The Nucleus of Active Asteroid 311P/(2013 P5) PANSTARRS

The prominent dust tails evident in earlier observations of 311P (Jewitt et al. 2013, 2015a) were absent in the new observations presented here. Figure 3 shows the apparent magnitude for both old and new data as a function of time, expressed as Day of Year (DOY), where DOY = 1 is defined as UT 2013 January 1. The symbols in the figure represent the averages of measurements taken within a given HST orbit. The associated photometric uncertainties, expressed as the 1σ errors on the means, are comparable to or smaller than the sizes of the plot symbols. Also plotted on the figure are curves showing the expected time-variation of the magnitude computed from

$$\begin{eqnarray}&&V=H+2.5{\mathrm{log}}_{10}({r}_{H}^{2}{{\rm{\Delta }}}^{2})-2.5{\mathrm{log}}_{10}({\rm{\Phi }}(\alpha ))\end{eqnarray} \tag{ 1 }$$

where r_H and Δ are the heliocentric and geocentric distances expressed in au, and Φ(α) ≤ 1 is the phase function at phase angle α. The absolute magnitude, H, defines the normalization of the curves to the data. It is the apparent magnitude that would be observed if the object were to be located at r_H = Δ = 1 au and α = 0°. The phase function of 311P is unmeasured but, over the range of phase angles at which 311P was observed, is unlikely to have a major effect on the interpretation of the photometry. To show this, we plot in Figure 3 two phase functions representing the nominal behavior of C-type (dashed line) and S-type (solid line) asteroids (characterized by assuming parameter G = 0.15 and 0.25, respectively, in the formulation by Bowell et al. 1989). The figure shows that variations in r_H and Δ dominate the long-term photometric variations of 311P and that the effect of phase angle, α, is comparatively modest at the phase angles sampled in our data. In the rest of this work, we assume the phase function of S-type asteroids, guided by the optical color of 311P (Hainaut et al. 2014) and by the location of this object in the inner region of the asteroid belt, where S-types are most abundant.

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By fitting the model curves to the data, we observe that most of the early measurements (specifically those from discovery at DOY = 230 to DOY ∼ 400) imply an H systematically brighter than later data (Figure 3). This is clearly seen in Figure 4, where we plot the absolute magnitude as a function of time. We attribute this fact to early dust activity in 311P and we use only the 2015 photometry from Table 2 to obtain our best estimate of H for the nucleus of 311P. The mid-light value, H = 19.14 ± 0.02, is consistent with a limit, H ≥ 18.98 ± 0.10, placed from an analysis of data taken in 311P's active phase (Jewitt et al. 2015a). Short-term variations about the mean, in the range 18.95 ≤ H ≤ 19.23, are likely due to light-curve effects.

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The absolute magnitude and the scattering cross-section, C_e, are related by

$$\begin{eqnarray}&&{p}_{V}{C}_{e}=2.25\times {10}^{22}\pi {10}^{-0.4(H-{m}_{\odot })}\end{eqnarray} \tag{ 2 }$$

where m_⊙ = −26.75 is the apparent V magnitude of the Sun (Drilling & Landolt 2000). We set p_V = 0.29 ± 0.09, which is the average geometric albedo of the Flora family asteroids as reported by Masiero et al. (2013), to find that C_e varies from C_min = 0.10 km² to C_max = 0.13 km², with a mid-light value C_e = 0.11 ± 0.04 km², where the error is dominated by the uncertainty on p_V. The mid-light cross-section corresponds to the area of a circle of effective radius r_e = (C_e/π)^1/2 = 190 ± 30 m (again, the uncertainty on the effective radius is dominated by the uncertainty on the albedo). This is consistent with an estimate based on earlier HST photometry (r_e ≤ 200 ± 20 m, Jewitt et al. 2015a) but considerably larger than the radius 15 ≤ r_e ≤ 65 m estimated by Moreno et al. (2014) based on a model of the motion of dust. Note that, if our albedo assumption is wrong, both the effective radius and the disagreement with the estimate by Moreno et al. (2014) would likely be larger (e.g., if p_V = 0.04, as for a C-type object, the inferred radius would be larger by the ratio (0.29/0.04)^1/2 = 2.7).

