Comparing Phoebe’s 2005 opposition surge in four visible light filters

Icarus 212 (2011) 819–834

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Comparing Phoebe’s 2005 opposition surge in four visible light filters C. Miller a,⇑, A.J. Verbiscer b, N.J. Chanover a, J.A. Holtzman a, P. Helfenstein c a

Box 30001, Dept. 4500, Department of Astronomy, New Mexico State University, Las Cruces, NM 88003, USA Department of Astronomy, University of Virginia, P.O. Box 400325, Charlottesville, VA 22904-4325, USA c Center for Radiophysics and Space Research, 320 Space Sciences Building, Cornell University, Ithaca, NY 14853, USA b

a r t i c l e

i n f o

Article history: Received 24 June 2010 Revised 15 December 2010 Accepted 19 December 2010 Available online 28 December 2010 Keywords: Regoliths Saturn, Satellites Satellites, Surfaces Kuiper Belt

a b s t r a c t We observed Phoebe for 13 nights over a period of 55 days before, during, and after the 2005 Saturn opposition with the New Mexico State University (NMSU) 1-m telescope at Apache Point Observatory (APO) in Sunspot, NM and characterized the width and magnitude of Phoebe’s opposition surge in BVRI filters. Our observations cover a phase angle range of 4.87° to 0.0509°. We use a Hapke reflectance model incorporating shadow hiding and coherent backscatter to investigate the wavelength dependence of Phoebe’s opposition surge. We find a significant opposition surge magnitude of 55–58% between phase angles of 5° and 0°. We find the strongest opposition surge for phase angles less than 2° in the I-band. The coherent backscatter angular width is on the order of 0.50°. We find Phoebe’s albedo to be spectrally flat within our error limits, with a B-band albedo of 0.0855 ± 0.0031, a V-band albedo of 0.0856 ± 0.0023, an R-band albedo of 0.0843 ± 0.0020, and an I-band albedo of 0.0839 ± 0.0023. We compare Phoebe’s albedo, color, and opposition surge magnitudes and slopes with those of other outer solar system bodies and find similarities to Centaurs, Nereid, Puck, and Comets 19P/Borrelly, 9P/Tempel 1, and 81P/Wild 2. We find that this comparison supports the idea that Phoebe originated in the Kuiper Belt. We also discuss the caveats of using results from a Hapke reflectance model to derive specific surface particle properties. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction During Saturn’s January 2005 opposition, the Earth transited the Sun as viewed from Phoebe. At that time, the Sun–Phoebe–Earth phase angle was less than the angular radius of the Sun as viewed from Phoebe, or 0.029°. Such a low phase angle will not occur again during a Saturn opposition until 2020. Airless objects such as Phoebe with a surface consisting of loose particles, or regolith, undergo a dramatic increase in reflectivity known as an opposition surge when observed at low phase angles (typically <2° for outer planet satellites). The angular width and amplitude of an opposition surge provides clues about the physical properties of a reflecting body’s surface particles. The variation of brightness with observed phase angle is referred to as the solar phase curve. Phoebe is an interesting object for several reasons. The fact that it is a retrograde orbiter of Saturn suggests that it is not native to the Saturn system. Degewij et al. (1980) characterized Phoebe as a possible C-type asteroid based on its UBV color. Kruse et al. (1986) also noted that the shape of Phoebe’s phase curve and Phoebe’s albedo of 0.084, extrapolated by Kruse et al. (1986), suggested classification as a C-type asteroid. More recent analysis of Phoebe’s phase curve in B and R filters by Bauer et al. (2006) sug-

⇑ Corresponding author. E-mail address: [email protected] (C. Miller). 0019-1035/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2010.12.024

gested that Phoebe may be a former Centaur that was captured by Saturn. Johnson and Lunine (2005) postulated that Phoebe is compositionally similar to Kuiper Belt Objects (KBOs) such as Pluto and Triton and likely formed independently of the Saturn system based on Phoebe’s bulk density of 1.63 ± 0.033 g cm1. Cassini Visual and Infrared Mapping Spectrometer (VIMS) infrared spectroscopy taken during Cassini’s Phoebe flyby on 11–12 June 2004 indicated the presence of water ice, amorphous carbon, tholins, CO2 ice, and iron compounds on Phoebe’s surface (Buratti et al., 2008). Verbiscer et al. (2009) showed Phoebe to be the source of a wide, diffuse ring of dark particles orbiting Saturn. Phoebe’s opposition surge was noted during previous Saturn oppositions. Kruse et al. (1986) published results of Phoebe’s opposition surge during the 1985 opposition in the V band. More recent studies of Phoebe’s opposition surge attempted to characterize the observed solar phase curves using Hapke photometric models, which mathematically describe the radiative transfer from a reflecting surface of discrete light scattering particles. Early Hapke models based on theory outlined in Hapke (1981, 1984, 1986), hereafter described as the Hapke (1986) model, described the opposition effect primarily in terms of shadow hiding. Shadow hiding occurs when shadows cast by individual particles disappear beneath the particles as seen by the observer when the phase angle goes to zero. Buratti et al. (2008) and Simonelli et al. (1999) used a Hapke (1986) model to describe Phoebe’s solar phase curve based on both ground-based and spacecraft data. Hapke (2002)

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introduced an enhanced photometric model, hereafter referred to as the Hapke (2002) model, that included the effects of coherent backscatter. Coherent backscatter occurs when multiply and singly scattered photons constructively interfere preferentially in the direction of the observer. According to Hapke’s (2002) theory, the contribution to opposition surge angular width from coherent backscatter should vary with wavelength. In this work, we applied a Hapke (2002) model incorporating coherent backscatter to Phoebe photometry acquired in four visible light filters to investigate the contribution of coherent backscatter to Phoebe’s opposition surge. 2. Observations and data reduction We observed Phoebe in four visible light filters during Saturn’s 2005 opposition with the New Mexico State University (NMSU) 1m telescope at Apache Point Observatory (APO) in Sunspot, NM. We imaged Phoebe on 13 nights during the period between 06 January 2005 and 02 March 2005 using Johnson B, V, and Kron–Cousins R and I filters. All Phoebe images were 150 s exposures acquired with an Apogee AP7P camera utilizing a 512  512 pixel CCD with 24 lm pixels and a read noise of 12.2 e pix1. The plate scale of the Apogee CCD on the NMSU 1-m telescope was 0.825 arcsec pix1, with a resulting field of view of 7 arcmin on a side. Table 1 summarizes our Phoebe observations. We needed to establish Phoebe’s apparent magnitude in all images as the initial step in producing solar phase curves (whole-disk mean hI/Fi vs. phase angle) for all four filters. However, none of our Phoebe images included standard photometric stars, and due to Phoebe’s rapid motion against the star background during opposition, images taken on non-consecutive nights did not have common stars. Therefore, we first established the apparent magnitudes of three reference stars in each Phoebe field for comparison to Phoebe’s magnitude. To do this, we re-imaged each Phoebe observation field and a set of Landolt standards under photometric conditions on the night of 10 February 2008 in BVRI filters. Each re-imaged field corresponded to the sky position of Phoebe on each of the 13 nights of our 2005 opposition observations. We used standard IRAF data reduction routines to process each image for bias and dark subtraction, and flat fielded the images using sky flats taken in each filter. The typical FWHM for stars in the 10 February 2008 images was 1.9–2.3 arcsec. We performed aperture photometry on a set of nine images of Landolt standards per filter to determine their instrumental magnitudes at varying airmasses using an aperture of 10 arcsec. We then performed linear fits for the difference between our

Table 1 Summary of observations. Date (2005)

Time (UT)

Phase angle (°)

No. of Phoebe observations

January 06 January 13 January 14 January 15 January 16 January 17 January 18 January 19 January 20 February 04 February 14 March 01 March 02

11:07:25–11:40:03 04:26:34–06:50:57 03:52:53–09:53:15 05:01:04–11:04:17 03:57:39–11:18:41 04:38:48–09:56:21 04:24:04–12:10:08 04:38:52–09:52:59 04:22:59–04:55:13 08:59:09–09:32:19 04:37:27–05:09:19 06:46:32–07:18:53 05:11:21–05:43:26

0.8560–0.8587 0.0509–0.0629 0.0532–0.0834 0.1777–0.2081 0.2913–0.3191 0.4134–0.4399 0.5307–0.5694 0.6502–0.6763 0.7669–0.7696 2.4966–2.4992 3.5057–3.5078 4.8008–4.8025 4.8691–4.8707

3 BVRI 6 BVR, 5 I 12 BVRI 11 B, 12 VRI 12 BVR, 9 I 9 BVRI 12 BVRI 9 BVRI 3 BVRI 3 BVRI 3 BVRI 3 BVRI 3 BVRI

0.0509–4.8707

89 B, 90 VR, 86 I

Total

instrumental and the published apparent magnitudes for the Landolt standards to derive a sky calibration relation with both extinction and color correction coefficients for each filter. For the B filter, we derived the following calibration relation:

B  b ¼ 2:170  0:208X þ 0:07ðB  VÞ

ð1Þ

where B is the instrumental magnitude, b is the apparent magnitude, X is the airmass, and (B  V) is the object’s instrumental color. The 1-r deviation for the free parameters were 0.008 magnitudes for the extinction coefficient, 0.005 magnitudes for the color correction coefficient, and 0.010 magnitudes for the zero-point offset magnitude. The remaining calibration relations for the other filters were as follows:

V  v ¼ 2:128  0:096X þ 0:02ðB  VÞ

ð2Þ

R  r ¼ 2:182  0:054X þ 0:00ðV  RÞ I  i ¼ 2:844  0:055X þ 0:058ðR  IÞ

ð3Þ ð4Þ

Next, we performed aperture photometry to determine the instrumental magnitude of three bright non-saturated field stars in each of the Phoebe fields taken on the night of 10 February 2008 using an aperture radius of 7 arcsec. We used our sky calibration relations in Eqs. (1)–(4) to derive the apparent magnitudes of each of these reference field stars. Using these apparent magnitudes, we then derived Phoebe’s apparent magnitude in each of the 2005 images through differential photometry. We determined the instrumental magnitudes of Phoebe and the three reference stars in each of our 2005 opposition images using an aperture radius of 4 pixels, or 3.3 arcsec. Typical FWHM of the 2005 images was 2.5–3.5 pixels, or 2.1–2.9 arcsec. We subtracted the error-weighted mean of the three reference stars in each image from their apparent magnitudes to get a magnitude correction for Phoebe in all frames. In total, we derived apparent magnitudes of Phoebe in 89 images in B filter, 90 images in the V and R filters, and 86 images in I filter. Our derived apparent magnitudes for Phoebe in all four filters are given in Table 2. We accumulated errors in each step of this differential photometry process. Systematic (non-random) errors were introduced from the uncertainty in the sky calibration coefficients based on the linear fits to the 10 February 2008 Landolt instrumental magnitudes. Photon noise, sky background, and read noise produced random errors in the derived instrumental magnitudes of the Phoebe field reference stars extracted from the 10 February 2008 Phoebe field images. These effects also produced random errors in both the Phoebe and field reference star instrumental magnitudes extracted from our 2005 opposition images. We accounted for all of these errors when estimating Phoebe’s apparent magnitude. After we derived best fit rotation and solar phase curves for Phoebe in each filter using the procedure described in Section 3.2, we calculated the standard deviation of the residuals with these curves subtracted and compared them to our estimated measurement errors. We found that the residual variance exceeded our estimated errors by about 30% due to unknown random effects. In light of this analysis, we adjusted our estimated magnitude errors to agree with our measured residual variance before fitting the data to our Hapke model. The magnitude errors reported in Table 2 are the adjusted errors. After establishing the apparent magnitudes of Phoebe for all phase angles, we converted these magnitudes to whole-disk mean reflectivity, or hI/Fi. Eq. (5) relates the derived apparent magnitudes of Phoebe to hI/Fi (Buratti et al. (1998)):

hI=Fi ¼

a2 D2 A2 r 2p

 10ð0:4Þðmp m Þ

ð5Þ

where I represents the scattered light intensity, pF is the incident solar flux, a is the Sun–Phoebe distance at time of observation, D

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C. Miller et al. / Icarus 212 (2011) 819–834 Table 2 Summary of Phoebe observations. JD - 2450000

Apparent Magnitude

Mag. Error

JD - 2450000

Apparent Magnitude

Mag. Error

B Filter 3376.964355 3376.966340 3376.968331 3383.685987 3383.687973 3383.689911 3383.766019 3383.767966 3383.769912 3384.662600 3384.664597 3384.666552 3384.737304 3384.739243 3384.741182 3384.813253 3384.815241 3384.817226 3384.890390 3384.892338 3384.894287 3385.709951 3385.713928 3385.786910 3385.788844 3385.790834 3385.862553 3385.864491 3385.866439 3385.939572 3385.941520 3385.943500 3386.665903 3386.667852 3386.669847 3386.795463 3386.797439 3386.799376 3386.875137 3386.877088 3386.879064 3386.955719 3386.957696 3386.959666 3387.694488

16.859 16.847 16.867 16.684 16.691 16.695 16.744 16.738 16.719 16.694 16.700 16.702 16.640 16.625 16.622 16.570 16.572 16.578 16.697 16.686 16.696 16.740 16.804 16.729 16.738 16.713 16.683 16.706 16.709 16.646 16.636 16.641 16.723 16.709 16.718 16.782 16.766 16.787 16.790 16.781 16.769 16.774 16.763 16.769 16.812

0.027 0.024 0.024 0.022 0.022 0.022 0.022 0.020 0.020 0.027 0.024 0.025 0.027 0.027 0.025 0.046 0.037 0.034 0.022 0.022 0.022 0.067 0.081 0.022 0.020 0.024 0.036 0.036 0.029 0.022 0.020 0.020 0.023 0.022 0.023 0.022 0.025 0.022 0.022 0.022 0.022 0.025 0.025 0.025 0.022

3387.696468 3387.698421 3387.789279 3387.791241 3387.793271 3387.891976 3387.894024 3387.895987 3388.684255 3388.686234 3388.688204 3388.766823 3388.768789 3388.770824 3388.850294 3388.852306 3388.854275 3388.985134 3388.987144 3388.989140 3389.694535 3389.696513 3389.698465 3389.787678 3389.789681 3389.791660 3389.890177 3389.892137 3389.894171 3390.683503 3390.685431 3390.687412 3405.875279 3405.877262 3405.879233 3415.693546 3415.695479 3415.697429 3430.783188 3430.785174 3430.787137 3431.717086 3431.719073 3431.721051

16.811 16.827 16.760 16.796 16.766 16.704 16.714 16.714 16.789 16.815 16.809 16.898 16.896 16.912 16.919 16.899 16.900 16.801 16.815 16.820 16.856 16.860 16.863 16.774 16.750 16.744 16.849 16.845 16.830 16.881 16.857 16.859 17.051 17.052 17.032 17.027 17.049 17.044 17.085 17.130 17.127 17.262 17.260 17.261

0.022 0.024 0.024 0.022 0.024 0.019 0.022 0.022 0.026 0.026 0.024 0.024 0.024 0.031 0.024 0.022 0.022 0.036 0.036 0.036 0.030 0.030 0.028 0.031 0.026 0.026 0.026 0.026 0.024 0.034 0.031 0.031 0.029 0.029 0.029 0.032 0.032 0.032 0.053 0.060 0.058 0.039 0.035 0.037

V Filter 3376.970471 3376.972491 3376.974470 3383.691976 3383.693947 3383.695895 3383.772013 3383.773965 3383.775904 3384.668622 3384.670612 3384.672575 3384.743244 3384.745188 3384.747131 3384.819299 3384.821255 3384.823204 3384.896559 3384.898528 3384.900535 3385.716017 3385.717985 3385.719950 3385.792995 3385.794944 3385.796910

16.190 16.193 16.197 16.061 16.053 16.053 16.049 16.050 16.040 16.042 16.060 16.061 15.873 15.992 15.980 15.935 15.949 15.957 16.087 16.079 16.083 16.102 16.110 16.094 16.075 16.064 16.077

0.020 0.017 0.020 0.017 0.017 0.017 0.017 0.017 0.017 0.023 0.020 0.020 0.023 0.023 0.029 0.031 0.026 0.023 0.020 0.020 0.020 0.061 0.047 0.047 0.020 0.023 0.020

3387.700714 3387.702672 3387.704613 3387.795509 3387.797495 3387.799457 3387.898209 3387.900178 3387.902147 3388.690362 3388.692322 3388.694312 3388.773008 3388.774976 3388.776919 3388.856380 3388.858346 3388.860323 3388.991341 3388.993307 3388.995340 3389.700551 3389.702502 3389.704515 3389.793795 3389.795748 3389.797743

16.167 16.167 16.169 16.114 16.123 16.110 16.063 16.064 16.068 16.124 16.115 16.121 16.202 16.213 16.215 16.207 16.204 16.222 16.150 16.148 16.141 16.200 16.204 16.185 16.116 16.107 16.103

0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.022 0.017 0.017 0.017 0.025 0.028 0.028 0.023 0.020 0.020 0.020 0.020 0.023 (continued on next page)

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C. Miller et al. / Icarus 212 (2011) 819–834

Table 2 (continued) JD - 2450000

Apparent Magnitude

Mag. Error

JD - 2450000

Apparent Magnitude

Mag. Error

3385.868640 3385.870606 3385.872565 3385.945782 3385.947735 3385.949693 3386.672001 3386.673985 3386.675933 3386.801575 3386.803535 3386.805542 3386.881197 3386.883186 3386.885148 3386.961965 3386.963934 3386.965921

16.038 16.045 16.043 15.974 15.967 15.961 16.061 16.059 16.056 16.122 16.121 16.121 16.122 16.134 16.125 16.119 16.119 16.125

0.023 0.023 0.020 0.017 0.014 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.020 0.020 0.020

3389.896269 3389.898235 3389.900222 3390.689506 3390.691485 3390.693464 3405.881632 3405.883581 3405.885538 3415.699532 3415.701476 3415.703417 3430.789311 3430.791256 3430.793214 3431.723153 3431.725083 3431.727015

16.198 16.187 16.198 16.231 16.224 16.237 16.382 16.392 16.376 16.375 16.362 16.380 16.509 16.523 16.520 16.634 16.617 16.621

0.017 0.017 0.017 0.022 0.022 0.022 0.022 0.020 0.022 0.024 0.024 0.022 0.045 0.045 0.048 0.028 0.039 0.034

R Filter 3376.976804 3376.978777 3376.980772 3383.698139 3383.700084 3383.702069 3383.778175 3383.780112 3383.782062 3384.674766 3384.676711 3384.678675 3384.749358 3384.751345 3384.753281 3384.825436 3384.827420 3384.829380 3384.902816 3384.904757 3384.906709 3385.722294 3385.724237 3385.726182 3385.799292 3385.801231 3385.803175 3385.874845 3385.876792 3385.878749 3385.951983 3385.953950 3385.955919 3386.678189 3386.680131 3386.682079 3386.807748 3386.809727 3386.811693 3386.887388 3386.889344 3386.891304 3386.968203 3386.970167 3386.972182

