Measuring comets with the Afρ quantity

Afρ, is a quantity introduced by Michael A'Hearn et al. in 1984 (AJ 89, 579, 1984) with the aim of comparing measurements concerning the dust continuum under different observing conditions, times and instruments.

The Afρ is simply the product between the albedo “A” (the reflectivity of grains), the filling factor “f” (optical density of the coma) and “ρ” (the Greek letter “rho”) that is the radius of the coma under investigation. The obtained number is a linear data (cm or metres) that is directly related to the total cross section of the grains and is often used as a proxy of dust production rate.

Assuming a constant dust reflectivity, the Af product (the power between the albedo and the filling factor) is not constant as the optical density of the coma gets its maximum close to the nucleus and then fades toward the outer edges. But the optical density of a theoretic isotropic coma made only of dust and where the grains have an uniform expansion speed (the so called “stationary coma model”) decreases with an inverse power law (proportional to 1/r, where r is the distance from the nucleus). Powering the product Af and the parameter ρ gives as a result a constant value theoretically at any indipendent from the adopted distance from the nucleus. The fact to have a constant value over a wide range of distances is a relevant characteristic of making the Afρ quantity because it is not affected (or very little affected) by instrumental characteristics and to the different image-scale or measuring window sizes used in photometry. This was very useful in the past reducing narrowband photoelectric photometry data, but also today in measuring CCD images. Of course the “stationary model” is a very rough approximation of a coma but we can easily check that many comets display a nearly constant Afρ value over a wide range of radius values ρ. Despite it is far away from the real situation, it represents an useful reference model for studying cometary comae.

Our goal is to observe the dust component of the comet and this can be done with proper filters as explained below in the specific sections. If we know the intensity of solar flux, the distance of the comet from the Sun and from our planet, we can get an accurate information on how much light is reflected from dust. We do this calculating the Afρ quantity.

About the filling factor ...

The filling factor, is a quantity related to the optical density of the coma, and strictly speaking, is how much the total cross section of grains fills the field of view. It is the ratio between the total area covered by dust in the considered field of view and the area of the field of view itself.
It is quite impressive to see that most relatively faint periodic comets, currently observed from amateur astronomers, display Afρ values of few tenths of centimetres, while many comets of average brightness fall around some meters values. A really great difference from the Hale-Bopp comet that indicatively approached to an Afρ quantity of 10 km (1,000,000 cm) around the epoch of its perihelion passage! Another recent impressive case was the Sun-grazer comet 1996 P1 (McNaught) that displayed an even higher Afρ value at perihelion.

It appears that these results are really amazing and allow us to well understand the basic meaning of the Afρ quantity.  It can tell us much more about comets, but , of course, with a bit more complex analysis. In this perspective it is important to remember that comets are not made of dust only and gas is always present. This make the Afρ analysis more difficult, and not always possible, except if appropriate narrowband filters for isolating selected spectral regions (with negligible gas contamination) are used or  with wide-band filters in the cases where the dust has really a dominant role in the brightening process of the comet and the gas contribution can be ignored.