In the previous chapter you found that mass was an important quantity for determining what stars are like. In fact, all of the other aspects of a star such as its luminosity, temperature, size, density, etc., can be explained using the fundamental property of a star: its mass. There is also a slight dependence of the luminosity, temperature, size, etc. on the composition of the star, but because stars are all mostly hydrogen and helium, the star's mass is the important quantity.
Stars shine because of nuclear fusion reactions in their core. The more luminous they are, the more reactions are taking place in their cores. Massive stars live shorter lives than the common small stars because even though they have a larger amount of hydrogen for nuclear reactions, their rate of consuming their fuel is very much greater. The massive stars are analogous to the big, gas-guzzling automobiles with big gas tanks of a few decades ago and the small stars are analogous to the small economy automobiles of today that are frugal with their gasoline.
It is a simple calculation to find out how long something can continue consuming fuel. The lifetime = amount of fuel/consumption rate. If your car has a full 15-gallon gas tank and it consumes 2 gallons/hour on the highway, then your car can travel for 15 gallons/(2 gallons/hour) = 7.5 hours. Stars are the same way. The amount of fuel for nuclear fusion is proportional to the total mass of the star when it first started producing energy from nuclear reactions, so the amount of fuel = k × initial mass. The consumption rate is simply the star's luminosity, so the star will live as a main sequence star for an amount of time = k × initial mass/luminosity. If the star masses and luminosity are in units relative to the Sun, then the star's lifetime = mass/luminosity × 1010 years. Recall that the Sun's will live for ten billion (1010) years before it runs out of hydrogen in its core.
In order to remain stable via hydrostatic equilibrium, a star's luminosity increases with mass as (the star's mass)p. The value of the exponent p varies between 3 and 4. For the rare massive stars (M* > 30 Msun), p = 3 and for the more common low-mass stars (M* < 10 Msun), p = 4. You can use the mass-luminosity relation to find the star's lifetime in terms of just its initial mass. The lifetime = mass/luminosity × 1010 years is simply = (star's mass)/[(star's mass)p] × 1010 years = 1/(star's mass)p-1 × 1010 years. Remember that the star's mass is in solar masses.
How do you do that?A 5 solar mass star has only five times more hydrogen fuel than the Sun,
but (the star's luminosity)/(the Sun's luminosity) = (5/1)4 = 625!
Its lifetime = 1/(5/1)(4-1) × 1010 years = (1/125) × 1010 years = 8.0 × 107 years.
Some representative lifetimes for other stars are given in the table below. Stars that have fewer elements heavier than helium in them compared to other stars, will have slightly shorter lifetimes than those given in the table.
|star mass (solar masses)||time (years)||Spectral type|
|1||10 billion||G2 (Sun)|
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last updated: 24 May 2001