Rotational Mass Shedding from Asteroid (6478) Gault

Gault appeared stellar in all our images. We measured the apparent magnitude in the NOT images using circular apertures with angular radii 5, 10, and 15 pixels (11, 21, and 32, respectively). Sky subtraction used an annulus having inner radius 15 pixels (32) and outer radius 50 pixels (108). The same apertures were used for field stars in order to obtain relative photometry. The apparent magnitudes were then converted to Johnson R magnitude (Jester et al. 2005). The data have a mean R magnitude ${\overline{m}}_{R}=17.64$, with rms error ± 0.01 mag. To compare with previous works, we convert the mean R magnitude to V, adopting m_V –m_R = 0.40 ±0.01 (Jewitt et al. 2019a). The mean V magnitude is thus ${\overline{m}}_{V}=18.04\pm 0.02$, and is related to the mean absolute magnitude, H, by

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

where r_H and Δ are the heliocentric and geocentric distances expressed in astronomical unit (au) and ${\rm{\Phi }}\left(\alpha \right)$ is the phase function measured at phase angle α. The absolute magnitude H_V (Equation (1)) is the magnitude Gault would have if observed from r_H = Δ = 1 au and α = 0°. We assume $2.5{\mathrm{log}}_{10}\left({\rm{\Phi }}\left(\alpha \right)\right)=-0.04\alpha$, broadly consistent with the measured phase function of asteroids. Equation (1) yields H = 15.0, 0.7 magnitudes fainter than the H = 14.3 absolute magnitude listed in the Jet Propulsion Laboratory (JPL) Horizons software catalog. Marsset et al. (2019) presents a summary of available photometry of Gault taken between 2019 January and May in their Figure 2. This figure shows that the faintest absolute magnitude recorded during that period was H = 14.8, obtained at the end of 2019 April. Our NOT images, obtained more than a year later, show Gault in an even fainter state, giving us confidence that we were observing the bare nucleus of Gault.

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As another way to determine whether our photometry might be contaminated by dust emission, we measured Gault's radial profile using both NOT and HST images. Figure 2 shows Gault's profile extracted from one of the NOT images, compared with the profile from the standard star Feige 11 (Landolt 2009) imaged using sidereal tracking rates. For both objects, the profiles were extracted the same way and have been normalized to the peak intensity. The two profiles have essentially the same FWHM of 4.6 pixels, or 10; the very slight discrepancy between the two profiles can be attributed to the different seeing of the two images. This result suggests that we observed the bare nucleus of Gault.

The HST images provided a higher-resolution measure of Gault's profile. In order to generate an HST point-spread function (PSF) for comparison with the Gault profile, we used the TinyTIM software (Krist et al. 2011), version 7.5. A PSF for the WFC3 camera and F350 LP filter combination was generated, then scaled to the same pixel scale as the Gault images. Figure 3 shows Gault's profile extracted from one of the HST images, compared with the model PSF. The two profiles are essentially identical, providing another confirmation that we were observing Gault's nucleus.

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To set a limit to the presence of coma we used the relation between the surface brightness Σ(θ) measured at angular radius, θ, and m_C , the apparent magnitude of the coma within a circle of radius θ, from Jewitt & Danielson (1984),

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

This relation is based on the assumption that Σ(θ) ∝ θ⁻¹, as expected for a steady-state flow, and as is commonly observed in the inner comae of comets before the effects of radiation pressure deflect the ejected particles from their initial paths.

From the HST surface brightness profile, we set a limit to Σ(02) ≥ 25.0 magnitudes arcsecond⁻². Substituting into Equation (2) gives m_C ≥ 23.5, at a time when the red magnitude of Gault was m_R = 17.64. We conclude that any contribution from a steady-state near-nucleus coma is fainter than the integrated magnitude by ∣m_C − m_R ∣ = 5.9 magnitudes. Such a coma could contribute no more than ${10}^{(-0.4| {m}_{C}-{m}_{R}| )}$ = 0.5% to the measured signal and thus cannot account for the 10-times larger photometric variations observed in Gault. However, we cannot use the surface brightness profile to constrain the presence of dust particles packed closely to the nucleus, such as, for example, sub-orbital particles moving in temporarily bound orbits close to the nucleus.