15.847 15.843 15.847 15.744 15.739 15.751 15.711 15.688 15.707 15.704 15.713 15.710 15.648 15.643 15.639 15.632 15.625 15.615 15.731 15.732 15.730 15.760 15.774 15.770 15.739 15.731 15.742 15.694 15.692 15.688 15.605 15.601 15.586 15.725 15.710 15.714 15.765 15.788 15.795 15.789 15.784 15.788 15.786 15.802 15.765

0.017 0.017 0.017 0.014 0.014 0.014 0.014 0.017 0.014 0.017 0.017 0.017 0.029 0.031 0.029 0.023 0.031 0.031 0.017 0.017 0.017 0.045 0.039 0.061 0.025 0.025 0.025 0.031 0.031 0.034 0.014 0.014 0.014 0.014 0.014 0.014 0.017 0.014 0.014 0.014 0.014 0.014 0.017 0.017 0.017

3387.707030 3387.709013 3387.710966 3387.801843 3387.803831 3387.805807 3387.904638 3387.906645 3387.908624 3388.696572 3388.698557 3388.700523 3388.779204 3388.781212 3388.783176 3388.862601 3388.864583 3388.866564 3388.997613 3388.999630 3389.001601 3389.706751 3389.708770 3389.710719 3389.799991 3389.802002 3389.803968 3389.902490 3389.904498 3389.906454 3390.695708 3390.697697 3390.699660 3405.888005 3405.889975 3405.891951 3415.705645 3415.707601 3415.709534 3430.795588 3430.797546 3430.799488 3431.729280 3431.731263 3431.733233

15.811 15.815 15.801 15.768 15.769 15.772 15.724 15.723 15.718 15.757 15.761 15.761 15.851 15.844 15.843 15.843 15.845 15.844 15.798 15.816 15.810 15.858 15.860 15.846 15.777 15.767 15.756 15.871 15.871 15.867 15.919 15.900 15.899 16.031 16.038 16.049 16.032 16.032 16.029 16.156 16.166 16.151 16.270 16.293 16.293

0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.017 0.017 0.014 0.014 0.017 0.017 0.025 0.025 0.025 0.017 0.017 0.017 0.017 0.017 0.017 0.014 0.014 0.014 0.020 0.017 0.020 0.017 0.020 0.020 0.021 0.019 0.019 0.034 0.034 0.036 0.023 0.025 0.023

I Filter 3376.983101 3376.985050 3376.987017 3383.704289 3383.706245 3383.708178 3383.784287 3383.786257 3384.680953

15.563 15.566 15.567 15.391 15.382 15.402 15.348 15.369 15.398

0.023 0.023 0.023 0.017 0.017 0.017 0.015 0.020 0.021

3387.810122 3387.812082 3387.910995 3387.913020 3387.915002 3388.702803 3388.704769 3388.706793 3388.785457

15.448 15.457 15.427 15.431 15.435 15.449 15.459 15.466 15.549

0.017 0.017 0.020 0.020 0.020 0.015 0.015 0.015 0.020

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C. Miller et al. / Icarus 212 (2011) 819–834 Table 2 (continued) JD - 2450000

Apparent Magnitude

Mag. Error

JD - 2450000

Apparent Magnitude

Mag. Error

3384.682896 3384.684814 3384.755476 3384.757417 3384.759395 3384.831616 3384.833575 3384.835530 3384.908950 3384.910905 3384.912852 3385.728477 3385.730410 3385.732378 3385.805469 3385.807403 3385.809380 3385.881115 3385.883060 3385.885009 3385.958233 3385.960186 3385.962183 3386.684350 3386.686303 3386.688252 3386.813948 3386.815956 3386.817910 3386.893572 3386.895537 3386.897553 3387.713368 3387.715310 3387.717310 3387.808117

15.381 15.377 15.363 15.329 15.330 15.310 15.363 15.356 15.407 15.415 15.416 15.554 15.464 15.472 15.445 15.452 15.436 15.390 15.410 15.378 15.283 15.293 15.288 15.399 15.384 15.392 15.462 15.455 15.464 15.459 15.466 15.455 15.517 15.522 15.509 15.475

0.018 0.018 0.035 0.032 0.029 0.037 0.029 0.024 0.023 0.024 0.029 0.089 0.076 0.048 0.031 0.034 0.037 0.034 0.028 0.026 0.020 0.020 0.020 0.015 0.017 0.015 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017

3388.787425 3388.789434 3388.868858 3388.870829 3388.872813 3389.003880 3389.005865 3389.007906 3389.713002 3389.714965 3389.716946 3389.806271 3389.808220 3389.810181 3389.908748 3389.910698 3389.912672 3390.701925 3390.703924 3390.705883 3405.894334 3405.896319 3405.898312 3415.711764 3415.713734 3415.715677 3430.801762 3430.803696 3430.805654 3431.735491 3431.737431 3431.739368

15.546 15.550 15.539 15.526 15.521 15.511 15.525 15.516 15.545 15.545 15.557 15.465 15.474 15.493 15.582 15.583 15.582 15.603 15.582 15.579 15.726 15.748 15.742 15.719 15.708 15.713 15.851 15.834 15.874 15.921 15.919 15.917

0.017 0.017 0.020 0.017 0.017 0.034 0.037 0.039 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.020 0.020 0.023 0.020 0.020 0.025 0.025 0.025 0.023 0.023 0.025 0.038 0.068 0.117 0.024 0.027 0.024

is the Earth–Phoebe distance, A is the Sun–Earth distance, or 1 AU, rp is the radius of Phoebe, mp is the apparent magnitude of Phoebe in a given filter, and m is the apparent solar magnitude in the same filter. We obtained Sun–Phoebe and Earth–Phoebe distances for each observation from the JPL HORIZONS ephemerides. We used an apparent solar magnitude for the Johnson V filter of 26.75 and derived the Johnson B filter apparent magnitude from the quoted solar color value of (B  V) = +0.65 (Cox, 2000). We derived solar magnitudes for the Kron–Cousins R and I filters from colors (V  R) = +0.35 and (V  I) = +0.69 established in Hartmann et al. (1990) and based on photometry of Landolt standard star SA 51-12. The resulting plots of hI/Fi vs. phase angle for each of the BVRI filters are shown in Fig. 1. The data points represent the initial hI/ Fi of Phoebe calculated from our images before correction for rotation. Our data covered Sun–Phoebe–Earth phase angles ranging from 0.0509° to 4.8707°. The absolute minimum phase angle of 0.029°, limited to the angular radius of the Sun as seen by Phoebe, occurred during daylight at APO. 3. Hapke model fitting procedure The apparent magnitude of Phoebe in our observations was modulated by two effects: the effect of Phoebe’s rotational light curve due to its 9.2735 h rotation period, and the solar phase curve due to the changing solar phase angle over our 55 day observation period. A preliminary fit of the Hapke (2002) model to our data showed the opposition surge to be on the order of 0.46 magnitudes from 0° to 5° phase angle. Bauer et al. (2004) reported a rotational light curve magnitude variation for Phoebe in the V band of 0.125 magnitudes, meaning Phoebe’s rotation curve amplitude was

significant compared to its opposition surge magnitude. We therefore needed to subtract Phoebe’s rotation curve from our data before making our final solar phase curve fits. 3.1. Subtraction of Phoebe’s rotation curve We modeled Phoebe’s rotational light curve from published V band data taken from 02 December 2003 to 16 March 2004 (Bauer et al., 2004). We fit an eight term Fourier series to the Bauer data and used this derived curve as our model for Phoebe’s rotational light curve. The Bauer et al. (2004) data and our derived rotation curve are shown in the top panel of Fig. 2. We assumed the rotational period of Phoebe to be 9.2735 h (Bauer et al., 2004) when fitting to the Fourier series function. We subtracted this model rotation curve from our data before fitting to our Hapke solar phase curve model. However, before doing do we needed to establish Phoebe’s rotational phase (sub-observer longitude) during our observational period. Phoebe is not tidally locked to Saturn and its rotation rate is not known to sufficient accuracy to allow subtraction based on a predicted sub-observer longitude, such as that from JPL HORIZONS. The uncertainty in the period reported by Bauer et al. (2004) of ±0.0006 h translates to a sub-observer longitude uncertainty of ±20° in the one year interval between our observations and those of Bauer et al. (2004). In order to determine Phoebe’s rotation phase for our observation interval, we subtracted a preliminary solar phase curve from our data in each of the BVRI filters. To subtract Phoebe’s light curve from our data, we adopted the iterative approach described by Verbiscer et al. (2005). We started by fitting our initial (nonrotation corrected) hI/Fi vs. phase angle data to our Hapke model to obtain a preliminary estimate for the Hapke parameters. We

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15

0.1

B filter 16

0.08

17

0.06

15

0.1

V filter 16

0.08

17

0.06

15

0.1

R filter 16

0.08

17

0.06

15

0.1

I filter 16

0.08

17

0.06 0.1

1

Phase angle (degrees)

0.1

1

Phase angle (degrees)

