Addendum 5: Stellar Evolution, Part 2 (Chapter 17)
First define some constants and dimensional units needed below
1. Mass-Luminosity relationship for main sequence stars. The luminosity of a main-sequence star is a very strong function of its mass. In the lecture notes, we approximated it as:
A more accurate fit is obtained by the epxonent 3.5 as shown on the plot at right.
Example 1: What is the luminosity (in solar luminosities) of a star with 0.5 solar masses?
Example 2: What is the mass of a main-sequence star with a luminosity 1,000x that of the Sun? Express in Solar masses.
2. Lifetime of a star of given mass. The lifetime of a star of a given mass can be estimated by taking the ratio of the mass (prop. to total fuel available) and the luminosity (proportional to rate of using the fuel). As usual we normalize to the Sun's lifetime, which is 1010 yrs.
Example: Estimate the mass and total lifetime of the star Sirius (spectral type A1V).
From the HR diagram at the bottom of this workshheet, we estimate the luminosity of Sirius as 25x Lsun. Hence, the mass of Sirius is
Hence, the lifetime of Sirius is:
3. Size and density of stars given temperature and luminosity. (review of Stephan-Boltzmann law). The relationship between the temperature, radius, and luminosity of a star is given by
where s is the Stephan-Boltzmann constant (listed at top of the worksheet). This formula can be applied to stars on the HR diagram to determine their sizes.
Example: What is the radius of Betelgeuse? (It is shown in the HR diagram at the bottom of the worksheet). If the Sun were replaced by Betelgeuse, would the Earth be inside the star? From the HR diagram, we estimate:
Solving for R:
Yes,inside! (The Earth-Sun distance is 1.0 AU < R)
Example 2: What is the mean density of Betelgeuse compared with the Sun?
4. Cool-down time for white dwarf. Since a white dwarf has no active fusion process occurring, its luminosity is entirely a result of its latent heat: It is simply a very large glowing piece of coal slowly cooling. We can roughly estimate the 'cooling time' by calculating the ratio of total heat energy stored to the luminosity.
where T is the mean temperature in the interior and N is the total number of atoms:
M is the mass of the star, mp is the mass of the average atom
So the cool-down time is:
Example: The white dwarf Procyon-B has a mass M = 0.9 Msun. Assume an initial mean interior temperature T = 107 K. Use the HR diagram to estimate its cool-down time
Assume the average atom is carbon.
From the HR diagram, we estimate the temperature and luminosity:
