THE ROLE OF KOZAI CYCLES IN NEAR-EARTH BINARY ASTEROIDS

We consider the contribution to pericenter precession caused by primary oblateness. Many NEAs have non-spherical shapes, and the level of oblateness can be described by a coefficient, J₂, as (Murray & Dermott 1999)

$$\begin{eqnarray} &&J_2 = \frac{C - (A + B)/2}{M_pR_p^2} \approx \frac{C-A}{M_pR_p^2}, \end{eqnarray} \tag{ 3 }$$

where A, B, and C are the primary's moments of inertia. M_p is the primary's mass and R_p is the primary's equatorial radius. The approximation is valid when A ≈ B.

The J₂ coefficient is an indirectly observable quantity that can be detected by its non-Keplerian effects induced on orbiting satellites. Such oblateness-induced precession on the orbits of satellites can be examined by the rate of change in the argument of pericenter ω (Vallado 2001):

$$\begin{equation} \frac{d\omega }{{dt}_{J_2}} = \frac{3}{2}\frac{nJ_2}{(1-e^2)^2}\left(\frac{R_p}{a}\right)^2\left(\frac{5}{2}\cos ^2I-\frac{1}{2}\right), \end{equation} \tag{ 4 }$$

where n is the binary's mean motion, e is the eccentricity, R_p is the primary's radius, a is the semimajor axis, and I is the binary's orbital inclination relative to the primary's equator. The J₂ effect is more relevant for asteroid binaries than their trans-Neptunian counterparts because asteroid binaries tend to be separated by several primary radii and trans-Neptunian binaries are typically much wider. The orbits of close-in satellites will be more perturbed by an oblate primary than the orbits of distant satellites, since Equation (4) shows that precession due to J₂ varies inversely as distance squared.

The primary's J₂ value for most NEA binaries is unknown; we calculate a range of values for apsidal precession due to J₂ coefficients ranging from 0.001 to 0.1. A few well-characterized NEA systems have known shapes and J₂ values, including 1999 KW4 (Ostro et al. 2006) with a primary J₂ of ∼0.06 and 1994 CC (Brozovic et al. 2011) with a primary J₂ of ∼0.01. In our calculations, we assume that the satellite is orbiting in the equatorial plane of the primary (support for this assumption comes from the generally accepted rotational fission formation model; Margot et al. 2002; Pravec et al. 2006; Walsh et al. 2008), and we use nominal values of their current separations and eccentricities. These numbers are presented in Table 2.

Table 2. Pericenter Precession Rates

System	Pericenter Precession
	(deg day−1)
	$\dot{\omega }_{\rm J_2}$	$\dot{\omega }_{\rm sat}$	$\dot{\omega }_{\rm tides}$
2002 CE26	0.22−22	<3.5e-6	5.0e-7−3.1e-5
2004 DC	0.069−6.9	<5.6e-3	5.7e-8−8.2e-5
2003 YT1	0.014−1.4	<2.8e-4	8.6e-9−1.0e-6
Didymos	0.25−25	<2.2e-4	1.3e-6−3.0e-4
1991 VH	0.027−2.7	<1.9e-4	3.3e-7−8.3e-6
HWBa	0.00094−0.094	<1.4e-4	1.5e-11−6.7e-9

Notes. Rates for the argument of pericenter ω are calculated for each binary and for three main sources of pericenter precession: primary's J₂, an additional satellite, and tidal bulges raised on the primary and secondary. A range of rates is given for the effects of oblateness, corresponding to J₂ values from 0.001 to 0.1. A range of rates is given for the effects of tides, corresponding to two different tidal Love number models (Goldreich & Sari 2009; Jacobson & Scheeres 2011). Precession due to an additional satellite assumes a size of 30 m. ^aHWB = hypothetical wide binary modeled after 1998 ST27.

