Answer to Question #173604 in Astronomy | Astrophysics for ANJU JAYACHANDRAN

Question

1b) What is meant by the apparent magnitude of a star? How is it related to the

brightness of the star? An object A has an apparent magnitude of –5. Another

object B has an apparent magnitude of –10. Calculate the ratio of their brightness.

Answer 1

The apparent magnitude $m$ is a measure of brightness $E$ (the illuminance is the measure of brightness):

$m = -2.5\log E + \mathrm{const}$, so the brighter is object, the smaller is the magnitude. The system of magnitudes was created in the Ancient Greece, the stars were divided into classes from 1 (the brightest) to 6 (the faintest, but still visible). But later the system was generalized, for example, Sirius has $m\approx -1.46.$

Worth noting, the brightness is related to the luminosity of the object (the electromagnetic power of its radiation) and to the distance from the observer to the object.

According to formula introduced by Pogson,

$m_1 - m_2 = -2.5 \log\dfrac{E_1}{E_2}$ .

In our case $m_1 = -5, \; m_2 = -10,$ so

$-2.5\log \dfrac{E_1}{E_2} = (-5) - (-10) = 5,$ so $\log \dfrac{E_1}{E_2} = -2,$ $\dfrac{E_1}{E_2} = 10^{-2} = 0.01.$