Fig. 1. Derived apparent magnitudes and hI/Fi as a function of phase angle for Phoebe during the 2005 opposition in Johnson BV and Kron-Cousins RI broadband filters. Observations were made from 06 January through 02 March 2005, with opposition occurring on 13 January. These data represent phase angles from 0.0509 - 4.8707°. Left panels plot Phoebe’s apparent magnitude in four wavelengths against phase angle. Right panels show the same data but with the phase angle on a log scale to better distinguish data points at phase angles < 1°. The data in these plots have not been corrected for Phoebe’s rotational light curve. Error bars are shown and are of the order of the size of the data points.

then generated an initial solar phase curve with our Hapke model based on these preliminary values and subtracted this solar phase curve from our Phoebe data to derive a set of residual magnitudes. Next, we fit these residual magnitudes to our model of Phoebe’s rotation curve defined by the eight term Fourier series fit to the data from Bauer et al. (2004) as described above. We fixed Phoebe’s period to 9.2735 h, set the rotation curve amplitude and rotation phase to be free parameters, and solved using a least-squares fitting routine. The resulting residuals for the BVRI filters and the fitted rotation curve are shown in the lower panel of Fig. 2. Our 55-day period of observation covered 142 rotations of Phoebe. Given a rotational period error of 0.0006 h as reported in Bauer et al. (2004), we estimate a maximum sub-observer rotational phase error of 3.3° throughout our observation period. We note that the general shape of our model Phoebe rotation curve matches our residuals, while many of the smaller ‘‘bumps’’ in the rotation curve do not match. However, we believe our model Phoebe rotation curve is a better match than a simple sinusoid. We used this model Phoebe rotation curve for all subsequent fitting to the Hapke solar phase curve as described in the next section. 3.2. Fitting to the Hapke (2002) model We used a Hapke (2002) model that incorporated both shadow hiding and coherent backscatter effects. There are many examples in the literature of fitting observational data to other phase curve functions. These include an empirical piece-wise linear function (Buratti et al., 1996; Kruse et al., 1986) and a Lumme–Bowell IAU

model (Bauer et al., 2006). Both approaches provide a qualitative method for characterizing solar phase curves that is useful for classifying objects by their surface reflectance properties. Many published studies of solar phase curves have employed a Hapke model, which is based on physical parameters of surface particles, such as the particle phase function and single scattering albedo. Some authors have used Hapke models based on shadow hiding only (Buie et al., 2010; Simonelli et al., 1999), while others have incorporated coherent backscatter in their Hapke models (Helfenstein et al., 1997; Schaefer et al., 2008; Verbiscer et al., 2005). Recent laboratory studies have cast some doubt on the legitimacy of using the results of Hapke model fitting to infer specific surface properties of planetary surfaces such as grain size and porosity (Nelson et al., 2000; Shepard and Helfenstein, 2007). These laboratory studies suggested that current Hapke bidirectional reflectance models may not adequately account for important features of a granular surface such as the aggregate nature of surface particles and the effects of porosity. In addition, parameter degeneracies can result when fitting to a Hapke model with data of a narrow phase angle range, e.g. data limited to phase angles less than 5°, as is typical for ground based observations of outer solar system bodies. Indeed, we encountered such degeneracies in our analysis as we describe below. However, we persisted with a Hapke model fit for several reasons. First, we found that our Hapke model generated solar phase curves well matched to our observational data. We therefore could extract useful parameters of Phoebe’s solar phase curve such as the magnitude and slope of the opposition surge and compare these values to previously published results for Phoebe and other solar system objects. Also, a

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Bauer et al., (2004) - V filter

16

16.1

16.2

16.3 -0.2

Circles - B filter Triangles - V filter Squares - R filter Crosses - I filter

-0.1

0

0.1

Phoebe 2005 opposition - BVRI filters 0.2

0

0.2

0.4

0.6

0.8

1

Rotational phase Fig. 2. Phoebe’s rotation curve taken by Bauer et al. (2004) (top panel) and this work (bottom panel). An eight term Fourier series was fit to the Bauer data to produce a model Phoebe rotation curve that was subtracted from our data before fitting to the Hapke model. The data points in the top panel were taken in V band between the nights of 02 December 2003 and 16 March 2004. The solid line is the best fit rotation curve. The data points in the bottom panel are the residuals after subtracting our best fit solar phase curves from our Phoebe data in all four filters. The solid line is the same rotation curve as in the top panel phase shifted for the best fit to the residuals.

Hapke model produces solar phase curve solutions that are more realistically physically constrained than those derived from nonparametric methods. Second, we wanted use our multiple filter data set to test the prediction made by the Hapke (2002) model that the coherent backscatter contribution to the opposition surge angular width varies by wavelength. Third, we wanted to compare our Hapke parameter solutions with previously published results for Phoebe and other bodies analyzed with similar Hapke models. Several previous studies of Phoebe’s solar phase curve included reflectance data over a wider range of phase angles than we observed. By using a physical Hapke model, we could extrapolate our solar phase curve solutions to phase angles greater than 5° to determine phase integrals for comparison to these previously published results. Given the laboratory results cited above, we felt that the values we derived for these model parameters were useful primarily for comparison to similar studies and may not necessarily represent absolute physical properties of Phoebe’s surface particles. Our Hapke model allowed both the fitting of Hapke free parameters to observed solar phase curve data and the generation of a solar phase curve given an input set of Hapke parameters. Up to seven free parameters were allowed in our Hapke model: the single scattering albedo -o, mean surface roughness  h, the single-term Henyey–Greenstein g parameter, and the Hapke parameters h and B0, corresponding to the opposition surge amplitude and width, respectively, for both shadow hiding (hSH and B0SH) and coherent backscatter (hCB and B0CB). The single-term Henyey–Greenstein phase function is given by Henyey and Greenstein (1941):

PðaÞ ¼

ð1  g 2 Þ 3

ð1 þ g 2 þ 2g cos aÞ2

ð6Þ

where a is the phase angle and g is the cosine of the scattering angle. Positive g values indicate forward scattering particles while negative g values indicate backscattering particles. We did not have sufficient phase angle coverage to accurately determine Phoebe’s mean surface roughness or derive a unique solution for the Henyey–Greenstein parameter, g. Both parameters require wider phase angle coverage, preferably up to 90°, to be constrained adequately with a photometric model. Buratti et al. (2008) established a mean surface roughness for Phoebe of 33° based on VIMS data covering phases angles from 27° to 92.5°. We therefore fixed the roughness parameter,  h, to 33° for all model fits. We also fixed the Henyey–Greenstein g parameter to four prechosen values assuming backscattering surface particles, specifically g = 0.45, 0.35, 0.25, and 0.15. These values covered the range of particle phase function values for Phoebe of g ¼ 0:24þ0:26 0:11 and g = 0.36 ± 0.11 reported in Simonelli et al. (1999) from data including Voyager 2 observations covering higher phase angles. We chose the lower value of g = 0.15 for comparison with other low albedo bodies with similar particle phase function values such as the Moon (Helfenstein et al., 1997) and Gaspra (Helfenstein et al., 1994). Finally, we chose a strongly backscattering value of g = 0.45 for comparison to the published opposition surge of Comets 19P/Borrelly, 9P/Tempel 1, and 81P/Wild 2. We adopted the iterative approach described above to subtract our model Phoebe rotational curve. We found preliminary fits for five Hapke free parameters, -0, hSH, B0SH, hCB, and B0CB, in each filter. Next, we generated preliminary solar phase curves for each filter based on these parameters. We subtracted these solar phase curves from our hI/Fi data to produce residuals, and fit our model Phoebe rotation curve to these residuals, fixing the rotational phase and allowing the magnitude to vary. We then subtracted

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the rotation curve solutions from our original hI/Fi data, and fit for a new set of Hapke parameters based the corrected hI/Fi data. We repeated this process until the rotational light curve was eliminated. This was defined as the point at which the difference in each Hapke parameter solution from one iteration to the next was less than the estimated Hapke parameter error. This error was initially estimated to be ±1.0% for -0, B0SH, and hCB and 10% for hSH and B0CB. Subsequent error analysis (described below) showed these initial error estimates to be well within our actual derived error limits. We allowed Phoebe’s rotation curve amplitude to vary in our rotation curve fitting for two reasons. First, the data from Bauer et al. (2004) were taken in the V filter while our data were taken in BVRI filters. We did not assume that the albedo variations contributing to Phoebe’s rotational light curve would be the same at all wavelengths. Second, Phoebe’s sub-observer latitude shifted from the time of the Bauer et al. (2004) observations (between 17.35° and 20.20°) to our observations (between 11.26° and 13.53°). This left open the possibility that different surface features would be visible at the time of our observations than were observed by Bauer et al. (2004). The results of our best fit amplitude for our modeled Phoebe rotation curve fit for each of the B, V, R, and I filters are shown in Table 3. Our amplitude was a simple multiplier to the rotation curve amplitude derived from Bauer et al. (2004). We found the best fit rotation curve amplitude to range from a low of 0.119 ± 0.007 magnitudes in the I filter to a high of 0.137 ± 0.005 magnitudes in the R filter. Our best fit V-band rotation curve amplitude of 0.128 ± 0.006 compares to the V-band amplitude of 0.125 magnitudes in the Bauer et al. (2004) data.