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We examine if the presence of an additional satellite (presumably undetected) in the asteroid system can suppress Kozai oscillations by causing the known satellite's orbit to precess. It is known that the presence of larger, detectable satellites can easily suppress Kozai cycles, as shown through numerical simulations by Fang et al. (2011) for NEA triple systems 2001 SN263 and 1994 CC and as discussed by Ragozzine & Brown (2009) for trans-Neptunian triple Haumea. Here, we investigate if a small, unobserved satellite can also damp Kozai cycles.

Since most NEA binaries are discovered by planetary radar, we adopt a fairly common value of the spatial resolution in radar images (15 m) as the typical radius of the additional satellite in our study. This is adopted for convenience and does not represent the finest spatial resolution available nor the radar detectability threshold. If a small, undetectable satellite can damp Kozai cycles, then it may be responsible for the survival of NEA systems that would otherwise undergo high oscillations in eccentricity and inclination.

In the presence of an additional satellite, both satellites will undergo time-averaged, secular changes in their orbital elements. For a coplanar system, their argument of pericenter rates are (Mardling 2007)

$$\begin{eqnarray} \frac{d\omega _{s1}}{{dt}_{\rm sat}} &=& \frac{3}{4} n_{s1} \left(\frac{M_{s2}}{M_p} \right) \left(\frac{a_{s1}}{a_{s2}}\right)^3 \big(1-e_{s2}^2\big)^{3/2}\nonumber \\ && \times \left[ 1 - \frac{5}{4} \left(\frac{a_{s1}}{a_{s2}}\right) \left(\frac{e_{s2}}{e_{s1}}\right) \frac{\cos (\omega _{s1} - \omega _{s2})}{1 - e_{s2}^2} \right] \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} \frac{d\omega _{s2}}{{dt}_{\rm sat}} &{=}& \frac{3}{4} n_{s2} \left(\frac{M_{s1}}{M_p}\right) \left(\frac{a_{s1}}{a_{s2}}\right)^2 \big(1-e_{s2}^2\big)^{-2}\nonumber \\ && \times \left[ 1 - \frac{5}{4} \left(\frac{a_{s1}}{a_{s2}}\right) \left(\frac{e_{s1}}{e_{s2}}\right) \frac{\big(1+4 e_{s2}^2\big)}{\big(1-e_{s2}^2\big)} \cos (\omega _{s1} - \omega _{s2}) \right]\nonumber \\ \end{eqnarray} \tag{ 6 }$$

and these equations are given to fourth power in a_s1/a_s2 and first order in e_s1. The subscripts s1 and s2 represent the inner and outer satellites, respectively, and the subscript p is for the primary. The equations also include the mean motion n, mass M, semimajor axis a, and eccentricity e. These equations are valid for M_s1 ≪ M_p, but there are no restrictions on M_s2.

We calculate the apsidal rates for each binary in Table 1, assuming a coplanar system. For each binary, the additional satellite is given a radius of 15 m and a typical rubble pile density of 2 g cm⁻³ with a circular orbit. For all binaries except the hypothetical wide binary modeled after 1998 ST27, we treat the observed satellite as the inner satellite and the additional, undetected satellite as the outer satellite with a semimajor axis of 15 primary radii (which is a typical separation for the outer satellite in an NEA triple, based on a sample of two known NEA triples). For the hypothetical wide binary, whose satellite is located at 6.66 km or 16 primary radii, we treat this satellite as the outer satellite and the undetected satellite as the inner satellite with a semimajor axis of 4 primary radii. In all cases, we use the observed satellite's current separation and eccentricity. Our calculated apsidal precession rates are shown in Table 2, which represent the pericenter precession due to a 30 m diameter satellite.

For completeness we investigate tidal bulges raised on both the primary and secondary, although we anticipate their contribution to pericenter precession to be small. The argument of pericenter rate is (e.g., Sterne 1939; Fabrycky & Tremaine 2007; Batygin & Laughlin 2011)

$$\begin{eqnarray} \frac{d\omega }{{dt}_{\rm tides}} &{=}& \frac{15}{16} n \left[\frac{8+12e^2+e^4}{(1-e^2)^5}\right]\nonumber \\ && \times\, \left[ k_p \left(\frac{R_p}{a}\right)^5 \left(\frac{M_s}{M_p}\right) + k_s \left(\frac{R_s}{a}\right)^5 \left(\frac{M_p}{M_s}\right) \right], \end{eqnarray} \tag{ 7 }$$

where the first term corresponds to the primary and the second term corresponds to the secondary. The equation also includes the mean motion n, eccentricity e, radius R, semimajor axis a, mass M, and tidal Love number k. Subscripts p and s stand for the primary and secondary, respectively.

