__Measuring comet magnitudes on CCD
images – Part II, Afρ__

Updated 2017 January 14

**Origins**

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 [1]. 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.
Müller and

**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

F_{sun}
= solar flux at 1 AU

r = sun –
comet distance in AU

The flux
falling on the Earth per unit area, F_{comet}, is given by the
equation;

F_{comet}
= _{} where F is the total
cometary flux and 4πΔ^{2} is the surface area of a sphere of
radius Δ (the Earth – comet distance).

**Figure 2**. Cometary
flux

Substituting
for F we get;

F_{comet}
= _{}which can be rearranged as;

A = _{}

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

Afρ = _{}

**Example calculation**

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.

MPC report

OBS R.Dymock

TEL 0.61-m f/10 reflector + CCD

AC2 roger.dymock@ntlworld.com

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

Multibox
report

OBS R.Dymock

CATALOG: USNO A2.0 /

10x10 20x20
30x30 40x40 50x50
60x60

OBJECT DATE TIME
+/- +/- +/-
+/- +/- +/-

------ ---------- --------
----- ----- -----
----- ----- -----
---- ---- ---

29P

29P

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 10^{8
}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 arcsecs^{2
}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 10^{8} 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 B_{1} and B_{2}, the
difference in their magnitudes, m_{1} and m_{2}, is given by
the equation;

_{} which can be
rewritten; m_{1} – m_{2} = - 2.5 log_{10}

Brightness is
a measure of the energy received, luminous flux or the rate of flow of photons,
and is measured in Lumens (

m_{1}
– m_{2} = - 2.5 log_{10 }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. C_{2, }C_{3}, 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

**Conclusion**

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.

References

1. A’Hearn
M. F. et. al., ‘Comet Bowell 1980b’, *The
Astronomical Journal*, **89**(4),
579–591 (1984)