THE ACTIVE ASTEROIDS

Since Whipple (1950), sublimation has been explored in great detail as the driver of mass loss from the classical comets. It need not be re-described in detail here. Although simple in concept, detailed studies of comets show that sublimation is a remarkably complex process when factors such as the porosity of the surface, nucleus rotation, the conduction of heat into the interior, and the development of a refractory mantle are considered (Guilbert-Lepoutre & Jewitt 2011). One simplification possible for the present objects is the assumption that asteroids contain no amorphous ice, since temperatures in the asteroid belt are too high for it to escape crystallization. Accordingly, we address only the highly idealized case of sublimation from an exposed crystalline ice surface in thermal equilibrium with sunlight.

The sublimation mass flux per unit area, dm/dt (kg m⁻² s⁻¹), from a patch of surface whose normal is inclined to the Sun-direction by angle θ is determined by solution of the energy balance equation

$$\begin{equation} \frac{F_{\odot } (1-A)}{R_{{\rm AU}}^2} \cos (\theta) = \epsilon \sigma T^4 + L(T) \frac{dm(T)}{dt} + f_c. \end{equation} \tag{ 2 }$$

Here, F_☉ = 1360 W m⁻² is the solar constant, R_AU is the heliocentric distance in AU, A is the Bond albedo, ∼ 0.9 is the emissivity of the surface, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴ is the Stefan–Boltzmann constant, T is the equilibrium temperature, and L(T) is the latent heat of sublimation. The temperature dependence of dm/dt may be estimated from thermodynamics and the Clausius–Clapeyron equation, or from direct measurement (Washburn 1926). The conduction term, f_c, requires knowledge of the thermophysical parameters and the spin vector of the nucleus and is small. We neglect it here.

The angle term, cos (θ), depends on the nucleus rotation vector, changes with the position on the nucleus, and varies with time as the nucleus rotates. Therefore, the total mass-loss rate must be obtained by integrating Equation (2) over the surface and with respect to time. To avoid sinking in complexity, we here focus on two limiting cases for the surface temperature and resulting sublimation rate. The maximum temperature, T_max, is reached at the subsolar point on a non-rotating body, where cos (θ) = 1. The minimum effective temperature, T_min, occurs when the surface is isothermal, corresponding to the largest possible radiating area. On a sphere, the average value of the angle term is $\overline{\cos (\theta)}$ = 1/4. Accordingly, we estimate the minimum and maximum temperatures and specific sublimation rates from Equation (2) for $\overline{\cos (\theta)}$ = 1 and 1/4.

We considered both low (A = 0.05) and high (A = 0.50) ice albedos and calculated sublimation mass fluxes from Equation (2). Clean (high albedo) ice sublimates too slowly to drive activity at asteroid belt distances, and so we present only the dirty (low albedo) ice solutions for the sublimation rate as a function of distance, R, in Figure 4. The curves are convergent toward small R, because the sublimation term in Equation (2) is then dominant, but separate dramatically at larger R as the sublimation term drops exponentially, leaving all the energy for radiation.

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Across the 2.0 ⩽R ⩽ 3.3 AU strip corresponding to the asteroid belt, sublimation rates from ∼10⁻⁴ kg m⁻² s⁻¹ (sublimation at T_max) to <10⁻⁸ kg m⁻² s⁻¹ (sublimation at T_min) are found. Mass-loss rates near 1 kg s⁻¹ (Table 2) require sublimating areas from 10⁴ m² (T_max) to 10⁸ m² (T_min). Since the latter exceeds the entire surface area of (for example) 133P, we can exclude the isothermal (T_min) model as the activity driver. The main conclusion to be drawn from Figure 4 is that exposed water ice can sublimate across the full range of heliocentric distances occupied by the main belt provided it is dirty.

Two other quantities are directly calculable once dm/dt is known. First, the rate of recession of the sublimating surface is given by

$$\begin{equation} \frac{d \ell }{dt} = \frac{1}{\rho } \frac{dm}{dt} \end{equation} \tag{ 3 }$$

and indicated on the right-hand axis of Figure 4. With ice density ρ = 1000 kg m⁻³, we find 350 μm yr⁻¹ ⩽dℓ/dt ⩽ 3 m yr⁻¹ over the 2–4 AU distance range swept by the majority of active asteroids (see right-hand axis of the figure).

Second, the size of the largest ejectable particle is estimated by balancing the gas drag force with the gravity, assuming spherical particles and nucleus and neglecting all other forces. We take the densities of the ejected grains and of the bulk nucleus, in the absence of contrary evidence, to be equal. The resulting maximum particle radius, computed from

$$\begin{equation} a_c = \frac{9}{16 \pi } \frac{C_D V_{{\rm th}}}{G \rho ^2 r}\frac{dm}{dt} , \end{equation} \tag{ 4 }$$

is plotted in Figure 5 as a function of heliocentric distance for low-albedo, subsolar sublimation. In Equation (4), C_D is the dimensionless drag coefficient, here taken to be unity, and V_th is the gas speed at the nucleus surface, taken from Biver et al. (2002). The equation offers, at best, a crude estimate of the critical radius; the highly anisotropic outflow of gas from a real nucleus can strongly affect dust acceleration and dynamics relative to the simple isotropic case used here (e.g., Crifo et al. 2005). Still, it is obvious from Figure 5 that sublimation gas drag can easily remove small solid particles across the full range of distances swept by the main-belt asteroids. Grains larger than a_c remain bound to the asteroid and contribute to the development of a refractory mantle, eventually leading to a stifling of the sublimation.

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While exposed ice sublimates quickly, ice protected from direct illumination by a covering or mantle of refractory debris can survive nearly indefinitely. A meter-thick layer of regolith can suppress sublimation sufficiently to retain ice for Gyr timescales (Schorghofer 2008). Then, to initiate sublimation, this protective mantle must be removed, which could occur naturally through impact.

Collisions between asteroids occur at characteristic speeds of several km s⁻¹ (Bottke et al. 1994) and are therefore highly erosive. Impact yields, defined as the ratio of the ejecta mass, m_e, to the projectile mass, M, are m_e/M ≫ 1. Material ejected by impacts may explain observed activity in some asteroids. In this section we aim to obtain a relation between the impactor properties and the resulting brightening caused by ejected material.

In an impact, the bulk of the ejecta travel at the lowest speeds. For equal target and projectile densities, the mass of ejecta, m_e, traveling faster than a given speed, v, can be roughly expressed by a power law

$$\begin{equation} m_e/M= A (v/U)^{\alpha } \end{equation} \tag{ 5 }$$

in which M and U are the impactor mass and speed and A ∼ 0.01 is a constant (Housen & Holsapple 2011). The index α depends slightly on the properties of the target but is reasonably well approximated for a range of materials as α = −1.5. Only ejecta traveling with v ⩾ v_e can escape from an asteroid to produce an increase in the scattering cross section, while the rest must fall back to coat the surface around the impact site. With the escape velocity given by

$$\begin{equation} v_e = \left(\frac{8 \pi G \rho }{3} \right)^{1/2} r, \end{equation} \tag{ 6 }$$

it is clear that Equation (5) defines, for a given target density ρ and impact speed U, a relation between the impact yield and the radius, r, of the impacted asteroid.

A relation between m_e and the scattering cross section of the ejecta and hence the change in brightness caused by impact can be established. The size distribution of the ejecta from NASA's Deep Impact mission to comet 9P/Tempel 1 has been modeled as a power law (Kadono et al. 2010). Their Figure 4 gives a differential power-law size index q = 3.7 over the 1 μm ⩽a ⩽ 100 μm size range, while other investigators have found slightly steeper distributions (Lisse et al. 2006). Separately, the size distribution of the ejecta from P/2010 A2 has been modeled as a power law with q = 3.3 ± 0.2 for 1 mm ⩽ a ⩽ 10 mm (Jewitt et al. 2010b). A wider range of power laws (from q = 3 to as steep as q = 6) has been reported in small-scale, hypervelocity impact experiments (Takasawa et al. 2011). For the sake of the present discussion, we adopt q = 3.5 and find that the relation between the scattering cross section, C_e, and the mass of particles having radii in the range a_min ⩽ a ⩽ a_max is

$$\begin{equation} m_e = \frac{4}{3} \rho \overline{a} C_e, \end{equation} \tag{ 7 }$$

where $\overline{a} = (a_{{\rm min}} a_{{\rm max}})^{1/2}$. For example, with a_min = 0.1 μm, a_max = 0.1 m, $\overline{a}$ = 10⁻⁴ m, or 0.1 mm.

We combine Equations (5)–(7) to compute the ratio of the cross section of the ejecta to the cross section of the target asteroid

$$\begin{equation} \frac{C_e}{\pi r^2}= \frac{A }{\overline{a} U^{\alpha } } \left(\frac{8 \pi G \rho }{3} \right)^{\alpha /2} r_p^3 r^{\alpha - 2}, \end{equation} \tag{ 8 }$$

where we have taken the projectile to be a sphere of the same density as the target and with a radius r_p. Substituting A = 0.01, $\overline{a}$ = 0.1 mm, U = 5 km s⁻¹, ρ = 2000 kg m⁻³, and α = −3/2, we obtain

$$\begin{equation} \frac{C_e}{\pi r^2} \sim 30 \left(\frac{r_p}{1 {\rm \,m}}\right)^{3}\left(\frac{r}{1 {\rm \,km}}\right)^{-7/2}. \end{equation} \tag{ 9 }$$

A 1 m projectile impacting a 1 km target asteroid will produce ejecta with a cross section 30 times the geometric cross section of the asteroid.

Lines of constant brightening, computed from

$$\begin{equation} \Delta m = 2.5 \log _{10}\left[1 + \frac{C_e}{\pi r^2}\right], \end{equation} \tag{ 10 }$$

are shown in Figure 6 as a function of asteroid and projectile radius. The curves are plotted only where the ejecta mass computed by Equation (5) is less than the target mass. The figure shows that, for example, the impact of a 1 m projectile into a 1 km asteroid would cause a brightening by Δm ∼ 3.5 mag, while impact of the same projectile into a 10 km asteroid would cause a brightening of only a few × 0.01 mag and would almost certainly escape detection. The published parameters of (596) Scheila and P/2010 A2, interpreted as impact ejecta, are also plotted in Figure 6. The location of Scheila overlaps the Δm = 1 mag brightening line, consistent with observations. P/2010 A2 was discovered nearly a year after disruption. The magnitude of its initial brightening is thus unknown, but Figure 6 suggests that it was a dramatic Δm ∼ 15. The curves in Figure 6 are uncertain, both because of the simplicity of the treatment and because of the many uncertainties in the impactor and dust grain parameters. For example, if U were larger by a factor of two, Equation (8) shows that the ejecta mass and cross section would be larger by a factor of ∼2.8 and the ejecta would be brighter by ∼1 mag. Nevertheless, Equation (8) and Figure 6 provide a useful first estimate of the observable effect of a given impact in the main belt. The figure leads naturally to the question of the impact rate in the main belt, which we discuss in Section 5.

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While some ejecta can be launched from the impact site at speeds comparable to the impact speed (i.e., v ∼ U ∼ 5 km s⁻¹), most material travels much more slowly, following Equation (5). The slowest escaping material leaves the target at a speed comparable to the gravitational escape speed, thus setting a relation between the size of the target and the duration of the brightening caused by impact. For example, consider two target asteroids of size 100 m and 100 km, observed from 1 AU distance using a typical charge-coupled device camera with a field of view 5' (corresponding to ∼10⁵ km half-width). Ejecta from the 100 km body will travel at ∼100 m s⁻¹ and cross the field of view in ∼10⁶ s (about 10 days). Ejecta from the 100 m body will travel at characteristic speed v_e ∼ 0.1 m s⁻¹ and take so long to cross the field (∼10⁹ s) that radiation pressure and Keplerian shear will dominate the distribution of the dust. The persistence of impact-produced coma is thus a measure of the size of the target body.

The equatorial, centripetal acceleration on a strengthless rotating object of mass density ρ (kg m⁻³) equals the gravitational acceleration at the critical rotational period, P_c, given by

$$\begin{equation} P_c = k \left[\frac{3 \pi }{G \rho }\right]^{1/2}. \end{equation} \tag{ 11 }$$

Here, k is a dimensionless constant dependent on the shape of the object. A sphere has k = 1. A prolate body with axes a ⩾ b in rotation about its shortest dimension has k ∼ a/b. Expressed in hours, Equation (11) may be written

$$\begin{equation} P_c\, [{\rm hr}] \sim 3.3 k \left[\frac{1000}{\rho }\right]^{1/2}. \end{equation} \tag{ 12 }$$

For example, a sphere with ρ = 2000 kg m⁻³ has P_c = 2.3 hr while an elongated body with the same density and a/b = 2 has P_c = 4.7 hr, by Equation (12). Although Equation (12) shows that P_c is independent of the object size, numerous measurements of asteroids show that only objects smaller than ∼0.1–0.3 km rotate with periods <2 hr (Pravec et al. 2002). The probable reason is that sub-kilometer asteroids have significant tensile strength and so can resist rotational disruption, while larger bodies are structurally weak and rotationally unstable when P ⩽ P_c. A weak (in tensile strength) aggregate or "rubble pile" structure could be produced by impact fracturing or, in the case of cometary nuclei, by the gentle settling together of icy planetesimals formed separately.

