TOOLS OF THE ASTRONOMER’S TRADE

Main-Sequence Lifetimes

Hydrogen fusion converts a portion of a star’s mass into energy. We can use Einstein’s famous equation relating mass and energy to calculate how long a star will remain on the main sequence.

Suppose that M is the mass of a star and f is the fraction of the star’s mass that is converted into energy by hydrogen fusion. During its main-sequence lifetime, the total energy E supplied by the hydrogen fusion can be expressed as

In this equation c is the speed of light.

This energy from hydrogen fusion is released gradually over millions or billions of years. If L is the star’s luminosity (energy released per unit time) and t is the star’s main-sequence lifetime (the total time over which the hydrogen fusion occurs), then we can write

(Actually, this equation is only an approximation. A star’s luminosity is not quite constant over its entire main-sequence lifetime. But the variations are not important for our purposes.) We can rewrite this equation as

From this equation and E = fMc², we see that

We can rearrange this equation as

Thus, a star’s lifetime on the main sequence is proportional to its mass (M) divided by its luminosity (L). Using the symbol ∝ to denote “is proportional to,” we write

We can carry this analysis further by recalling that main-sequence stars obey the mass-luminosity relation (see Section 17-9, especially the Cosmic Connections figure). The distribution of data on the graph in the Cosmic Connections figure in Section 17-9 tells us that a star’s luminosity is roughly proportional to the 3.5 power of its mass:

Substituting this relationship into the previous proportionality, we find that

This approximate relationship can be used to obtain rough estimates of how long a star will remain on the main sequence. It is often convenient to relate these estimates to the Sun (a typical 1-M_⊙ star), which will spend 1.2 × 10¹⁰ years on the main sequence.

EXAMPLE: How long will a star whose mass is 4 M_⊙ remain on the main sequence?

Situation: Given the mass of a star, we are asked to determine its main-sequence lifetime.

Tools: We use the relationship t ∝ 1/M^2.5.

Answer: The star has 4 times the mass of the Sun, so it will be on the main sequence for approximately

times the Sun’s main-sequence lifetime

Thus, a 4-M_⊙ main-sequence star will fuse hydrogen in its core for about (1/32) × 1.2 × 10¹⁰ years, or about 4 × 10⁸ (400 million) years.

Review: Our result makes sense: A star more massive than the Sun must have a shorter main-sequence lifetime.