All the discussion of luminosity, brightness, and energy flux (Section 17.4) can equally well be written in terms of magnitudes. Here we present a little more detail on the “magnitude versions” of two important topics—stellar luminosity and the inverse-square law. Recall that a star’s absolute magnitude is the apparent magnitude we would measure if the star were located at a standard distance of 10 pc from us. Absolute magnitude is equivalent to luminosity—an intrinsic property of a star. Given that the Sun’s absolute magnitude is 4.85, we can construct a “conversion chart” (shown) relating these two quantities. Since an increase in brightness by a factor of 100 corresponds, by definition, to a reduction of a star’s magnitude by five units, it follows that a star with luminosity 100 times that of the Sun has absolute magnitude 4.85 - 5 = -0.15, and a 0.01 solar luminosity star has absolute magnitude 4.85 + 5 = 9.85. We can fill in the gaps by realizing that one magnitude corresponds to a factor of 100^{1/5} 2.512, two magnitudes to 100^{2/5} 6.310, and so on. A factor of 10 in brightness corresponds to 2.5 magnitudes. You can use this chart to convert between solar luminosities and absolute magnitudes in many of the figures in this and later chapters. EXAMPLE: Let’s calculate the luminosity (in solar units) of a star having absolute magnitude M (the conventional symbol for absolute magnitude, not to be confused with mass!). The star’s absolute magnitude differs from that of the Sun by (M - 4.85) magnitudes so, following the above reasoning, the luminosity L differs from the solar luminosity by a factor of 100 ^{- (M- 4.85)/5}, or 10 ^{-2 (M- 4.85)/5.} We can therefore write L (solar units) = 10^{ -0.4 (M - 4.85)}. Plugging in some numbers (taken from Appendix 3), we find that the Sun, with M = 4.85 of course has L = 10° = 1. Sirius A, with M = 1.4 has luminosity 10^{1.38} = 24 solar units, Barnard’s Star, with M = 13.2 has luminosity 10^{-3.34} = 4.5 10^{-4} solar units, Betelgeuse has M = -5.5 and luminosity 14,000 Suns, and so on. We can also invert the above relationship and write M = 4.85 - 2.5 log_{10} L (solar units) (where the logarithm function is defined by the property that if a = log_{10}(b), then b = 10^{a}). Thus, a star (Vega) of luminosity 55 times that of the Sun has M = 4.85 - 2.5 log_{10} 55 = 4.85 - 2.5(1.74) = 0.5, one of 0.3 solar luminosities ( Eridani) has M = 4.85 - 2.5(20.52) 5 6.1, and so on. The inverse-square law can also be recast in these terms. Increasing the distance to a star by a factor of 10 decreases its energy flux by a factor of 100 (by the inverse-square law) and hence increases its apparent magnitude by five units. Increasing the distance by a factor of 100 increases the apparent magnitude by 10. Since absolute magnitude is a measure of the energy flux received from a star at a distance of 10 pc, denoting the apparent magnitude by m and the distance by D, we can write: The difference m - M between the apparent and absolute magnitudes of an object is usually referred to as its distance modulus. Notice once again that, if D is greater than 10 pc, then m is greater than M, and vice versa. In terms of distance modulus, we can rearrange the preceding equation and rewrite it as:
This doesn’t look much like the inverse-square law presented in the text! Nevertheless, it contains exactly the same information, and it is in widespread use in this form throughout astronomy. EXAMPLE: Again using stars listed in Appendix 3, the star Rigel is observed to have apparent magnitude m = 0.14 and has a measured distance (using parallax) D = 240 pc. From the preceding equation, its absolute magnitude then is M = 0.14 - 5 log_{10 }24 = -6.8, corresponding to a luminosity 46,000 times that of the Sun. Conversely, the star Rigel Kentaurus (Alpha Centauri) has absolute magnitude M = 4.4 and is observed to have apparent magnitude m = 0. Its distance must therefore be D = 10 pc 10^{ (0-4.4)/5} = 1.3 pc (in agreement with the result obtained by parallax!). |