Considerations regarding the colors and low surface albedo of comets using the Hapke methodology

Icarus 252 (2015) 32–38

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Considerations regarding the colors and low surface albedo of comets using the Hapke methodology Uwe Fink Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, United States

a r t i c l e

i n f o

Article history: Received 12 August 2014 Revised 11 December 2014 Accepted 15 December 2014 Available online 5 January 2015 Keyword: Comets

a b s t r a c t The single scattering albedos (SSA’s) determined for 9P/Tempel 1 are interpreted in terms of the Hapke model of irregular particle scattering efficiencies. Absorption coefficients versus wavelength from 0.31 to 2.5 lm are obtained. It is shown that the colors and exceedingly low reported SSA’s in the UV region of the spectrum below 0.4 lm cannot be reproduced with the geometric Hapke scattering model for irregular particles. However, by increasing the reported SSA’s by a small amount, absorption coefficients for particle radii of 10–100 lm versus wavelength from 0.31 to 2.5 lm can be fitted. Several reasons are given for slightly increasing the SSA’s, such as neglect of the effects of porosity, having a more complex phase function for the particles, uncertainties in the absolute calibration and the uncertainties associated with the complex treatment of surface roughness. The absorption coefficients determined show good agreement with potential surface constituents, Mg rich olivine and pyroxene with some amount of darkening iron or organic component. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction The Deep Impact mission to 9P/Tempel 1 (e.g. A’Hearn et al., 2005) has made it possible to obtain disk resolved bidirectional reflectivities as a function of wavelength from 0.31 to 2.5 lm. The portion of these data in the CCD range have been analyzed by Li et al. (2007) to obtain Hapke (1981, 1993) scattering parameters such as the surface single scattering albedo (SSA), the asymmetry parameter of the phase function, the surface roughness parameter, and the opposition width and amplitude. While Li et al. provided an excellent detailed treatment of the scattering properties of the surface in terms of Hapke parameters, he did not continue with the next step trying to relate single scattering albedos to optical constants of potential cometary surface constituents. We feel that the ultimate goal of analyzing the surface reflectivity and removing the effects of the varying conditions of illumination and phase angles must be to obtain physical single scattering albedos as a function of wavelength which can then be related to the optical constants of the scattering medium so that conclusions about the composition of the surface can be reached. If this cannot readily be accomplished, the analysis of surface reflectivity remains essentially a parameterized description. In this paper we shall attempt the determination of the optical constants that match the derived SSA’s, but the task is not straightforward since the physical mechanisms of reflection from a porous http://dx.doi.org/10.1016/j.icarus.2014.12.018 0019-1035/Ó 2015 Elsevier Inc. All rights reserved.

cometary surface are quite complex. Thus the present treatment must be considered as a first step in that process. Hence, for example, we do not consider multi component systems which can be applied later using either linear areal or intimate mixtures. For particles lying on a surface in close contact with each other, the standard approach for isolated particles using Mie scattering, or methods such as discrete dipole approximations for more complex particle conglomerates, will not work. Hapke in his treatment thus neglects diffraction, especially the large forward scattering lobe of large particles, and uses geometric optics approximations to calculate the surface scattering properties of particles. Our analysis presented in this paper is thus applicable only under this restriction. The complete original Hapke model (1981, 1993) includes a number of effects, such as surface roughness, particle phase function and the opposition effect, all of which affect and complicate the extraction of the desired SSA as a function of wavelength. The latter is the paramount parameter that relates the surface scattering to the optical constants and composition of the surface. Hapke (1986) considered porosity (i.e. the fraction of the scattering volume of the medium not occupied by particles) in his treatment and showed that to first order, porosity should have no effect on the SSA. Since this disagreed with observations which indicate that more porous surfaces are darker, Hapke (2008) updated his treatment of porosity and showed that increasing the porosity will lower the surface reflectivity. This formulation