Given the evidence from the distribution of asteroid rotation rates, it is reasonable to conjecture that rotational instability might be a cause of observable mass loss in the main belt. A number of torques are capable, in principle, of driving an asteroid into rotational instability. For example, collisions between asteroids lead to a random walk in angular momentum that can end in an unstable state. Anisotropic radiation of photons can lead to a net torque (the so-called YORP effect) that can do the same (Marzari et al. 2011; Jacobson & Scheeres 2011). An object losing mass through sublimation can, depending on the rate and angular dependence of the mass loss, experience a torque orders of magnitude larger than torques due to either collisions or YORP, quickly being driven toward rotational breakup (cf. Drahus et al. 2011). While, in principle, rotational instability can affect any asteroid regardless of its size, in practice the spin-up times vary inversely with radius and the objects most likely to experience rotational bursting are the smallest (exactly as observed in the near-Earth population). Could some active asteroids be rotationally disrupted objects?

The action of electrostatic forces in moving dust particles across a planetary surface is best established on the Moon. Images from the Surveyor lunar lander spacecraft showed an unexpected "horizon glow," caused by forward scattering from dust particles ∼10 μm in size located ∼1 m above the surface (Rennilson & Criswell 1974). Later, the Lunar Ejecta and Meteorites (LEAM) dust detection experiment of Apollo 17 recorded the impact of low-velocity (⩽100 m s⁻¹) dust particles and showed particularly intense fluxes of such particles near local sunrise and sunset (Berg et al. 1976). The measured dust particle fluxes are ∼7 orders of magnitude too large to be associated with churning of the surface by micrometeorite bombardment.

Instead, electrostatic levitation was suggested as the source of these particles. On the dayside, positive charging of the lunar surface occurs by ejection of electrons owing to the photoelectric effect from UV and X-ray solar photons. In the absence of a discharging current, the potential can rise to the energies of the most energetic photons (10³ V, or more; cf. De & Criswell 1977), but, in practice, the measured dayside potential is about +10 V (Colwell et al. 2007). On the unilluminated side, or in shadows, the surface becomes negatively charged as a result of the greater flux of solar wind electrons (the solar wind is electrically neutral, but the low-mass electrons travel much faster than solar wind protons, leading to a net flux of electrons onto the surface). The typical energy of solar wind electrons is ∼10 eV, leading to a negative potential of order 10 V. Near the terminator, the same processes lead to positive and negative charging of the surface, but on much smaller spatial scales corresponding to the scale of the local topography, leading to locally high electric field gradients. Near a shadow edge, the photoelectrons produced in sunlight can travel to and stick in unilluminated regions, causing local electric field gradients estimated at E ∼ 10 to 100 V m⁻¹ (Colwell et al. 2007; Farrell et al. 2007). The positive, sunlit surface attracts a cloud of electrons, effectively neutralizing the gradient on length scales ℓ ≳ 1 m.

The electrostatic processes that move dust particles on the Moon presumably operate also on the asteroids. The charging time on the Moon is ∼10²–10³ s (de & Criswell 1977); photoelectron charging currents will be nine times weaker at 3 AU and the charging times will be nine times longer, but the potentials attained, for a given dielectric constant, will remain the same. The principal difference in the asteroid belt is that, whereas levitated lunar dust is retained by the gravity of the Moon, dust ejection speeds on small bodies can exceed the escape velocity, v_e. Therefore, electrostatic processes are potentially capable of leading to mass loss from asteroids.

The charge on a spherical grain of radius a is related to the potential on the grain, V, by q = 4π₀Va, where ₀ = 8.854 × 10⁻¹² F m⁻¹ is the permittivity of free space. The force on a charged particle exposed to an electric field E (V m⁻¹) is just F_e = qE. As our criterion for dust ejection, we demand v > V_e, where v is the grain speed achieved by accelerating across the shielding distance ℓ and V_e is the gravitational escape speed at the surface. Assuming that the grain and the asteroid are spherical, of radius a and r, respectively, and that both have density ρ, this criterion gives the critical grain size for electrostatic ejection as

$$\begin{equation} a_e= \left(\frac{18 \epsilon _0 V E \ell }{4 \pi G \rho ^2 r^2}\right)^{1/2}. \end{equation} \tag{ 13 }$$

Substituting the lunar values, V = 10 V, ℓ = 1 m, E = 10–100 V m⁻¹, and using ρ = 2000 kg m⁻³ as the canonical asteroid density, Equation (13) gives a_e = 1.5–5 μm for an r = 1 km asteroid (cf. Figure 5). These sizes are somewhat smaller than the ∼10 μm sized particles inferred from observations of some active asteroids (e.g., 133P; Hsieh et al. 2004), but, given the many uncertainties in both the model and the interpretation of observations, perhaps the differences are acceptable. In contrast, Equation (13) shows that on the Moon (r = 1600 km), only nanometer-sized grains can be ejected, meaning that the process is irrelevant there. Even for asteroid (596) Scheila (r = 56 km and a_e = 0.04–0.1 μm), any electrostatically ejected particles would be smaller than a wavelength and inefficient optical scatterers. Figure 5 shows that a_e ∼ 10⁻⁴a_c for a given asteroid radius. We conclude that electrostatic ejection of particles large enough to scatter optical photons is a plausible mass-loss mechanism only for smaller asteroids.

