Recurrent Perihelion Activity in (3200) Phaethon
RRevised 2013 April 1
Recurrent Perihelion Activity in (3200) Phaethon
Jing Li and David Jewitt , (1) Department of Earth and Space Sciences, University of California at Los Angeles(2) Department of Physics and Astronomy, University of California at Los Angeles [email protected] ABSTRACT
We present a study of planet-crossing asteroid (3200) Phaethon at three successiveperihelia in 2009, 2010 and 2012, using the NASA STEREO spacecraft. Phaethon isclearly detected in 2009 and 2012, but not in 2010. In both former years, Phaethonbrightened unexpectedly by ∼ ∼ ∼ > ,to explain the brightening. Neither can ice survive on this body, ruling out comet-likesublimation. Our preferred explanation is that brightening occurs as a result of dustproduced and ejected from Phaethon, perhaps by thermal fracture and/or thermal de-composition of surface minerals when near perihelion. A contribution from promptemission by oxygen released by desiccation of surface minerals cannot be excluded. Weinfer an ejected mass of order 4 × a mm kg per outburst, where a mm is the mean dustradius in millimeters. For plausible dust radii, this mass is small compared to the esti-mated mass of Phaethon ( ∼ × kg) and to the mass of the Geminid stream (10 kgto 10 kg) with which Phaethon is dynamically associated. Perihelion mass-loss eventslike those observed in 2009 and 2012 contribute to, but do not necessarily account forthe Geminids stream mass. Subject headings: asteroids, comets
1. Introduction
Object (3200) Phaethon (formerly 1983 TB) is dynamically associated with the Geminid me-teor stream, suggesting that it is the long-sought parent of this stream (Whipple 1983). Unlike a r X i v : . [ a s t r o - ph . E P ] A p r ◦ ), with anasteroid-like Tisserand parameter with respect to Jupiter (Kresak 1980) of T J = 4.5. Phaethon isabout 5 km in diameter (V-band geometric albedo = 0.17, Veeder et al. 1984) and appears to bedynamically associated not just with the Geminids, but also with at least two smaller (kilometer-scale) asteroids, namely 2005 UD (Ohtsuka et al. 2006; Jewitt & Hsieh 2006; Kinoshita et al. 2007)and 1999 YC (Kasuga & Jewitt 2008; Ohtsuka et al. 2008). Together, these objects constitute theso-called “Phaethon-Geminid Complex” (PGC) presumably formed by the disruption of a largerparent body. Physical observations show that the PGC members share similar neutral-blue colorsthat are relatively rare in the main-belt population and suggesting a common composition. ThePGC objects all possess small perihelion distances (0.14 AU in the case of Phaethon) resulting inhigh surface temperatures.Physical observations of Phaethon when far from perihelion have consistently failed to show ev-idence for on-going mass loss, either in gas or in scattered continuum from entrained dust (Cochran& Barker 1984; Chamberlin et al. 1996; Hsieh & Jewitt 2005; Wiegert et al. 2008). Neither do theassociated bodies 2005 UD and 1999 YC (references above) show evidence for on-going mass loss.However, observations of Phaethon at very small solar elongations using the STEREO spacecraftrevealed anomalous photometric behavior near perihelion in 2009 (Jewitt & Li 2010, hereafter,Paper I). Specifically, Phaethon was observed to brighten with increasing phase angle (near ∼ ◦ )when near perihelion, a behavior inconsistent with scattering from any macroscopic solid bodyand opposite to the phase functions of known solar system objects (Lane & Irvine 1973; Li et al.2007a,b, 2009). We interpreted the brightening as caused by an increase in the scattering cross-section following the ejection of dust from Phaethon. The required mass of dust is ∼ × a mm kg, where a mm is the unmeasured size of the dust grains, expressed in millimeters (Paper I). Theseobservations are potentially important both for showing that Phaethon is an active source of matterfor the Geminid stream, and for illuminating physical processes induced on bodies when close tothe Sun.In this paper, we report new STEREO observations of Phaethon in 2010 and 2012 combinedwith a re-analysis of measurements from 2009 earlier reported in Paper I. The principal question weseek to address is whether the anomalous brightening detected in 2009 is recurrent at subsequentperihelia.