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was not available to Li et al. (2007) so that his reported SSA’s do not include porosity effects. Unfortunately the Hapke (2008) porosity model does not converge to Hapke’s original (1981, 1993) formulation, but instead the reflectivity goes to infinity as the porosity goes to zero. It is thus not possible to connect the new porosity model quantitatively with the older model that does not include this effect (cf. our discussion in Section 3). We find in this paper that the Hapke algorithms using geometric optics for the surface scattering properties of the particles cannot reproduce the very low large area or global SSA’s reported by Li et al., in the UV region the spectrum. We therefore suggest in this paper a slight increase in these values, which results in a reasonable fit to optical constants of potential surface materials. Our rationale for slightly increasing the SSA’s is detailed in Section 3 and is based mainly on the unknown influence of porosity, as mentioned above, and uncertainties introduced into the SSA’s by simultaneously modeling surface roughness and phase effects, as well as possible uncertainties in the absolute calibration. Our slightly increased SSA’s could be considered as the local or physical SSA’s which are more closely related to the optical constants of the surface materials. Because of the limitations of the present data we do not consider changes in the phase function with wavelength which may affect the wavelength dependent SSA’s. Li et al. report that they ‘‘could not confirm’’ any phase reddening. Such effects may become apparent when a detailed analysis of the Rosetta spacecraft data is carried out. We note that our solution presented cannot be considered unique, especially if the Rosetta spacecraft lander results show that the geometric calculation of the SSAs is not a realistic approach over the wavelength region considered in this paper. Davidsson and Skorov (2002), before any spacecraft provided detailed reflection data of a comet nucleus, presented an excellent and detailed treatment of potential local or physical surface scattering layers of cometary nuclei. They used Mie scattering and discrete dipole approximation codes to investigate the scattering properties and phase functions of a variety of single particles as well as clusters of such particles. Their particles consisted of cores and mantles using various silicate, organic and icy constituents. Their results yielded SSA’s that are considerably higher than those reported by Li et al. and they had difficulty reproducing the red slope over the extended wavelength range from 0.31 to 2.5 lm as well as the backscatter asymmetry parameter reported by Li et al.

Considering the complex nature of the scattering process for the dusty comet surface, we feel that the analysis we will present is a reasonable first step in the treatment of the surface scattering of a comet, in relation to the optical constants of likely surface materials. While a low resolution, relatively smooth spectrum without unique mineral features cannot positively identify a specific dust component, it can still be used as a general constraint on the surface composition. When ices or organics with distinctive specific absorption features are observed on a comet’s surface (e.g. Sunshine et al., 2006; Capaccione et al., 2015), it can also provide a general background scattering formulation to which the ices are added either as an intimate mixture or as separate surface units. We hope that the extensive measurement which the Rosetta space mission (e.g. Glassmeier et al., 2007) to Comet 67P/Churyumov–Gerasimenko will return, will allow a much more extensive and sophisticated analysis. 2. Observed single scattering albedos for Comet 9P/Tempel 1 from 0.31 to 2.5 lm During the encounter, the high resolution instrument (HRI) and the low resolution instrument (MRI) of the Deep Impact mission (e.g. A’Hearn et al., 2005) acquired disk resolved images of the surface of 9P/Tempel 1 through a number of narrow and broad band filters. These extensive photometric data were analyzed by Li et al. (2007) using Hapke scattering theory. The derived SSA’s are plotted in Fig. 1. The SSA’s showed a fairly steep red slope of a factor of 2.4 from 0.31 to 0.95 lm. This agrees quite closely with the disk integrated spectral slope of 12.5 ± 1% per 1000 Å at an approach phase angle of 63°. The slope for the disk integrated spectra at the look-back phase angle of 117° of 16 ± 3% per 1000 Å was even steeper (Li et al., 2007). The derived SSA’s were very low indeed and probably represent the darkest objects in our Solar System. The derived single scattering albedos are listed in Table 1. How such dark surface reflectivities can be related to the surface composition is difficult to explain via scattering theories, but we shall explore this in the following sections. A spectrometer for the wavelength range 1.05–4.8 lm provided data for the infrared region. This instrument yielded surface temperature maps whose analysis is described in Groussin et al. (2007, 2013). Unfortunately there was an error in the absolute calibration of the instrument so that the radiance values published in the first paper, Groussin et al. (2007) are about a factor of two too

