ACTIVITY IN GEMINID PARENT (3200) PHAETHON

To quantify the apparent brightening in Phaethon, we obtained photometric measurements. For our purposes, coronal structures in the STEREO images represent an unfortunate, time-dependent distraction. As a first step, we computed the median of all images taken in the above period and subtracted it from the individual images in the sequence. The resulting images show Phaethon and the field stars superimposed on a much-reduced background consisting of coronal structures whose position and/or surface brightness changed appreciably during the five days of measurement.

We used aperture photometry to measure the data, paying particular attention to the selection of the optimum aperture size in view of the large pixels and the structured coronal background. Small apertures exclude a significant fraction of the light from astronomical images while large apertures suffer from excessive uncertainties due to noise and gradients in the sky background. The optimum aperture size is, in part, a compromise between these effects. Very small apertures cannot be used, even though the pixel size is very large (70 arcsec), because light from a point source occupies more than 1 pixel. Moreover, the distribution of light amongst pixels varies from image to image as the center of light moves with respect to the pixel grid.

Another factor limiting the use of very small apertures is image trailing, caused by the motion of the HI-1 field of view as it follows the Sun at a fixed solar elongation of 14°, and also by the Keplerian motion of Phaethon. The motion of the spacecraft tracks the CCD at about 150 arcsec hr⁻¹ east and 50 arcsec hr⁻¹ north. The motion of Phaethon relative to the fixed stars varied from 390 to 180 arcsec hr⁻¹ east and 100 arcsec hr⁻¹ south to 150 arcsec hr⁻¹ north, in the June 17–22 period. In the 1800 s required to accumulate a single HI-1 camera image, Phaethon moved relative to the CCD by at most 120 arcsec (1.7 pixels) in R.A. and 100 arcsec (1.4 pixels) in declination. By experimentation, we selected a photometry box 3 × 3 pixels (210 × 210 arcsec) in size for all objects. Sky subtraction was obtained using the median of the 16 contiguous pixels within a box having dimensions 5 × 5 pixels (350 × 350 arcsec). The use of the median provides some protection from noise fluctuations and from sky pixels contaminated by faint field stars, a significant problem given the large pixel scale. We rejected images in which visual inspection revealed the close proximity of a field star brighter than Phaethon. Sky boxes larger than those employed were found inappropriate because they tend to sample spatial gradients in the coronal surface brightness and so have reduced relevance to the background brightness in the pixels containing the object.

Photometric calibration was achieved using field stars in the images containing Phaethon. We used measurements of the field stars HD 92184, 91667, 88724, 88680, 87739, 87178, and 86340 (all spectral classes FGK) taken using the same photometry box sizes as used on Phaethon. The V magnitudes and V–R colors of these stars were taken from the SIMBAD online database and used to convert instrumental magnitudes in the STEREO data to apparent magnitudes in the HI-1 (approximately R) bandpass. We estimate that the uncertainty of this transformation is about ±0.1 mag. Repeated measurements of the field stars show that the photometry is precise, with 1σ ∼ ±0.01 to ±0.03 mag, depending on the brightness of the star (Figure 2).

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Figure 3 shows the sky-subtracted Phaethon photometry (black circles) as a function of time. A running-mean smoothed line has been drawn through the data to guide the eye, and the data are presented as counts (not magnitudes) so that negative excursions in noisy parts of the figure can be shown. Phaethon's light curve is divided into two portions. For DOY < 171.2 ± 0.2, the brightness is approximately constant with time. For DOY > 171.2, Phaethon shows a brightness enhancement, peaking at DOY = 171.8 ± 0.2 and declining steadily thereafter (Table 2). Also shown in the figure is the sky background determined at the location of Phaethon. The brightening in the sky background in the period 170.6 <DOY< 172.2, with a maximum near DOY = 171.3, is caused by a large-scale coronal streamer (see Figure 1 highlighted by the cone) across which Phaethon moved. While at first we thought that the background might be implicated in the apparent brightening of Phaethon, the background rises ∼0.5 day before Phaethon and is already subtracted from the plotted Phaethon photometry; it cannot be the cause of variations seen there.

