Star Magnitude and Flux

Luminosity

The Luminosity(L) of an object is the amount of energy emitted in one second in the form of light. This is measured in Joules($sec^{-1}$) or Watts For the majority of stars light is emitted isotropically(in all directions) with a few exceptions such as pulsars.

Flux

The flux(f) is the amount of light an observer measures at a distance from the object this can also be called the flux density. This is the power observed at a distance(d) in one second in an area of $1m^2$.

Units are given in: Watts($m^{-2}$)
This formula can be derived by the fact that the light spreads over a sphere in equal proportion(the inverse square law).

Comparing Flux

The energy passing through a unit area decreases with the square of d. Thus we can can compare flux and distance as below.

Discussion of Scales

The units of Watts($m^{-2}$) may be inconvenient to use on an astronomical scale. Sometimes a unit called the Jansky(Jy) $= 10x10^{-26} Wm^{-2}Hz^{-2}$. These are linear scales which is useful for very small numbers but is not very convenient when dealing with large ranges. For this reason astronomers often use a logarithmic scale in terms of magnitudes.

The Magnitude Scale

Stars are divided into 5 classes ( $1 \rightarrow 6$ ) with class 1 been the brightest. This scale was derived from visual observations by classical astronomers therefore the classes $1 \rightarrow 6$ represent most the visible stars we can see from earth. When this system was devised the concept of 0 was not known but nowadays we use 0 as the reference class (Vega is an example of a class 0 star).

The Difference of 5 classes is 100x in flux but the human eye sees logarithmically for example going from class $1 \rightarrow 2$ is going to a star about half as bright.
1 magnitude $\equiv$ factor of $100^{1/5}$ difference in flux or 2.5 this ratio is called Pogson's Law

Apparent Magnitude

Based on the above we can compare stars apparent magnitudes or flux of 2 stars.

Note: log is used as log to the base 10 ln will be used to represent the natural logarithm.
Linear scales make sense for systems which detect light linearly like image sensors. Logarithmic scales are useful for systems like our eyes or photographic film which record light logarithmically.

Absolute Magnitude

We can define an an absolute magnitude(M) to be the apparent magnitude of a star at a distance of 10 parsecs from us. This is useful as it will allow us to compare simmilar stars based on their luminosity rather than what we percieve their brightness to be.

This equation is called the distance modulus equation and we can use this to get the absolute magnitude of an object at a known distance(d).