Single Scaering Albedos for 9P/Tempel 1 0.120

0.100

SSA

0.080

0.060

HRI values MRI values

0.040

Using Av slope (Groussin)

0.020

0.000 0.00

500.00

1000.00

1500.00

2000.00

2500.00

Wavelength nm Fig. 1. The single scattering albedo for 9P/Tempel 1. HRI values and MRI values are those derived by Li et al. (2007) from the high resolution and medium resolution camera. The values from 1000 to 2500 nm are from Groussin et al. (2013). Groussin only gives normalized values at 1.8 lm fitted to the region 1.5–2.2 lm, and reports an average slope of 3.25%/100 nm. To get absolute values of the SSA we have fitted his curve at 1000 nm to the data of Li et al. (2007).

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Table 1 Reported and scaled single scattering albedos and calculated aD values for 9P/Tempel 1. Wavelength, lm

Reported SSA

Scaled SSA

Calculated aD

0.31 0.35 0.38 0.45 0.55 0.62 0.65 0.74 0.84 0.95 1.00 1.50 2.00 2.50

0.026 0.028 0.030 0.035 0.039 0.430 0.044 0.051 0.057 0.062 0.063 0.074 0.087 0.102

0.038 0.041 0.044 0.051 0.057 0.063 0.064 0.074 0.083 0.091 0.092 0.108 0.127 0.149

6.70 4.45 3.83 3.11 2.76 2.50 2.46 2.16 1.96 1.82 1.80 1.58 1.37 1.20

high. A re-interpretation of the thermal IR region with corrected calibration was published by Groussin et al. in 2013, both for Comet 9P/Tempel 1, and for Comet 103P/Hartley 2. Neither paper reported SSA’s by analyzing the measured infrared radiance in the region from about 1–3 lm in terms of the bi-directional reflectance. The second paper gives a normalized surface reflectivity as a function of wavelength with a ‘‘typical’’ red slope of 3–3.5% per 1000 Å for both comets. This does not include regions with exposed water ice which exhibit somewhat bluer slopes. Using an average slope value of 3.25% per 1000 Å results in a factor of 1.62 increase in the SSA from 1.0 to 2.5 lm. (Both the general red slope of 3.25% per 1000 Å and the resulting SSA values agree with an earlier analysis by the present author who took the radiance values from Fig. 2 of Groussin et al. (2007), lowered them by a factor of 2, and by applying the solar flux at the quoted heliocentric distance used them to obtain I/F values, and from these estimated SSA’s in the region 1.0–2.5 lm.) To get absolute values of the SSA in the infrared we have taken the reported value of 0.064 by Li et al. at 0.95 lm and then used the reported slope of 3.25% per 1000 Å to extrapolate SSA values to 2.5 lm. Using both the visible photometric data published by Li et al. and the infrared data allows us to get SSA values over a considerable wavelength range of a factor of eight, from 0.31 to 2.5 lm. This extrapolation assumes that the other parameters such as phase reddening, discussed earlier, are not changing. The SSA values from 0.31 to 2.5 lm are listed in Table 1 and plotted in Fig. 1. We note that early results from the VIRTIS imaging spectrometer on board the Rosetta mission from 0.40 to 5.0 lm (Capaccione et al., 2015) show roughly the same slope from 0.4 to 1.0 lm as presented above as well as the change to a shallower slope beyond 1 lm. The spectra show an organic absorption feature from about 2.8 to 3.2 lm. The region beyond 2.5 lm was severely compromised by thermal emission in the Deep Impact data and was thus not considered in the present paper. 3. The Hapke approximation to the single scattering albedo The surface reflectivities of comets were analyzed by Li et al., as mentioned, using the Hapke methodology. We thus continue using this model and will try to explain the derived SSA’s in terms of the optical constants of likely cometary surface constituents. The Hapke surface scattering model is based on principles of radiative transfer as developed e.g. in Chandrasekhar (1960). However, it poses the problem how to define or determine the single scattering ~ for particles lying on a surface in contact with each other, albedo x rather than being isolated particles.

For such particles Hapke approaches this problem by neglecting diffraction and using geometric optics to calculate the SSA. This requires that the sizes of the particles (effective radius a) be considerably larger than the wavelength (k) of the scattered radiation. In other words, x > 10, where x = 2pa/k, the parameter used in Mie scattering calculations. For x = 20 this means that for the UV wavelength of 0.31 lm the effective radius must be larger than 1 lm while for the wavelength of 2.5 lm it must be larger than 10 lm. Should the cometary surface be made up of submicron particles, the Hapke model will no longer apply since such particles are too small to neglect diffraction. Most of their scattering is caused by diffraction rather than the surface geometrical reflection. There is presently, however, no surface scattering model available for this case. We note that such small particles can produce very low SSA’s. The author has considered this case but the steep red slope of the colors are very difficult to reproduce. A scattering theory would have to be developed that takes into account the interaction of the electromagnetic radiation between hundreds of interacting surface particles. For this calculation, an assumption about their shapes, sizes, distributions and orientation of the particles, as well as porosity would have to be made. Such a calculation could probably be done at some time with high speed computers using algorithms such as the discrete dipole approximation, but might be valid only for a specifically chosen type of surface. It is probably unlikely, though, that the surface layers of comets are made up of isolated submicron particles in contact since electrostatic and van der Waals forces very likely lead to clumping and aggregation. We give a brief review of the geometrical approximations developed by Hapke for the SSA’s of particles in terms of the optical constants of the surface material. The single scattering albedo ~ ðn; k; aÞ of an isolated particle, with an index of refraction n, a x radius a, and at wavelength k, is defined in terms of Q sca ; Q abs and Q ext , the scattering, the absorption and the extinction efficiencies respectively of the individual particles by

~ ðn; k; aÞ ¼ x

Q sca Q ¼ sca Q sca þ Q abs Q ext

For particles lying in close proximity on the surface of an object, Hapke makes the approximation that scattering of the particles by diffraction can be neglected. He gives several reasons for the validity of that approach. If the particles are considerably larger than the wavelength of the incident radiation, the diffracted light is limited to small forward scattering angles, which for the short distances involved to get from the topmost surface layer to the next layer, is indistinguishable from the incident light. Furthermore, particles in contact do not have space around them that is the cause of the diffracted light in the Mie scattering theory. When the particles are much larger than the wavelength, the scattering efficiency due to diffraction becomes one and the extinction efficiency approaches an asymptote of 2 (van de Hulst, 1957; Hapke, 1981). Thus, for large particles, if diffraction is neglected Q ext becomes 1 and

~ ðn; k; aÞ ¼ x

Q sca ¼ xðn; k; aÞ ¼ Q sca Q ext

To differentiate the two formulations, we stick to convention ~ for the single scattering albedo of isolated particles and use x but x for particles on a surface. Because of the above approximation for large particles, we use the terms single scattering albedo (SSA), xðn; k; aÞ, and scattering efficiency, Q sca , interchangeably in this paper. From geometrical optics, the following approximations to Q sca were developed by Hapke (1981, 1993):

xðn; k; aÞ ¼ Q sca ¼ Se þ ð1  Se ÞH

ð1  Si Þ ð1  Si HÞ

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U. Fink / Icarus 252 (2015) 32–38

Here Se is the surface reflection coefficient, or the total fraction of the incident light externally reflected into all directions, for which Hapke (1993) gives a good approximation as:

Se ffi

" ðnr  1Þ2 þ n2i ðnr þ 1Þ2 þ n2i

# þ 0:05

The index of refraction of a medium n ¼ nr þ ni i is made up of a real part nr and an imaginary part ni . The imaginary part of the index of refraction deals with the absorption of light by the medium. It is related to the absorption coefficient a (cm1) and the wavelength of light (expressed in cm) by

ni ¼

ak 4p

The quantity Si is the surface reflection coefficient for internally incident light. For spherical particles, Se ¼ Si . For irregular particles, as long as ni  1; Si can be approximated by

