Magnitudes, or, Stupid Astronomical Systems

For this post I'm going to talk a little about the astronomical magnitude system. First, let me say that I think this system is really stupid and if I could I would eliminate it from astronomy. The only reason it exists is because it has been used for hundreds of years, but it is backwards and ridiculous.

Anyways, a star emits light. This is measured as luminosity L coming from the star, in units of energy per time, or W. Any point in space is going to receive a flux F of this light, in units of energy per time per area, or W/m^2.

For example, the Sun's luminosity L_sun is 3.839×10^26 W. The Earth is at a radius of 1 AU, or 1.5x10^11 m. This means the sphere at the radius of the Earth is 4*pi*(1 AU)^2 = 2.83x10^23 m^2. So, 3.84x10^26 W is reaching an area of 2.83x10^23 m^2, giving a flux of (3.84x10^26 W)/(2.83x10^23 m^2) = 1357 W/m^2.

This brings me to the magnitude scale, which basically tells you how bright something is compared to a reference 0 magnitude. The general equation is this:

'm' is known as the apparent magnitude of an object. This can be simplified if object 2 is considered the reference object (m_2 = 0):

A common reference is the star Vega, which has a flux in the V (visible) band of 3.55x10^-9 erg/s/cm^2/Angstrom or 3.55x10^-12 W/m^2/Angstrom. The problem is that through better measurements it has turned out that Vega actually has a magnitude of 0.035, so the actual zeropoint flux is 3.67x10^-12 W/m^2/Angstrom.

Now we can calculate the apparent magnitude of, say, the Sun. Though we calculated the Sun's flux to be 1357 W/m^2 above, that is the total flux from the Sun. The flux per wavelength in the V band for the sun is 0.184 W/m^2/Angstrom. So:

The apparent magnitude of the Sun is -26.75. The more negative the magnitude, the brighter the object. Each 5 magnitudes corresponds to a flux that is 100 times brighter, so 25 magnitudes would be 100*100*100*100*100=10 billion times brighter.

In addition to apparent magnitude there is also absolute magnitude, which is defined as the magnitude of an object if it were 10 pc away. This is an absolute measure of every object's brightness. We can calculate the absolute magnitude M of an object if we know its apparent magnitude and its distance. The relationship, known as the 'distance modulus,' can be easily derived by making m_2 in the magnitude equation the absolute magnitude, and remembering that F=L/area:

Now, for example, we can calculate the Sun's absolute magnitude. Its distance is 1 AU = 1.5x10^11 m = 4.85x10^-6 pc.

The absolute magnitude of the Sun is 4.82. Clearly, the apparent magnitude is so bright purely because we are so close to it.

Next post I'll cover AB magnitudes, so look forward to that ;)