Answer to Question #87670 in Electricity and Magnetism for Shivam Nishad

Question

Explain what you understand 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

Apparent magnitude of a star is a measure of its brightness in logarithmic units. By conventional definition, apparent magnitude $m$ is related to the observed flux density $F$ of light from the star (which is its brightness) by the formula

$$m = - 5 \log_{100} \frac{F}{F_0} = - 2.5 \log_{10} \frac{F}{F_0} \, ,$$

where $F_0$ is the reference flux density corresponding to zero apparent magnitude. Since the flux density (brightness) for a star can be measured in different spectral bands (ultraviolet, visible, infrared etc.), one can speak of the apparent magnitude in the corresponding spectral band. For two objects A and B with apparent magnitudes $m_A$ and $m_B$, respectively, we have

$$m_A - m_B = - 5 \log_{100} \frac{F_A}{F_0} + 5 \log_{100} \frac{F_B}{F_0} = 5 \log_{100} \frac{F_B}{F_A} \, .$$

Thus, for $m_A = - 5$ and $m_B = - 10$, we obtain $5 \log_{100} \left( F_B / F_A \right) = 5$, or $\log_{100} \left( F_B / F_A \right) = 1$. Hence, the ratio of their brightness is $F_B / F_A = 100$.