We sought spatially resolved evidence for dust in the data of Table 2 by comparing photometry within concentric apertures. Specifically, we used δV = V_0.2 − V_0.4 to measure the excess contribution from dust in the 02–04 radius range. Light in this annulus includes contributions from the wings of the point-spread function, together with additional light scattered by near-nucleus dust. In all but the UT 2014 November 17 images, the value of δV is consistent with expectations based on modeling the point-spread function of HST, for which we used the TinyTim software. Considering all of the data, we find a median δV_m = 0.092 mag. On UT 2014 November 17, we find median δV = 0.15 mag and interpret the difference, δV − δV_m = 0.06 mag, as caused by light scattered from near-nucleus dust, with an average surface brightness in the 02–04 annulus of Σ(02−04) = 25.5 mag arcsec⁻².

The relation between the surface brightness of a steady-state dust coma measured at radius θ'' and the integrated brightness, V_Tot, measured within an aperture is (Jewitt & Danielson 1984)

$$\begin{eqnarray}&&{V}_{\mathrm{Tot}}={\rm{\Sigma }}(\theta )-2.5{\mathrm{log}}_{10}(2\pi {\theta }^{2}).\end{eqnarray} \tag{ 3 }$$

Substituting into Equation (3), we find coma magnitude V_Tot = 26.12, compared with the measured magnitude of 311P on UT 2014 November 17 of V = 23.47 ± 0.01 (Table 2). If the model coma brightness is subtracted from the measured brightness, we find a bare nucleus magnitude V = 23.57, about 0.1 mag fainter than in the table. The exact value of the coma correction is uncertain because, for instance, we do not know that the coma is in steady state, as assumed. Steeper (flatter) surface brightness profiles would lead to larger (smaller) coma contributions to the photometry. Nevertheless, it appears that ∼10% of the signal measured on UT 2014 November 17 results from dust while, in all other epochs in Table 2, we find no evidence for coma.

Large particles tend to be ejected slowly and their motions are confined to follow along the projected orbit, forming low surface brightness, nearly parallel-sided "trails" (e.g., Kim et al. 2017). We find no evidence for such a large-particle trail, even in our deepest imaging data (Figure 7). Particles released from the nucleus in 2013 with zero initial velocity would be pushed by radiation pressure to an angular distance ∼10'' in our 2015 data if their sizes were ≲1 cm. The apparent absence of such particles is consistent with the cross-section weighted mean particle size (3.4 mm) inferred in the tails in earlier images (Jewitt et al. 2015a). Particles of this size would have been dispersed by radiation pressure in the two years since the 2013 outbursts. The low surface brightness excess in the 02 to 04 annulus, described above, is most likely an indicator of small dust grains expelled by late-stage activity.

A ∼190 m sized body has a very small region of gravitational control, given by the Hill radius,

$$\begin{eqnarray}&&{R}_{H}=a\left(\displaystyle \frac{{r}_{e}}{{r}_{\odot }}\right){\left(\displaystyle \frac{\rho }{3{\rho }_{\odot }}\right)}^{1/3}\end{eqnarray} \tag{ 4 }$$

where a is the semimajor axis, r_⊙ and ρ_⊙ are the radius and the density of the Sun, respectively, and r_e and ρ are again the effective radius and the density of 311P. The latter is unmeasured and so we adopt the mean density of small S-type asteroids, ρ_S = 2700 ± 500 kg m⁻³ as determined by Carry (2012). For comparison, the LL chondrite meteorites are thought to originate from the inner asteroid belt, in the orbital vicinity of 311P. Their average density has been measured at ρ = 3200 ± 200 kg m⁻³ (Consolmagno et al. 2008). The (small) density difference likely results from the existence of macroporosity in the measured asteroids and from a selection effect whereby denser, stronger asteroid fragments better survive passage through the Earth's atmosphere than do less dense, weaker fragments. We assume that the density of 311P is well represented by ρ_S.

