## Stars

### Star Magnitudes

The apparent magnitude m is a measure of the amount of light arriving on Earth from a star or other celestial objects. The brighter object has a smaller apparent magnitude. This curious property of "less is more" is a result of the work of Hipparchus (c130 BC) who classified stars into six magnitudes. The ‘first magnitude stars’ were the brightest in the heavens, which included Capella (alpha Aurigae), Sirius (alpha Canis Majoris), Vega (alpha Lyrae) and the like. Hipparchus categorized the other stars according to their relative brightness, down to the dimmest that the naked eye could see, which were called sixth magnitude.

In simple mathematic the apparent magnitude is defined by the formula:

m = -2.5xlog(I/I_{o}) ---------------------------------------- (1)

where I = L/(4D^{2}) is the intensity (apparent brightness), L is the luminosity (intrinsic brightness), D is the distance to the object, and I_{o} = 2.52x10^{-5} erg-sec^{-1}-cm^{-2} is the intensity corresponding to m = 0.

Thus the apparent magnitude can be simplified to:

m = -2.5xlog(I) - 11.5 ---------------------------------------- (2)

Since the absolute magnitude M is defined as the magnitude of an object at a distance of 10 parsecs by the formula:

M = 4.8 - 2.5xlog(L/L_{sun}) ---------------------------------------- (3)

where L_{sun} = 3.86x10^{33} erg/sec; the apparent magnitude in Eq.(2) can be rewritten in terms of the absolute magnitude M and distance D (in cm):

m = M - 97.5 + 5xlog(D) ---------------------------------------- (4)

Conversely, the distance D can be deduced from Eq.(4) if both the absolute and apparent magnitudes M and m are known.

Substituting I = 1.4x10^{6} erg-sec^{-1}-cm^{-2} into Eq.(2) for the Sun, we obtain the Sun's apparent magnitude m = -26.8; the same result can be derived by applying Eq.(4) with M = 4.8 and D = 1.5x10^{13} cm. ( 1 AU) for the Sun.

Followings is the apparent magnitude for some common objects:

m_{Full Moon} = -12.6,

m_{Venus} = -4.4,

m_{Sirius} = -1.46 (the brightest star).

While naked-eye limit is about m = +6, the faintest object detectable by HST (Hubble Space Telescope) extends to an apparent magnitude +30.
.