Triton's Evolution with a Primordial Neptunian Satellite System

We use disruption scaling laws, derived by Movshovitz et al. (2016) for non-hit-and-run impacts between two gravity-dominated bodies, to estimate whether the impacts recorded by the N-body code are disruptive. The scaling laws identify impacts that would disperse half or more of the total colliding material, regarded hereafter as disruptive collisions. For head-on impacts, the minimum required kinetic energy (K*) to disrupt the target has a linear relation with the gravitational binding energy of the colliding bodies (U; Movshovitz et al. 2016),

$$\begin{eqnarray}&&{K}^{* }={c}_{0}U\end{eqnarray} \tag{ 1 }$$

where c₀ is the slope derived for head-on impacts (${c}_{0}\,=5.5\pm 2.9$).

Higher impact angles require higher energies to disrupt a body, as the velocity is not tangential to the normal plane. A modified impact kinetic energy is required to incorporate the geometric effects,

$$\begin{eqnarray}&&{K}_{\alpha }^{* }=\left(\displaystyle \frac{\alpha {M}_{1}+{M}_{2}}{{M}_{1}+{M}_{2}}\right){K}^{* }\end{eqnarray} \tag{ 2 }$$

where α is the volume fraction of the smaller body (M₂) that intersects the second body (M₁; Movshovitz et al. 2016; Leinhardt & Stewart 2012). The disrupting relation transforms to

$$\begin{eqnarray}&&{K}_{\alpha }^{* }={cU}\end{eqnarray} \tag{ 3 }$$

where c is the geometrical factor derived from the collision outcomes with three tested angles. The collisions were tested on a limited number of impacting angles (direction of velocity relative to the line connecting the centers at contact, 0°, 30°, and 45°); therefore, we assume that the relation of c on the impact angle (θ) is given by the following step function:

$$\begin{eqnarray}c={c}_{{}_{0}}\left\{\begin{array}{cc}1, & \theta \lt 30^\circ ,\\ 2, & 30^\circ \leqslant \theta \lt 45^\circ \\ 3.5, & \theta \geqslant 45^\circ \end{array}\right..\end{eqnarray} \tag{ 4 }$$

It should be noted that ejected material from a satellite collision can escape the gravitational well of the colliding objects if it has enough velocity to expand beyond the mutual Hill sphere. For the typical impacts observed, the required velocity to reach the Hill sphere is $\sim 0.9{V}_{\mathrm{esc}}$, where ${V}_{\mathrm{esc}}$ is the two-body escape velocity. The binding energy used in Equation (3) to determine the disruption scales as $\sim {V}_{\mathrm{esc}}^{2}$; therefore, the required energy to disrupt orbiting bodies may be reduced $\sim 0.8{K}_{\alpha }^{* }$. Hence, the disruptive scaling laws used may somewhat overestimate the disrupting energy required, but not too substantially.

In the 200 simulations with the Uranian satellite system mass ratio (${M}_{\mathrm{USats}}\equiv 1.04\times {10}^{-4}$), we find the overall likelihood of Triton's survival after ${10}^{7}\,\mathrm{years}$ is ∼40%. Different sets of initial conditions have different probabilities for Triton's loss either by escaping the system or falling onto Neptune. For example, a highly inclined initial Triton (175°) does not survive more than ${10}^{4}\,\mathrm{years}$, due to the near alignment of its orbit with Neptune's equatorial plane, which contains the prograde satellites. In this case, after a final Triton-satellite collision, the orbital angular momentum of the merged pair is small, leading to collapse onto Neptune. However, with a lighter satellite system ($0.3{M}_{\mathrm{USats}}$), the post-impact angular momentum is high enough and Triton survives. In the cases in which Triton survives, it is usually the last survivor (or one of two remaining satellites if Triton's initial pericenter is large), reproducing the low number of Neptunian satellites. Overall, Triton's dynamical survivability decreases with increasing mass of pre-existing satellite system, 12% for the heavier satellite system with $3\,{M}_{\mathrm{USats}}$, and 88% for the lighter satellite system with $0.3\,{M}_{\mathrm{USats}}$.