Substituting orbital semimajor axis a = 2.189 au, ρ = 2700 kg m⁻³, ρ_⊙ = 1400 kg m⁻³, r_e = 190 m and r_⊙ = 7 × 10⁸ m, we find R_H = 80 km. When observed from minimum distance Δ ∼ 1.3 au (as in 2015 March, see Table 1), the Hill radius subtends an angle θ_H = R_H/Δ ∼ 008, comparable to the ∼008 resolution corresponding to Nyquist (2 pixel) sampling of the WFC3 images. Therefore, any existing gravitationally bound companions to 311P fall beneath the resolution of the HST data.

Table 2.  Average Nucleus Photometry

Date	DOYa	V0.2b	H0.2(S)c	Nd
2014 Nov 17	686	23.47 ± 0.01	18.92 ± 0.04	5
2015 Mar 03	792	22.12 ± 0.03	19.23 ± 0.03	5
2015 Mar 19	808	21.67 ± 0.03	19.03 ± 0.03	19
2015 Apr 07	827	21.98 ± 0.01	18.95 ± 0.01	5
2015 May 04	854	22.71 ± 0.01	19.04 ± 0.01	4
2015 Jun 29	910	23.85 ± 0.04	19.17 ± 0.04	5
2015 Jul 27	938	24.01 ± 0.01	19.07 ± 0.01	5

Notes.

^aDay of Year, UT 2013 January 01 = 1. ^bApparent V magnitude within 5 pixel (02) radius aperture. Quoted uncertainty is the statistical error, only. ^cAbsolute V magnitude, H_0.2 computed from V_0.2 assuming an S-type asteroid phase function and Equation (1). ^dNumber of images used to compute the magnitudes on each date.

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In some models of rotational breakup (e.g., Jacobson & Scheeres 2011; Walsh et al. 2012; Boldrin et al. 2016, discussed later), temporarily stable companion objects are launched from orbit around the primary by complex gravitational interactions, eventually creating asteroid pairs. With radius r_e = 190 m and density ρ = 2700 kg m⁻³, a non-rotating and spherical 311P would have a gravitational escape speed V_e ∼ 0.23 m s⁻¹. If traveling at V_e, an escaping body would spend 4 × 10⁶ s (48 days) within 1'' of the primary nucleus and ∼2 × 10⁸ s (7 years) within 50'' of it. Objects ejected in 2013, ∼400 days before the present observations, would have traveled ∼10⁷ m (10'') from the nucleus at speed V_e and would be well within the WFC3 field of view. Therefore, although we cannot hope to resolve bound companions inside the Hill sphere, it remains worthwhile to search the HST images for evidence of unbound objects that might be slowly leaving the vicinity of the primary.

To this end, we examined the WFC3 images from UT 2015 March 19 (when Δ = 1.283 au) in search of co-moving field objects near 311P. We first aligned the images on 311P and then formed image composites using different combinations of the 20 separate images. Each composite image consisted of 10 separate 400 s exposures, for an effective exposure of 4000 s. Pairs of image composites were then visually compared in order to distinguish spurious sources formed by chance noise clumps and overlapping background trail residuals from potentially real ones. Because of the large parallax motions in these data, background stars and galaxies were not a significant source of confusion except to the extent that they contribute to the background noise. Numerous cosmic-ray tracks in the CCD images were successfully removed in the computation of the composites.