Si ¼ 1 

4 nðn þ 1Þ2

Several points are noteworthy in this plot. Once aD exceeds about 5 the particles become black and Q sca no longer responds to changes in their optical constants. The lowest Q sca that can be achieved for irregular particles is 0.0376, which is considerably higher than the lowest value of 0.026 reported by Li et al. For spherical particles the lowest value that can be reached is 0.109. Also for particles, with aD > 5, the lowest value for Q sca actually increases as ni (or the absorption coefficient) increases, rather than becoming darker as one might desire by increasing the absorption. The surfaces of these particles become more reflective or shiny. The above considerations pose some serious problems for interpreting the low SSA’s determined by Li et al. in terms of the Hapke model. To illustrate the problem, we show an expanded version of Fig. 2 in Fig. 3. On this figure we have marked the range of SSA’s reported by Li et al. It can be seen that it is not possible to model the SSA’s for the wavelength range 0.31–0.45 lm (SSA range 0.026–0.035). Below 0.45 lm the particles would be black and not show a red slope as reported. 3.2. Suggested SSA’s

The quantity H, is the internal transmission factor and, if internal scatterers are neglected, is related to the absorption coefficient of the medium and the particle size D by H ffi eaD . The average particle size D is closely equal to 2a. Including the internal scattering factor would add one additional parameter to be fitted and will actually make the single scattering albedos larger while we need them to be smaller to match the very low SSA’s reported by Li et al. Thus we will not include internal scattering in our treatment. 3.1. The problem of reproducing the low cometary single scattering albedos and their colors In Fig. 2 we present calculations of Q sca versus aD for a real index of refraction of 1.75 and several imaginary indices. A real index of 1.75 was chosen because it is roughly in the middle range of the index of refraction of plausible cometary surface materials, whose optical constants are discussed in a later section (cf. Table 2). The calculations presented are relatively insensitive to the real index of refraction in the range 1.5–2.5.

To interpret the colors and SSA’s we are therefore suggesting that the modeled SSA’s by Li et al. be increased by a factor of 1.46 to bring the lowest value of 0.026 to a value of 0.038, just within the range where increasing alpha will produce a red slope for the SSA’s reported. By using a constant factor for all of the SSA’s the red slope of 12.5% per 1000 Å is preserved. This is clearly a first order simplification since we ignore phase reddening and the potential invalidation of the Hapke geometrical treatment in the infrared, where substantially larger particle sizes are required. The increased suggested values are listed in Table 1. We offer several suggestions why the very dark SSA’s published by Li et al. might be higher. A major rationale for increasing the SSA’s reported by Li et al. is his omission of the effects of porosity for which a treatment by Hapke (2008) became available only after his analysis was completed. In his original treatment of porosity, Hapke (1986) derives the volume scattering SðkÞ and extinction efficiency EðkÞ for a closely packed medium with porosity w. His formulas are:

1.2000

Q sca Hapke Approx Irregular Parcles n=1.75

i=1.0E-05 i=0.10

1.0000 i=0.40 i=0.80 i=1.60

0.8000

Qsca

Spherical i= 0.01

0.6000

0.4000

0.2000

0.0000 0.01

0.10

1.00

10.00

100.00

1000.00

αD Fig. 2. The single scattering albedo plots using the Hapke model for irregular particles with a real index of 1.75 and various imaginary indices of refraction. For aD values above 5 the SSA becomes essentially irresponsive to changes in ni. The lowest Qsca that can be achieved for irregular particles with a very low ni is about 0.0376. For spherical particles the lowest achievable value is 0.109.