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To further test the accuracy of the background removal procedure, we obtained photometric measurements of blank sky at several positions following Phaethon but offset from it by a fixed amount. The offsets were close enough to ensure a common coronal background. As expected, the measurements of blank sky are consistent with zero, albeit with small fluctuations caused by field stars and residual coronal structures in the CCD image. One such measurement is shown in Figure 4 (small crosses, again with a smoothed line overplotted) and the light curve of Phaethon is included for comparison. From this and Figure 3, we conclude that Phaethon's brightness increased after DOY = 171.2 ± 0.2 by a factor of two, and that this increase is not coincident with the coronal brightening but occurred ∼0.5 day after it. From this and other measurements of the sky having different offsets from Phaethon, we are confident that Phaethon is strongly detected and that large-scale variations in the brightness of Phaethon are not caused by imperfect subtraction of the coronal sky background.

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The surface brightness profiles computed before (76 images in the period UT 2009 June 17.90 to 20.25) and after (55 images in the period UT 2009 June 20.25 to 21.90) brightening are shown in Figure 5. Insets in Figure 5 show the image composites produced by shifting individual images of Phaethon to a common center, subtracting the coronal background and computing the median of the set. To make the profiles, we measured the data counts within a set of concentric annuli each 0.5 pixel (35 arcsec) wide and extending out to radius 7 pixels (490 arcsec), from the image centroid. The profiles, normalized to unity at the central pixel, are similar in shape, with the post-brightening profile being slightly wider than the pre-brightening profile (Figure 5). A fit to the profiles gives FWHM equal to 1.66 pixel and 1.83 pixel, for the pre- and post-brightening images. However, the very angular resolution of the data introduces considerable uncertainty into the image profile, depending on the precise location of the image centroid with respect to the pixel grid. Furthermore, fluctuations in the background affect the profile (the diagonal band in the post-brightening image is residual structure in the scattered light field significantly above local fluctuations due to noise on the sky). Within these pixellation and background uncertainties, we conclude that the measured profiles are consistent with the point-spread function in the images (i.e., with the profiles of field stars) and that there is no convincing evidence for either a coma or a tail on Phaethon.

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The brightening of Phaethon at α∼ 80° occurs too suddenly to be caused by phase-angle-dependent scattering effects. For example, in Figure 6 we show phase curves for the Moon (Lane & Irvine 1973; red line) and for the nucleus of comet P/Tempel 1 (Li et al. 2007; blue line). Both curves are corrected to the heliocentric and geocentric distances of Phaethon using the inverse square law so that they should provide a match to the Phaethon data if its surface is either Moon-like or comet-like. Evidently, the Lunar phase function does match the data for α<80° but the agreement is very poor at larger angles, even given the wide scatter in the Phaethon data. Other solid-body phase dependences, for example, those shown by C-type and S-type asteroids (Bowell et al. 1989) were tested and likewise fail to match the Phaethon data. Opposite to the asteroids, diffraction from dust particles with 2πa/λ> 1 causes comets to be forward scattering. However, the cometary phase function also provides an unconvincing match to the Phaethon photometry because the forward-scattering peak begins at α> 100° and increases steadily up to α = 180° (Kolokolova et al. 2004), whereas Phaethon's brightness decreases for α> 90° (Figure 6). For all these reasons, we discount the possibility that Phaethon's brightness variations are caused by simple phase-related effects.

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Nevertheless, given the large and changing phase angles of Phaethon (Table 1), it is appropriate to make an allowance for phase function effects in the interpretation of the photometry. We note that the Lunar phase function provides a reasonable match to the data for α< 80° (Figure 6). In Figure 7, we show Φ, the ratio of the signal from Phaethon to the Lunar phase curve, normalized such that the median value of Φ in the range 30° <α< 80° is unity. The figure shows that the brightness of Phaethon grows dramatically relative to that of the Moon for larger phase angles. For angles α> 110° the noise in the ratio becomes excessive, since Phaethon appears very faint. The right-hand axis in Figure 7 shows the scattering cross section, C (km²), computed from the brightness on the assumption that Φ = 1 corresponds to C = πr²_e = 20 km². This cross section only has meaning if the phase function of the scatterers is equal to the phase function of the nucleus. For example, if Phaethon ejected small, forward-scattering particles at α> 80°, then a smaller geometric cross section would be indicated than is given by the right-hand axis in Figure 7. We possess too little information to place useful constraints on any change in the phase function caused by the outburst event. Figure 7 shows, however, that the brightening of Phaethon corresponds to an increase in the scattering cross section at least comparable to, and perhaps many times larger than, the geometric cross section of the bare nucleus of Phaethon. For definiteness in the following discussion, we assume from Figure 7 that the brightening of Phaethon corresponds to an increase in the effective cross section by C = 100 km², while accepting that this value is uncertain (because of the unknown phase function) by a factor of a few.