2. Observations
The present observations were made with the Heliospheric Imagers (HI) which are part ofthe Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI) package (Howard etal. 2008; Eyles et al. 2009) onboard the STEREO spacecraft. The HI instruments consist of twowide-angle visible-light imagers, HI-1 and HI-2, with field centers offset from the solar center by14.0 ◦ and 53.7 ◦ , and fields of view 20 ◦ and 70 ◦ , respectively. The HI detectors are charge-coupled 3 –devices (CCDs) with 2048 × × (cid:48)(cid:48) for HI-1 and 4 (cid:48) for HI-2. The very large pixels subtendsolid angles 10 to 10 times those of pixels commonly used on night-time telescopes, but this is anadvantage for the intended detection of large scale, diffuse structures in the solar corona.We used the standard HI-1 camera Level 1 images in our study. The data are publicly availablevia the UK Solar System Data Center (UKSSDC) web site. Multiple short-exposure images aretaken before a 1024 × ∼ (cid:48)(cid:48) hr − ) is comparable to the binned pixel size. The quantum efficiency ofthe CCD camera and the absolute transmission efficiencies of the optics are nearly constant acrossthe 6300 to 7300 ˚A wavelength range (Eyles et al. 2009). The short orbit period (1.43 yr) offers frequent opportunities to observe Phaethon at perihelion.The orbit period of STEREO A is ∼
346 days so that three STEREO orbits (2.84 yr) are almostexactly equal to twice the orbit period of Phaethon (2.86 yr). Therefore, observations in 2009(perihelion June 20) and 2012 (perihelion May 02) share similar perspectives viewed from STEREOA. We obtained the celestial coordinates of Phaethon from NASA’s HORIZONS Web-Interfaceand transformed them to pixel coordinates on the HI cameras in the “AZP” Zenithal projection(Calabretta & Greisen 2002; Thompson 2006, Thompson, private communication 2012). The sky-plane trajectories of Phaethon are shown in Fig. (1). As expected, they follow similar pathsin 2009 and 2012. The angular speed of Phaethon in the images is the result of the combinedspacecraft parallactic motion and object Keplerian motion. Viewed from STEREO A, Phaethonmoved relative to the field center at a speed varying between 0 (cid:48)(cid:48) hr − and 175 (cid:48)(cid:48) hr − from east towest, and at a roughly constant speed 180 (cid:48)(cid:48) hr − (in 2009) and 200 (cid:48)(cid:48) hr − (in 2012) from south tonorth. Phaethon was readily apparent even in a cursory visual examination of the 2009 and 2012data. The observing geometry in the intervening orbit in 2010 is completely different from that in2009 and 2012. Phaethon was in the field of view of STEREO A from November 11-19 and fromDecember 15-25 (upper panel in Figure 2). It left the HI-1 field of view six days before its perihelion 4 –on November 25 and only re-entered the field about twenty days after perihelion in the second timeperiod. Phaethon was not detected in either observing window. During the first time interval, thephase angle was above 120 ◦ making an unfavorable condition for detection. During the second timeinterval, the phase angle was a modest 10 ◦ , but the heliocentric distance had increased to ∼ ∼
15, making Phaethon too faint tobe detected in STEREO data.Phaethon stayed within the field of the STEREO B HI-1 camera for almost three monthsduring its 2010 return (bottom panel in Fig. 2) but still contrived to leave the field of view six daysbefore perihelion. Between mid-August and mid-November, the phase angle varied from ∼ ◦ to amaximum 30 ◦ , while the heliocentric distance decreased from 1.8 to 0.3 AU. The apparent visualmagnitude calculated assuming solid body reflection decreased from 18.5 to 12.5. At its predictedbrightest, Phaethon should have been marginally within reach of STEREO HI-1 camera detection,but in practice the object was not detected.Presumably as a result of these different observational circumstances, the anomalous bright-ening detected in 2009 and 2012 was not found in 2010. Activity specifically at perihelion couldnot be detected because Phaethon was outside the fields of both the A and B cameras when atperihelion. We conclude that differences in the observational geometry prohibited the detection ofPhaethon and its activity at perihelion in 2010. To supplement the near-Sun photometry from STEREO, we also observed Phaethon when farfrom the Sun using the Keck 10-m telescope on UT 2012 October 14.5. We used the Low ResolutionImaging Spectrometer (LRIS, see Oke et al. 1995) and a broadband R filter (central wavelength λ c = 6417˚A, full-width at half maximum, FWHM = 1185˚A) in seeing of FWHM = 1.0 (cid:48)(cid:48) . The LRISpixel scale is 0.135 (cid:48)(cid:48) pixel − . Flat field images were obtained using an illuminated patch on theinside of the Keck dome. The data were photometrically calibrated using solar-colored standardstars from Landolt (1992). We measured R = 17.25 ± α = 18.1 ◦ . This measurement refers to anunknown rotational phase of Phaethon.
3. Photometry
With pixel sizes 70 (cid:48)(cid:48) , images of HI-1 are sensitive to the diffuse background coronal emissionbut under-sample the point-spread function of the telescope, and lead to frequent confusion withbackground sources as Phaethon moves across the sky. In these data, the relatively uniformlydistributed background corona overwhelms both field stars and Phaethon. To suppress the coronalbackground, we grouped images over timespans from a few hours to a few days. Within each 5 –group, the background corona was relatively constant, and was calculated using the minimum filtertechnique (an IDL code “min filter” in the SolarSoft IDL package was used) and then subtractedfrom the images in the group. The coronal filtering leaves rapidly varying background structuresvisible in the images, but they are muted in intensity compared to those present in the raw data.Sample corona-subtracted images are shown in Figures (1) and (2).During the 30 minutes required to accumulate a single HI-1 image, Phaethon moved a maxi-mum of 1.8 pixels (126 (cid:48)(cid:48) ) relative to the CCD. We experimented with different photometry aperturesin order to examine the effects of trailing and background subtraction. In particular, we checkedthat the measured brightness variations are not related to the angular speed of Phaethon andtherefore to trailing of the images. Very small apertures are sensitive to the trailing, while verylarge ones achieve poor signal-to-noise ratios owing to uncertainty in the large background signalfrom the corona. By trial and error, we chose a photometry box of 5 × (cid:48)(cid:48) × (cid:48)(cid:48) ). Thesky background was obtained from the median count in 56 sky pixels defined by a box 9 × (cid:48)(cid:48) × (cid:48)(cid:48) ) wide, surrounding the 5 × ◦ to 130 ◦ in 2009, and from 32 ◦ to 105 ◦ in 2012. In both years, Phaethon shows an apparentbrightness surge by a factor of ∼ ±
10% due to passing stars (blue circles in Figure 3). For these reasons, we areconfident that the brightening in 2009 and 2012 is real and associated with Phaethon, not causedby background object contamination or sky subtraction errors.To photometrically calibrate the measurements of Figure (3), we chose field stars near theprojected path of Phaethon across the CCD. Field stars were chosen to be within 20 pixels ( ∼ . (cid:48) )of Phaethon, with magnitudes 8 ≤ V ≤
10 and of spectral types FGK. Eleven standard stars in2009, and fifteen stars in 2012 were available for the Phaethon flux calibration (see Table 2).Reference star photometry was obtained in the same manner as for Phaethon. The 6300 ˚A to 6 –7300 ˚A passband of the STEREO camera is close to astronomical R-band. However, Phaethon isnearly neutral and the reference stars were selected to be similar in color to the Sun. Therefore,our measurements of the brightness ratio are equivalent to V-band measurements save for a small,color-correction offset (cid:46)
4. Discussion
To correct for the variations in the heliocentric and Phaethon-STEREO distances, R au and∆ au , respectively (both expressed in AU), we use the inverse-square law, written m (1 , , α ) = m obs − R au ∆ au ) . (1)Here m obs is the apparent magnitude and m (1 , , α ) is the resulting magnitude which would beobserved from R au = ∆ au = 1 and phase angle α . Equation (1) is plotted against α in Figure(5). Since the individual measurements are very scattered (Figure 4), the plotted curves show therunning means of 10% of the data. Also plotted in the Figure is m (1 , , . ◦ ) from our Keckobservation on UT 2012 October 14, and m (1 , , . ◦ ) from Hsieh & Jewitt (2005). The lattertwo measurements are plotted with error bars of ± m (1 , , α )upto α ∼ ◦ . Specifically, Phaethon fades by ∼ α = 18.1 ◦ to α = 60 ◦ , giving alinear phase coefficient, ∼ − , that is unremarkable when compared with otherasteroids. At larger α , Phaethon in Figure (5) shows sudden distance-corrected brightening startingat α = 80 ◦ in 2009 and α = 65 ◦ in 2012. These phase angles correspond to perihelion in each year,while maximum brightness is reached at α = 100 ◦ in 2009 and α = 80 ◦ in 2012, about 0.5 daylater. Phaethon fades to invisibility at larger phase angles. As noted in Paper I, this brightnessvariation is unexpected for a solid body viewed in scattered