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U. Fink / Icarus 252 (2015) 32–38

Table 2 Optical constants of plausible surface materials. Wavelength (lm)

Reference

0.50

1.00

2.00

Mg rich pyroxene (Enstatite) Mg0.7Fe0.3SiO3

nreal nimag Abs. coeff. (cm1)

1.64 0.0057 1430

1.62 0.0027 338

1.59 0.0017 108

Dorschner et al. (AA 300, 503–520, 1995)

Mg rich olivine (Forsterite; San Carlos) Mg1.96Fe0.16Si0.89O4

nreal nimag Abs. coeff. (cm1)

1.63 2.00E05 5.86

1.63 2.30E05 24.60

1.63 5.90E05 3.80

Zeidler et al. (AA 526, A68–A78, 2011)

Carbon (Graphite; CDE theory) (Rough av. || and \ Polr.)

nreal nimag Abs. coeff. (cm1)

1.97 0.48 12,000

2.18 0.57 72,000

2.67 0.66 42,000

Edoh (1983) (Dissert. Univ. of Arizona)

Tholin

nreal nimag Abs. coeff. (cm1)

1.71 0.045 11,400

1.65 0.0012 150

1.63 0.001 45

Khare et al. (Icarus 60, 127–137, 1984)

0.2000 0.1800

Qsca Hapke Approx. Irregular Parcles n=1.75

i=1.0E-05

i=0.10 i=0.40

0.1600

i=0.80

i=1.60

0.1400

SSA published SSA suggested

Qsca

0.1200 0.1000 0.0800 0.0600 0.0400 0.0200 0.0000 0.01

0.10

1.00

10.00

100.00

1000.00

αD Fig. 3. An expanded portion of Fig. 2 emphasizing the single scattering albedo (Q sca ) values below 0.10. Also plotted on this graph are the single scattering values reported for 9/P Tempel 1 from our Table 1 for wavelengths of 0.31, 0.55 0.95 and 2.5 lm, as well as our suggested slightly higher values. It can be seen that the low SSA’s reported by Li et al. (2007), for wavelengths below 0.40 lm cannot be reproduced by the Hapke calculations for irregular particles.

SðkÞ ¼

3 ln w 3 ln w Q ðkÞ and EðkÞ ¼ Q ðkÞ 4amean sca 4amean e

Since the single scattering albedo of the medium xðmediumÞ is the ratio of the above quantities, the effect of porosity cancels out to first order. It seemed to us however physically counterintuitive that the porosity would have no effect on the scattering properties of the surface. We note that increasing porosity should affect the phase function since porous holes can only diffract light in the forward direction. In his updated approach, Hapke (2008) shows that increasing the porosity of the medium will lower the reflectivity of the surface. Thus if porosity is included in the surface scattering mechanisms, the local physical SSA values will be higher. The equations given in his paper can unfortunately only be used as a guide for the effects of porosity since his equations give unphysical results for both zero porosity and very high porosity where no scattering particles are left. A typical radiance factor or I/F value for a cometary surface at 0.50 lm is 0.014 (Capaccione et al., 2015), where I is the observed reflected intensity and pF is the incident solar flux. For such an I/F value using Hapke’s equations, which are plotted in his Fig. 3, a surface with a porosity of 0.476 yields a SSA of

0.055 while a porosity of 0.80 yields a SSA of 0.085. Thus increasing the porosity by a factor of 1.7 requires an increase in the SSA by a factor of 1.5. This is very close to the increase we used for the reported Li et al. SSA values with no porosity. Unfortunately Hapke’s (2008) model does not converge to his old model as the porosity approaches zero. For porosity values of 0.248 or lower, his equations become indeterminate and yield the nonsensical result that the surface reflectivity goes to infinity, as Hapke himself notes in his paper. Thus this porosity model probably needs further examination and re-evaluation. A second effect may be the treatment of the surface roughness. As Hapke (1993) points out, surface roughness and accompanying shadowing effects makes the observed surface darker. Hapke (1981, 1993) has developed a model that modifies his basic bidirectional reflectivity to take account of surface roughness. Li et al. have included this effect in their Hapke parameter analysis. However, the roughness model appears to be rather complex and it is unsure whether it accounts properly for the roughness effects of a complicated surface such as a comet. Thus the local SSA of a cometary surface element unaffected by roughness might be higher than a global SSA determined from a large surface area or