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Table 1. Observational Geometry for HI-1A Data

UT Date	DOYa	R (AU)b	Δ (AU)c	α (deg)d	(deg)e
2009 Jun 17	168	0.197	1.131	24.7	4.9
2009 Jun 18	169	0.171	1.090	35.7	6.0
2009 Jun 19	170	0.151	1.042	51.9	7.1
2009 Jun 20	171	0.141	0.989	72.8	8.1
2009 Jun 21	172	0.143	0.932	95.6	8.6
2009 Jun 22	173	0.158	0.877	115.8	8.5

Notes. ^aDay of year. ^bHeliocentric distance in AU. ^cObject to spacecraft distance in AU. ^dPhase angle (Sun–Phaethon–spacecraft) (deg). ^eElongation angle (Sun–spacecraft–Phaethon) (deg).

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Table 2. Averaged Photometry

Phase Anglea	$\overline{R}$b	$\overline{\Delta }$c	Nd	mRe
30 ⩽α< 40	0.172	1.094	19	11.35 ± 0.19
40 ⩽α< 50	0.157	1.061	19	11.32 ± 0.11
50 ⩽α< 60	0.148	1.034	19	10.91 ± 0.11
60 ⩽α< 70	0.143	1.008	18	11.08 ± 0.10
70 ⩽α< 80	0.140	0.982	16	11.32 ± 0.14
80 ⩽α< 90	0.140	0.958	18	10.41 ± 0.08
90 ⩽α< 100	0.143	0.933	15	10.35 ± 0.07
100 ⩽α< 110	0.148	0.907	19	10.73 ± 0.06

Notes. ^aPhase angle range over which photometry was averaged. ^bAverage heliocentric distance, AU. ^cAverage geocentric distance, AU. ^dNumber of measurements used to compute the average. ^eAverage apparent magnitude and standard deviation.

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What is the origin of the brightening of Phaethon? The probability that we are witness to dust ejected from Phaethon by recent impact is vanishingly small. The fact that the main brightening of Phaethon occurs soon (∼1/2 day) after a brightening of the corona hints at a causal relationship, perhaps through solar wind excitation of surface materials. We first address this possibility using an energy argument. The increase in brightness of Phaethon by 2 mag (relative to the phase-darkened Lunar curve in Figure 6) corresponds to an increase in the photon flux density at Earth by f_λ = 2 × 10⁻¹⁶ W m⁻² Å⁻¹ (Drilling & Landolt 2000). Assuming isotropic emission, this corresponds to an emitted optical power P_o = 4 πΔ²f_λΔλ (W), where Δ is the geocentric distance and Δλ = 1400 Å is effective width of the spectral response function of the camera. The total energy of all the solar wind particles striking Phaethon, per second, is of order P_w = πr²_eΔVN₁kT, where ΔV is the speed of the wind sweeping past Phaethon, N₁ is the number density of particles in the wind, k is the Stefan–Boltzmann constant and T is the temperature of the particles in the wind. We set P_o = ζP_w, where ζ is the efficiency with which solar wind energy can be converted into optical photons. This gives

$$\begin{equation} N_1 = \frac{4 \Delta ^2 f_{\lambda } \Delta \lambda }{r_e^2 \zeta \Delta V k T} \end{equation} \tag{ 1 }$$

as the critical density needed for fluorescence to power the optical emission. We take Δ = 1 AU (Table 1), ΔV = 500 km s⁻¹, radius r_e = 2.5 km, ζ = 1, k = 1.38 × 10⁻²³ J K⁻¹, and T = 10⁶ K to find N₁ = 5 × 10¹⁴ m⁻³. The estimate is very crude in the sense that the emission is unlikely to be isotropic, emission outside the filter bandpass is neglected and all realistic processes have ζ< 1, tending to increase N₁ still further. The required density is far higher than the typical densities in streamers when scaled to 0.14 AU (N₁≪ 10¹⁰ m⁻³, Li et al. 1998) or CMEs (N₁< 10⁹ m⁻³; Reinard 2008) so that we can dismiss the possibility that brightening is stimulated by charged particle impact. By a similar argument we reject fluorescent emission caused by solar X-ray or ultraviolet photons.