There are major uncertainties concerning electrostatic effects on the Moon, especially regarding the tendency for small particles to stick to each other and to surfaces, through the action of Van der Waals and other contact forces. These uncertainties are magnified in importance on the asteroids because, unless the forces of cohesion can be overcome, electrostatic levitation and ejection will be impossible (Hartzell & Scheeres 2011). Unfortunately, ignorance of grain cohesion limits our ability to know whether electrostatic ejection is important on the asteroids, although flat, pond-like structures on Eros have been interpreted this way (Hughes et al. 2008). Sonnett et al. (2011) report that ∼5% of main-belt asteroids show ultra-faint comae that cannot be detected individually but which are collectively significant. If these comae are real, electrostatic ejection of sub-micron grains might offer a plausible source mechanism that is approximately independent of distance from the Sun.

Thermal fracture can occur when the stress associated with a change in the temperature of a material exceeds the tensile strength of the material. The thermal stress is S ∼ αYδT, where α (K⁻¹) is the thermal expansivity, Y (N m⁻²) is the Young's modulus, and δT (K) is the responsible temperature change. The expansivities of common rocks are α ∼ 10⁻⁵ K⁻¹ (Lauriello 1974; Richter & Simmons 1974). Young's moduli Y = (10–100) × 10⁹ N m⁻² are typical for rock (Pariseau 2006, p. 474). If converted into kinetic energy with efficiency η, thermal strain energy can generate fractured material with speeds (Jewitt & Li 2010)

$$\begin{equation} v \sim \alpha \delta T \sqrt{\frac{\eta Y}{\rho }}. \end{equation} \tag{ 14 }$$

Substituting η = 1 gives an upper bound to the ejection speeds of fracture fragments as v ∼ 20 m s⁻¹, for fracture at δT = 1000 K. This is sufficient to launch particles above the gravitational escape speed from asteroids up to radius r ∼ 20 km. Thus, thermal fracture is a potential source of small particles and an agent capable of ejecting these particles from asteroids, but only those approaching the Sun very closely.

Some carbonaceous chondrite classes (probable fragments of outer-belt asteroids) contain 10%–20% water by weight, bound into hydrated minerals (Jarosewich 1990). If it could be liberated, this bound water might drive observable mass loss by entraining small grains in the gas flow produced as the water escapes into space.

Laboratory experiments show that thermal dehydration of hydrous phyllosilicates of planetary relevance (e.g., serpentine, brucite, muscovite, talc) is characterized by activation energies in the range 325–400 kJ mol⁻¹ (Bose & Ganguly 1994). Thermal dehydration is accompanied by a net volume change that can crack the dehydrating material and provide a source of small particles. However, the above activation energies correspond to temperatures (∼1000 K) much higher than normally found in the asteroid belt. For this reason, thermal dehydration is unlikely to play a role in a majority of the active asteroids since they have perihelia ⩾2 AU and temperatures too low to trigger dehydration.

Dehydration can be caused by shock pressure waves in impacts. For example, shock dehydration of brucite (Mg(OH)₂) begins near 12 GPa and is complete by 56 GPa (50% of the trapped water is lost at pressures 17 GPa; Duffy et al. 1991). Pressures this high are confined to a region comparable in size to the projectile diameter, so that the volume of shock-dehydrated target material is small compared with the volume of the resulting impact crater. Nevertheless, water released by impact could sustain mass loss from an asteroid for a timescale longer than that on which the impact ejecta dissipates. For example, a 10 m radius projectile would dehydrate ∼10⁷ kg of target. With a water fraction of 10%, some 10⁶ kg of water could potentially leak from the impact site, enough to sustain a dusty flow at 0.1 kg s⁻¹ for ∼100 days. While shock dehydration must be considered, at best, a process of secondary importance, it does raise the prospect that gaseous products detected following an impact might, in the future, be misinterpreted as evidence for sublimating ice.