No co-moving objects were found in a region (limited by the position of 311P on the WFC3 CCD) extending ∼40'' to the south, 120'' to the north, and ∼80'' east and west of the nucleus. The limiting magnitude of the image composites was estimated in two ways. First, we digitally added sources of known brightness to the data and searched for them in the same way as we searched for real companions. An example grid of digital stars is shown in Figure 5 labeled by their V magnitudes. It may be seen that the visibility of the faintest sources is limited by small variations in "sky" noise caused by the residual images of trailed galaxies. Using the digital stars, we estimate an effective limiting magnitude V = 28.3, applicable at distances from 311P greater than ∼1''. At smaller separations, the surface brightness from the wings of the point-spread function reduces the sensitivity. Separately, we used the on-line exposure time calculator for the WFC3 camera (http://etc.stsci.edu/) to obtain an independent estimate of the limiting magnitude. For 10 exposures of 400 s each, on a source with G2V spectral type observed using the 350LP filter, a 3σ detection is expected at V = 27.8. This is slightly poorer than found empirically, a fact which we attribute to our ability to detect objects with formal significance <3σ in blinked-pair data. However, in order to be conservative, we take V = 27.8 as the upper limit to the allowable brightness of any point-like, co-moving companion.

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By Equation (2), V = 27.8 corresponds to a spherical effective radius r_e = 11 m (again, assuming p_V = 0.29) and sets a practical upper limit to the size of any objects ejected from 311P and still remaining within the vicinity of the nucleus. Such a body would have a mass ∼2 × 10⁻⁴ times that of the central body.

Figure 6 shows the individual light curves from the single-orbit visits to 311P. Each visit is limited to ≲40 minutes duration by the orbital motion of HST, but even during this short time, the apparent brightness of 311P is observed to vary by an amount large compared to the ±0.01 mag uncertainty of measurement. For example, on UT 2015 March 3 and June 29, the brightness changes by ∼0.2 mag, or 20×  the measurement uncertainty. The plotted photometry is extracted from a 02 radius photometry aperture, corresponding to ∼190 km at Δ = 1.3 au. Within this aperture, the signal is dominated by light from the nucleus and so we presume that most or all of the measured photometric variation is due to rotational modulation of the scattered light, owing to the aspherical shape of the nucleus.

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Unfortunately, the majority of the light-curve segments in the figure are so widely separated in time that it is not possible to combine them in order to reconstruct a unique rotational light curve. Instead, the strongest light-curve constraint is provided by the data from four consecutive orbits on UT 2015 March 19. These four orbits span a period of 5.4 hr (Figure 7), and, as the features of the light curve do not repeat, we conclude that a limit to the light-curve period (which is not necessarily equal to the rotation period) may be set at P_ℓ ≳ 5.4 hr.

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If the brightness variations on 311P are due to azimuthal albedo variations, then the rotational period, P, and the light-curve period are equal, P = P_ℓ. However, azimuthal albedo variations on most small bodies are modest to the point of being undetectable. While we have no specific evidence in the case of 311P, we consider it unlikely that the light curve in Figure 7 is caused by surface albedo variations.

The light curves of most small bodies in the solar system reflect variations in the projected cross-section caused by rotation. Such geometrically produced light curves are doubly periodic (two maxima and two minima per rotation), meaning that P = 2P_ℓ. In this interpretation, the rotation period of 311P would be constrained by the data to be P ≥ 10.8 hr.

The critical period at which a weak body becomes rotationally unstable depends on both its density and its shape. Consider a prolate (American football shaped) body with a long axis of length, 2b and two, equal short axes of lengths 2a. In rotation about one of the short axes, the gravity at the tips equals the centripetal acceleration there at critical period

$$\begin{eqnarray}&&{P}_{c}={\left(\displaystyle \frac{3\pi }{G\rho }\right)}^{1/2}\left(\displaystyle \frac{b}{a}\right).\end{eqnarray} \tag{ 5 }$$

The axis ratio, b/a, can be estimated from the measured rotational brightness variations, ΔV using b/a = 10^0.4ΔV. Substituting ΔV = 0.3 mag (Figure 7), we find b/a = 1.3, and this is formally a lower limit to the axis ratio because of the effects of projection (i.e., the rotation axis might not be perpendicular to the line of sight).