We next consider the possibility that the brightening results from scattering from dust particles ejected from Phaethon and having a combined cross section C = 100 km², as noted above. The mass corresponding to C is dependent on the mean radius of the scatterers, $\overline{a}$, and given by $M \sim \rho \overline{a} C$ (kg). Substituting ρ = 2500 kg m⁻³ and C = 10⁸ m² we find M = 2.5 × 10⁸a₁ kg, where a₁ is the particle radius expressed in millimeters. For example, if a₁ = 1 (the nominal Geminid meteoroid size), the 2.5 × 10⁸ kg mass is equivalent to the loss of a monolayer from the surface. The Geminid stream has a mass M_s = 10¹²–10¹³ kg (Hughes & McBride 1989; Jenniskens 1994) and, therefore, the mass of dust implied by our photometry with a₁ = 1 is only 10⁻⁴M_s. (If the emitted particles are much smaller than 1 mm, their mass would be correspondingly reduced.) The stream age is ∼10³ yr (Gustafson 1989; Ryabova 2007). We conclude that mass-loss events like the one observed cannot account for the Geminid stream mass unless they are frequent (∼10 events per orbit). Given the limited nature of published constraints on mass loss from Phaethon, it is difficult to rule out mass-loss events with this frequency. For example, mass loss could occur only close to perihelion, when ground-based observations are practically impossible, rendering all previously published limits irrelevant.

Sustained loss of particles from a small solar system body requires both a source of particles and a means to eject them from the body. Without a source, ejection processes will deplete the dust to a bare surface. Without an ejection mechanism, the surface will clog with dust, impeding the production. On comets, dust particles are exposed at the surface by the sublimation of entrapping ice, while drag forces from the sublimated gas deplete dust grains by launching them to the interplanetary medium. Could this process operate on Phaethon?

High surface temperatures are expected on the day side of Phaethon as a result of its small perihelion distance. The peak temperatures cannot be exactly calculated because they depend on many unknown or poorly known quantities (including the obliquity and orientation of the nucleus spin vector relative to the line of apsides, and the thermal parameters of the surface). We assess the approximate range of perihelion dayside temperatures as follows. An isothermal, spherical blackbody in equilibrium with sunlight would have T = 746 K at Phaethon's 0.14 AU perihelion distance. Since real objects sustain a day–night temperature contrast (i.e., they are not isothermal), this may be taken as an effective lower limit to the dayside temperature at perihelion. An effective upper limit is given by the subsolar temperature on a non-rotating body (or on one having a spin vector aligned with the Sun), which could reach T = 1050 K at q = 0.14 AU. The peak dayside temperatures are thus contained in the range 746 <T< 1050 K, in agreement with more complicated thermal models (Ohtsuka et al. 2009). For comparison, dayside aphelion temperatures will be smaller by the factor ((1+e)/(1−e))^1/2∼ 4, corresponding to 180 K and 256 K for the low and high temperature limits, respectively. Thus, any element of the surface of Phaethon must experience large temperature variations, δT∼ 500 K, in the course of an orbit. Comparably large variations may be experienced even in the course of a nucleus rotation.

Such high temperatures preclude the existence of water ice near the surface of Phaethon and thus eliminate comet-like water ice sublimation as a driving mechanism. Could the activity instead be caused by the sublimation of ice buried at depth? The timescale for the conduction of heat from the surface to the core of a spherical body of radius r_e is roughly τ_c = r²_e/κ, where κ is the thermal diffusivity of the material. Common dielectrics have κ∼ 10⁻⁶ m² s⁻¹. Substituting r_e = 2.5 km, we obtain τ_c = 2 × 10⁵ yr. The dynamical lifetime of Phaethon in its present orbit is unknown and can only be guessed statistically; the median dynamical lifetime of bodies moving in similar orbits is τ_d∼ 26 Myr (de León et al. 2010). With τ_d ≫ τ_c, we can assume that heat deposited on the surface of Phaethon has conducted to the deep interior. The core temperature calculated as in Jewitt & Hsieh (2006), T_c = 300 K, is too high for water ice to survive in the core. Accordingly, we consider it unlikely that ongoing activity in Phaethon could be driven by the sublimation of deeply buried water ice; Phaethon is simply too hot, even in its core. A separate argument against deeply rooted activity of any kind is the observation that the activity in Phaethon coincides with perihelion. This coincidence can only be explained if the source of the activity lies less than a few thermal skin depths (a few × 0.1 m) beneath the physical surface, otherwise the activity source would be thermally (and temporally) decoupled from the heat of the Sun.

