A level / IB PhysicsAstrophysicsDistance measurement by parallax
Wien's Displacement Law

^{ } L = σAT^{4} 
L = Luminosity i.e the amount of
energy radiated per unit time i.e.power (W). 
b = L / 4πd^{2} 
b = brightness i.e. energy received per unit time per
unit area (Wm^{2}) d = distance d (m) from the source, where I stands for Intensity and b stands for apparent brightness, L = luminosity (W) (see above) A more general form of this law is I = P / 4πd^{2} , where I stands for Intensity (Wm^{2}) and P stands for Power (W) 
r^{5 } = 100 ⇔ r = 100^{1/5} ≈ 2.512  r = the brightness ratio between two objects that differ by one magnitude. (An irrational number.) 
'BRIGHTNESS' 
'MAGNITUDE' (logarithmic 'brightness') 

The apparent brightness (b) of a celestial body is how much energy is coming from the body per unit area per unit time, as measured on Earth. The units are watts per square metre (Wm^{2}). 
The apparent magnitude (m) of a celestial body is another measure of how bright it is as seen by an observer on Earth. There are no units. See the formulae below. 

The absolute brightness (B) is the apparent brightness a celestial body would have were it to be placed at a distance of 10 parsecs from Earth.  The absolute magnitude (M) is the apparent magnitude a celestial body would have were it to be placed at a distance of 10 parsecs from Earth. 
(magnitude is a logarithmic measure of 'brightness'):
Exponential form  Equivalent log form  


Comparing two stars: 
b = b_{0} r^{m} 
m = –2.5 log (b/b_{0}) 
One star only: b_{0} = 2.52 x 10^{8}Wm^{2} is the reference brightness of Vega, a zero magnitude star (b has replaced b_{1} , b_{0} has replaced b_{2} , m has replaced m_{1} and m_{2} is zero) 
Exponential form  Equivalent log form  
d_{2}^{2}/d_{1}^{2} = r^{m2m1}  m_{2}m_{1} = 2.5
log (d_{2}^{2}/d_{1}^{2}) ≡ 5 log (d_{2}/d_{1}) ≡ 5 log d_{2} – 5 log d_{1} 
[1] From the Radiation Inverse Square
Law above, and the fact that L is constant for the one star, we
have [2] If you decide to make m_{1
}the absolute magnitude (M) of the object, then its distance
d_{1} will be 10pc. The log form, with
m and d the apparent magnitude and d the distance of that object,
simplifies to: If you decide to make m_{2 }the absolute magnitude, then
d_{2} will be 10pc and things are switched
round: 