Core Temperatures for Stars

Simple estimates indicate that for stars the core temperature rises in proportion to the cube root of the core density as long as the gas of the core behaves normally:

T ~ constant x (density)1/3

Thus, protostars get hotter as they contract and their density increases. However, as we discuss later in conjunction with white dwarfs and neutron stars, at very high density the gas switches from a normal gas to a special kind of gas called a degenerate gas. For a degenerate gas the temperature no longer rises with density.

For a contracting protostar, what is the temperature when the core density reaches the critical density for forming a degenerate gas? For the Sun, this is about 10 million K, more than enough to initiate fusion reactions. For less than about 1/12 solar mass, the core becomes degenerate before the temperature rises high enough to start fusion. These failed stars become the brown dwarfs discussed in the next section.

Lower Mass Limit for the Main Sequence

How large and how small can the mass of a main sequence star be? Let's consider first the lower limit. The limiting lower mass is set by the smallest mass that can produce the core temperatures and densities necessary to sustain steady core hydrogen fusion. Obviously, the Sun does this and Jupiter does not, so the limit must lie between the masses of these two objects. However, that is still a rather large range since the Sun is about 1000 times more massive than Jupiter.

The Lightest Stars
Detailed calculations and observations suggest that this lower limit is about 1/12 (that is, 0.08) of a solar mass. We certainly see little evidence for stars any less massive than this, but that is not a definitive test of our theoretical ideas because these low-mass stars are very difficult to find (see the right panel).

Testing the Theoretical Limit
As we discuss shortly, these ideas may be tested by studying objects intermediate between stars and planets that are called brown dwarfs. The observation of brown dwarfs gives us considerable confidence that the lower limit for the mass of a star is indeed about 0.08 solar masses.