The tidal Love number is a poorly known quantity for small rubble pile asteroids, and there are currently two different rubble pile models that describe the Love number's dependence on size (or radius R). Goldreich & Sari (2009) give the relation k_rubble ∼ 10⁻⁵ (R/1 km), and Jacobson & Scheeres (2011) find that k_rubble ∼ 2.5 × 10⁻⁵ (1 km/R). Using both tidal Love number models and the binaries' current separations and eccentricities, we calculate a range of possible apsidal precession rates due to tidal bulges for each NEA binary in Table 2.

Examination of the values in Table 2 indicates that even the smallest amount of primary oblateness (J₂ of 0.001) will dominate over all other perturbations; the contribution from each perturbation is quantified in Table 2 for all NEA binaries in our sample. Therefore, the numerical integrations described in the next section only include the effect of primary oblateness out of the three sources of pericenter precession considered here.

We extrapolate from our simulation results to the observed NEA binary population. Numerical integrations in Section 3 demonstrated that four out of five well-characterized NEA binaries in our sample (Table 1) showed no signs of Kozai cycles and the exception, 2003 YT1, only showed evidence of Kozai cycles when the primary had a very minimal level of oblateness (J₂ = 0.001). All five of these well-characterized binaries have semimajor axes less than 8 primary radii. Only the hypothetical wide binary (at 16 primary radii) modeled after 1998 ST27 exhibited signs of Kozai cycles at a range of J₂ values from 0.001 to 0.1. The effects of primary oblateness are strongest for close-in binaries (pericenter precession due to J₂ increases as the semimajor axis squared; Equation (4)), where an oblate primary can cause the satellite's orbital plane to precess fast enough to thwart any Kozai oscillations.

If we consider all well-characterized NEA binaries as compiled by Fang & Margot (2012b), none of them have separations greater than 8 primary radii. In fact, the semimajor axes of all of these well-characterized NEA binaries are within the range of semimajor axes of the binaries in our sample, which did not show Kozai cycles at a range of plausible J₂ values. If we consider this list of well-characterized NEA binaries to be representative of the observed population of NEA binaries, then typical NEA binaries do not appear to be affected by Kozai cycles in their orbital evolution. We point out a mild selection effect in that most well-characterized NEA binaries have made close enough approaches to Earth to be detected by radar, and this subset of the population may have fewer wide binaries than the general population. One could argue that we mostly observe binaries immune from Kozai effects because the Kozai-acting binaries have disrupted on short Kozai timescales (see following subsections on disruptions and contact binary formation). However, our simulations show that the requirements on primary oblateness and component separation are quite stringent, and we conclude that Kozai cycles are unlikely to be an important effect in most NEA binaries.

Orbit pole orientations can be constrained for binaries that undergo Kozai cycles. In Section 3, we explored a hypothetical wide binary modeled after 1998 ST27 that exhibited signs of Kozai oscillations and disruptions, and in this subsection we illustrate this binary's stable orbit pole orientations. Stable orbit poles are defined as initial orientations that do not end in a collision after 10⁵ years of numerical integrations; these results are based on the ensemble of simulations performed in Section 3 as well as additional simulations to sample the full range of inclinations. The range of inclinations corresponding to stable binaries allows us to calculate the corresponding J2000 ecliptic coordinates (latitude β and longitude λ) of binary orbits for stable binaries. Since the hypothetical wide binary under consideration is modeled after 1998 ST27, we use 1998 ST27's well-known heliocentric orbit poles (ecliptic longitude of the ascending node = 1975842 and ecliptic inclination = 2105458) for our calculations.

