Pogson's Law

Pogson's Law

This post is one in three of a series on Astronomy. It determines the distances of galaxies by looking at their magnitude

Magnitude

Stellar brightness is based on a convention first devised by Hipparchus. The brightest stars that can be seen with the eye have a magnitude of 1.0; the faintest have a magnitude of 6.0. Due to the invention of the telescope, stars with negative apparent magnitudes can be seen. It is important for us to note that the more negative the apparent magnitude, the brighter a star appears.

This is for the following reason: the human eye perceives equal ratios of brightness at equal intervals. This means the difference in brightness of a 100W light bulb to a 200W light bulb would appear the same as the difference.

On the original scale, the flux (W/m^2) from stars of the first magnitude was about 100 times greater than that from stars of the sixth magnitude. Thus a difference of 5 magnitudes gives a flux ratio of 100. A magnitude difference of 1 would therefore correspond to a flux ratio of (100)^(1/5) = 2.512.

This can be re-formulated (as was by Pogson) into the mathematical definition: 

m(1) - m(2) = -2.5 log [f(2)/f(1)]

Where: m(1) and f(1), and m(2) and f(2) are the apparent magnitudes and fluxes of star 1 and star 2, respectively.

To estimate stellar and galactic distances, an absolute scale must be established. This can be done by looking at the consequences of having all stars equidistant from Earth, meaning the difference in stellar and galactic luminosity would be the only factor affecting the luminosity. The absolute magnitude, M, is therefore the apparent magnitude a star or galaxy as if it were 10 parsecs from Earth.

Assuming that the light is spread evenly by the star or galaxy in question, then the flux can be determined by the equation: f(10) = L/4(pi)(r^2), where r is 10 parsecs, and f(10) is the flux a star would have at that distance, and L is the luminosity of the star.

Thus: m-M = -2.5 log [f/f(10)], and f/f(10) = ((10^2)/(r^2)) = (10/r)^2

We can now re-write Pogson's law as: m-M = -2.5 log[(10/r)^2]
= -5 log(10) + 5 log r

Thus: m - M = 5 log r - 5

Which can be re-arranged to: r = 10^[(m+27)/5]

QED

Therefore, we can use the apparent magnitude of distant stars to show their distance from Earth in parsecs. For example, the galaxy coma2 has the apparent magnitude of 12.55 and thus is 81.28 Mega-parsecs from Earth.