Dust particles can be swept from the surface of an asteroid through the action of radiation pressure. To see this, we consider a grain located just above the surface and compare the gravitational acceleration toward the nucleus with the acceleration in the anti-sunward direction caused by radiation pressure, the latter expressed as βg_☉, where g_☉ is the gravitational acceleration to the Sun. A dust particle can be swept away by radiation pressure if

$$\begin{equation} \beta \frac{G M_{\odot }}{R^2} > \frac{4}{3} \pi G \rho r \end{equation} \tag{ 15 }$$

in which we have assumed that the nucleus is at heliocentric distance R and is spherical, of density ρ, radius r, and non-rotating. Also, G is the gravitational constant and M_☉ is the mass of the Sun. The dimensionless quantity β depends on the grain shape, composition, and porosity but is principally a function of particle size, a. As a useful first approximation, we take β ∼ 1/a_μ, where a_μ is the grain radius expressed in microns (Bohren & Huffman 1983). Then, the condition for a grain to be lost to radiation pressure becomes

$$\begin{equation} a_{\beta } < \frac{3 \,M_{\odot }}{4 \pi \rho r R^2}. \end{equation} \tag{ 16 }$$

Substituting ρ = 2000 kg m⁻³ and expressing the nucleus radius in km and the heliocentric distance in AU, we find the critical radius for grain loss to be

$$\begin{equation} a_{\beta } = 10 \left(\frac{1 {\rm \,km}}{r}\right)\left(\frac{1 {\rm \,AU}}{R}\right)^2. \end{equation} \tag{ 17 }$$

Equation (17) (also plotted in Figure 5) shows that, for example, a 1 km radius object at R = 3 AU would lose dust grains smaller than about a ∼ 1 μm. Optically active particles (a ⩾ 0.1 μm) can be swept away throughout the asteroid belt provided their parent bodies are smaller than about 10 km. Larger particles can be lost if the nucleus is rotating and has an aspherical shape.

The above considerations are simplistic in that they take no account of the direction of the radiation pressure acceleration relative to the nucleus. Grains released on the dayside, for instance, will be pushed back into the nucleus by radiation pressure from above. On the other hand, grains released near the terminator will feel a net force in a direction determined by the vector sum of the local gravitational acceleration and radiation pressure acceleration. These grains have the potential to escape. The principal limitation to efficacy of radiation pressure sweeping is, as with electrostatic launch, set by contact forces that hold small particles to the asteroid surface. If these forces can be overcome, then radiation pressure sweeping can play a role across the asteroid belt for 10 km–sized asteroids and smaller.

The key observational property of these objects is that the activity is recurrent (Hsieh et al. 2004, 2010), qualitatively consistent with sublimation in the same way that mass loss from the classical comets is modulated by the varying insolation around the orbit. Other mechanisms in Section 3 either are inconsistent with recurrent mass loss (impact, rotational instability) or can be reconciled with it only in the most contrived way. With equilibrium sublimation rates ≲(3–5) × 10⁻⁵ kg m⁻² s⁻¹ at R = 2.6–3.0 AU (Figure 4), the most stringent production rates ≲0.01 kg s⁻¹ (Table 2) imply mass loss from areas of exposed ice ≲200–300 m², corresponding to circles of radius ∼10 m (Hsieh et al. 2004).

The eccentricities of 133P and 238P are small (0.165 and 0.253, respectively), and over the range of heliocentric distances swept, the expected sublimation mass flux varies by less than an order of magnitude (upper curve in Figure 4). This is a problem because a 10-fold reduction in activity at aphelion relative to perihelion would still produce detectable dust, whereas none is observed. Instead, mass loss might be confined to a window near perihelion because of a shadowing effect of local topography on the nucleus, perhaps caused by recession of the sublimating surface, forming a pit (Equation (3)). The surface recession rate at R = 2.6–3.0 AU is (3–5) × 10⁻⁸ m s⁻¹, so that a sublimation pit meters deep should develop in only a few orbits. However, that the ice must still be close to the physical surface is shown by the appearance of comae near and soon after perihelion. Ice more than a few centimeters deep would be thermally decoupled from the deposition of heat on the surface.

The evidence for ice sublimation, although self-consistent, remains indirect. Detection of gaseous products of sublimation would provide definitive evidence for the role of sublimation in active asteroids. Just as in the classical comets, practical concerns render the daughter species (molecular fragments) optically more visible than the parent molecules sublimated from the ice. The most prominent band in the optical spectra of comets is due to the trace species CN, itself a product of dissociation of the parent molecule HCN. As noted in Section 2, several attempts to detect CN λ3889 in active asteroids have been reported, typically resulting in limits to the total gas production rate 0.1–1 kg s⁻¹, about two to three orders of magnitude smaller than found in bright comets near 1 AU.