We use the limit to the coma obtained from the surface brightness profile to set a limit to the production of dust. The cross section of Gault is ${C}_{n}=\pi {r}_{n}^{2}$, where r_n = 2 km is the estimated radius (Sanchez et al. 2019). The dust cross section, C_d , inside a circle having radius θ_i = 02 is roughly ${C}_{d}\lt {C}_{n}{10}^{(-0.4| {m}_{C}-{m}_{R}| )}$, or C_d < 63,000 m². We suppose that the dust has average radius $\bar{a}$ and is moving radially outward at speed V. The time taken for dust to travel angular distance θ is τ ∼ θΔ/V, with θ expressed in radians, and the mass supply and loss rates needed to maintain steady state are $\dot{M}\sim 4\rho \bar{a}{C}_{d}/(3\tau )$, or

$$\begin{eqnarray}&&\frac{{dM}}{{dt}}=\frac{4\rho \bar{a}{C}_{d}V}{3\theta {\rm{\Delta }}}.\end{eqnarray} \tag{ 3 }$$

We set $\bar{a}$ = 200 μm and V = 0.15 m s⁻¹ (Jewitt et al. 2019a), Δ = 1.32 au (Table 1) and assume ρ = 1700 kg m⁻³ (see below), to find $\dot{M}\lesssim 0.02\,\mathrm{kg}$ s⁻¹. This compares with peak mass-loss rates $\dot{M}=$ (20–40) kg s⁻¹ during the maximum tail formation phase (Jewitt et al. 2019a).

Except for small-amplitude objects, most asteroid lightcurves show two clear maxima and minima over one rotation period, caused by the changing cross section of an irregularly shaped body. In contrast, Gault's brightness variations show multiple (four or five) small peaks, all with very small (≤0.025–0.05 mag) amplitudes (Figure 4). These small, short-term brightness variations could be due to albedo variations, to local topographical deviations from symmetry (i.e., "lumps" on the surface), or to a combination of both.

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To look for periodicity in the brightness variations, we use the phase dispersion minimization (PDM) algorithm (Stellingwerf 2011). The correct period would result in a minimum theta statistic, as defined in Stellingwerf (1978). The result of the PDM algorithm is shown in Figure 5, and shows a minimum at 2.55 hr, with a shallower minimum near ∼5.3 hr. For reasons explained below, we believe the 2.55 hr is the correct rotation period, with the 5.3 hr most likely due to subharmonics. To verify the 2.55 hr rotation period, Figure 6 shows the phase plot for this period; the phased data, which cover nearly two full rotations, show good overlap over the full rotational phase. To guide the eye we also plot the median of the data divided into 20 phase bins. This median line suggests that Gault has four or five peaks with amplitudes of a few ×0.01 magnitude. Periods outside the range 2.45 and 2.65 hr fail to generate convincing phase plots.

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As noted above, several groups reported no detectable rotation period for Gault during the 2019 apparition (Moreno et al. 2019; Sanchez et al. 2019, and Lin et al. 2020), but we believe that this was due to the dust emission that existed throughout 2019. The dust velocity was measured at ∼0.15 m s⁻¹ (Jewitt et al. 2019a); at this velocity, assuming geocentric distance Δ = 1.4 au (as in Sanchez et al. 2019), it would take ∼8 months for the dust to traverse a 3'' radius photometry aperture. The dust contribution would obscure any rotational modulation due to the nucleus, leaving the slight modulation of the brightness due to nucleus rotation undetected in the 2019 observations. We note that our result is longer than the 2 hr rotation period reported by Kleyna et al. (2019), and inconsistent with those of Ferrín et al. (2019) and Carbognani & Buzzoni (2020), who reported a ∼3.36 hr rotation period.

The 2.55 hr rotation period is near the empirical P ∼ 2.2 hr critical period for asteroids (Pravec & Harris 2000), and suggests rotational instability as the simplest explanation for Gault's mass loss. The critical period for rotational instability of a spherical fluid (i.e., strengthless) body in rotation at period P occurs for density

$$\begin{eqnarray}&&{\rho }_{c}=\displaystyle \frac{3\pi }{{{GP}}^{2}}\end{eqnarray} \tag{ 4 }$$

where G = 6.67 × 10⁻¹¹ N kg⁻² m² is the gravitational constant. Substituting P = 2.55 hour gives ρ_c = 1700 kg m⁻³ which is comparable to, but slightly higher than, the densities of (101955) Bennu, ρ = 1190 ± 13 kg m⁻³ (Scheeres et al. 2019) and of (162173) Ryugu, ρ = 1190 ± 20 kg m⁻³ (Watanabe et al. 2019). The higher density, taken at value, would imply that Gault is less porous than either of these two asteroids, perhaps consistent with its larger size (Bennu and Ryugu are, respectively, about 0.5 km and 1 km in diameter, compared with Gault at ∼4 km). However, density estimates using Equation (4) are crude, because Gault is unlikely to be either perfectly spherical or strengthless, and we do not wish to over-interpret the result.

The small lightcurve amplitude of Gault may result from a spin axis pointing nearly along the line of sight, or from a nearly symmetric shape. We cannot rule out the former hypothesis, but we prefer the latter for the following reasons. We notice that Gault shares several similarities with asteroid (101955) Bennu, the target of NASA's OSIRIS-REx mission (see Table 2). Both asteroids display lightcurves with very small amplitudes: 0.025–0.05 mag for Gault, 0.03–0.06 mag for Bennu (Hergenrother et al. 2019). (We note that there is an earlier version of Bennu's lightcurve with larger amplitudes (Hergenrother et al. 2013), but here we refer to the lightcurve measured by OSIRIS-REx, as the 4°–18° phase angles of these measurements are much more comparable to ours). Both asteroids lose mass episodically, although at ∼10⁻⁷ kg s⁻¹ (Lauretta et al. 2019a), Bennu's mass-loss rate is 8 orders of magnitude smaller than Gault's 40 kg s⁻¹ at its peak (Jewitt et al. 2019a). Both asteroids eject large and slow-moving particles: 1–10 cm diameter and 0.07–1 m s⁻¹ for Bennu (Lauretta et al. 2019a), ∼0.4 mm diameter and 0.15 m s⁻¹ for Gault (Jewitt et al. 2019a; Kleyna et al. 2019). Images of Bennu reveal a rubble-pile asteroid with a top-like shape (Lauretta et al. 2019b). The top shape is also shared by (162173) Ryugu, although mass loss has not been reported for this object (Watanabe et al. 2019). Given the preponderance of the top shape among small asteroids (e.g., Nolan et al. 2013), and Gault's similarities with Bennu, we suspect Gault is also top-shaped.

The top-like shape has been seen in several small asteroids, such as 1994 KW4 Alpha (the primary of a binary system, Ostro et al. 2006), (65803) Didymos (Pravec et al. 2006), 2008 EV5 (Busch et al. 2011), and most recently in Bennu and Ryugu. Several hypotheses have been proposed for their shape. Small asteroids can be spun up or down by the Yarkovsky–O'Keefe–Radzievskii–Paddack (YORP) effect on timescales 10⁵−10⁶ yr (Rubincam 2000), so that over the lifetime of the asteroid, material from higher latitudes is driven toward the equator by centrifugal forces (Walsh et al. 2012). Rubble-pile asteroids like Bennu and Ryugu are believed to have formed from the re-accumulation of fragments produced in a catastrophic asteroid collision (Michel et al. 2001, 2020). Support for an early origin for the top shape comes from images showing craters imprinted on the equatorial ridge at both Bennu and Ryugu (Walsh et al. 2019; Hirata et al. 2020). Alternatively, fast-spinning rubble-pile asteroids are prone to deformation, both internally and at the surface, and may have acquired the top shape that way (Hirabayashi et al. 2020). Bennu, Ryugu, and Gault thus could have acquired their shape at the time of formation, or soon after.

The surfaces of Bennu and Ryugu provide a glimpse of what Gault's surface might look like. Bennu's surface is much rougher than expected, with hundreds of 10 m size boulders and even more at the 1 m scale (Lauretta et al. 2019b). Fine grains exist, but in limited amounts. Gault dust tail measurements also suggest a surface lacking in small particles (Jewitt et al. 2019a), and the same was also reported for (3200) Phaethon (Ito et al. 2018). Small particles are apparently scarce on small rubble-pile asteroids, whether they are active or not; the reason for the scarcity of small particles is still unclear.

The level of mass shedding in fast-spinning rubble-pile asteroids is a competition between centrifugal force and gravity, complicated by the effects of inter-particle friction and cohesion. To investigate the different regimes of rotational mass shedding, we calculate the Froude number Fr for those active asteroids suspected to be losing mass due to rotational instability. For example, Fr is often used to investigate granular flow regimes, such as in rotating drums (Mellmann 2001). The Froude number is defined here as the ratio between the centrifugal and gravitational forces, i.e., Fr = ω² R/g, where ω is the angular rotation speed of the granular system, R its radius and g the gravitational acceleration. Fr is proportional to the square of rotation speed, reflecting the fact that the rotation speed is the most important factor in controlling the flow of the particles.

Table 3 lists the active asteroids for which diameter, rotation period, mass-loss rate data are available; it is a subset of Table 2 of Jewitt et al. (2015). The density is available for most; we indicate where the density is assumed. In Figure 7 we plot the normalized mass-loss rate, $\left({dM}/{dt}\right)/(4\pi {R}^{2}$), as a function of the Froude number. We note the following.

• 1.  
  For fast-spinning rubble-pile asteroids, the onset of mass shedding starts at Fr ∼ 0.5. Particles can be lost at Fr < 1 because particle escape is aided by other factors not included in the Froude number, such as surface slopes, and radiation pressure forces (McMahon et al. 2020).
• 2.  
  Asteroids Bennu, (3200) Phaethon, and 133P/(7968) Elst-Pizarro all have Fr ∼ 0.5 but Bennu's mass loss rate is 6 orders of magnitude smaller than the others. The large gap between Bennu and the others is no doubt due to the different sensitivity limits of ground-based versus spacecraft measurements, and as more data become available, we expect the gap to be filled in. The clustering of Fr ∼ 0.5 indicates that once mass shedding starts at Fr ∼ 0.5, a single Fr could be associated with a wide range of mass-loss rates.
• 3.  
  Beyond the Fr ∼ 0.5 threshold, mass loss generally increases with Fr, consistent with rotation being the main driver of mass loss. As Fr grows, there must be a critical Fr beyond which the asteroid is completely blown apart by centrifugal forces. This is most likely what happened to asteroid 2013 R3 (Jewitt et al. 2014a; Hirabayashi et al. 2014), but there is no measurement of 2013 R3's rotation rate, so it was not included on this Figure. Hopefully, future observations will reveal more insight into this extreme rotation regime.

Finally, Figure 7 suggests that rotational instability can contribute to mass loss in some asteroids where the cause of activity has been ambiguous. For example, activity in asteroid (3200) Phaethon has been tentatively attributed to thermal fracture and/or desiccation cracking, while thermal fracturing, volatile release by dehydrated rocks, and meteoroid impacts have been postulated for Bennu. While the extreme temperature cycling in Phaethon may contribute dust ejection, Figure 7 suggests that rotation may also play a role. Lauretta et al. (2019a) rejected rotational disruption as the origin of Bennu's ejected particles due to the fact that the particles were in retrograde orbits. McMahon et al. (2020) pointed out that the inclusion of non-Keplerian forces like solar tides and radiation pressure could change the orbital elements significantly over a timescale of ∼100 days. Such fast evolution of the orbits could explain the retrograde orbits of Bennu's dust particles, allowing rotational instability to be reconsidered as a possible contributor of dust ejection.

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