Measuring comet magnitudes on CCD images – Part II, Afρ
Updated 2017 January 14
The parameter Afρ (pronounced Af-rho) was introduced in 1984 by Michael A'Hearn et. al. to describe the brightness of the cometary dust comae . This quantity widely used to compare measurements of the dust produced by and surrounding a comet made under different geometric circumstances, at different times and using different instruments. It is useful for long-term studies of the behavior of the dust comae because it removes variations introduced by changing geocentric and heliocentric distances and aperture sizes.
Afρ is defined as the product of the albedo of the dust grains (A) within the coma, a filling factor (f) of the grains within the photometric aperture and the radius (ρ) of that aperture projected to the distance of the comet (In Part I in the 2010 April issue of The Astronomer I incorrectly stated that ρ was the density of the dust grains – a reversion to schooldays when I believe ρ was the symbol for density). The filling factor is the total cross section of the dust grains divided by the area within the aperture. The quantity Afρ defines the height, usually measured in cm, of a cylinder of base area equivalent to the projection of the aperture used for photometry at the distance of the comet from Earth completely filled with dust particles – Figure 1. An Afρ measurement of 100 centimeters equates to about 100 kilograms of dust produced per second.
Figure 1. Parameters used to calculate Afρ (based on a similar diagram by M.
Derivation of the equation for Afρ
The total cometary flux, F, as shown in Figure 2, is given by the equation;
F = where;
A = the albedo of the dust grains
N = number of dust grains within the aperture
σ = cross sectional area of a single grain
Fsun = solar flux at 1 AU
r = sun – comet distance in AU
The flux falling on the Earth per unit area, Fcomet, is given by the equation;
Fcomet = where F is the total cometary flux and 4πΔ2 is the surface area of a sphere of radius Δ (the Earth – comet distance).
Figure 2. Cometary flux
Substituting for F we get;
Fcomet = which can be rearranged as;
The filling factor, f, is the total cross section of the dust grains within the aperture divided by the area of the aperture i.e. f = Therefore;
Af = x = and
This example uses data from observations of comet 29P/ Schwassmann-Wachmann with the Sierra Stars Observatory Network 0.61m reflector sited on the east side of the Sierra Mountains in Alpine County, California, USA The report below was generated using Astrometrica and then FoCAs as previously described.
TEL 0.61-m f/10 reflector + CCD
COM USNO-A.2 used for photometry
0029P C2010 03 22.18161 09 19 27.48 +13 17 35.0 15.88N G68
0029P C2010 03 22.20221 09 19 27.14 +13 17 36.2 15.90N G68
0029P C2010 03 22.22304 09 19 26.82 +13 17 37.0 15.89N G68
0029P C2010 03 22.24388 09 19 26.47 +13 17 38.1 15.88N G68
0029P C2010 03 22.26471 09 19 26.13 +13 17 39.1 15.88N G68
CATALOG: USNO A2.0 /
30x30 40x40 50x50
OBJECT DATE TIME
+/- +/- +/-
+/- +/- +/-
------ ---------- -------- ----- ----- ----- ----- ----- ----- ---- ---- ---
FoCAs II -
At the time of the observations;
Earth – comet distance (Δ) = 5.4377 AU = 5.4377 x 149,597,870 kms = 8.1347 x 108 km
Sun – comet distance (r) = 6.2010 AU
ρ – aperture size (radius) at the distance of the comet. Originally FoCAs used a square 10x10 arc sec aperture (as previously described) giving an area of 100 arcsecs2 but now use a circular aperture. The angle, α, subtended by the radius of a circle of the same area is given by the equation;
α = = 5.64 arc secs. Therefore ρ = 8.1347 x 108 x tan 5.64 = 22243 km
The magnitude of the comet used in this example = 15.89 which is the average of the five observations and shown in the Multibox report above under the 10x10 heading.
The sun’s apparent visual magnitude = -26.7.
To convert magnitudes to flux we must seek the help of N. R. Pogson who proposed that a difference of 5 magnitudes should correspond to a difference in brightness of 100 between the two objects concerned. If the brightness of two stars is B1 and B2, the difference in their magnitudes, m1 and m2, is given by the equation;
which can be rewritten; m1 – m2 = - 2.5 log10
a measure of the energy received, luminous flux or the rate of flow of photons,
and is measured in Lumens (
m1 – m2 = - 2.5 log10 and therefore so;
= 9.1868 x 10-18
Using the above values;
Af ρ = = = 0.04203 km = 4203 cms
Log Afρ = Log 4203 = 3.623
As mentioned previously, this calculation of Afρ is based on an observational method developed by a group of Spanish amateur astronomers and the data, an example of which is shown in Figure 3, can be viewed on-line at http://astrosurf.com/cometas-obs/
A cautionary note – close in to the Sun active comets show very strong Swan band emissions (named after the Scottish physicist William Swan), e.g. C2, C3, CN and CO, and the use of unfiltered photometry in such instances can lead to AFRho values that are too large. This problem could be overcome by imaging bright comets using an R filter but for the vast majority of comets observed by amateurs Swan emission is not a problem since it is not strong enough to affect photometry obtained from unfiltered images.
Figure 3. Magnitude and AfRho measurements for comet 29P/Schwassmann-Wachmann
The actual dust production rate can be calculated knowing Afρ and the velocity, bulk density, geometric albedo and scattering fuction of the dust grains. Comets not only throw off dust but all manner of molecules and even quite large pieces of their surface material. Some cease such activity temporarily or permanently and become dormant or dead comets and some, for example sun-grazers, disintegrate completely but all of that is another story.
1. A’Hearn M. F. et. al., ‘Comet Bowell 1980b’, The Astronomical Journal, 89(4), 579–591 (1984)