To set this on a quantitative basis, we show in Figure 7 the expected CN band flux as a function of R for several outgassing rates, as marked, computed as described in Jewitt et al. (2009). The fluxes in the figure were evaluated for opposition (i.e., Δ = R − 1) and assume the Haser model and a spectroscopic slit 1'' × 5''. The Swings effect (in which the resonance fluorescence cross section is affected by the heliocentric velocity through the Doppler effect) amounts to less than a factor of two and has been neglected for simplicity. Plotted on the figure are published limits to the CN band flux in active asteroids. State-of-the-art measurements, indicated by the horizontal gray band in the figure, reach CN fluxes ∼(2–3) × 10⁻¹⁶ erg cm⁻² s⁻¹, corresponding to Q_CN ∼ 10²³ s⁻¹ at R = 2 AU, but by R = 3 AU, the limits barely reach Q_CN ∼ 10²⁴ s⁻¹, showing how difficult is the spectroscopic detection of gas at larger distances. The steep distance dependence is made worse by the rapidly declining sublimation rates from 2 AU to 3 AU (Figure 4). For the most favorable (high temperature) case in that figure (albedo 0.05) the sublimation rate drops by a factor of four from 2 AU to 3 AU. For these reasons, spectroscopic detection of gas at main-belt distances is challenging even in many classical comets, where sublimation is the undoubted driver of the observed activity. Insofar as spectroscopic emission detections of gas in active asteroids are concerned, absence of evidence is not evidence of absence.

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The brightening of Scheila was sudden, the scattering cross section declined smoothly with time, and the morphology evolved in a way consistent with the expansion of an impulsively ejected coma under the action of radiation pressure (Bodewits et al. 2011; Jewitt et al. 2011b; Moreno et al. 2011b; Hsieh et al. 2012; Ishiguro et al. 2011). All these signatures are consistent with impact production but are difficult or impossible to reconcile with the other mechanisms of Section 3. Scheila is also alone among the active asteroids, so far, in being very large (113 km diameter). It presents a substantial cross section for impact, but the resulting strong gravity precludes the action of electrostatic ejection, while the slow rotation eliminates any possibility of rotational instability. Sublimation of water ice can eject small grains from Scheila, but the short coma fading time is difficult to understand given the sublimation rates in Figure 4.

The outburst brightness was ∼1 mag brighter than the bare asteroid (Larson 2010; Jewitt et al. 2011b). Figure 6 is consistent with the conclusion that the projectile was between 35 m and 60 m in diameter (Jewitt et al. 2011b; Ishiguro et al. 2011; cf. Bodewits et al. 2011).

Impact may account for the dust tail of P/2010 A2, but the interpretation in this case is non-unique. This body is comparatively tiny (only 120 m in diameter versus 113 km for Scheila), with a tail mass suggesting collision with a projectile only a few meters in size (Jewitt et al. 2010b; Snodgrass et al. 2010). A key observational difference is that, whereas (596) Scheila faded on a timescale of one month, the tail of P/2010 A2 remained prominent ∼15 months after formation, despite the addition of no new dust (Jewitt et al. 2010b, 2011b). This difference in fading timescales arises because the slowest particles to escape from a body leave the target with a speed comparable to the gravitational escape speed. For P/2010 A2 this is ∼0.1 m s⁻¹, allowing large, slow grains to persist for >1 yr after impact, while, for Scheila, particles ejected at less than the 60 or 70 m s⁻¹ escape speed simply fell back on the surface (Jewitt et al. 2011b; Ishiguro et al. 2011). Furthermore, most grains traveling fast enough to be ejected were also small enough to be deflected by solar radiation pressure, providing another mechanism to quickly clear the coma. There are no direct measurements of the total, early-time brightening, only a limit Δm < 19 (Jewitt et al. 2011b). However, the model suggests that if the P/2010 A2 were hit by a 2 m–sized projectile, the brightening would be Δm ∼ 15 (Figure 6), which is consistent with the empirical constraint. Extreme brightening caused by impact has been suggested as an explanation of "guest stars" recorded in ancient Chinese records (Reach 1992).

Rotational instability offers another explanation for mass loss from P/2010 A2 (Jewitt et al. 2010b). The small diameter of the primary body (120 m) corresponds to a short spin-up timescale under the action of YORP. In fact, for sub-kilometer main-belt objects the YORP spin-up timescale is less than the timescale for collisional disruption (Marzari et al. 2011; Jacobson & Scheeres 2011). While the rotation period of the primary body in P/2010 A2 is unknown, it would not be surprising to find that it had exceeded the centripetal limit. Unfortunately, there are few published predictions for the detailed appearance of a rotationally disrupted body (Richardson et al. 2005).

Object (3200) Phaethon is distinguished from the other active asteroids by its small perihelion, q = 0.14 AU, allowing it to reach extreme subsolar temperatures (∼1000 K). These are sufficient to trigger thermal fracture and to dehydrate water-bearing minerals, if they are present (Jewitt & Li 2010). It has been suggested that Phaethon was recently scattered inward from a Pallas-like orbit (a = 2.771 AU, e = 0.281, i = 33°; de León et al. 2010), where a large proportion of the asteroids are volatile-rich B and C types. If Phaethon originated in the vicinity (or as part) of Pallas, then finding hydrated minerals would not be surprising, since Pallas itself contains hydrated minerals. The small asteroids associated dynamically with Phaethon (1999 YC and 2005 UD) should have similar compositions, but they have not been observed close to perihelion and have shown no evidence for mass loss.

In addition to the creation of dust and fragments by thermal fracture and dehydration shrinkage, we conjecture that the escape of particles produced by these thermal mechanisms is facilitated by radiation pressure sweeping. This is especially important in (3200) Phaethon near perihelion relative to other asteroids. According to Equation (17), particles up to 300 μm in diameter can be swept from its surface when at q = 0.14 AU. Particles produced by thermal fracture or dehydration cracking, or lifted from the surface by electrostatic repulsion, would be rapidly swept away from it by radiation pressure (Jewitt & Li 2010).

The mechanisms behind the mass loss in other active asteroids remain less clear. Activity in 176P can be interpreted in terms of a sublimation model that is able to fit the observed coma morphology (Hsieh et al. 2010). Models involving impulsive ejection (e.g., by impact) reportedly provide less good fits to the imaging data, but, given the many unknown parameters in the models (e.g., the relation between dust particle size, scattering efficiency, and speed), the identification of sublimation in 176P cannot be regarded as definitive. The same must be said for observations of P/2008 R1, where sublimation can fit the data (Jewitt et al. 2009) but where an impact origin may also apply. We also regard the nature of activity in P/2008 R1 as unknown. The coma isophotes of P/La Sagra are consistent with dust emission over an extended period (Moreno et al. 2011b), as is integrated light photometry (after correction for the unmeasured phase function of the comet; cf. Hsieh et al. 2012). Again, isophote models involve the assumption of numerous, unknown parameters and provide solutions that are non-unique. This is especially true for objects viewed at small angles to their orbital planes, as are most of the active asteroids. Clear evidence of the weakness of models is the fact that Moreno et al. (2010), using limited data, were able to fit the isophotes of P/2010 A2 with a model in which particles were released over an eight-month period, inconsistent with the finding of an impulsive origin based on more and better imaging data (Jewitt et al. 2010b; Snodgrass et al. 2010). With this as a cautionary reminder, we regard the evidence concerning activity in 176P, P/2008 R1, P/La Sagra, and 300163 as insufficient to diagnose the cause. Evidence concerning 107P and (2201) Oljato is even more fragmentary, and the cause of their activity is also unknown.

The results of this and the previous section are summarized for convenience in Table 3 and shown schematically in Figure 8. The figure shows the plane of object radius, r, versus heliocentric distance, R, with the regions of action of the different physical processes considered in Section 3 marked as color fields. The 11 reported active asteroids are plotted at the heliocentric distances at which activity was observed. The figure shows that, for example, (596) Scheila lies in a region of the r versus R plane in which only impact and sublimation are capable of ejecting particles. (As described above, detailed studies of the morphology and time dependence of the coma of Scheila show that impact is the true explanation.) Conversely, a majority of the active asteroids lie in regions of the r versus R plane where many processes are potentially important. For example, in 238P, sublimation, electrostatic ejection, rotational instability, radiation pressure, and impact process are all potentially active. Only through detailed physical investigation is it possible to discriminate between these possibilities (in favor of sublimation, in the case of 238P, based principally on the repetition of the observed mass loss). For many active asteroids, the physical observations needed to discriminate among mechanisms do not exist.

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Table 3. Summary of Mechanisms

Name	Sublimation	Impact	Electrostatics	Rotation	Thermal
(3200) Phaethon	×	?	?	?	$\checkmark$
P/2010 A2	×	$\checkmark$	×	$\checkmark$	×
(2201) Oljato	?	?	?	?	×
P/2008 R1 (Garradd)	?	?	?	?	×
(596) Scheila	×	$\checkmark$	×	×	×
300163 (2006 VW139)	?	?	?	?	×
133P/Elst–Pizarro	$\checkmark$	×	?	?	×
176P/LINEAR (118401)	?	?	?	×	×
238P/Read	$\checkmark$	×	×	?	×
P/2010 R2 (La Sagra)	?	?	?	?	×
107P/Wilson–Harrington	?	?	?	×	×

Notes. $\checkmark$: evidence exists consistent with the process; ×: evidence exists inconsistent with the process; ?: insufficient evidence exists.

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