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for a whole hemisphere of an object from ground based observations. A third factor is the effect of the phase function. This may be illustrated by using calculations of the geometric albedo. The geometric albedo can be determined from whole disk observations using either ground based or spacecraft observations. For low reflectivity objects, Hapke (1981, 1993) gives a simple expression for the geometric albedo, pv , in terms of the medium single scattering albedo and the phase function P(0) at 0° phase angle.

pv ¼

to 10% but in the UV the uncertainty rises to 20%. It is in the UV range that we have difficulty in matching the very low observed SSA’s with the Hapke SSA geometric approximations. As an alternative to raising the SSA’s at all wavelengths, one could posit that below 0.40 lm the SSA becomes flat or ‘‘black’’ with a value of <0.038. In our present approach, however, we raise the very low SSA’s by a constant factor which will shift them into a range where they become sensitive to changes in the absorption coefficient and will thus allow an interpretation in terms of the optical constants of potential surface constituents, as discussed in the next section.

xPð0Þ 4

Li et al. reviews the observations for Comet 9P/Tempel 1 and reaches a most likely value for the geometric albedo of 0.056 ± 0.007 at 0.55 lm. This value is readily reproduced using his reported SSA of 0.039 and his single term Henyey–Greenstein phase function with an asymmetry parameter of 0.49 (P(0) = 5.73). The same pv can however also be obtained with our slightly higher visible SSA of 0.057 and an asymmetry parameter of 0.40 (P(0) = 3.93). For bi-directional reflectivity calculations at a phase angle of 63°, where most of the SSA’s by Li et al. were determined, the phase function is quite flat and changing the asymmetry parameter of the Henyey–Greenstein phase function unfortunately does not substantially change the resulting reflectivities (I/F values), or conversely the SSA’s for an observed I/F. Li et al. state that a 4% change in the asymmetry value will cause a 12% change in the SSA. However, a single term Henyey–Greenstein backscattering phase function may not be a sufficient approximation to the real phase behavior of a complex cometary surface. Finally there is the uncertainty in the absolute calibration. Li et al. state that in the visible region the calibration should be good

4. Comparison of absorption coefficients with optical constants of plausible surface components Using the suggested increased SSA’s we can apply the equations given in the previous section to determine aD values from 0.31 to 2.5 lm. We list these values in Table 1. We note that we can only determine the product of the absorption coefficient times the size of the particles. It is reasonable, though, to make the assumption that the particles on the surface of the comet are of the order of 10’s of lm. In any case, for the Hapke model to be applicable at a wavelength of 2.5 lm the particle radius must be about 10 lm or larger. We thus plot in Fig. 4 the absorption coefficient derived for D = 20 and D = 200 lm (a = 10 and 100 lm), and compare these values with optical constants of potential surface constituents to investigate whether a reasonable match with optical constants of potential surface constituents can be achieved. To make this comparison, we plot in Fig. 4 the absorption coefficients of pyroxene (Dorschner et al., 1995), olivine (Zeidler et al., 2011), graphite (Edoh, 1983) and organic tholins (Khare et al., 1984). We give a précis of the optical constants of these materials in Table 2. There is a fair amount of evidence that the major

1000000

Carbon Olivine Fe rich Fe=1.2 Mg= 0.8 Pyroxene Fe rich Fe = 0.60 Mg= 0.40 Titan Tholin Pyroxene Fe poor Fe= 0.05 Mg= 0.95 Olivine Fe poor Fe= 0.16 Mg=1.96 a = 10 um a = 100 um

Absorpon coefficients for potenal cometary surface constuents

Absorpon Coeff. α (cm-1)

