2. LIFETIMES OF MAIN SEQUENCE STARS 

Armed with an understanding of how main sequence stars burn by fusing hydrogen into helium, astrophysicists were ready to return to the old problem that Eddington addressed so eloquently in The Internal Constitution of the Stars. How long can stars run?

This problem is analogous to determining how long a vehicle can run on a tank of gas. The answer is simple: divide the fuel available by the rate of energy consumption. For example, if your Subaru gets 30 miles to the gallon, it can go 300 miles on a 10 gallon tank. That means you can drive 10 hours at 30 miles per hour. But now suppose that you are driving a "sports utility vehicle," capable of crushing a Subaru without noticing. It has a bigger tank, say, 20 gallons. But it only gets 10 miles per gallon, so it can only go 200 miles, or less than 7 hours at 30 miles per hour.

From Einstein's formula E = mc2 and Aston's measurement of the masses of hydrogen and helium, Eddington knew how much energy would be released by the fusion of hydrogen into helium, and he understood that the Sun could continue to produce its present luminosity for about 1010 (10 billion) years before all the hydrogen in its burning core (about 10% of the total mass of the star) would be converted into helium.

But how long will the more massive main sequence stars live? For example, consider a luminous blue giant star in the Pleiades cluster (the cosmic Subaru), which has a mass about 6.3 times that of the Sun. and a luminosity about 630 times that of the Sun. Well, the fuel supply is up by a factor 6.3, but the energy consumption rate (the observed luminosity) is up by a much bigger factor (6.3)3.5 = 630. So the lifetime of this star will be less than that of the Sun (1010 years) by a factor of about 6.3/630 = 0.01. That brings its lifetime down to 0.01 x 1010 = 108 years. Evidently, the cosmic Subaru is more like a sports utility vehicle than a fuel-efficient compact!

Using the mass-luminosity relationship, L(M) = M3.5 (see Lesson 4), it's easy to generalize this argument into an equation that gives the main-sequence lifetime, tMS, of any star in terms of its mass:

tMS = tSun M/L(M) = 1010 years M/M3.5 = 1010 years M-2.5 ,

where masses and luminosities are measured in Solar units. Check it! Put in M = 10 Solar masses and see whether the answer agrees with our estimate for the blue giant star in the Pleiades. How long will a red dwarf star with M = 0.1 live?

We know now that the Sun has a main sequence lifetime of 10 billion years and that it has already lived for about 4.5 billion years. So it has already converted about half the hydrogen in its core into helium. In another 5.5 billion years, the core of the Sun will be pure helium. What happens then? 

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Last modified September 23, 2000
Copyright by Richard McCray