Substituting ρ = 2700 kg m⁻³, b/a = 1.3 into Equation (5) we find P_c = 2.6 hr, which matches the "spin barrier" detected in the rotations of most asteroids larger than ∼150 m in size, C- and S-type alike (e.g., Carbognani 2017, and references therein). Given that P ≫ P_c, we conclude that 311P, interpreted as a single, rotating prolate body, is centripetally stable. Forcing P_c = 10.8 hr in Equation (5) gives ρ < 160 kg m⁻³ for centripetal instability, which we consider implausibly small. The light curve (Figure 7) is thus inconsistent with centripetal instability models of the activity in 311P (Jewitt et al. 2013, 2015a; Hirabayashi et al. 2015) if the nucleus is prolate.

Radar observations (e.g., Ostro et al. 2006; Busch et al. 2011; Naidu et al. 2015) show that many rapidly rotating objects possess an oblate, discus or muffin-like, body-shape likely produced by equatorward migration of material from higher latitudes. A perfectly oblate body in rotation about its minor axis is rotationally symmetric, would have no rotational light curve and so could not be identified using time-series photometry. While radar-observed asteroids are not perfectly oblate, the typical rotational variation in the cross-section is just a few percent (e.g., Pravec et al. 2006), again challenging detection in our data. We cannot appeal to such a body to explain the large photometric variations in 311P. Specifically, the 0.3 mag brightness variation in Figure 7 must have another cause.

With equal long equatorial axes, 2b, and short axis, 2a, the centripetal and gravitational accelerations are equal in magnitude at the period

$$\begin{eqnarray}&&{P}_{c}={\left(\displaystyle \frac{3\pi }{G\rho }\right)}^{1/2}{\left(\displaystyle \frac{b}{a}\right)}^{1/2}.\end{eqnarray} \tag{ 6 }$$

For modest b/a = 1 to 2, consistent with the radar-inferred shapes of critically rotating asteroids (e.g., Ostro et al. 2006; Busch et al. 2011; Naidu et al. 2015), and ρ = 2700 kg m⁻³, Equation (6) gives P_c = 2.0–2.8 hr for the critical period below which mass could be lost centripetally. This again matches the "spin barrier" detected in the rotations of asteroids (Carbognani 2017) and is substantially smaller than the limit on the period, P ≥ 10.8 hr.

The V-shaped drop in the brightness (at rates up to ∼0.4 mag hr⁻¹) shown in Figure 7 is reminiscent of the light curve of an eclipsing binary (e.g., Pravec et al. 2006, 2016; Lacerda & Jewitt 2007; Scheirich & Pravec 2009). In this case, the V-shaped minimum would be caused by the transit or eclipse of a secondary body in an unequal pair. Representing both bodies as spheres having the same albedo and with radii r_p (primary) and r_s (secondary), we write

$$\begin{eqnarray}&&{C}_{\max }=\pi ({r}_{p}^{2}+{r}_{s}^{2})\end{eqnarray} \tag{ 7 }$$

at maximum light and

$$\begin{eqnarray}&&{C}_{\min }=\pi {r}_{p}^{2}\end{eqnarray} \tag{ 8 }$$

at minimum light, assuming a full transit. With C_max = 0.13 km² and C_min = 0.10 km², from Section 3.1, we solve Equations (7) and (8) to find r_p = 178 m and r_s = 98 m for the radii of the two components. The mass ratio of these bodies, if they are of the same density, is m_s/m_p = (r_s/r_p)³ ∼ 1/6. The observed duration of the supposed transit event, from first contact to last, is ΔT = 2 hr (Figure 7). From this, we compute the sky-plane velocity of the secondary as it crosses the face of the primary, using

$$\begin{eqnarray}&&V=2({r}_{p}+{r}_{s})/{\rm{\Delta }}t,\end{eqnarray} \tag{ 9 }$$

which gives V = 0.077 m s⁻¹. Assuming, for lack of evidence, that the orbit of the secondary is a circle, Kepler's law relates V to the component separation, r, through

$$\begin{eqnarray}&&r=\displaystyle \frac{4\pi G\rho ({r}_{p}^{3}+{r}_{s}^{3})}{3{V}^{2}}.\end{eqnarray} \tag{ 10 }$$

Substituting, we find r = 840 m or, in units of the primary radius, r/r_p = 4.2. The separation is a small fraction of the Hill radius (Equation (4)), r/R_H = 10⁻². These properties are broadly similar to those of a large number of binary asteroids thought to have been formed by rotational fission (Pravec et al. 2006).

