Homework 9. Main-Sequence Stars

Name: ________________________ Due 10/31 ID Number: ________________________

The HR diagram below shows ``surface'' temperatures and and luminosities for a sample of fifteen stars. Here, temperature is measured in °K, and luminosity is measured in units of the Sun's luminosity, L. Each star is assigned a letter (a through o); for example, star a has a surface temperature of about 10000 °K and a luminosity of about 80 L.

 

As usual, this HR diagram uses logarithmic spacing along both axes. Each step leftward is factor of 2 increase in T, and each step upward is a factor of 10 increase in L. For example, a star which is plotted halfway between 102 and 103 on the vertical axis has a luminosity of roughly 3.2×102 L, because a half-step up represents a factor of 10 = 3.162... increase.

1. List these fifteen stars in order of increasing temperature:


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2. List these fifteen stars in order of increasing luminosity:


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3. The two lists you've just made should be nearly identical, but there will be a few stars (exactly three, in fact), which don't appear in the same order in the two lists. Eliminate these stars from the sample, and list the remaining twelve stars in order of increasing temperature and increasing luminosity:


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4. List the stars on the main sequence in order of increasing luminosity:


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5. The next part of this assignment asks you to calculate the lifetime of each main-sequence star. To do this, you'll need to know the mass M of each star in units of the Sun's mass, M. This information is given in the following table:

star M     star M     star M     star M     star M
a 3.5     b 4.2     c 0.77     d 1.38     e 10.8
f 0.27     g 3.4     h 25.2     i 0.63     j 6.2
k 1.08     l 1.69     m 0.14     n 16.9     o 0.52

To calculate the lifetimes, begin by listing the main-sequence stars, in order of increasing luminosity, in the ``star'' columns of the table below. Next, use the HR diagram on the other side of this paper to find the luminosity L of each star you've just listed, and fill in the L columns of the table. Now suppose, for a moment, that each star has exactly the same amount of fuel as the Sun; the star will use up this amount of fuel in a ``reference'' time tref = tlife, ÷ L, where the Sun's lifetime is tlife, = 1010 yr, and L is expressed in units of the Sun's luminosity, L. Finally, use the values of tref to compute each star's main-sequence lifetime tlife = tref × M, where the star's mass M is taken from the table above and expressed in units of the Sun's mass, M. For example, star a has L = 80 L and M = 3.5 M, so tref = (1010 yr) ÷ 80 = 1.25×108 yr, and tlife = (1.25×108 yr) × 3.5 = 4.4×108 yr.

star L tref tlife star L tref tlife
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6. Use the numbers you just worked out to determine how stellar lifetimes vary along the main sequence. Which stars have the longer lifetimes: high-mass main sequence stars or low-mass main sequence stars?









Joshua E. Barnes (barnes@ifa.hawaii.edu)
Last modified: October 23, 2006
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