light. For such an object, the brightness 7 –decreases monotonically as the phase angle increases both because a progressively smaller fractionof the surface is illuminated and because the scattering efficiency of the surface decreases as thescattering angle grows.This is emphasized in Figure (5), which compares Phaethon with the scaled phase function ofthe Moon, taken from Lane & Irvine (1973). The plotted lunar curve is an average of the phasefunctions at wavelengths 6264 ˚A and 7297 ˚A in Table V of their paper. We show the differencebetween m (1 , , α ) and the scaled magnitude of the Moon in Figure (6). The difference plot showsthat Phaethon’s phase dependence is Moon-like before the onset of the anomalous brightening eventin each year. While the apparent brightness of Phaethon (Figure 4) increased by slightly more thana magnitude near α = 80 ◦ to 100 ◦ , the brightening relative to the phase-darkened nucleus (Figure5) is a much larger ∼ ∼
6. The figure also shows that theapparent, post-peak fading in Figure (4) is consistent with the dimming expected from the phasefunction, not necessarily to loss of the scattering cross-section.
We first consider and reject several mechanisms that might be implicated in the observedanomalous brightening at perihelion.The intrinsic brightening is too large (2 mag.) and too long-lived (2 days) to be plausibly at-tributed to rotational variation of the projected cross-section of the aspherical nucleus of Phaethon.Lightcurve observations show a variation ≤ ∼ ≥ yr), leaving no explanation for why the brightening is correlated withperihelion.The solar wind kinetic energy flux onto the surface of Phaethon is E KE = ρv / − ),where ρ is the solar wind mass density, and v is the solar wind speed. At the Earth’s orbit, the solarwind number density is about 10 m − . Scaled by the inverse-square law to perihelion at 0.14 AU,the density is about N =5 × m − . The solar wind speed varies with time and radius, but is oforder v = 500 km/s. Substituting ρ = µm H N , where µ = 1 (for protons), and m H = 1 . × − E KE ∼ − . This is tiny compared with the solar radiation flux at perihelion, F (cid:12) /R AU = 70 ,
000 W m − . Consequently, we conclude that the solar wind is a negligible source ofenergy and cannot account for the anomalous brightening by impact fluorescence.Could part or all of the measured excess optical brightness be thermal emission resulting fromPhaethon’s high surface temperatures when at perihelion? We obtain a rough lower limit to thetemperature by considering the case of an isothermal, spherical blackbody in equilibrium withsunlight, namely T BB = 278 R − / au . At perihelion, R au = 0.14, we find T BB = 743 K. A practicalupper limit is given by the sub-solar temperature on a non-rotating body (or a rotating one whoserotation axis points at the Sun), namely T SS = √ T BB , giving T SS = 1050 K for Phaethon atperihelion. This temperature range is in good agreement with independent estimates ( ∼ ≤ T ≤ T SS , the maximum possible surface equilibrium temperature. Under thisassumption, the flux density in the STEREO bandpass is calculated from f λ = πr n ∆ (cid:34) (cid:82) λ λ B λ ( T ) dλ ∆ λ (cid:35) (2)in which B λ ( T ) is the Planck function evaluated at T = T SS , r n = 2.5 km is the effective circularradius of Phaethon, ∆ is the Phaethon-to-observer distance, the integration is taken over the filtertransmission from λ = 6300 ˚A to λ = 7300 ˚A, and ∆ λ = λ − λ .For comparison with Equation (2), we convert the apparent magnitudes, m obs , into flux den-sities, f oλ , using f oλ = 3 . × − (9+ m obs / . [erg s − cm − ˚A − ] (Drilling and Landolt 2000). Theresults are shown in Figure (7), where solutions to Equation (2) (blue) are compared with theobservations (red). Evidently, thermal emission even at the peak sub-solar temperature is orders ofmagnitude too small to account for the anomalous brightening of Phaethon observed in our data.To see this a different way, we substituted the measured peak flux densities, f oλ ∼ − erg s − cm − ˚A − (c.f. Figure 7), into the left-hand side of Equation (2) and solved for thetemperature. We find that values T ∼ D stateby photodissociation of a parent molecule, for example water (Festou and Feldman 1981). As 9 –noted earlier, although water ice cannot survive, water might be bound within hydrated mineralsin Phaethon and released by desiccation. In the case of water, the photodissociation timescale atperihelion is about half an hour, so that any water molecules released from Phaethon would bedestroyed within a single pixel of the STEREO camera. We estimate the flux density produced byprompt emission in oxygen, averaged over the passband of the camera, from f [ OI ] λ = αQhc π ∆ λ ∆ λ (3)in which α ∼
