Answer to Question #87080 in Astronomy | Astrophysics for AMIT

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

The apparent magnitude of a star is a value that represents the brightness of this star as seen from Earth. The apparent magnitude alone can't be used for determination of luminosity of the star but can be used for determination of the relative brightness of two objects. The difference of the apparent magnitudes equal to 1 corresponds to the difference in brightness equal to about 2.5 times. Which means:

$$m_1 - m_2 =-2.512log\frac {I_1} {I_2}$$

where m₁ and m₂ - apparent magnitudes, log - decimal logarithm, I₁ and I₂ - brightness of the stars 1 and 2 respectively. As we can see, the apparent magnitude scale is logarithmic and have reversal dependence: brighter star will have lower apparent magnitude value.

Let's use this formula for given values of apparent magnitudes:

$$(-5) - (-10) =-2.512log\frac {I_1} {I_2}$$ $$-1.99 = log\frac {I_1} {I_2}$$

According to the definition of decimal logarithm, we can transform the above equation into:

$$\frac {I_1} {I_2} = 10^{-1.99} \approx 0.01;$$

Answer: The object A is 100 times dimmer than object B.