Answer to Question #122788 in Astronomy | Astrophysics for pikacho

Question

Question 3. Magnitude of Stars
The star Sirius has an apparent magnitude of -1.46 and appears 95-times brighter compared to the more distant star Tau Ceti, which has an absolute magnitude of 5.69.

(Distance Orbital Period)
Mercury 0.39 AU 88 days
Venus 0.72 AU 225 days
Earth 1.00 AU 365 days
Mars 1.52 AU 687 days


(a) Explain the terms apparent magnitude, absolute magnitude and bolometric magnitude.
(b) Calculate the apparent magnitude of the star Tau Ceti.
(c) Find the distance between the Earth and Tau Ceti.

Answer 1

(a) Apparent magnitude is a measure of brightness of an astronomical object (see https://en.wikipedia.org/wiki/Apparent_magnitude). It depends not only the luminosity of object, but also on the distance between the object and observer.

Absolute magnitude is the measure of the luminosity of the object. It is equal to the apparent magnitude from the distance of 10 parsecs.

These magnitudes can be defined for different wavelength ranges and they are measured on an inverse logarithmic scale.

The bolometric magnitude depends on the radiation at all wavelengths (see https://en.wikipedia.org/wiki/Absolute_magnitude#Bolometric_magnitude)

(b) We may apply Pogson's ration for fluxes from stars (see https://en.wikipedia.org/wiki/Apparent_magnitude):

$\dfrac{F_S}{F_T} = 10^{0.4(m_T-m_S)}$ , $m_T = m_S +2.5\lg\dfrac{F_S}{F_T} = -1.46+2.5\lg95 = 3.48.$

(c) The absolute and apparent magnitudes are related by the formula

$m=M-5+5\lg r, \; \lg r = 0.2(m-M+5), \;\; r = 10^{0.2(m-M+5)} = 10^{0.2(3.48-5.69+5)} = 3.6\,\mathrm{pc}.$