10% is the fraction of water dissociations leaving oxygen in the excited D state, Q is the production rate of water molecules, h is Planck’s constant, c is the speed of light, ∆ isthe Phaethon to STEREO distance, λ is the wavelength and ∆ λ is the filter FWHM, expressedin Angstroms. Setting f [ OI ] λ = f oλ , we find that a water production rate Q ∼ s − (3 × kgs − ) would be needed to account for the measured excess flux density of Phaethon. Although therequired rate of production (which is similar to that of comet 1P/Halley at perihelion) seems high,we cannot rigorously rule out the possibility that some fraction of the excess perihelion emissionis caused by prompt emission from oxygen. However, the observation that the fading of Phaethonafter peak brightness follows the phase function of a solid object (Figure 5) suggests that gas is notthe dominant cause of the anomalous brightness. The remaining alternative is that Phaethon has ejected dust particles with a combined cross-section larger than that of the solid nucleus, as earlier concluded in Paper I. In this scenario, the risein brightness in Figure (3) then corresponds to the ejection of dust, while the subsequent declinein brightness is naturally explained as fading owing to the ever-growing phase dimming, perhapsaided by grain sublimation or disintegration. The natural test of this hypothesis would be to searchfor coma scattered by the ejected dust. Unfortunately, as also noted in Paper I, the limited angularresolution and high background surface brightness in the STEREO data make the detection ofresolved coma impossible.A temperature-controlled mechanism for the ejection of dust is strongly suggested; the activityis observed at the highest (perihelion) temperatures, and is absent in observations of Phaethon takenat substantially larger heliocentric distances and lower temperatures. The reflection spectrumof Phaethon has been interpreted in terms of thermally modified hydrated minerals (Licandroet al. 2007; Ohtsuka et al. 2009; de Le´on et al. 2010) while the depletion of sodium in someGeminids provides independent evidence for thermal alteration (Kasuga et al. 2006). The periheliontemperatures on Phaethon exceed those needed to break-down phyllosilicates (Akai 1992), and aresufficient to induce thermal fracture (Paper I, Jewitt 2012). In this sense, thermal disintegrationand fracture are plausible sources of the anomalous brightening and Phaethon may be accuratelylabeled a “rock comet” (Paper I). An additional requirement is that dust must be cleared from the 10 –surface in order for these processes to operate. A regolith of fine particles built up in previous orbitswill inhibit thermal fracture, since small grains are unable to sustain large temperature differences.Likewise, surface materials dehydrated by baking in previous perihelion passages must be clearedaway from the surface in order for dehydration cracking to remain a persistent dust source.In both 2009 and 2012, the apparent brightness increased by ∼ ∼
6. Given that the cross-section of the nucleus of Phaethon is C n = πr n ∼
20 km , the cross-section of added dust is then C = 100 km , in both years. The mass in spherical particles of mean radius a having cross-section, C , is M d ∼ ρaC , (4)where ρ is the grain density. With ρ = 3000 kg m − , we find M d ∼ × a mm kg, where a mm isthe grain radius expressed in millimeters. The mass of the nucleus, represented as a 2.5 km radiussphere of the same density, is a much larger 2 × kg.The Geminid stream mass is 10 ≤ M s ≤ kg (Hughes & McBride 1989; Jenniskens 1994),while the Geminid stream lifetime estimated on dynamical grounds is τ ∼ N , given by N ∼ (cid:18) M s τ (cid:19) (cid:18) PM d (cid:19) , (5)where P = 1.4 year is the orbital period. Substituting, we obtain N ∼ a − mm to 40 a − mm . If theparticles are millimeter-sized, a mm = 1, then the Geminids could be supplied in steady-state by 4 ≤ N ≤
