Background/history:

As has long been known, dust particles in the comae of comets (along with zodiacal light and interstellar dust) exhibit differing scattering efficiencies with phase or scattering angle. In all cases, there is a strong increase in the forward direction of scattering, and a much smaller peak at small backscattering angles. Except for sun-grazing comets, nearly all comets are observed between phase angles of 0-110°, but even within this range the phase effect varies up to a factor of 3. [We define θ as the phase angle, where this is the angle between the Sun, the comet, and the observer; thus, 0° corresponds to direct backscattering seen at opposition.]

To intercompare measurements of the dust flux or A(θ)fρ values (see A'Hearn et al. 1984, AJ 89, 579) at various phase angles, either for an individual comet or a group of objects, a normalization should be applied to adjust for this phase effect. While the shape of this function depends upon a variety of properties of the dust particles, including the distribution in size, roughness, and albedo among the ensemble of grains, to first order measured phase curves are fairly similar in shape, and so a single curve can provide a good correction if comet specific information is not available.

Previous phase functions:

An early phase function for cometary dust was constructed by Divine (1981, ESA SP-174), which appears to provide a reasonable match to comet observations between phase angles of ~15-70°. However, there is considerable evidence that Divine's curve is too shallow at smaller phase angles (Ney and Merrill 1976, Science 194, 1051; Hanner and Newburn 1989, AJ 92, 254; Schleicher et al. 1998, Icarus 132, 397), and does not increase sufficiently fast at large phase angles (Marcus 2007a, ICQ April 2007). Note that the Marcus curve is also too shallow at small (<15°) phase angles, while the curve derived by Schleicher et al. for Comet Halley is not useful for large (>55°) phase angles.

For our own use (in particular to normalize all dust measurements in our narrowband photometry database of comets), I first created a composite curve by splicing together the Halley function with Divine's curve. Shortly thereafter I learned of Marcus' results, and following discussions with him to clarify some issues, I created a revised composite curve in 2010 May (the original composite with Divine was never used in publications), splicing the Halley curve at smaller phase angles with a Marcus curve at larger angles.

The Halley curve:

The phase function derived to match Halley photometry by Schleicher et al. (1998) is a quadratic fit in log Afρ vs phase angle space (see their Figure 5 and associated text): log A(θ)fρ = -0.01807 θ + 0.000177 θ², with θ in degrees.

The Marcus curve:

The phase functions produced by Marcus (2007a,b) are based on the Henyey-Greenstein function. The basic equation is given as Equation 1 in Marcus (2007b, ICQ Oct 2007, 119), while the data are shown in Figure 15 in Marcus (2008a, ICQ Apr 2007, 39) along with two model curves. These two curves use the following values for the basic parameters in the H-G function (as given in the same paragraph as the equation): g_f=0.9; g_b=-0.6; and k=0.95. There is also a parameter, δ₉₀, that is the effective dust/gas ratio in the broadband observations, and Marcus shows the curve using two values for δ₉₀ — 1 and 10. Note that the H-G function is self-normalized to a value of unity at a phase (and scattering) angle of 90°. Also note that Marcus uses the variable θ as the scattering angle, while we call the phase angle θ, maintaining our convention (see Schleicher et al. 1998).

Splicing the phase function curves:

Several steps and decisions were required to splice together the phase function curves derived for Halley and by Marcus.

Starting with Marcus' H-G curves: As discussed, Marcus used two values for the dust-to-gas ratio, δ₉₀, because of possible gas contamination in the broadband data shown in his Figure 15. Here, we just want the pure dust phase function (for instance, our narrowband filters effectively eliminate gas contamination); therefore I used a value of δ₉₀=100 when solving his Equation 1, i.e. effectively making gas contamination negligible in the derived phase function. I also converted the equation to using phase rather than scattering angle. [Note that I confirmed my implementation of the Henyey-Greenstein phase function by checking the results for δ₉₀=10.] Resulting values were computed at 1° intervals from 0° to 180°.

For the Halley phase function curve, we first directly computed values (in log space) at 1° intervals from 0° to 60° from the quadratic function discussed previously, and these were then converted to linear values.

The actual splicing was done by scaling the resulting Halley curve so as to (initially) preserve the H-G normalization. The splice point was set to 42°, a somewhat arbitrary choice since the curves are similar but different in detailed shape — the Halley quadratic should not be used past about 55°, but putting the splice point at too small a phase angle won't fit Halley as well. Ultimately, 42° was the mid-point of the viable range of scaled, overlapping curves.

The combined phase function curve:

As noted, the combined curve was normalized to 90° phase angle. However, one can scale the entire curve to normalize at any other phase angle, depending on the circumstances (for instance, Schleicher 2007, Icarus 190, 406, normalized to 41° for Comet 9P/Tempel 1 because this was the value at the time of the Deep Impact encounter). Overall, most comet observations are obtained at much smaller phase angles than 90°; therefore, the most obvious alternative normalization point is at 0°, and we also provide this version of the curve. [This renormalization is identical to that used in the original Halley phase function, and so is the equivalent of having scaled the H-G curve to Halley rather than vice versa.]