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Johannesburg Centre,
Astronomical Society of Southern Africa
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THE AGES OF THE STARS
Many of the stars in the globular clusters are
red giants, stars that have pushed off their upper layers and left their positions in the
main sequence of the Hertzsprung-Russell diagram. These stars must have ages in excess of
that of the Sun. While in the main sequence a star converts hydrogen in its nucleus into
heavier atoms: helium, carbon, oxygen, etc. The energy set free by the fusion reactions,
is given by Einstein's equation E = mc2 , where m is the mass
converted in grams; c is the speed of light in centimetres per second and E
is in ergs.
If a star consumes a fraction f, of its total mass while it is
in the main sequence, the total amount of energy liberated will be given by E = f x mc2,
f is, of course less than 1. The total energy E, is equal to
the product of luminosity L, and the total time T, that the star has spent
in the main sequence, so that E = L x T. Therefore LT = f m c2,
so that From the relationship A star of 2 times the mass of the Sun will consume its available
hydrogen 22,5 times faster than the Sun, i.e. 5,657 times faster so that it
will reside in the main sequence In the case of a star of 5 times the mass of the Sun, its hydrogen will be consumed in 52,5 times less than in the case of the Sun. This is 55,9 times faster and its lifetime in the main sequence will be 10 x 109 ¸ 55,9 = 179 x 106 years - only 179 million. A star of 10 times the mass of the Sun ( eg Sanduleak -69°202 in the Large Magellanic Cloud which ended its life in a supernova explosion in 1987 ) consumes its hydrogen fuel 102.5 times or 316 times faster than the Sun. Its lifetime will thus be 10 x 109 ¸ 316 = 31,2 x 106 years. This explosion actually took place 166 000 years ago because the Large Magellanic Cloud is 166 000 light years distant from Earth. So we see that the most massive stars have lifetimes of less than one three-hundredth of that of the Sun. By contrast a star of one half (0,5) times the mass of the Sun will consume its hydrogen (0,5)2.5 times, or 0,17678 times as leisurely as does the Sun. Its lifetime on the main sequence is therefore 10 x 109 ÷ 0,17678 = 56 x 109 years - more than 5 times as long as the Sun!
Jan Eben van Zyl |
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. Now, f is approximately the same
for all stars and since c is constant, we have 
, i.e. T
is proportional to 
. According to
the mass-luminosity law, first developed by A S Eddington in 1924, the luminosity L
of a star is proportional to the (3,5)th power of the mass m, of the star.
Therefore we can write m3,5 for L so that the lifetime of a star T
is proportional to 
, i.e. proportional to 
.
This means that the lifetime of a star in the main sequence is 
we see that the greater the mass of a star, the shorter is its lifetime in the main
sequence because 1 divided by a quantity more than 1 is less than 1. 
times the
time the Sun will reside there. 10 x 109 ¸ 5,657 =
1,768 x 109, i.e. a star of 2 times the mass of the Sun will reside on the main
sequence for only 1,768 times 10 to the power 9 years - 1/6 of that of the Sun. 