4. Characterization of Phoebe’s solar phase curves Our final solar phase curves for BVRI filters are shown in Fig. 3. These curves were generated using our best fit Hapke model after rotation curve subtraction. The solar phase curves in Fig. 3 correspond to our best-fit solutions for an assumed Henyey–Greenstein value of g = 0.35. Note that in Fig. 3, the open circles represent the Phoebe hI/Fi data before rotation curve subtraction and the dark filled circles correspond to the same data points after rotation curve subtraction. The rotation curve subtracted data were used as input to the Hapke model fitting routine. To determine the error limits for our derived Hapke parameters, we performed a sensitivity analysis for four free Hapke parameters, -0, hSH, hCB, and B0CB. Because the Hapke model parameters are strongly coupled to one another (Helfenstein and Veverka, 1989), standard statistical methods that treat them as independent variables are not applicable. We followed the approach of Helfenstein et al. (1997) and Shepard and Helfenstein (2007) to investigate how rapidly our model fit errors increased when individual Hapke parameters were perturbed from their best-fit solutions. For reasons discussed earlier, we fixed the values for mean surface roughness,  h, to 33° and Henyey–Greenstein value, g, to 0.45, 0.35, 0.25, and 0.15 and set B0SH to 1.0 as this was the best fit for all cases we analyzed. We then employed a step-wise gridsearch approach to incrementally adjust one Hapke parameter at a time while solving for the best-fit solution for the other three free Hapke parameters with a v2 gradient following algorithm. For each grid-search increment we recorded the best-fit results and

Table 3 Phoebe rotation curve amplitudes in four filters.

a

Parametera

B

V

R

I

Amplitude (magnitudes)

0.121 (0.008)

0.128 (0.006)

0.137 (0.005)

0.119 (0.007)

Errors are in parentheses.

corresponding v2 values. We then determined our Hapke parameter error limits by establishing a meaningful v2 error envelope to constrain the allowable parameter ranges that fall within it according to the methods outlined in Helfenstein and Shepard (in preparation) and described next. We plotted the incremental v2 values as a function of each gridsearched parameter to map out the sensitivity of v2 to mutually adjusting changes in each variable. To define our confidence envelope, we computed the variance, rv, of all data fit errors about our minimum v2 and required that the envelope extend by 1rv beyond the largest measured error in the cumulative distribution of v2 values. The v2 threshold corresponding to the 1rv envelope as a function of the minimum v2 value is given by:

v2limit ¼

v2min  ðv2max þ 1rv Þ v2max

ð7Þ

where v2limit is the v2 threshold value corresponding to the Hapke parameter 1r error, v2min is the local v2 minimum at the best-fit parameter value, and v2max is the value at which the cumulative distribution of error reaches 100%. Based on this analysis, we found that only two parameters, -0 and hCB had well defined v2 minima. The other two free parameters, hSH and B0CB, were coupled in that increasing the value of one could be compensated by a corresponding increase in the other to produce nearly identical results. This dependency is likely the result of the fact that our observational phase angle range was limited to 5°. As a result, we were unable to determine meaningful error ranges for hSH and B0CB. We ran our sensitivity analysis for values for hSH and B0CB from 0 to 0.6; we considered any values higher than 0.6 to be physically unrealistic for both parameters. Once we had determined our error limits for each filter, we generated a set of model solar phase curves using the value of g corresponding to the Hapke parameter solutions at the error limits. Limits for hSH were set from 0.03 to 0.60 and for B0CB from 0.05 to 0.60 based on our sensitivity analysis results. An example of four sets of solar phase curves for BVRI filters given g = 0.35 is shown in Fig. 4. From each set of solar phase curves, we derived a family of possible opposition surge parameters and determined their error limits from the range of values extending over all curves. 4.1. Magnitude of Phoebe’s opposition surge We found that Phoebe’s brightness increased between 0.472 and 0.501 magnitudes between 5° and 0°, or Dm05, depending on the filter and value of g. The opposition surge, i.e. the increase in brightness in the non-linear region of the phase curve between 2° and 0°, or Dm02, ranged from 0.311 to 0.351 magnitudes. Table 4 shows the magnitude of the opposition surge for each of the four filters and for four values of g. The opposition surge magnitude increase was consistent across all values of g, with maximum variation of 1.2% from g = 0.45 to g = 0.15 for a given filter. 4.2. Phase coefficients We report phase coefficients, the slope of the solar phase curve with respect to phase angle, or b, in Table 5 for two phase angle regimes: the first for the steepest region of the opposition surge, from phase angles of 0° to 1°, b01, and the second for the linear region to the endpoint of our observational data, from 2° to 5°, b25. Table 5 shows an average of the opposition surge magnitudes and phase coefficients across all values of g for each filter. The opposition surge magnitude, Dm02, was greatest for the I filter with an average value of 0.351 ± 0.037 magnitudes, and smallest for the R filter with an average value of 0.312 ± 0.028 magnitudes. We find no clear trend of the magnitude of opposition surge with

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0.1

B filter

0.08 0.06 0.1

V filter

0.08 0.06 0.1

R filter

0.08 0.06 0.1

I filter

0.08 0.06 0.1

B filter - Solid V filter - Dashed R filter - Dotted

0.08

I filter - Dot-dash 0.06 0

1

2

3

4

Phase angle (degrees)

5

0.1

1

Phase angle (degrees)

Fig. 3. Phoebe’s opposition surge in four filters with best fit solar phase curves generated by our Hapke model for the case of g = 0.35. The left and right panels show the same data, with the right panels displaying the phase angle on a log scale to better distinguish data points for phase angles < 1°. The bottom panel shows a composite plot of the solar phase curves for all four filters. Open circles represent the Phoebe hI/Fi data before correction for Phoebe’s rotation curve. Filled dark circles are the Phoebe data after rotation curve subtraction, which were used for the Hapke model fitting. Error bars are shown for the rotation curve corrected data.

wavelength, although we note that the opposition surge is greater in B filter than in R filter, as was also reported by Bauer et al. (2006) and Verbiscer et al. (2006). The average V-band opposition surge was 0.319 ± 0.027 magnitudes, which compares to values reported by Simonelli et al. (1999) in Fig. 2a of 0.31 (solution 1) and 0.35 (solution 2) magnitudes. S0 in Table 5 is the slope of the solar phase curve at a phase angle of 0°. S1 is the slope of the solar phase curve at a phase angle of 1°. We derived averaged values of b01 ranging from a low of 0.209 ± 0.034 mag deg1 in R to a high of 0.254 ± 0.036 mag deg1 in I. Our average b01 in V of 0.220 ± 0.033 is greater than the value of 0.180 mag deg1 reported by Kruse et al. (1986) for phase angles <1.3°. Simonelli’s Hapke model fits resulted in phase coefficients of 0.162 (solution 1) to 0.214 mag deg1 (solution 2). 4.3. Geometric albedos and colors We determined Phoebe’s geometric albedo, or average wholedisk hI/Fi at a phase angle of 0°, for all four filters based on our Hapke model generated solar phase curves. These values are reported in Table 4. The geometric albedo values are essentially unchanged with respect to values of g in the range of 0.15 to 0.45. We find Phoebe’s geometric albedo to be essentially flat within the error limits with a slight albedo peak in the B filter (0.0855 ± 0.0031) and V filter (0.0856 ± 0.0023), decreasing to 0.0839 ± 0.0023 in the I filter. This wavelength dependence is consistent with spectra of Phoebe taken with the Palomar 200-inch telescope, which show an essentially flat spectrum with a slight negative slope in albedo from 0.5 to 0.9 lm (Buratti et al., 2002, 2008, Fig. 5). Bauer et al. (2004) report a V-band geometric albedo of 0.0821 ± 0.0015 and Kruse et al. (1986) reported a geometric albedo of 0.084 ± 0.003

in V band. Simonelli et al. (1999) derived geometric albedo values by fitting the Kruse data and Voyager 2 data to a Hapke model without a coherent backscatter component of 0.081 ± 0.002 (solution 1) and 0.078 ± 0.003 (solution 2). From our solar phase curve, we derive the following colors for Phoebe at a phase angle of 0°: (B  I) = 1.320 ± 0.050, (B  V) = 0.652 ± 0.051, and (V  R) = 0.333 ± 0.040. Our (B  R) value of 0.985 ± 0.049 compares to the value of 0.95 ± 0.06 reported by Bauer et al. (2006), which is also based on observations made during the 2005 opposition. 4.4. Comparisons to other solar system bodies Color, geometric albedo, and the magnitude and slope of solar phase curves are parameters often used to classify objects in the outer solar system. Table 6 provides a comparison of color, albedo, and opposition surge parameters to previous Phoebe results and published values for selected solar system bodies. Phoebe’s geometric albedo in the V band of 0.0856 is similar to that of Centaurs, KBOs, Jupiter family comets, C-type asteroids, and icy satellites such as Nereid, Charon, Puck, and the Portia group (comprising Uranus’ seven innermost satellites). Schaefer et al. (2009) collected and sorted the properties of 52 icy bodies in the outer solar system to categorize objects by type and assess the role of coherent backscatter in their opposition surge. Based on albedo, Phoebe most closely fits into their categories labeled ‘‘Small/Red’’ (mostly H2O and tholin covered surfaces) and the darker ‘‘Small/Gray’’ objects (Centaurs and KBOs). However, Phoebe is bluer than Centaurs as reported by Rabinowitz et al. (2007) and Bauer et al. (2003). In fact, Phoebe’s color is a better match to the Haumea collisional family, and the icy satellites Charon, Nereid, and Puck.

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0.1

B filter

V filter

R filter

I filter

0.08

0.06

0.1

0.08

0.06

0.1

1

0.1

Phase angle (degrees)

1

Phase angle (degrees)

Fig. 4. Set of solar phase curves generated by our Hapke model for the range of errors of the Hapke parameters x0 ; hSH ; hCB ; and B0CB reported in this paper for the case of g = 0.35. The dark solid line is the best fit solar phase curve. The dotted lines are the fits given variations for each Hapke parameter including the error limits. Open circles represent the Phoebe hI/Fi data before correction for Phoebe’s rotation curve. Filled dark circles are the Phoebe data after rotation curve subtraction, which were used for the Hapke model fitting. Error bars are shown for the rotation curve corrected data.