100000

10000

1000

100

10

1 0

0.5

1

1.5

2

2.5

Wavelength μm Fig. 4. Plots of absorption coefficients of materials likely to be found on the surface of a comet. Remarkable is the large change of absorption coefficient in the visible from low to high iron content, a factor of 1000. Also plotted are sample curves of our derived absorption coefficients from Section 3 in our paper for irregular particles with radii of 10 and 100 lm. Our derived absorption coefficients fall in the middle of the range between Fe rich and Fe poor olivine and pyroxene, but favor the Fe poor end members corroborating other investigations as described in the text. Darkest materials are carbon (Edoh, 1983), iron rich olivine (Mg0.80Fe1.20SiO4) and pyroxene (Mg0.40Fe0.60SiO3) both from Dorschner et al. (1995). Iron poor olivine (San Carlos olivine Mg1.96Fe0.16Si0.89O4) from Zeidler et al. (2011), and iron poor pyroxene Mg0.95Fe0.05SiO3 (Dorschner et al., 1995) exhibit quite low absorption coefficients. Titan tholin has high absorption coefficients at short wavelengths, but drops to fairly low values in the IR. (Not shown in this wavelength range is the organic ‘‘tholin’’ absorption from about 2.8–3.5 lm, recently observed on 67P by the VIRTIS instrument on the Rosetta spacecraft (Capaccione et al., 2015).)

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components of cometary dust and thus surface constituents are the silicates olivine and pyroxene with the addition of organic tholins and graphite. This evidence comes from observations in the 10 lm region where silicate emission peaks have been observed for quite some time (e.g. Hanner,1983). More recently both amorphous and crystalline pyroxene and olivine features have been detected (e.g. Crovisier et al., 1997; Lisse et al., 2006, 2007; Wooden, 2008). The olivine and pyroxene appear to be of the Mg rich (Fe poor) type. The same conclusion is reached from the analysis of Interplanetary Dust Particles (IDP’s), especially the so called chondritic porous IDP’s, which are most likely cometary material (e.g. Bradley, 2003; Keller and Messenger, 2011). These data also indicate that pyroxene appears to be more abundant than olivine. Very noticeable in Fig. 4 is the very large increase in absorption coefficient by about a factor of 1000 or more in going from the Fe poor end member of pyroxene (enstatite) and olivine (forsterite) which are transparent to visible radiation to the Fe rich end members, which are totally opaque. Graphite has a very high absorption coefficient making carbon dust and naturally occurring bitumens (Moroz et al., 1998) spectrally black. Its high index of refraction also makes it quite reflective and thus unable to produce the low reported SSA’s. Organic materials, exemplified by tholins, have a high absorption coefficient in the visible but decreasing to rather low values in the IR. There is, however, considerable divergence by a factor of ten or more in the absorption coefficients of tholins when they are prepared by different methods (e.g. SciammaO’Brien et al., 2012). It is thus difficult to pick a specific material or curve of absorption coefficient versus wavelength to which the measured absorption coefficients should be matched. A small amount of iron will quickly increase the absorption coefficient of enstatite or forsterite by a factor of 10 or more. Additionally, Zeidler et al. (2011) note that any small amount of metal ion impurities such as Fe, Ni, Cr, Mn, will turn a transparent mineral from originally colorless to red and then opaque or black. Zeidler et al. (2011) have performed quantitative experiments on the mineral spinel by adding increasing small amounts of Cr. A quantitative analysis of the fraction of iron versus visible absorption coefficient for olivine or pyroxene is however presently not available, but would be useful because it could be used to constrain the iron content of the cometary components. From the above considerations, we expect that our deduced absorption coefficients should fall somewhere in the middle of the range of Fe poor and Fe rich pyroxene and olivine. The plots in Fig. 4 show that this is indeed the case. Our absorption coefficient versus wavelength shows a steep upturn in the blue to UV region and then a slow fall off towards the IR. Our upturn in the UV is not as steep as indicated by the fast rise in absorption coefficient of pure enstatite or forsterite. (This rapid upturn below 0.4 lm is caused by the onset of the intrinsic Fe2+ ions, Zeidler et al., 2011.) This could be accounted for if the SSA values in the UV region are flatter than reported, as discussed earlier, so that the particles become opaque and no longer respond to changes in the absorption coefficient nor display a steep red slope. Considering the complexity of the mineral mixture that makes up a comet’s surface and dust coma, we feel that the agreement between our determined absorption coefficient versus wavelength and the optical constants of plausible surface constituents is reasonable. A determination of the comet surface composition from broad continuum measurements can however not be made. This can only be done if distinct narrow absorption features exist as for example exhibited by ices or organics. Interestingly, Zeidler

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