Part 4: Luminosity of a star#
Intrinsic energy radiated per second#
Luminosity is an intrinsic value which does not depend upon distance. It is a measure of the total energy radiated by a star at a given wavelength per second. The total energy radiated at all wavelengths is called bolometric luminosity.
If you know the distance to a star, and how bright it appears be (magnitude), you can calculate its luminosity.
Luminosity of Tau Cygni#
As an illustration, according to SIMBAD, the star Tau Cygni has a parallax value of \(0.04916\) arcseconds, which corresponds to a distance of \(D=1/0.04916=20.34\) parsecs, and has an apparent magnitude flux of \(3.730\) measured through a V filter (visual).
Luminosity and absolute magnitude are related as follows:
Equation 1: \(M=-2.5\log_{10}(\frac{L}{L_0})\)
where \(M\) is the absolute magnitude, \(L\) is luminosity, and \(L_0\) is the zero point luminosity recommended by the International Astronomical Union. It is the luminosity at which a detector reads 1 photon count per second at 10 parsecs: \(L_0=3.0128\cdot 10^{28}\; W\)
Absolute magnitude can be calculated from distance and apparent magnitude as follows:
Equation 2: \(m=M-5+5\log_{10}(D)\)
\(3.730=M-5+5\log_{10}(20.34)\)
\(3.730-5\log_{10}(20.34)=M-5\)
\(M=3.730-5\log_{10}(20.34)+5\)
\(M=3.730-6.54+5\)
\(M=2.19\)
Plugging \(M=2.19\) into the first equation, we get the following luminosity:
\(M=-2.5\log_{10}\left(\frac{L}{L_0}\right)\)
\(2.19=-2.5\log_{10}\left(\frac{L}{3.0128\cdot 10^{28}}\right)\)
\(\frac{2.19}{-2.5}=\log_{10}\left(\frac{L}{3.0128\cdot 10^{28}}\right)\)
\(10^{\frac{2.19}{-2.5}}=\frac{L}{3.0128\cdot 10^{28}}\)
\(L=10^{\frac{2.19}{-2.5}}\cdot 3.0128\cdot 10^{28}\)
\(L=4.01\cdot 10^{27}\; W\)
Comparing this to the luminosity of the Sun, Tau Cygni produces approximately 10.5 times the energy per second as the Sun.
\(\frac{L}{L_s}=\frac{4.01\cdot 10^{27}}{3.828\cdot 10^{26}}=10.47\)
It is also possible to combine both equations:
