}}_{\mathrm{H}}}{\bar{a}}.$$

$$K_{\mathrm{enc}}=\frac{{\hat{r}}_{\mathrm{H}}}{\pi }{\int }_{{\bar{a}}_{\min }}^{{\bar{a}}_{\max }}\Delta e(\bar{a}){\bar{a}}^{-\frac{3}{2}}\mathrm{d}\bar{a}=\frac{2{\hat{r}}_{\mathrm{H}}^2}{3\pi }\left(1+\frac{1}{2}{e}_{\mathrm{p}}^2\right)\left({\bar{a}}_{\min }^{-\frac{3}{2}}-{\bar{a}}_{\max }^{-\frac{3}{2}}\right),$$

$$K_{\mathrm{tc}}=\frac{1}{T_{\mathrm{p}}}{\int }_0^{T_{\mathrm{p}}}{\int }_{{\bar{a}}_{\min }}^{{\bar{a}}_{\max }}\Delta e_{\mathrm{tc}}\frac{\Delta {\theta }_{\mathrm{tc}}}{2\pi }{\bar{a}}_{\mathrm{tc}}^{-\frac{3}{2}}\mathrm{d}{\bar{a}}_{\mathrm{tc}}\mathrm{d}t.$$

$$K_{\mathrm{tc}}$$

$$\frac{\mathrm{d}{\bar{a}}_{\mathrm{tc}}}{\mathrm{d}\nu }$$

$$\frac{\mathrm{d}{e}_{\mathrm{tc}}}{\mathrm{d}\nu }$$

$$\frac{\mathrm{d}{\theta }_{\mathrm{tc}}}{\mathrm{d}\nu }$$

$$\frac{\mathrm{d}{e}_{\mathrm{tc}}}{\mathrm{d}{\bar{a}}_{\mathrm{tc}}}$$

Full text

planets and satellites: formation

\AuthorCallLimit

=2

1 Introduction

Irregular satellites around giant planets, which are small and with elliptical and inclined orbits, are usually thought to be captured passing asteroids (e.g., Jewitt & Haghighipour, 2007; Nicolson et al., 2008). The objects captured temporarily 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 and semimajor axis of the host planet, and clarified the range of semimajor axes of field particles for prograde and retrograde capture.

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: Phobos and Deimos. Two major theories of the origin of these satellites are (1) in situ formation through accretion of an impact-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 characteristics of the satellites are similar to those of primitive asteroids. Spectral observations of Phobos and Deimos suggest that the material of the satellites is best modeled as a primitive material, which may not be easily explained by the large impact origin (Fraeman et al., 2014). Future sample return missions, such as MMX (Mars Moon eXploration), will provide important clues about the Martian satellite origin. It is important to explore the possibility of the capture origin 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 the temporary capture probability through analytical and numerical calculations. We derive the probability of temporary capture from encounters with the planet’s Hill sphere as a function of planetary eccentricity $e_{\rm p}$ and mass $m_{\rm p}$. If the encounter frequency is given by other simulations, we can evaluate the probability of temporary capture throughout the history of the solar system. Most of our analysis and orbital calculations assume planar orbits, but some calculations are done with small finite inclinations. Temporary capture is a necessary condition for permanent capture. The relation 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 define and derive the efficiency of temporary capture in Section 3. The methods and results of numerical calculations are presented and compared with the analytical prediction in Section 4. In Section 5, we summarize the results and comment on the origin of Martian satellites.

2 ANALYTICAL DERIVATION OF TEMPORARY CAPTURE EFFICIENCY BY AN ECCENTRIC PLANET

We first derive analytical formulae for temporary capture by an eccentric planet. These formulae give orbital elements of the asteroids that can be captured, as functions of the mass, eccentricity, and true anomaly of the host planet. As we will 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 independent two-body problems (Sun-particle and planet-particle). The particles are candidates that are captured by the planet to become satellites. Hereafter, we refer to particles as ”asteroids,” although the particles do not necessarily originate from the asteroid belt. We identify the relative velocity between the asteroid and the planet in heliocentric orbits with the satellite velocity orbiting around the planet (condition [1]) at the capture point. The capture points are assumed to be the $L_{1}$ and $L_{2}$ points (condition [2]). The distance of the points from the planet is the Hill radius,

[TABLE]

where $m_{\rm p}$ is the planet mass and $M_{\odot}$ is the solar mass. The instantaneous heliocentric distance is given by

[TABLE]

where $a_{\rm p}$, $m_{\rm p}$, $e_{\rm p}$, and $f_{\rm p}$ are semimajor axis, mass, eccentricity, and true anomaly of the planet, respectively. We also assume that the geometric condition that the two elliptic orbits are touching at a capture point; the velocity vectors of 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 toward the 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 heliocentric distance, semimajor axis, eccentricity of the asteroid, and true anomaly, respectively.

Condition [2] reads as

[TABLE]

where $\theta=f_{\rm p}-f$ and

[TABLE]

The heliocentric velocity of the asteroid at capture is

[TABLE]

where $v_{\rm p}=\sqrt{GM_{\odot}/a_{\rm p}}$ and

[TABLE]

Condition [1] reads as

[TABLE]

where ${\bf v}$ and ${\bf v}_{\rm p}$ are heliocentric velocities of the asteroid and the planet and ${\bf v}_{\rm s}$ is the planetocentric velocity of the asteroid as a satellite at the capture. The velocity of the satellite at the planetocentric distance $r_{\rm s}=r_{\rm H}$ is

[TABLE]

where $a_{\rm s}=r_{\rm H}/\Phi_{\rm s}$ is the planetocentric semimajor axis of the satellite,

[TABLE]

and $e_{\rm s}$ and $f_{\rm s}$ are the planetocentric eccentricity and true anomaly. Since $v_{\rm H}$ is a circular velocity around the planet at the planetocentric distance $r_{\rm s}=r_{\rm H}$,

[TABLE]

is related to the planetocentric orbital eccentricity (which is equivalent to $\kappa^{2}$ appearing in Higuchi & Ida (2016)); $\nu=1$ corresponds to a circular orbit with the semimajor axis $a_{\rm s}=r_{\rm H}$ and the orbit is hyperbolic for $\nu>\sqrt{2}$.

Condition [3] is expressed by $\alpha=\alpha_{\rm p}$, where $\alpha$ and $\alpha_{\rm p}$ are the angles between the position and velocity vectors of the asteroid and those of the planet, which are given by

[TABLE]

These angles are given geometrically, applying the law of cosines to a triangle composed of ${\bf r}$, the $x$ axis, and the tangent line of the orbit at $r$. Another way to derive $\alpha$ 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 temporarily captured asteroids.

2.3.1 Derivation of Heliocentric Orbital Elements for Temporary Capture

Semimajor axis.

Using ${\bf v}\parallel{\bf v}_{\rm p}$ (condition [3]), $v_{\rm s}=\nu v_{\rm H}$ (Eq. 11), and $v=v_{\rm p}\chi$ (Eq. 14), condition [1] (Eq. 10) becomes

[TABLE]

Substituting Equations (3),(6), and (8), into Equation (17), we obtain the heliocentric semimajor axis of the asteroid at temporary capture as

[TABLE]

Note that Equation (18) has four values corresponding to a combination of prograde or retrograde and $L_{1}$ or $L_{2}$. If the sign in front of $\sqrt{3}$ is $"+"$, the temporary capture is prograde. The $"-"$ sign represents retrograde capture. The sign in $A_{\mp}$ represents $L_{1}$-type or $L_{2}$-type (Eq. (6)).

Eccentricity.

The heliocentric orbital angular momentum of the asteroid is

[TABLE]

Substituting Equations (19) into condition [2] given by Eq. (3) with $\alpha=\alpha_{\rm p}$, we obtain the heliocentric eccentricity at temporary capture,

[TABLE]

Angle of perihelion $\theta$.

The perihelion angle at temporary capture is easily obtained from Equation (3),

[TABLE]

where Equation (20) is substituted at the end.

Inclination.

If the asteroid has non-zero heliocentric inclination $i$, the relative velocity is modified. Since the relative velocity is equal to $v_{\rm s}$,

[TABLE]

which is reduced to

[TABLE]

For this equation to have a solution, the inclination must satisfy

[TABLE]

The maximum value of $i$ for capture is obtained with $f_{\rm p}=180^{\circ}$ ($\Phi_{\rm p}=1+e_{\rm p}$).

2.3.2 Dependence on $f_{\rm p}$ and $e_{\rm p}$

Higuchi & Ida (2016) found that capture is mostly retrograde for asteroids near the planetary orbit and is prograde for those from distant orbits. We found that this property does not change for a planet in an eccentric orbit. The solutions to Equation (18) and are plotted against $f_{\rm p}$ with $e_{\rm p}=0.2$ and a Jovian mass planet for $\nu$ from $\nu=0$ (planetocentric circular orbit case) to $\nu=\sqrt{2}$ (parabolic orbit cases) in Figures 1a and b. For $f_{\rm p}=0$, the plot shows the following:

[TABLE]

The asteroids from these regions to the planet’s Hill sphere have orbital eccentricities given by Eq. (20). As seen in Figure 1a and b, the boundaries of individual regions depend on $f_{\rm p}$. The planet can capture asteroids from further regions near perihelion ($f_{\rm p}$=0/360∘) than near aphelion. During a planet’s orbital period, the instantaneous Hill radius $r_{\rm H}$ and $v_{\rm H}$ change. At its perihelion, $v_{\rm H}$ has the largest value, so that the planet captures asteroids from distant regions that have large relative velocity. Equation (18) suggests that the range of encounters, $\bar{a}_{\rm max}-\bar{a}_{\rm min}$, increases with $m_{\rm p}$ and $e_{\rm p}$, because $\hat{r}_{\rm H}\propto m_{\rm p}^{1/3}$ and $\Phi_{\rm s}\propto e_{\rm p}$ (for $e_{\rm p}^{2}\ll 1$). Numerically obtained values of $\bar{a}_{\rm max}$ and $\bar{a}_{\rm min}$ are plotted in Figure 2.

Figure 1c and d show the solutions to Equation (22) with $\nu=\sqrt{2}$ for different values of $e_{\rm p}$. For $e_{\rm p}\sim 0$, $\theta_{\rm tc}$ covers all the range ($0^{\circ}$-$360^{\circ}$) as $f_{\rm p}$ changes from $0^{\circ}$-$360^{\circ}$. The whole range is covered for small values of $\theta_{\rm tc}$ with slight modulation. However, for $e_{\rm p}$ larger than a threshold value ($e_{\rm p}^{\rm c}$), the coverage of $\theta_{\rm tc}$ is only a part of $0^{\circ}$ to $360^{\circ}$. We will show that capture probability decreases with the increase in $e_{\rm p}$ when $e_{\rm p}>e_{\rm p}^{\rm c}$. Since we found that $e_{\rm p}^{\rm c}$ is the largest for $\nu=\sqrt{2}$, we define the value for $\nu=\sqrt{2}$ with a given $m_{\rm p}$ as $e_{\rm p}^{\rm c}$ for $m_{\rm p}$.

2.3.3 The dependence of $e_{\rm p}^{\rm c}$ on the planetary mass

The values of $e_{\rm p}^{\rm c}$ are obtained numerically, by finding if the point satisfying ${\rm d}\theta_{\rm tc}/{\rm d}f_{\rm p}=0$. Figure 3 shows $e_{\rm p}^{\rm c}$ for four types of temporary capture for $\nu=\sqrt{2}$ as a function of $m_{\rm p}$. The dependence of $e_{\rm p}^{\rm c}$ on $m_{\rm p}$ is approximately given by $e_{\rm c}\simeq 5\hat{r}_{\rm H}\propto m_{\rm p}^{1/3}$.

In the figure, the current values of the eccentricities of the eight planets of the solar system are also plotted. The bars attached to the points show the maximum variation ranges over past 10 Myr, calculated by following the method developed by Ito et al. (1995) which is based on the secular perturbation theory of Laskar (1988). As we will show later, the analytically derived values of $e_{\rm p}$, beyond which the temporary capture probability drops, agree with the results obtained by numerical orbital integration. Jupiter, Saturn, and Neptune always have $e_{\rm p}<e_{\rm p}^{\rm c}$. This means that their rates of temporary capture have remained relatively high. The maximum $e_{\rm p}$ values for Venus, Earth, and Uranus are slightly higher than $e_{\rm p}^{\rm c}$ but the current values and most of the error-bar ranges of $e_{\rm p}$ are below $e_{\rm p}^{\rm c}$.

Mars, which has relatively high $e_{\rm p}$, apparently has less chance to capture asteroids with its current orbit. However, the bar of $e_{\rm p}$ for Mars shows that the Martian $e_{\rm p}$ can have the values of $e_{\rm p}$ much smaller than $e_{\rm p}^{\rm c}$ during orbital variations. Mercury never has $e_{\rm p}<e_{\rm p}^{\rm c}$.

3 THE EFFICIENCY OF TEMPORARY CAPTURE BY AN ECCENTRIC PLANET

Now we estimate the dependences of the probability of temporary capture on $e_{\rm p}$ and $m_{\rm p}$ of the host planet. We define the probability as $K_{\rm tc}/K_{\rm enc}$, where $K_{\rm enc}$ and $K_{\rm tc}$ are the phase space volume that satisfies the conditions for encounters with the planet’s Hill sphere, and that for temporary capture, respectively. Encounters wth the Hill sphere are defined as those with minimum distance to the planet less than their instantaneous Hill radius $r_{\rm H}$. For simplicity, we here set $r_{\rm p}=a_{\rm p}(1+e_{\rm p}^{2}/2)$. For example, we consider a close encounter orbit with $\bar{a}<1$. The maximum eccentricity $e_{1}$ is required for an orbit with its aphelion at the $L_{2}$ point;

[TABLE]

In a similar way, the minimum eccentricity $e_{2}$ satisfies

[TABLE]

Then, the range of eccentricity for close encounters is given by

[TABLE]

The range of eccentricity for close encounters with $\bar{a}>1$ is the same. The range of the angle of perihelion for close encounters is $\Delta\theta/2\pi$, where we can set $\Delta\theta=2\hat{r}_{\rm H}$. Then we obtain $K_{\rm enc}$ as the phase space volume by integrating $\Delta e\cdot\Delta\theta/2\pi$ over $\bar{a}$ with the time weight ($\propto\bar{a}^{-3/2}$),

[TABLE]

where we assumed a uniform $a$-distribution of asteroids. We use $\bar{a}_{{\rm tc,min},L_{1}}$ and $\bar{a}_{{\rm tc,max},L_{2}}$ for $\bar{a}_{\rm min}$ and $\bar{a}_{\rm max}$, which are obtained from Equation (18). We set the upper limit of $\bar{a}_{\rm max}=3$ to avoid the divergence in the calculation of $K_{\rm enc}$. This is used only in cases of Jovian mass planets. Assuming $e_{\rm p}\ll 1$ and $\hat{r}_{\rm H}\ll 1$, one can find that $K_{\rm enc}\propto\hat{r}_{\rm H}^{3}$.

The phase space volume for temporary capture is much more restricted than for the encounters. In a similar way as we defined $K_{\rm enc}$, the phase volume of temporary capture is given by

[TABLE]

Because $a_{\rm tc}$, $e_{\rm tc}$, and $\theta_{\rm tc}$ are correlated, it is useful to rewrite $\Delta e_{\rm tc}$, $\Delta\theta_{\rm tc}$, and ${\rm d}a_{\rm tc}$ as $\Delta e_{\rm tc}=({\rm d}e_{\rm tc}/{\rm d}\nu_{\rm tc})\Delta\nu_{\rm tc}$, $\Delta\theta_{\rm tc}=({\rm d}\theta_{\rm tc}/{\rm d}\nu_{\rm tc})\hat{r}_{\rm H}\Delta\nu_{\rm tc}$, and ${\rm d}\bar{a}_{\rm tc}=({\rm d}\bar{a}_{\rm tc}/{\rm d}\nu_{\rm tc})d\nu_{\rm tc}$. Using these relations, we change the integral of $K_{\rm tc}$ by ${\rm d}a_{\rm tc}$ to that by $\Delta\nu_{\rm tc}$. For $e_{\rm p}=0$, we set $\Delta\theta_{\rm tc}=\Delta\gamma\cdot\hat{r}_{\rm H}$, where $\Delta\gamma\ll 1$. Because the integrands depend on $f_{\rm p}$, we also added time averaging over an orbital period of the planet ($T_{\rm p}=1$).

Thereby, the temporary capture rate is given from Equations (18) as

[TABLE]

For $e_{\rm p}=0$,

[TABLE]

Assuming $e_{\rm p}\ll 1$ and $\hat{r}_{\rm H}\ll 1$, one can find $({\rm d}a_{\rm tc}/{\rm d}\nu_{\rm tc})\propto\hat{r}_{\rm H}$, and $({\rm d}e_{\rm tc}/{\rm d}\bar{a}_{\rm tc})$, $({\rm d}g/{\rm d}e_{\rm tc})$, and $g$ are independent of $\hat{r}_{\rm H}$. This leads to $K_{\rm tc}\propto\hat{r}_{\rm H}^{3}$, which is the same as $K_{\rm enc}$, implying that $K_{\rm tc}/K_{\rm enc}$ is independent of $m_{\rm p}$ for $e_{\rm p}\ll 1$.

The integration range, $\nu_{\rm min}<\nu_{\rm tc}<\nu_{\rm max}$, 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 $\nu_{\rm min}$. A planetocentric temporarily captured orbit has its apocenter distance at $a_{\rm s}(1+e_{\rm s})\simeq r_{\rm H}$. The pericenter distance, $a_{\rm s}(1-e_{\rm s})$, must be larger than the physical radius of the planet, $R_{\rm p}$, to avoid a collision. From these two equations,

[TABLE]

Since $\nu=\sqrt{\kappa}=\sqrt{1-e_{\rm s}}$ for $f_{\rm s}=0$,

[TABLE]

The simplest assumption for the maximum value in the framework of the two-body problem is $\nu_{\rm max}=\sqrt{2}$, which is the 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 $\nu_{\rm min}=0.25$ and $\nu_{\rm max}=2$ for a prograde trap and $\nu_{\rm min}=0.5$ and $\nu_{\rm max}=\sqrt{2}$ for a retrograde trap from the orbital calculations described below. We adopt these integration ranges.

Figure 4 shows the ratio $K_{\rm tc}/K_{\rm enc}$ as a function of $e_{\rm p}$ 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 total ratio (black) is almost constant or rather gradually increases with $e_{\rm p}$. until $e_{\rm p}$ exceeds $e_{\rm p}^{\rm c}$. The asymptotic values of $K_{\rm tc}/K_{\rm enc}$ at $e_{\rm p}\rightarrow 0$ are independent of planetary mass ($m_{\rm p}$) and semimajor axis ($a_{\rm p}$), as we predicted. As shown in Figure 4, $e_{\rm p}^{\rm c}\simeq 5\hat{r}_{\rm H}$ where $\hat{r}_{\rm H}=4.8\times 10^{-3}$ for Mars, $\hat{r}_{\rm H}=0.068$ for Jupiter, $\hat{r}_{\rm H}=0.01$ for Earth, and $\hat{r}_{\rm H}=0.026$ for Neptune. For $e_{\rm p}>e_{\rm p}^{\rm c}$s, $K_{\rm tc}$ decays with $e_{\rm p}$ approximately as $\propto e_{\rm p}^{-1}$. As will be shown in the next section, the functional form of the predicted $K_{\rm tc}/K_{\rm enc}$ agrees very well with the results of numerical orbital integrations, while the allowance for temporary capture $\Delta\nu$ and $\Delta\gamma$ cannot be estimated by analytical arguments here. Because $\nu$ expresses the satellite orbital energy at the Hill radius, it is expected that the allowance $\Delta\nu$ is independent of $m_{\rm p}$ and $a_{\rm p}$ as well. Also the independence of $\Delta\gamma$ is expected since the angle $\Delta\theta_{\rm tc}$ would be a function only of $\hat{r}_{\rm H}$. From comparison with the numerical simulations, we empirically set $\Delta\nu\sim 0.025$ and $\Delta\gamma\sim 0.05$.

4 Comparison with Numerical Results

We perform numerical calculations for the temporary capture of bodies by planets with Mars, Jupiter, Earth, and Neptune 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 a circular 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 $5\times 10^{6}$. Asteroids are initially uniformly distributed on the $a,e$-plane between $\bar{a}_{{\rm tc,min},L_{1}}<\bar{a}<\bar{a}_{{\rm tc,max},L_{2}}$, $e_{\rm min}<e<e_{\rm max}$, which are derived analytically and numerically in Section 2.3.2 and summarized in Figure 2. The parameter $\theta$ is also uniformly distributed between 0 and $2\pi$. We set the upper limit of $\bar{a}_{{\rm tc,max},L_{2}}=3$. In most runs we assume $i=0$ for the asteroids. In several additional runs, we give $i$ with a uniform distribution for $0<i<i_{\rm tc,max}$ where $i_{\rm tc,max}$ is given by Equation (25) for $f_{\rm p}=180^{\circ}$ and $\nu=\sqrt{2}$. We regard asteroids as temporarily captured bodies if they stay within $r_{\rm H}$ from the planet longer than one orbital period of the planet $T_{\rm p}$.

Using the planetocentric location and the relative velocity vector to the planet, at the moment when an asteroid enters the $r_{\rm H}$ region around the planet for the first time, we define the type of temporary capture: [prograde-$L_{1}$], [retrograde-$L_{1}$], [retrograde-$L_{2}$], and [prograde-$L_{2}$].

In this paper, we focus on the equilibrium state where the ratio of temporary capture and encounter rates becomes constant with time. To obtain this state, we first perform several long-time calculations with $10^{5}$ particles for $10^{5}T_{\rm p}$ and choose the time range where the ratio is constant with time. Note that Higuchi & Ida (2016) presented the cumulative number of captured bodies over $10^{6}$ years, which is not directly compared with the results presented here.

4.2 Results

Figure 5 shows $\theta$ of the temporarily captured bodies against $f_{\rm p}$ for a Martian mass planet with various $e_{\rm p}$. The analytical prediction (eq. (21)) for $0\leq\nu\leq\sqrt{2}$ is also plotted. The analytical prediction, which determines the critical eccentricity for temporary capture ($e_{\rm p}^{\rm c}$), agrees well with the numerical results.

Figure 6 shows the ratio ($n_{\rm tc}/n_{\rm enc}$) of the temporary capture and encounter rates as a function of $e_{\rm p}$ for planets with Martian, Jovian, Earth, and Neptunian mass, respectively. The ratio drops beyond the predicted values of $e_{\rm p}^{\rm c}\simeq 5\hat{r}_{\rm H}$, which are 0.02, 0.27, 0.04, and 0.1 for Martian, Jovian, Earth, and Neptunian mass. This drop of $n_{\rm tc}/n_{\rm enc}$ is well reproduced by the analytical prediction in Figure 4.

The value of $n_{\rm tc}/n_{\rm enc}$ for $e_{\rm p}<e_{\rm p}^{\rm c}$ is $\sim 10^{-3}$, which is almost independent of the planetary mass, as predicted. We performed additional numerical calculations using particles with $i<i_{\rm max}$ for $\nu=\sqrt{2}$ given by Equation (25). The results show that the values of $n_{\rm tc}/n_{\rm enc}$ for the 3D calculations are similar to those for the 2D calculations (within a factor of 2).

5 SUMMARY AND DISCUSSION

In order to explore the origins of irregular or minor satellites around the planets in the solar system, we have investigated the probability of temporary capture through semi-analytical arguments and numerical integration. We extended 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 derived the capture probability as a function of planetary mass ($m_{\rm p}$) and eccentricity ($e_{\rm p}$). Analytical formulae reproduce the numerical integrations very well.

We found that the temporary capture occurs at $\sim 0.1\%$ of encounters that enter Hill sphere of a planet, independent of $m_{\rm p}$, $a_{\rm p}$ (semimajor axis) and $e_{\rm p}$ up to a critical value $e_{\rm p}^{\rm c}\simeq 5(m_{\rm p}/3M_{\odot})^{1/3}$. For $e_{\rm p}>e_{\rm p}^{\rm c}$, the probability decays with increasing $e_{\rm p}$ as $\propto e_{\rm p}^{-(1-2)}$.

The current eccentricity of Mars is several times larger than $e_{\rm p}^{\rm c}$, so that the capture origin of Phobos and Deimos looks unfavored. However, as shown in Figure 3, the Martian eccentricity changes with time and can be lower than $e_{\rm p}^{\rm c}$ for some fraction of time, and temporary capture may have been available in the past. Note again that temporary capture is a necessary condition for permanent capture and their respective probabilities are not necessarily proportional to each other. As will be discussed in a separate paper, tight capture could be found in the cases where $e_{\rm p}>e_{\rm p}^{\rm c}$. In a subsequent paper, we will discuss the probability of permanent capture and the possibility of the capture origin of Phobos and Deimos.

We thank an anonymous referee for his/her useful comments that helped to improve the paper. This work was supported by JSPS KAKENHI grant Number 23740335 and 15H02065. Data analyses were in part carried out on the PC cluster at the Center for Computational Astrophysics, National Astronomical Observatory of Japan.