Table 4 Phoebe solar phase curve parameters in BVRI for all values of g. Parameters

Filters (kcentral) B (480 nm)

V (541 nm)

R (649 nm)

I (830 nm)

g = 0.45 Dm02, magnitude b01, mag deg1 b25, mag deg1 Geometric albedo, p

0.325 (0.053) 0.232 (0.058) 0.054 (0.017) 0.0855 (0.0031)

0.318 (0.032) 0.220 (0.036) 0.061 (0.014) 0.0857 (0.0021)

0.311 (0.036) 0.209 (0.037) 0.061 (0.009) 0.0843 (0.0020)

0.349 (0.036) 0.253 (0.032) 0.042 (0.021) 0.0839 (0.0016)

g = 0.35 Dm02, magnitude b01, mag deg1 b25, mag deg1 Geometric albedo, p

0.326 (0.043) 0.232 (0.048) 0.053 (0.015) 0.0855 (0.0031)

0.318 (0.028) 0.220 (0.033) 0.060 (0.012) 0.0857 (0.0022)

0.312 (0.026) 0.209 (0.028) 0.060 (0.008) 0.0843 (0.0016)

0.351 (0.047) 0.253 (0.044) 0.041 (0.019) 0.0839 (0.0021)

g = 0.25 Dm02, magnitude b01, mag deg1 b25, mag deg1 Geometric albedo, p

0.327 (0.036) 0.232 (0.042) 0.053 (0.013) 0.0855 (0.0027)

0.319 (0.021) 0.220 (0.021) 0.059 (0.016) 0.0856 (0.0016)

0.312 (0.020) 0.209 (0.034) 0.059 (0.011) 0.0843 (0.0017)

0.352 (0.040) 0.254 (0.041) 0.041 (0.019) 0.0839 (0.0023)

g = 0.15 Dm02, magnitude b01, mag deg1 b25, mag deg1 Geometric albedo, p

0.327 (0.021) 0.232 (0.044) 0.053 (0.013) 0.0855 (0.0023)

0.320 (0.028) 0.220 (0.042) 0.058 (0.012) 0.0856 (0.0023)

0.313 (0.028) 0.209 (0.037) 0.059 (0.009) 0.0843 (0.0020)

0.353 (0.023) 0.254 (0.026) 0.040 (0.019) 0.0839 (0.0017)

Dm02 is measured between phase angles a = 0–2°. b01 is measured between phase angles a = 0–1°. b25 is measured between phase angles a = 2–5°. Errors are in parentheses.

Comparing the opposition surge magnitudes and slopes of different bodies is complicated by the fact that these values can change dramatically at low phase angles. Therefore, reported

opposition surges are likely underestimated for objects which have not been observed at phase angles less than 0.1°. Nevertheless, comparing opposition surge values has some value in rough

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C. Miller et al. / Icarus 212 (2011) 819–834 Table 5 Phoebe solar phase curve parameters averaged over all g. Parameters

B

Average Average Average Average Average Average

0.486 0.326 0.232 0.053 0.338 0.128

Dm05, magnitude Dm02, magnitude b01, mag deg1 b25, mag deg1 S0, mag deg1 S1, mag deg1

V (0.052) (0.038) (0.048) (0.015) (0.256) (0.025)

R

0.498 0.319 0.220 0.060 0.319 0.128

(0.033) (0.027) (0.033) (0.014) (0.198) (0.021)

0.491 0.312 0.209 0.060 0.270 0.134

I (0.034) (0.028) (0.034) (0.009) (0.137) (0.020)

0.475 0.351 0.254 0.041 0.345 0.143

(0.041) (0.037) (0.036) (0.019) (0.156) (0.033)

Dm05 is measured between phase angles a = 0–5°. Dm02 is measured between phase angles a = 0–2°. b01 is measured between phase angles, a = 0–1°. b12 is measured between phase angles a = 1–2°. b25 is measured between phase angles a = 2–5°. S0 is the phase curve slope at a = 0°. S1 is the phase curve slope at a = 1°. Errors are in parentheses.

Fig. 5. Plots of Phoebe’s solar phase curves to a phase angle of 180° generated by our Hapke model for sixteen combinations of filter and Henyey-Greenstein phase function parameter, g. The left panels compare solar phase curves for each of the filters for a given value of g. The right panels compare solar phase curves for each value of g for a given filter.

classification of objects. Phoebe’s V-band opposition surge magnitude between 0° and 1°, Dm01, of 0.220 ± 0.033 magnitudes is similar to that of Uranian satellites Puck and the Portia group (Karkoschka, 2001). Phoebe’s V-band surge from 0° to 2° of 0.319 ± 0.027 is similar to that of Charon (Buie et al., 2010) although it should be noted that the value for Charon is based on phase angles greater than 0.3°. Nereid exhibits an opposition surge greater than Phoebe. Opposition surge magnitudes of the ‘‘Small/Red’’ and ’’Small/Grey’’ groups as reported by Schaefer et al. (2008) cover a wide range that brackets the value for Phoebe.

5. Hapke (2002) model parameter fit analysis Our best-fit solutions for the Hapke free parameters are shown in Table 7, which presents sets of Hapke parameter fits for sixteen solutions of solar phase curves – solutions for BVRI filters for each of our four fixed values of Henyey–Greenstein parameter. We discuss the results of our Hapke model fits in the following sections.

5.1. Single scattering albedo, -o During our initial fitting, when we set the single scattering albedo as a free parameter, we found a strong degeneracy between the single scattering albedo and the single-term Henyey–Greenstein parameter, g. We observed that strong backscattering solutions (g = 0.45) with low single scattering albedo matched our data as well as a weaker backscattering (g = 0.15) solution with higher single scattering albedo. This most likely arose from the fact that our data set extended only to a phase angle of 5°. We therefore derived best-fit solutions for four pre-selected values of g. The dependence of final single scattering albedo best-fit solutions with chosen g value can be seen in the results in Table 7. Previous studies using a Hapke model incorporating shadow hiding only (a Hapke (1986) model) found a best fit single scattering albedo of 0.062–0.070 in Voyager clear filter (Simonelli et al., 1999) and with VIMS data (Buratti et al., 2008). Our closest V band solution (0.0712 ± 0.0022) corresponded to a strongly backscattering regolith with g = 0.45. For a solution with a phase function value closer to other published results, at g = 0.35, we found a single scattering albedo of 0.104 ± 0.003.

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Table 6 Color, albedo, and opposition surge parameters of Phoebe and selected solar system bodies. Objects

(B  I)

(B  V)

(B  R)

(V  R)

(V  I)

Dm01(V)

Dm02(B)

Dm02(V)

Dm02(R)

pv

Phoebe Phoebeb Phoebec

1.320

0.652

0.985 0.95

0.333

0.668

0.220

0.326

0.319

0.312 0.27

0.0857

Nereidd Charone Puckf Portiaf

1.43

a

⁄

0.71 0.73 0.66

1.51

19P/Borrellyg 9P/Tempel 1h 81P/Wild 2i

a b c d e f g h i j k l

0.84

C-type asteroidsj Centaursk ‘‘Small/Grey’’l

1.32–1.95

‘‘Small/Red Group’’l ‘‘Huamea Group’’l

1.97–2.53 1.33–1.51

1.15

0.44

0.72

0.54 0.55

0.203 0.265 0.22 0.23

0.85 0.88

0.35#

0.31 0.35⁄⁄

0.081⁄ 0.078⁄⁄

0.43 0.32##

0.18 0.104 0.080 0.072 0.056 0.059

0.50

0.691

0.058 0.58

0.044–0.14 0.02–0.36

0.23 0.03–0.06

0.12–0.42

0.05–0.16 0.84

0.11

This work. Bauer et al. (2006). Simonelli et al. (1999). ⁄Solution 1, ⁄⁄Solution 2. Schaefer et al. (2008), Thomas and Veverka (1991). Buie et al. (2010). #HST F435W filter, ##HST F555W filter. Karkoschka (2001). Li et al. (2007a). Li et al. (2007b). Li et al. (2009). Lumme and Bowell (1981). Bauer et al. (2003), Campins et al. (1994), Davies et al. (1993), Jewitt and Kalas (1998). Rabinowitz et al. (2007), Schaefer et al. (2008); Dm01 is measured between phase angles a = 0–1°, Dm02 is measured between phase angles a = 0–2°.

