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# Luminosity

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Luminosity

 Problem (A) Find the number which, when raised to the fifth power, equals 100. Then derive the formula relating magnitudes to fluxes. (B) Find the absolute magnitude of Sirius and the Sun. (C) Calculate the bolometric magnitude of Antares. Calculate its radius, given the corrected bolometric magnitude. Calculate the temperature of Rigil Kent. (D) Find the bolometric correction of Sirius.     Hints: For problem (B), use log to the base 10.

Data

Apparent Magnitude of Sirius

 (2.1)

Bolometric Magnitude of Sirius

 (2.2)

Distance to Sirius in Parsecs

 (2.3)

Solar Apparent Magnitude

 (2.4)

Distance to Sun in Parsecs

 (2.5)

 (2.6)

Temperature of Antares in Kelvins

 (2.7)

Corrected Bolometric Magnitude of Antares

 (2.8)

Bolometric Magnitude of Rigil Kent

 (2.9)

 (2.10)

 Useful Equations increment   (x is the number which, when multiplied by the log of 100, gives the increment.)   Absolute Magnitude from Apparent Magnitude M and Distance in PC, d     Bolometric Magnitude     Bolometric Radius     Bolometric Temperature   Bolometric Correction

Solution (A) Apparent Magnitude, m

A range of five orders of magnitude equals a difference in luminosity of 100. Find the number which, when raised to the fifth power, equals 100.

 (4.1)

 (4.2)

For base 10,

 (4.3)

 (4.4)

 (4.5)

Therefore, the difference between two apparent magnitudes (m,n) is related to their fluxes (Fm, Fn) by

Solution (B) Absolute Magnitude, M

Absolute magnitude is equal to the apparent magnitude a star would have at a distance of ten parsecs.

 (5.1)

To find the absolute magnitude of Sirius (apparent magnitude = -1.44; distance = 2.64 pc):

 (5.2)

 (5.3)

The Sun's apparent magnitude is -26.8 and it is 1/206265 parsec from Earth.

 (5.4)

 (5.5)

The Sun's absolute magnitude is 4.8.

Solution (C) Bolometric Magnitude

The magnitude calculated over all wavelengths is the bolometric magnitude. The formula for the absolute bolometric magnitude in terms of the solar radius is

Example: Calculate the absolute bolometric magnitude for Antares, a star with 800 times the solar radius and a temperature of 3,500 K.

 (6.1)

The absolute bolometric magnitude of Antares is given here as -7.5. (Antares varies in luminosity, and its absolute bolometric magnitude is usually taken to be about -7.2.)

Example: Calculate the radius of Antares, given the bolometric magnitude (-7.2) and the temperature (3500).

 (6.2)

This is an under-estimate. The radius of Antares is actually about 800 times the radius of the Sun. The outer layers of the star are being blown off, which makes calculating the radius difficult.

Example: Calculate the temperature of Rigil Kent (Alpha Centauri A), given the bolometric magnitude (4.35) and the radius (1.227 times the solar radius).

 (6.3)

This is a slight overestimate. The temperature of Rigil Kent is generally taken to be 5,790 K.

Solution (D) Bolometric Correction

The bolometric correction is a number added to the apparent magnitude to compensate for discrepancies caused by significant amounts of luminosity outside the visible range. Therefore, the correction is largest for stars that radiate primarily in the infrared or ultraviolet.

 (7.1)

Example: Sirius

Apparent magnitudes:

 (7.2)

 (7.3)

 (7.4)