While ice sublimation is unlikely, other heat-triggered processes might operate at the extreme perihelion temperatures on Phaethon. The surface is not hot enough for rock itself to significantly sublimate, but the perihelion temperatures exceed those needed to thermally decompose some rocks. For example, experiments with the hydrated mineral serpentine show structural changes on laboratory timescales (hours and days) beginning at T∼ 600 K (Akai 1992; Nozaki et al. 2006; Nakato et al. 2008). These changes become extensive by 900 K, comparable to the peak temperatures anticipated on Phaethon. Chemically bound water is progressively lost as the temperature rises, and there are associated changes in the crystal structure and general shrinkage of the rock leading to internal stress and cracking. The loss of water from dehydration could provide a source of gas, although it is not obvious that gas-drag forces produced this way would be sufficient to eject solid particles against nucleus gravity. Dehydration of phyllosilicates will result in shrinkage and cracking, with associated dust production, just as occurs in sun-baked mud flats on Earth.

Based on a numerical model, Bottke et al. (2002) found that Phaethon has a 20% likelihood of originating in the central main belt (2.5 <a< 2.8 AU) and an 80% likelihood of originating in the inner main belt (a< 2.5 AU). A specific association with asteroid (2) Pallas (a = 2.77 AU) has been recently proposed (de León et al. 2010). Asteroids in this region of the belt include a large proportion of volatile-rich C types that may contain phyllosilicates (clays) or other hydrated minerals. Indeed, these materials have been specifically suggested as components of Phaethon (Licandro et al. 2007; Ohtsuka et al. 2009). Thus, there exist good reasons to think that the surface of Phaethon might be susceptible to the high perihelion temperatures by cracking and the production of dust through thermal decomposition.

Another potential producer of particles is cracking resulting from stresses induced by differential thermal expansion (Oliva et al. 1971; Lauriello 1974). It is well known from everyday experience that thermal fracture can be induced by strong temperature gradients (imposed by heating a body on a timescale shorter than its thermal conduction time, as when hot water is poured into, and cracks, a cold glass). On Phaethon, strong diurnal temperature variations (δT∼ 500 K) are expected in association with the ∼3.6 hr rotation period (Krugly et al. 2002). More generally, thermal fracture occurs even when the temperature changes very slowly, because rocks have a granular structure and the expansivities of grain and inter-grain material are typically different. Thermal cracking has been studied experimentally using sound waves produced when cracks open around mineral grains in heated rocks (Chen & Wang 1980). Experiments in the laboratory show that thermal fracture proceeds down to the grain size in rocks. In terrestrial rocks and meteorites, the natural grain size is commonly near 1 mm, providing an approximate but intriguing match to the characteristic sizes of Geminid meteoroids.

In the elastic regime, the stress on a material caused by thermal expansion is just

$$\begin{equation} S = \alpha Y \delta T, \end{equation} \tag{ 2 }$$

where Y is Young's Modulus, α is the thermal expansivity, and αδT is the strain resulting from thermal expansion from a temperature change δT. Values Y = (10–100) × 10⁹ N m⁻² are typical for rock (Pariseau 2006), while the thermal expansivities of common rocks are within a factor of a few of α = 10⁻⁵ K⁻¹ (Lauriello 1974; Richter & Simmons 1974), which we take as our best-guess estimate of the expansivity of Phaethon. Substituting δT = 500 K (from the previous section) in Equation (2) we obtain thermal stress S = (5–50) × 10⁷ N m⁻² (500–5000 bars) corresponding to the temperature changes experienced on Phaethon as it moves around its orbit. Depending on the thermal diffusivity, comparable temperature variations and thermal stresses may also be experienced on a much shorter timescale as Phaethon rotates. The thermal stress is larger than the ∼100 bar yield strength characteristic of rocks in tension (Lauriello 1974). Thus, thermal cracking should be expected and could be an important mechanism for producing small particles on Phaethon.

The highest temperatures and thermal stresses will be concentrated near the surface, in a layer having thickness comparable to the diurnal thermal skin depth, d ∼ (κP)^1/2, where κ (m² s⁻¹) is the thermal diffusivity and P the rotation period of Phaethon. Substituting κ = 10⁻⁶ m² s⁻¹ and P = 3.6 hr (Krugly et al. 2002), we estimate d = 0.1 m. Representing Phaethon as a sphere with effective radius r_e = 2.5 km, the mass of material accessible to thermal fracture at any time is ΔM = 4πr²_eρd, where ρ is the density, here taken to be ρ = 2500 kg m⁻³. Substituting, we find that strong thermal gradients can affect ΔM∼ 10¹⁰ kg of material on Phaethon. This is larger than the inferred dust mass but two to three orders of magnitude too small to be consistent with the Geminid stream mass. However, thermal fracture will continue so long as fresh material is exposed and the resulting debris cleared away. As with dust produced by dehydration of hydrated minerals, the rate of progression depends on how quickly particulate material on Phaethon can be ejected against self-gravity.

As noted above, many dust production mechanisms become self-limiting if there is no process by which freshly produced particles can be removed from the surface. Here, we briefly consider three processes that might, singly or in combination, lead to the escape of dust from Phaethon.

Thermal fracture itself is expected to launch freshly produced particles with a non-zero velocity. To see this, we consider a flat plate of cross section A and thickness ℓ, with a temperature difference impressed between the two sides of the plate, δT. The thermal expansion distance between the two flat sides is δℓ = αℓδT. The force exerted by the expansion is F = Y(δℓ/ℓ)A. The approximate work done by the expansion is the product of these two quantities, W ∼ Y(δℓ/ℓ)(δℓ)A. Substituting, we obtain W = Yα²δT²ℓA. Suppose that some fraction, η, of this thermal strain energy can be converted into kinetic energy of the fractured material, E ∼ ρℓAV². Setting E = ηW, obtain

$$\begin{equation} V \sim \alpha \delta T \sqrt{\frac{\eta Y}{\rho }} \end{equation} \tag{ 3 }$$

as a crude estimate of the characteristic speeds of particles produced by fracture. We adopt the same values for α, δT, Y, and ρ as in Section 3.4. The magnitude of the energy conversion efficiency, η, is unknown and reflects a competition between energy used to fracture the material, energy radiated away as vibrations, and thermodynamic and other effects. We assume η = 1 to give an upper bound to the particle velocity. Then, Equation (3) gives V∼ 10–30 m s⁻¹ as the characteristic particle speed. This is an order of magnitude larger than the gravitational escape speed from Phaethon. Therefore, thermal fracture should be capable of launching particles above the escape speed if the efficiencies η> a few percent. While η is unknown, it would not be surprising to find that a significant fraction of the particles produced by thermal fracture can be launched directly from the surface into interplanetary space.

Electrostatic forces offer another mechanism to remove dust. Evidence from the Moon shows that dust particles are periodically lifted from the surface at speeds ∼1 m s⁻¹ by steep electric field gradients that develop in the terminator regions (de & Criswell 1977; Colwell et al. 2007). Strong electric field gradients should be present on any low-conductivity surfaces exposed to the solar radiation and the solar wind. Indeed, electrostatic levitation has been implicated in, for example, mobilizing dust and forming "ponds" on 16 km mean diameter asteroid (433) Eros (Colwell et al. 2005). Even neglecting centripetal effects from its rapid rotation, the gravity on Phaethon is ∼3 times smaller than on Eros and ∼700 times smaller than on the Moon; the dynamical effects of electrostatic charging should be proportionally greater. Dust speeds sufficient to produce only ballistic jumps on the Moon may be sufficient to produce escape from the gravitational field of a body as small as Phaethon. The small heliocentric distance of Phaethon at perihelion will not enhance the magnitude of the electrostatic forces (which depend ultimately on the photoelectric effect, not the distance) but the photocurrent rates should be ∼50 times larger than on the Moon because of Phaethon's smaller distance.