Detection of a transit would only be possible given a specific geometrical alignment of the binary system. Is this alignment likely? In order for a partial transit to be observed, the Earth must lie closer to the orbital plane of the secondary than an angle tan(θ) = ±(r_p + r_s)/r. We find θ = ±0.36 radian ( ± 21°). Given a random distribution of possible orbital poles, the likelihood of finding this alignment by chance is ∼62%. To observe a full transit (in which the silhouette of the secondary lies totally within that of the primary) requires tan(θ) = ±(r_p − r_s)/r, which gives θ = ± 12°. The probability of finding even this more stringent alignment by chance, given a random distribution of planes, is still ∼40%. Hence, it is not statistically remarkable that we would detect mutual events given the small separation of this binary.

The orbit period of such a system is P_K = 2πr/V, or 6.8 × 10⁴ s (about 19 hr or 0.8 days), meaning that transits and mutual occultations are to be expected every ∼9.5 hr. There is therefore a high probability that we would detect one minimum in our 5.4 hr observational window. Specifically, given that the HST observing windows are each δt ∼ 40 minutes (2/3 hr) in duration, the probability that a randomly sampled observing window will sample a deep light-curve minimum is 4δt/P_K ∼ 1/7. In the six single-orbit observing windows shown in Figure 6, we should expect to find ∼1 deep light-curve minimum. The observations from March 03 definitely record the deep minimum, while observations from June 29 probably just missed it, apparently sampling the rise toward peak brightness.

We conclude that mutual events are quite likely to be detected in a binary with $r/{r}_{p}=4$ and P_K = 19 hr, given a randomly selected spin-pole and the temporal sampling of our HST observations. However, while mutual events in a binary can explain the 0.3 mag dip in the light curve of 311P, they do not provide an explanation for the episodic mass loss indicated by the multiple tails ejected in 2013 (Jewitt et al. 2013, 2015a).

Our suspicion is that the ("landsliding") mechanism indeed drives the mass loss from one or both of the components, but that the rapid rotation required for the instability contributes little to the measured light curves because the unstable body has the rotationally symmetric (muffin-shaped) morphology described above. For example, measurements from individual HST orbits show short-term variations (Figure 6) that are large compared to the errors of measurement (but small compared to the deep minimum recorded best on UT 2015 March 19; Figure 7). These short-term variations, which we have been unable to phase into a convincing light curve because of the strong aliasing in our widely spaced data, could result from small deviations from rotational symmetry in a critically rotating (P ∼ 2 hr) muffin-shaped body.

Hainaut et al. (2014) conjectured that dust emission from 311P might be due to friction at the contact point of a "rubbing binary." However, it is not clear, in the context of a rubbing binary, why the interval between dust events should be measured in weeks and months (nine events spread over ∼250 days; Jewitt et al. 2013, 2015a) rather than being comparable to the orbital period, P_K = 19 hr. Furthermore, in the absence of a continuous exciting force, and for any plausible values of the friction coefficient or the coefficient of restitution, we expect that relative motion between binary components would be quickly damped-out. These concerns deserve exploration in a future, quantitative treatment of the physics of a binary rubble pile merger.

A final possibility is that 311P is in an excited rotational state, as a result of its small size and the long damping time due to internal friction. Either or both components of 311P could be non-principal axis rotators. Excited rotation is common in small asteroids because the damping times are long (Harris 1994), but typically much more data than we possess is required in order to identify the rotational state. Therefore, we must leave this possibility open.

A leading model for the formation of close asteroid binaries is by the rotational disruption of a weakly cohesive primary (e.g., Walsh et al. 2012; Boldrin et al. 2016; Sánchez & Scheeres 2016, and see reviews by Margot et al. 2015, Walsh & Jacobson 2015). Spin-up is driven by radiation torques (the YORP effect) for which the characteristic time at 2 au (estimated from published observations of asteroids in which YORP torque has been detected) is τ_Y(Myr) ∼ 2 (r_p/1 km)² (Jewitt et al. 2015b). Beyond the ${r}_{p}^{2}$ size-dependence in this equation, the YORP timescale is very uncertain for any particular object because it is dependent upon unknown details of the shape, spin vector and surface thermal properties. Nevertheless, for an object as small as 311P (r_p ∼ 0.2 km gives τ_Y ∼ 0.1 Myr) it is reasonable to expect that YORP torque has affected the spin. Binaries form when slowly launched debris is captured and reassembled through dissipative collisions in orbit (e.g., Jacobson & Scheeres 2011; Walsh et al. 2012). The orbits of binaries so-formed are initially chaotic, with some reimpacting the primary and others escaping to infinity (forming asteroid pairs, Pravec et al. 2010; Polishook 2014). Stabilization of the binaries can occur by the transfer of energy from orbital motions into the rotations of the components and by the ejection of mass (and energy and angular momentum) from the system. Jacobson & Scheeres (2011) found that stable binaries formed only when the mass ratio is m_s/m_p < 0.2. This is consistent with our estimate m_s/m_p ∼ 0.15 for 311P, but the modeling of disrupting systems is very sensitive to assumptions about the physical properties, most of which are unmeasured, and the agreement may not be significant. A wide binary active asteroid (288P/(300163), with a separation r/r_p ∼ 100) has been discovered but was probably not formed by rotational fission and in-orbit reassembly alone (Agarwal et al. 2017).

We show in Figure 8 a comparison between the inferred secondary/primary object diameter ratio of 311P with the same ratio for known binary asteroids having primary diameters ≤20 km. For the latter, we have used the compilation by Johnston (2016) and retained only those objects for which uncertainties on the primary and secondary object diameters are quoted. The uncertainties on the parameters of 311P are unknown: we show ±50% errors in diameter and diameter ratio, for illustrative purposes only. While there are relatively few known sub-kilometer binary asteroids, the figure shows that the component diameter ratio of 311P is not atypical. Similarly, in Figure 9 we compare 311P with the known binaries in the diameter versus orbital radius/primary radius plane, again taking data from Johnston (2016) and plotting only objects ≤20 km in diameter for which uncertainties in both diameter and orbit radius/primary radius are quoted. The dashed horizontal line at r/r_p = 2.6 in Figure 9 shows the Roche limiting radius for equal density spheres. It roughly coincides with the lower boundary of the measured binary separations, consistent with the expected very weak nature of secondaries formed by agglomeration in orbit. Again, 311P would be among the smallest binary objects known (this is an artifact of its transient prominence caused by dust ejection) but it is not otherwise remarkable. We conclude that the eclipsing binary interpretation of the 311P light curve is plausible.

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Numerical simulations show that the evolution of rotationally disrupted aggregate bodies can be both complex and protracted (Boldrin et al. 2016). Collisions and excitation of the spin of the secondary body can both dissipate orbital energy and so help to stabilize the asteroid as a close binary. The timescales for system evolution can be surprisingly long compared to the orbit period of the system, measured in 100s and even 1000s of days. If the detection of 311P in 2013 coincided with its initial breakup, then it seems entirely possible that the dynamical evolution may not yet be finished. The multiple tails, for example, could result from equatorward landsliding on a secondary being spun-up by mutual interactions, as well as from dust escaping from the unstable primary. Radiation pressure-swept dust, kicked up from the primary by boulders infalling from temporary orbit, could also supply the episodic ejections. Although no departing fragments larger than ∼10 m were detected in our search to date (Section 3.2), it is possible that such objects have yet to be launched and may be detected in future observations. These intriguing possibilities make 311P an object highly deserving of continued study.