5.2. Shadow hiding angular width, hSH, and Amplitude, B0SH We allowed the shadow hiding amplitude term, B0SH, to vary as a free parameter in fitting to our Hapke model, but found that the final best fit value remained fixed at a value of 1.00 in all cases. A value of 1 for B0SH indicates perfectly opaque surface particles (Hapke, 1986). This is not an unusual result. B0SH values of 1 have been noted in fits of Hapke (2002) models incorporating coherent backscattering to the reflectivity curves of the Moon (Helfenstein

et al., 1997) and Enceladus (Verbiscer et al., 2005). We note that several previous studies with fits to Hapke (1986) models reported that B0 values exceeding unity were required to match the solar phase curve of dark bodies with strong opposition surges even though such a B0 value is not physical (Helfenstein et al., 1994, 1996; Simonelli et al., 1999). Our model did not allow values B0SH greater than 1.0. As noted above, our sensitivity analysis showed that the shadow hiding angular width, hSH, was poorly constrained by our

Table 7 Fits to the Hapke (2002) photometric model. Parametersa

B

V

R

I

0.0720 (0.006) 0.106 0.0112 (0.0178) 0.276

0.0712 (0.0022) 0.0648 0.0099 (0.0101) 0.205

0.0704 (0.007) 0.0787 0.0137 (0.0133) 0.235

0.0745 (0.0065) 0.400 0.0141 (0.0069) 0.421

0.104 (0.004) 0.0959 0.0111 (0.0189) 0.269

0.104 (0.003) 0.0586 0.0097 (0.0133) 0.194

0.102 (0.004) 0.0757 0.0141 (0.0119) 0.236

0.105 (0.007) 0.366 0.0143 (0.0067) 0.425

0.141 (0.011) 0.0924 0.0112 (0.0188) 0.269

0.143 (0.007) 0.0576 0.0100 (0.0090) 0.197

0.140 (0.009) 0.0692 0.0139 (0.0122) 0.226

0.138 (0.004) 0.360 0.0145 (0.0075) 0.429

0.185 (0.016) 0.0800 0.0107 (0.0183) 0.254

0.189 (0.016) 0.0562 0.0104 (0.0126) 0.200

0.184 (0.016) 0.0650 0.0139 (0.0161) 0.223

0.172 (0.017) 0.395 0.0148 (0.0072) 0.439

g = 0.45

-0 hSH hCB B0CB g = 0.35

-0 hSH hCB B0CB g = 0.25

-0 hSH hCB B0CB g = 0.15

-0 hSH hCB B0CB

a In all cases,  h was fixed at 33° as per Buratti et al. (2008). B0SH was a free variable, but remained at its initial value of 1.00 after fitting in all cases. hSH and B0SH, were linked such that increases in one could be compensated by a corresponding increase in the other. Errors are in parentheses.

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model given the range of phase angles represented by our data. We found that hSH and the coherent backscatter amplitude, B0SH, were linked in that increases in one could be compensated by a corresponding increase in the other to produce solar phase curves equally well fit to the data. As such, we report our best fit values in Table 7 but cannot make meaningful conclusions about any wavelength dependence. 5.3. Coherent backscatter angular width, hCB, and Amplitude, B0CB We found the angular width for coherent backscatter to be an order of magnitude narrower than that for shadow hiding. This is a similar result to that found by fitting the lunar solar phase curve to a Hapke model incorporating both shadow hiding and coherent backscatter (Helfenstein et al., 1997). The value of hCB was relatively unchanged with respect to the Henyey–Greenstein parameter for a given filter. We found a range of average hCB from 0.0100 ± 0.0113 for the V filter to 0.0144 ± 0.0071 for the I filter. The values cannot be distinguished as a function of wavelength due to the relatively large uncertainties in the final fits. The results do provide an upper bound of 0.030 over all wavelengths. We calculated the coherent backscatter FWHM angular width, Dac(1/2), by combining the relations Eq. (34) and Eq. (36) from Hapke (2002) to get:

Dacð1=2Þ ¼ 0:72hCB

ð8Þ

We found the coherent backscatter angular width to be 0.42° ± 0.55 to 0.59° ± 0.29. The angular width for our upper bound of 0.030 over all wavelengths comes to 1.24°. As noted earlier, we could not place error limits on the coherent backscatter amplitude, B0CB. Our sensitivity analysis indicates that valid values of B0CB given our model and the phase angle coverage of our data range from 0 to at least 0.6. Therefore, we can make no inference about the wavelength dependence of B0CB based on these results. 5.4. Phase integral and spherical albedo We generated solar phase curves extending to a phase angle of 180° and calculated the phase integral for B, V, R, and I filters. Fig. 5 shows a comparison of the solar phase curves to a = 180°. The left panels compare Phoebe’s solar phase curves across filters for each given Henyey–Greenstein parameter, g, and the right panels compare solar phase curves across values of g given a specific filter. With these generated curves, we calculated the phase integral, q,

for each of our sixteen scenarios. The phase integral is defined by the following relation (Horak (1950)):

q¼2

Z p

UðaÞ sinðaÞda

ð9Þ

0

where a is the phase angle and U(a) is the disk-integrated flux normalized such that U(0)° = 1. Care must be taken when interpreting phase curves that are extrapolated beyond the phase angle coverage of our data. As the phase integral contains a sin(a)da term, the value of the phase integral is particularly sensitive to the shape of the solar phase curve around a phase angle a = 90°. We found that the shapes of our solar phase curves at a = 90° varied significantly with the Henyey–Greenstein g parameter, more so than the choice of filter. The right panels of Fig. 5 illustrate the divergence of solar phase curves across values of g in all filters between phase angles of 5–180°. We have no data at phase angles in this range and we therefore cannot distinguish between these solutions. However, we can compare our calculated phase integral with other published results for Phoebe that are based on wider phase angle coverage to constrain our solution for g. Our derived phase integrals are shown in Table 8. The phase integrals increased with decreasing values of g, which corresponds to increasing backscatter. Phase integrals ranged at the low end from 0.238 ± 0.065 (V filter) to 0.291 ± 0.056 (I filter) for g = 0.45 and at the high end from 0.471 ± 0.102 (V filter) to 0.528 ± 0.058 (I filter) for g = 0.15. Simonelli et al. (1999) reported a phase integral for Phoebe of 0.24 ± 0.05 in the V band using data from Voyager 2 at phase angles that extended to 33.2°. Buratti et al. (2008) reported a value of 0.29 ± 0.03 from VIMS data in the 0.83– 5.1 lm spectral range with data at phase angles from 27.3° to 92.5°. Our best match phase integral results are those for strong backscattering with g = 0.45 when comparing our both our V-band results to Simonelli’s and our I-band results to the VIMS results. Strong backscattering (g <  0.45) solutions have also been reported for Comets 19P/Borrelly, 9P/Tempel 1, and 81P/Wild 2 (Li et al., 2007a,b, 2009). We calculated the bolometric Bond albedo based on our derived phase integrals for the BVRI filters given the same caveat that applied to our phase integral estimates, namely that both are derived from model solar phase curves extrapolated beyond our data set phase angle range. The bolometric Bond albedo, AB, is defined as:

R1 AB ¼

0

qðkÞpðkÞFðkÞdk R1 FðkÞdk 0

ð10Þ

Table 8 Derived phase integrals and spherical albedos for Phoebe in BVRI.

a

Parametersa

B

V

R

I

g = 0.45 Geometric albedo, p Phase integral, q Spherical albedo, A = pq

0.0855 (0.0031) 0.247 (0.073) 0.0211 (0.0063)

0.0857 (0.0021) 0.238 (0.065) 0.0204 (0.0056)

0.0843 (0.0020) 0.241 (0.079) 0.0204 (0.0067)

0.0839 (0.0016) 0.291 (0.056) 0.0241 (0.0047)

g = 0.35 Geometric albedo Phase integral Spherical albedo

0.0855 (0.0031) 0.320 (0.091) 0.0274 (0.0079)

0.0857 (0.0022) 0.311 (0.096) 0.0267 (0.0083)

0.0843 (0.0016) 0.314 (0.093) 0.0265 (0.0079)

0.0839 (0.0021) 0.370 (0.059) 0.0310 (0.0050)

g = 0.25 Geometric albedo Phase integral Spherical albedo

0.0855 (0.0027) 0.396 (0.101) 0.0339 (0.0087)

0.0856 (0.0016) 0.391 (0.103) 0.0334 (0.0088)

0.0843 (0.0017) 0.391 (0.094) 0.0330(0.0083)

0.0839 (0.0023) 0.449 (0.058) 0.0380 (0.0051)

g = 0.15 Geometric albedo Phase integral Spherical albedo

0.0855 (0.0023) 0.471 (0.102) 0.0403 (0.0088)

0.0856 (0.0023) 0.471 (0.087) 0.0403 (0.0075)

0.0843 (0.0020) 0.470 (0.086) 0.0396 (0.0073)

0.0839 (0.0017) 0.528 (0.058) 0.0430 (0.0048)

Errors are in parentheses.

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Table 9 Hapke parameter fits and phase function for Phoebe and selected solar system bodies. Objects a

Phoebe

a

c d e f g h i j k l

B0SH

Das(1/2) B0CB

Phoebec

0.0712 (V) 1.0 7.4 0.0745 (I) 1.0 45.8 0.104 (V) 1.0 6.7 0.105 (I) 1.0 41.9 ⁄ 0.062 (V) 3.4 2.8 0.068⁄⁄ (V) 2.0 4.4 0.07 (IR) P1 4.6

Nereidd Pucke Portiae Titanf C-type asteroidsg

0.14 0.09 0.62 0.18

19P/Borrellyh 9P/Tempel 1i 81P/Wild 2j

0.057 0.039 0.034

1.0 1.0 1.0

Enceladusk Moonl

0.995 0.279

1.0 18.1 1.0 18.4

Phoebeb

b

-0

1.0 14 4.6 5.7 1.0

Dac(1/2) g

q

0.205 0.421 0.194 0.425

0.41 0.58 0.40 0.59

0.238 0.291 0.311 0.370 0.23 0.24 0.29

0.45

0.47

0.45 0.45 0.35 0.35 0.36 0.24 0.36

0.28 0.34 0.28 0.33 0.03 0.04 0.33

4.5 1.1 1.1

0.46 0.26 0.49 0.13 0.53 0.16 0.36

0.12 2.3

0.38 0.66 0.23

This work. Simonelli et al. (1999). ⁄Solution 1, ⁄⁄Solution 2. Buratti et al. (2008). Schaefer et al. (2008). Karkoschka (2001). Schröder and Keller (2009). Lumme and Bowell (1981). Li et al. (2007a). Li et al. (2007b). Li et al. (2009). Verbiscer et al. (2005). HST F555W filter. Helfenstein et al. (1997).

where q(k) is the phase integral as a function of wavelength, p(k) is the geometric albedo as a function of wavelength, F(k) is the solar flux intensity as a function of wavelength and k is wavelength. Through numerical integration of Eq. (9), we get a bolometric Bond albedo of 0.022 ± 0.006 for g = 0.45 and 0.028 ± 0.007 for g = 0.35. These compare to a value of 0.023 ± 0.007 reported by Buratti et al. (2008) based on VIMS data.