In Figure 3, we map out constraints on the orbit orientations for this hypothetical wide binary, showing results for three different J₂ values for the primary: 0, 0.005, and 0.05. Color-coded lines (for different J₂ values) separate the "Kozai stable" and "Kozai unstable" regions. Regions where at least half of simulations resulted in stable binaries are called "Kozai stable," and regions where over half of simulations resulted in unstable binaries are called "Kozai unstable." This orbit orientation map provides constraints on allowable orientations because binaries that will disrupt under the effect of Kozai are unlikely to be observed with orbit ecliptic coordinates in the unstable regions of Figure 3. Most Kozai-induced disruptions occurred within 100 years, shorter than evolutionary timescales due to tides, BYORP, and close planetary encounters. Accordingly, for this hypothetical wide binary we can predict that its orbit does not lie in the region bounded by the color-coded J₂ lines. This analysis is performed here for this hypothetical wide binary and can similarly be applied to other binaries that can be affected by Kozai perturbations. We note that a different binary (presumably with a different heliocentric orbital pole) would result in a different layout of stable and unstable zones (i.e., stability islands).

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Figure 3 also shows how different J₂ values can affect the stability map, discussed here for the hypothetical wide binary. For discrete J₂ values of 0, 0.001, 0.005, 0.01, and 0.05, we find that an increasingly larger J₂ value results in an increase in the number of unstable orbit orientations in our simulations. These higher J₂ values cause collisions that can occur at a larger range of initial inclinations by increasing the maximum eccentricity acquired during a sequence of Kozai cycles. A larger maximum eccentricity therefore decreases the pericenter distance to allow for more frequent collisions (e.g., Nesvorný et al. 2003). We do not continue to observe this trend at high J₂ values greater than ∼0.05, where substantial primary oblateness prevents additional unstable orientations from forming.

Here we discuss how disruptions due to the Kozai effect will result in collisions for typical NEA binaries. We only consider potential cases where the Kozai effect operates and can produce instabilities, and we assume these instabilities occur on much shorter timescales (on the order of tens to hundreds of years) compared to other perturbations such as planetary encounters, tidal evolution, and BYORP. We define disruptions as dynamical instabilities due to the Kozai effect, whose outcomes include (1) collisions between binary components and (2) ejections where the binary becomes unbound. The Kozai effect is capable of increasing a binary's eccentricity, and very high eccentricities can cause the binary to become unbound or cause a collision between the binary components. An ejection occurs when an orbit's apocenter Q, which is related to the semimajor axis a and the eccentricity e as Q = a(1 + e), grows to a distance greater than the binary's Hill radius R_Hill. A collision occurs when an orbit's pericenter q, where q = a(1 − e), is less than the primary radius R_p. R_Hill and R_p are known quantities given in Table 1. Therefore, for given R_Hill, R_p, and a, whether a gradually increasing eccentricity (i.e., due to the Kozai effect) causes a binary to undergo a collision or ejection is solely determined by the value of the eccentricity.

For all NEA binaries studied here (Table 1), values for their maximum Q (by assuming an e of 1) are much smaller than their Hill radii. In order for Q to be as large as the Hill radius, if we assume a maximum e of 1 then any binary's semimajor axis needs to be at least half as large as the Hill radius in order for an ejection to occur. Thus, for any typical binary with a semimajor axis smaller than 0.5 R_Hill, Kozai cycles will not cause ejections. On the other hand, collisions will occur at an eccentricity less than 1. Collisions will occur (by setting q = R_p) at the following e values for the binaries in our sample: 0.64 for 2002 CE26, 0.77 for 2004 DC, 0.94 for the hypothetical wide binary, 0.86 for 2003 YT1, 0.66 for Didymos, and 0.82 for 1991 VH. The specific case of the hypothetical wide binary is most interesting because we have already shown that the other binaries do not generally undergo Kozai cycles, let alone Kozai-induced instabilities. But for any wide NEA binary such as the hypothetical case examined here, if Kozai cycles are acting with sufficiently high inclinations to induce high eccentricities, collisions between binary components are the only possible instability outcome. The binary's pericenter will shrink below the primary radius before its apocenter can increase beyond the Hill radius and become unbound. Since collisions are the only instability outcome, this means that the Kozai effect cannot directly form asteroid pairs. It is possible that the Kozai effect may indirectly form asteroid pairs if a collision occurs but the components separate again and become unbound, or if the binary fortuitously makes a close planetary encounter during the short Kozai timescale.