Finally, the small perihelion distance of Phaethon can help to sweep dust particles from its surface through the action of radiation pressure. To see this, we assume that the nucleus can be approximated by an ellipsoid with axes a > b = c, in rotation about the c axis. Measurements of the rotational light curve define the effective cross section, πr²_e, and set a limit to the ratio of the equatorial axes, f = a/b. The data give r_e = 2.5 km and f = 1.4. Setting πr²_e = πab we may write a = r_ef^1/2 and b = r_ef^−1/2. The volume of the ellipsoid, V = 4/3πab² can then be written in terms of the two measurable quantities r_e and f as V = 4/3πr³_ef^−1/2. Rotation about the c-axis with period, P, gives rise to a centripetal acceleration which is largest at the apex of the figure. There, the net acceleration toward the center may be written as

$$\begin{equation} g = \frac{4 \pi G \rho r_e}{3 f^{3/2}} - \frac{4 \pi ^2 r_e f^{1/2}}{P^2}, \end{equation} \tag{ 4 }$$

where the first term is the gravitational attraction and the second is the centripetal acceleration. Equation (4) may be compared with the acceleration due to radiation pressure,

$$\begin{equation} g_{pr} = \beta g_{\odot } / R_{au}^2, \end{equation} \tag{ 5 }$$

where g_☉ = 0.006 m s⁻² is the gravitational acceleration to the Sun at 1 AU, and R_au is the heliocentric distance expressed in AU. The dimensionless quantity β is principally a function of particle size, a, but also depends upon the grain shape, composition, and porosity. As a useful first approximation, we take β ∼ 1/a, where a is expressed in microns. Comparing Equation (4) to Equation (5) gives the critical value of radiation pressure efficiency, β_c = a⁻¹_c, above which a particle will be unbound to the nucleus. We obtain

$$\begin{equation} \frac{1}{a_c} = \frac{4 \pi R_{au}^2 f^{1/2} r_e}{3 g_{\odot }}\left[\frac{G \rho }{f^2} - \frac{3 \pi }{P^2}\right], \end{equation} \tag{ 6 }$$

in which a_c is expressed in microns.

Equation (6) is presented graphically in Figure 8. Substituting the nominal values R_au = 0.14, f = 1.4, r_e = 2.5 km, ρ = 2500 kg m⁻³, and P = 3.6 hr into Equation (6) we obtain a_c = 10³μm = 1 mm. Particles smaller than a_c experience radiation pressure forces larger than their weight and can be stripped from the nucleus if they are briefly detached from the surface and the net force vector acting upon them aims away from the nucleus. The latter condition is naturally satisfied at the terminator, where electric field gradient forces are the largest (Colwell et al. 2007). Equation (6) is approximate in the sense that we have assumed a prolate nucleus (b = c) whereas a triaxial approximation may be more accurate. We have also assumed the density, ρ, and that f measured from the light curves represents the true axis ratio whereas in fact it is only a lower limit to the ratio because of the effects of projection. Given these many approximations and unknowns, we cannot prove that radiation pressure sweeping, in combination with electrostatic ejection and the expected fracture launch speeds, is responsible for cleaning particles from Phaethon's surface. However, these effects combined do constitute a plausible path by which particles can be removed from Phaethon. The small size, r_e, small perihelion distance, R_au, and short rotation period, P, all maximize a_c relative to larger, more distant and more slowly rotating asteroids.

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To conclude, we speculate that dust production and loss rates are both high as consequences of the very high surface temperatures on Phaethon at perihelion. The comet-like activity does not imply that sublimating ice is responsible in this object. Instead, it is appropriate to think of Phaethon as a rock comet in which thermal fracture, dehydration cracking, radiation pressure sweeping, and electrostatic effects may all play roles in producing and removing particles from the surface. Several of these processes decline precipitously with increasing distance from the Sun. For example, if displaced from q = 0.14 AU to the main belt (3 AU), thermal fracture would become unimportant and the critical dust size for radiation pressure sweeping on Phaethon would decrease by nearly a factor of 10³, from ∼1 mm to ∼1 μm (Figure 8). Of the 19 known asteroids having perihelia smaller than Phaethon's, few have been searched for evidence of mass loss or, indeed, subjected to any meaningful physical study. An important observational next-step will be to examine these objects near perihelion in search of comet-like activity analogous to that observed in Phaethon. Unfortunately, none is bright enough to be observed using STEREO, so that relevant observations will be difficult to obtain. It will also be important to examine Phaethon at future perihelion passages, to determine the frequency of mass-loss events and so to decide whether the meteoroid stream is in steady state.