40 outbursts like the one observed, each orbit.However, there is no compelling physical reason to assume that Geminid stream production isin steady state. While a mm = 1 may approximately represent the radii of the Geminid meteors,much larger examples up to 5 kg in mass (equivalent radius ∼ yr?) scale. We conclude that, whilecontinuing mass-loss near perihelion may contribute to actively replenishing the smaller Geminids,the PGC complex as a whole is likely the product of a more ancient and catastrophic breakup(c.f. Jewitt and Hsieh 2006, Kasuga and Jewitt 2008). 11 – Many puzzles remain in understanding the anomalous brightening of Phaethon and its relationto the ejection of dust and to the Geminids. We list the following key questions:1. What is the value of the effective dust radius, a mm ? This radius directly affects estimatesof the ejected mass through Equation (4) and so determines the extent to which on-goingactivity in Phaethon contributes to the Geminid stream.2. What is the origin of the ∼
5. Summary
We report new observations of planet-crossing asteroid and Geminid meteoroid parent (3200)Phaethon, using the NASA STEREO solar spacecraft. We find that1. (3200) Phaethon exhibited anomalous brightening when at perihelion in 2009 and 2012,but not in 2010 (the latter likely owing to unfavorable observing geometry). The distance-corrected apparent brightness increased near phase angle 100 ◦ in 2009 and 80 ◦ in 2012, inboth years ∼ M d ∼ × a mm kg,where a mm is the effective dust radius in millimeters. A contribution from prompt emissionby atomic oxygen cannot be excluded.3. Thermal fracture and the decomposition of hydrated silicates are two plausible mechanismsof dust production at the ∼ ≤ REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
15 –Table 1. Photometry Statistics and Detection CharacteristicsYear Target Mean Std. Deviation Threshold † Date ‡ Phase Angle § · · · · · · · · · Phaethon 0.44 0.37 0.27 June 17.5-23.0 29 ◦ -130 ◦ · · · · · · · · · Phaethon 0.36 0.40 0.30 April 30.5 - May 4.3 32 ◦ -105 ◦† The adopted threshold for Phaethon photometry, equal to three times the sky mean. ‡ Time interval during which Phaethon’s light curves are above the threshold. § Phase angle ranges corresponding to these date ranges. 16 –Table 2. Standard StarsHIC HD SAO [R.A., Dec.] Magnitude Spectra · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ×
700 pixels(7.8 ◦ × ◦ ). 18 –Fig. 2.— The path of Phaethon (white circles) across the STEREO A (upper) and B (lower) HI-1fields of view in 2010. Numbers along the path show the date in month/day format. Perihelionoccurred on UT 2010 November 25 18:00, at which time Phaethon was not within the field of eitherSTEREO camera. Panels show the full size HI-1 images of 20 ◦ × ◦ . The sun is on the right in theupper panel, and on the left in the lower panel. 19 –Fig. 3.— Raw photometry of Phaethon (red circles) as a function of time near perihelion in 2009and 2012. The median sky brightness surrounding Phaethon is shown with blue circles. Smoothedcurves (red and blue lines) have been plotted to guide the eye. Horizontal lines represent thephotometry thresholds that are three times of the sky mean (see Table 1). Above the levels,Phaethon was detected. 20 –Fig. 4.— Apparent V magnitude of Phaethon (red circles) in 2009 and 2012. The magnitudespredicted by NASA’s ephemeris software are shown for reference (blue lines). Letter “P” andarrows mark times of perihelion, date = 20.3 for June 2009 and date = 02.3 for May 2012. Thered curve is a smoothed fit to the data added to guide the eye. The time ranges correspond to thevalid Phaethon photometry measurements (see Table (1)). 21 –Fig. 5.— Magnitude at R = ∆ = 1 AU, vs. phase angle ( α ) in 2009 and 2012. The reducedPhaethon magnitudes are plotted in red (2009) and blue (2012) curves, and are smoothed fits tothe actual data points. Two Keck data points are from a new measurement in 2012 for m(1,1,18 . ◦ );and from Hsieh & Jewitt (2005) for m(1,1,37 . ◦ ). The lunar phase function is over-plotted fromLane & Irvine (1973) with the thick solid black curve. For α > ◦ , the lunar phase function isextrapolated (dashed curve). The letters “P” indicate phase angles α = 79 ◦ (red) and α = 66 ◦◦