For Kozai cycles in the absence of other perturbations, these collisions occur when the starting inclination results in a maximum eccentricity (Equation (1)) that is greater than the limiting eccentricities listed in the previous paragraph. Thus, for these starting inclinations, collisions will occur over the course of one Kozai cycle and repeated cycles will not occur. These disruption timescales are fast; for the hypothetical binary modeled after 1998 ST27, our longer-term (10⁵ years) simulations show that from a starting eccentricity of 0.3, collisions occur ranging from ∼3 years to >10⁴ years later with the majority of collisions occurring within 100 years.

We discuss contact binary formation from large-amplitude Kozai oscillations. For wide binaries such as 1998 ST27 that can potentially undergo Kozai cycles even in the presence of primary oblateness, high starting inclinations can lead to high eccentricities (Equation (1)). During a single Kozai oscillation, these high eccentricities can drive the orbital pericenter distance to very low values and can cause the binary components to collide (see previous section for discussion on collisions). If a collision does not occur, tidal friction may play an important role and decrease the semimajor axis as well as the eccentricity. When the pericenter distance decreases due to high eccentricities, tides can more efficiently circularize the orbit by dissipating orbital energy during each pericenter passage (e.g., Perets & Naoz 2009; Fabrycky & Tremaine 2007). This leads to a circular orbit and a smaller semimajor axis.

If Kozai cycles and possibly tidal effects can cause binary components to come into contact with each other, this may be a mechanism to create near-Earth contact binaries, which constitute ∼10% of NEAs larger than 200 m in diameter (Benner et al. 2006). If this process occurs, we would expect the observed contact binaries to have high obliquities preferentially between ∼40° and ∼140°, assuming no significant obliquity evolution has occurred after formation of the contact binary. Obliquity is defined here as the inclination between the contact binary's rotational angular momentum vector and the vector normal to its heliocentric orbital plane. This obliquity angle is equivalent to the inclination angle defined in Section 1 and used throughout the paper; we assume that the direction of an initially detached binary's orbital angular momentum vector is the same as the direction of a then-collided contact binary's rotational angular momentum vector.

Observed near-Earth contact binaries include Castalia (Hudson & Ostro 1994), Bacchus (Benner et al. 1999), Mithra (Brozovic et al. 2010), 1996 HW1 (Magri et al. 2011), and Itokawa (e.g., Ostro et al. 2004); all of which have high obliquities (M. W. Busch et al., in preparation). In the inner main belt, small binaries are observed with obliquities concentrated toward 0° and 180° and there is a lack of obliquities near 90° (Pravec et al. 2011). The concentration of binaries with low inclinations and the concentration of contact binaries near high inclinations hints that the Kozai effect may be effective at disrupting high-inclination binaries, but Pravec et al. (2011) have shown that the Kozai effect cannot completely explain the observed concentration of binaries with obliquities near 0° and 180°.

Although the Kozai effect is not responsible for the observed binary pole concentrations of small main belt binaries near 0° and 180°, it can be effective for more rarely observed NEA binaries with wide separations. Our simulations show that widely separated binaries can escape the dominating influence of primary oblateness to be significantly affected by solar perturbations. As a result, we suggest that wide binaries with eccentric or highly inclined orbits may become unstable under the effects of Kozai, leading to the formation of near-Earth contact binaries. Alternate theories of contact binary formation include low-velocity collisions between components in an unstable binary due to planetary encounters or radiative effects such as YORP and BYORP (e.g., Taylor & Margot 2011).