5.5. Comparisons to other Hapke model analyses There are several examples in the literature of fitting observed solar phase curves of airless bodies to Hapke (2002) models incorporating both shadow hiding and coherent backscatter. These include results for the Moon, (a dry, low albedo regolith), Enceladus (a high albedo, H2O-dominated surface), Titan, and Nereid. Table 9 compares Hapke parameter fit results for Phoebe and selected solar system bodies. The results for Nereid, Titan, Enceladus, and the Moon are from fits to a Hapke (2002) model incorporating coherent backscatter, while the results for the other objects are based on a Hapke (1986) model that included only shadow hiding. Our results for Phoebe’s coherent backscatter angular width, on the order of 0.5°, were similar to that of Nereid, and between that of Enceladus and the Moon. Nereid’s coherent backscatter amplitude was significantly larger than Phoebe’s, consistent with the fact that Nereid’s reported opposition surge magnitude was 0.11 magnitudes greater than that we found for Phoebe (0.43 vs. 0.32 magnitudes from 0° to 2°) in the V band. The results for analyses involving Hapke (2002) models all converge to the maximum shadow hiding amplitude value, B0SH, of 1, indicating perfectly opaque surface particles for all bodies. The objects in Table 9 cover a wide range of surface particle single scattering albedos, with -0 ranging from 0.034 to 0.995, and so likely

represent surfaces with very different particle compositions. It is not clear whether this indicates similar surface particle transparency properties for all bodies studied or is rather an issue with the current model. All bodies shown in Table 9 exhibit significant backscatter, indicated by negative values of g, except the Huygens’ Probe landing site on Titan. As noted earlier, our data did not provide for a unique solution of g. However, our solutions with the strongest backscatter (g = 0.45), resulted in phase integral values, q, in agreement with previous Phoebe studies using data covering a wider phase angle range. This g value matches the strong backscattering values found for Comets 19P/Borrelly, 9P/Tempel1, and 81P/Wild 2 (Li et al., 2007a,b, 2009). As noted earlier, Shepard and Helfenstein (2007) used a Hapke (2002) model to analyze laboratory measurements of reflectance phase curves from a variety of sample surfaces. Our best fit Hapke parameter solutions for Phoebe’s phase curve were a close match to one surface in particular – that of manganese oxide. This sample consisted of dark, spectrally flat particles with grain diameters of 125 ± 56 lm and a very rough grain surface on a scale of 5 lm with a porosity of 67%. Hapke (2002) model fitting to laboratory measurements of this material taken at 700 nm (labeled MO 700 in Shepard and Helfenstein (2007)), yielded the following best fits: -0 = 0.15, hSH = 0.09, B0SH = 1.00, hCB = 0.01, B0CB = 0.45, and g = 0.30. While Shepard and Helfenstein (2007) caution that their laboratory results suggest that the Hapke (2002) model may not be a useful diagnostic tool for determining specific physical properties, we speculate that the sample particle dimensions (165 lm diameter particles with smaller scale 5 lm particle roughness) may provide a clue to the dimensional structure of Phoebe’s surface particles. Nelson et al. (2000) speculated that strong coherent backscattering effects may be due to small-scale irregularities on the surface of large aggregate particles. 5.6. Comparison of I-band solar phase curves to BVR curves As discussed previously, Fig. 5 (left panels) shows an overlay of solar phase curves for BVRI filters for each value of g. The solar phase curves for the BVR filters were in close agreement in general shape out to phase angles of 180°. However, the best fit solar phase curves for I-filter data showed a consistent deviation from the other filters. Phoebe’s opposition surge in the I-band was stronger than that in the BVR filters, though still within our error limits. For all cases studied, the I-filter curve was lower, or ‘‘bluer’’ below phase angles of 5°, and higher, or ‘‘redder’’ for phase angles above 5°. We note that the solar phase curves seem to match our I-band data well at phase angles of 2.5–3.5°; thus we cannot easily discount the validity of this result. This ‘‘phase reddening’’ effect has been observed on several other airless bodies including C-type asteroids (Lumme and Bowell, 1981), the Moon (Buratti et al., 1996), and Mercury (Warell and Bergfors, 2008). 6. Discussion Our analysis of 355 ground-based images of Phoebe in BVRI filters revealed a strong opposition surge on the order of 55% from phase angles of 0–5° with a significant coherent backscattering angular width on the order of 0.5°. Phoebe’s coherent backscatter angular width was between that of Enceladus (0.12–0.24°) and the Moon (2.3°) and was a good match to that found for Nereid (Verbiscer et al., 2005; Helfenstein et al., 1997; Schaefer et al., 2008). We found Phoebe to be spectrally gray, possibly slightly blue, with a geometric albedo ranging from 0.0856 ± 0.0023 in V band to 0.0839 ± 0.0023 in I-band. Phoebe’s opposition surge magnitude and slopes matched several outer solar system objects with

C. Miller et al. / Icarus 212 (2011) 819–834

H2O spectral signatures such as Centaurs (Bauer et al., 2003), Charon (Buie et al., 2010), and Nereid (Schaefer et al., 2008), while Phoebe’s (B  I) color of 1.320 ± 0.050 was bluer than most Centaurs and was a closer match to Charon, Nereid, and the Haumea collisional family (Schaefer et al., 2009). The interpretation of Hapke (2002) model fitting is complicated by the fact that Phoebe’s surface is not covered with a uniform regolith. Disk-resolved visible light Voyager 2 and Cassini Imaging Science Subsystem (ISS) imagery unambiguously showed that Phoebe has a variegated surface with a 4:1 ratio of normal reflectance from the darkest to brightest surface areas (Simonelli et al., 1999; Porco et al., 2005). The disk-integrated data we used to fit to our Hapke (2002) model represented the globally averaged reflectance from various terrains on Phoebe. In addition, laboratory results by Nelson et al. (2000) and Shepard and Helfenstein (2007) cast doubt on the reliability of using Hapke (2002) models to infer basic properties of a granular surface such as particle phase functions. While we presented an analysis of surface particle properties based on Hapke model fitting in this work, we acknowledge that these results are tentative subject to improvements in the Hapke model. Nevertheless, comparing phase curve Hapke parameter fits is useful for categorizing solar system objects in a general sense, especially when combined with other spectral and reflectance information such as color and albedo. Phoebe shares photometric similarities to a variety of other outer solar system bodies. Phoebe shows similarities to Centaurs based on geometric albedo and opposition surge magnitudes. Phoebe has similar color and geometric albedo to the icy satellites Nereid, Charon, Puck, and Portia. Finally, Phoebe resembles the Jupiter family Comets 19P/Borrelly, 9P/Tempel 1, and 81P/Wild 2 in albedo and the suggestion of strong surface particle backscattering, with g = 0.45. The suggestion of strong backscattering for Phoebe’s surface particles may be explained by aggregate particles with features at scales from greater than 100 lm to less than 5 lm. This is supported by results from comparing spectral mixing models by Buratti et al. (2008) for both the bright and dark regions on Phoebe to data acquired by Cassini VIMS. They reported a best fit mix of 12% 5.2 lm H2O and 88% 140 lm amorphous carbon to match the spectral characteristics of Phoebe’s bright terrain and a best fit mix of 39% 1.4 mm H2O, 2% 39 lm amorphous carbon, 41% 58 lm serpentine, 13% 70 lm Triton tholin, and 4% 207 lm CO2 for the darker terrain. Numerical simulations of the long-term evolution of cometary orbits suggest that short period Jupiter family comets originated from the Kuiper Belt (Fernandez and Gallardo, 1994). The fact that Phoebe shares photometric properties with Jupiter family Comets 19P/Borrelly, 9P/Tempel1, and 81P/Wild 2 lends credence to the theory that Phoebe was formed in the Kuiper Belt and later captured by Saturn. Phoebe’s surface has been altered by collisions, outgassing, and interactions with solar wind and radiation. The presence of a dust ring associated with Phoebe’s orbit implies that Phoebe has had a significant history of collisions (Buratti et al., 2008; Verbiscer et al., 2009). Cassini ISS imagery shows Phoebe to be heavily cratered (Porco et al., 2005). Spectroscopic analysis of Centaurs and irregular satellites suggests that the amount of surface reprocessing due to the migration history of KBO-like objects played a significant role in determining their current surface composition (Brown, 2000). Therefore, surface similarities between Centaurs, KBOs, comets, Phoebe, Nereid, Puck and the Portia group support the contention that Phoebe, and possibly Nereid, Puck, and Portia are captured KBOs. Acknowledgments We gratefully acknowledge the support of James Bauer of the Jet Propulsion Laboratory for providing us with the observational data that we used to derive our model Phoebe rotation curve. We

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