Measuring comet magnitudes on CCD images – Part II, Afρ

 

Updated 2017 January 14

 

Contents

 

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 Afrho parameters

 

Figure 1. Parameters used to calculate Afρ (based on a similar diagram by M. Müller and E. Grün)

 

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

 

Figure 2. Cometary flux

 

Substituting for F we get;

 

Fcomet = 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

 

COD G68

OBS R.Dymock

MEA R.Dymock

TEL 0.61-m f/10 reflector + CCD

AC2 roger.dymock@ntlworld.com

ACK G68_2010_03_22-1

COM USNO-A.2 used for photometry

 

NET USNO-B1.0

 

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

 

COD G68

OBS R.Dymock

CATALOG: USNO A2.0 / CMC-14 - BAND: R

 

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

OBJECT    DATE      TIME     +/-    +/-    +/-    +/-    +/-    +/-     N   FWHM  CAT

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

29P    22/03/2010 05:21:15  15.89  14.99  14.39  13.95  13.60  13.34  23.2  19.9  G68

29P    22/03/2010 05:21:15*  0.01   0.01   0.02   0.02   0.02   0.02     5   3.5  USN

 

FoCAs II - 17/03/2010

 

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

Brightness is a measure of the energy received, luminous flux or the rate of flow of photons, and is measured in Lumens (Watts in old money) so the above equations can also be written;

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 Mag and Afrho plots

 

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)

 

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