Out of the surviving cases, Triton usually ($\sim 90 \%$) experienced at least one impact. Due to Triton's initial retrograde orbit, mutual impacts between the primordial satellites (Figure 1(a)) have a significantly lower velocity compared to the collisions between Triton and a primordial satellite (Figure 1(b)). The mean velocity for non-Triton impacts decreases with increasing mass of the pre-existing satellite system, as Triton is less able to excite the more massive system.

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It should be noted that the number of Triton impacts decreases with increasing satellite mass, because Triton is lost earlier as the mass pre-existing satellites increases, and therefore, fewer events are recorded.

For satellite systems with a Uranian mass ratio, impacts onto Triton are more disruptive (18% of impacts; pink circles with black dots, see Figure 2(b)) than mutual collisions among the primordial satellites (3%; green triangles with black dots, see Figure 2(b)). With decreasing satellite mass, disruption of Triton is inhibited as the mass ratio decreases. Overall, 33% of tested cases with ${M}_{\mathrm{USats}}$ (81% with $0.3{M}_{\mathrm{USats}}$ and 10% with $3{M}_{\mathrm{USats}}$) resulted in a stable Triton that did not encounter any disrupting impacts throughout the evolution. Although Triton does experience disruption in some cases, a satellite system of $0.3\mbox{--}1\,{M}_{\mathrm{USats}}$ has a substantial likelihood for Triton's survival without the loss of Triton's initial inclination by disruption and reaccretion into the Laplace plane.

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With decreasing satellite mass, disruption for non-Triton bodies increases somewhat as the typical impact velocity increases (8% for $0.3{M}_{\mathrm{USats}}$ and none for $3{M}_{\mathrm{USats}}$). The low rate of primordial satellite disruption calls into question the formation of a debris disk from the primordial satellites envisioned by Ćuk & Gladman (2005) to rapidly circularize Triton. Moreover, assuming that a disruptive impact between the primordial satellites leads to the formation of a debris disk, the rubble will quickly settle onto the equatorial plane and re-accrete to form a new satellite. The timescale of reaccretion can be estimated by the geometric mass accumulation rate (Banfield & Murray 1992)

$$\begin{eqnarray}&&{\tau }_{{acc}}\sim \displaystyle \frac{m}{\pi {r}^{2}{\sigma }_{s}{\rm{\Omega }}}\end{eqnarray} \tag{ 5 }$$

where m is the mass of the satellite, r its radius, ${\sigma }_{s}$ is the surface density of the debris disk, and Ω its orbital frequency. For typical debris ejection velocities $\sim {V}_{\mathrm{esc}}$ (e.g., Benz & Asphaug 1999), the width of the debris ring created by a disrupting impact is estimated by ${\rm{\Delta }}a\sim {ae}\sim a\tfrac{{V}_{\mathrm{esc}}}{{V}_{\mathrm{orb}}}$, where a is the semimajor axis of the debris, e is the eccentricity of the debris, ${V}_{\mathrm{orb}}$ is the orbital velocity, and $\tfrac{{V}_{\mathrm{esc}}}{{V}_{\mathrm{orb}}}$ is proportional to the eccentricity of the debris induced by the impact. Rearranging the equation above for a disrupted satellite of mass m dispersed across an area of $2\pi a{\rm{\Delta }}a$, we obtain:

$$\begin{eqnarray}&&{\tau }_{\mathrm{acc}}\sim \displaystyle \frac{2{a}^{2}}{{r}^{2}{\rm{\Omega }}}\cdot \displaystyle \frac{{V}_{\mathrm{esc}}}{{V}_{\mathrm{orb}}}.\end{eqnarray} \tag{ 6 }$$

Assuming an impact between two primordial satellites equivalent to the mass of Oberon and Titania at a distance of $10{R}_{\mathrm{Nep}}$, the timescale for reaccretion is $\sim {10}^{2}\,\mathrm{years}$. The reaccretion timescale is smaller than the evaluated eccentricity decay time for Triton of ${10}^{4}\mbox{--}{10}^{5}\,\mathrm{years}$ (Ćuk & Gladman 2005). Thus, the debris disk, even if it formed, would likely re-accrete before Triton's orbit circularized.

In order to ensure stabilization of Nereid (and Nereid-like satellites), Triton's apoapse needs to decrease to within Nereid's orbit in ${10}^{5}\,\mathrm{years}$ (Ćuk & Gladman 2005; Nogueira et al. 2011). Otherwise, Neptune's irregular satellites must be formed by an additional subsequent process (e.g., Nogueira et al. 2011). Torques produced by the Sun and Neptune's shape are misaligned by Neptune's obliquity (30°), causing a precession of the argument of pericenter (Kozai oscillations). For large orbits, the Kozai mechanism induces oscillations (period $\sim {10}^{3}\ \mathrm{years}$) in the eccentricity and inclination such that the z-component of the angular momentum (${H}_{z}\,=\sqrt{1-{e}^{2}}\cos I$, where I is the Triton's inclination with the respect to the Sun) is constant. For retrograde orbits, the eccentricity and inclination oscillate in phase; therefore, the maximum inclination occurs at the maximum eccentricity. For small orbits ($\lt 70{R}_{\mathrm{Nep}};$ Nogueira et al. 2011), the torque induced by Neptune's shape is larger than the Kozai cycles induced by the Sun, and Triton's tidal decay occurs with approximately constant inclination. Here, we seek to identify Triton analogs with final apoapses smaller than Nereid's pericenter ($55\,{R}_{\mathrm{Nep}};$ Jacobson 2009) and inclinations close to Triton's current inclination of 157°, as this will remain constant in subsequent tidal circularization.

As seen previously, the smallest pre-existing satellite system mass ratio has a high rate of survival, and about half of the Triton's analogs are within Nereid's orbit with minimal inclination change (Figure 3(a)). As the primordial satellites mass increases, the percentage of Triton analogs inside Nereid's orbit increases, although Triton's inclination change is larger. Moreover, due to the small survivability rate, only a small number of cases with $3{M}_{\mathrm{USats}}$ fulfilled all of the required conditions to be regarded as successful Triton analogs that may allow Nereid-like satellites to survive (Triton's dynamic survival with no disruption, on a small and inclined orbit). Typically, Triton analogs that did not experience any impact have larger final orbits, as scattering alone does not effectively decrease the orbital apoapse.

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In addition, we roughly estimated the effect of solar perturbations on Triton's orbital evolution by performing additional simulations that include Triton, Neptune, and the Sun (positioned on an inclined orbit relative to Neptune's equatorial plane in order to mimic the planet's obliquity; see Appendix B for more details). Due to induced eccentricity oscillations, Triton spends only $\sim 10 \%$ of the time in the region populated by primordial satellites (Ćuk & Gladman 2005, Figure 1). Out of the recognized successful cases in Figure 3, ∼30%–50% are already within Nereid's orbit after ${10}^{4}\ \mathrm{years}$ (see Figure 6). Even though Kozai could lengthen the time of orbit contraction by a factor of 10, Triton's orbit in these cases would still decrease within Nereid's orbit in $\leqslant {10}^{5}\ \mathrm{years}$. Therefore, even if solar perturbations were included in the numerical scheme, Nereid would still likely be stable in these cases. Additional studies are needed to determine the specific details induced by the Kozai mechanism in the first part of Triton's evolution, when the semimajor axis is still large, in order to fully evaluate the stability of pre-existing irregular moons, including the effects of a pre-existing prograde satellite system.