Questions: Main Sequence Lifetime and the Mass-Luminosity Relation

Question 1

A star has 4 times the Sun's mass. Using the mass-luminosity relation (L ∝ M^3.5) and the lifetime scaling (t ∝ M^−2.5), approximately how does its main sequence lifetime compare to the Sun's?

• A. About 4 times longer — more mass means more fuel and a proportionally longer life
• B. About 2 times longer — the higher luminosity is partially offset by greater fuel supply
• C. About 1/32 as long — its luminosity is ~128 times solar, so it burns through its greater fuel supply about 32 times faster
• D. About 1/4 as long — lifetime scales inversely with mass in a simple ratio

Answer: (C)

Applying t ∝ M^(−2.5): for M = 4 solar masses, t ∝ 4^(−2.5) = 1/32. The luminosity is 4^3.5 ≈ 128 times solar, but the fuel supply is only 4 times greater — so the star exhausts its hydrogen about 32 times faster than the Sun. This counterintuitive result is the key insight: more mass does not mean a longer life because the luminosity (burn rate) grows much faster than the fuel supply. A 4-solar-mass star lives only ~300 million years versus the Sun's ~10 billion years.

Question 2

A star cluster's HR diagram shows the main sequence 'turning off' at a point corresponding to stars of approximately 2 solar masses. What does this tell you about the cluster?

• A. The cluster is very young — stars of 2 solar masses have not yet had time to reach the main sequence
• B. All stars in the cluster formed with 2 solar masses and are in the process of becoming red giants
• C. The cluster is old enough that 2-solar-mass stars have exhausted their hydrogen; using t ∝ M^(−2.5), the cluster age is roughly 2^(−2.5) × 10 billion years ≈ 1.8 billion years
• D. Stars more massive than 2 solar masses are still forming in this cluster

Answer: (C)

Stars in a cluster form at roughly the same time. As the cluster ages, progressively less massive (and less luminous) stars exhaust their hydrogen and leave the main sequence. The turnoff point marks the mass of stars currently finishing main-sequence life. Applying t ∝ M^(−2.5) to 2 solar masses: 2^(−2.5) ≈ 0.18, so the cluster is about 0.18 × 10 billion ≈ 1.8 billion years old. This is how astronomers determine cluster ages without being present at their birth — the main sequence turnoff is a built-in clock.

Question 3

A star with twice the Sun's mass is more than twice as luminous and therefore burns through its hydrogen in less than half the Sun's main sequence lifetime.

• T. True
• F. False

Answer: True

Both parts of this statement follow from L ∝ M^3.5. A 2-solar-mass star has luminosity 2^3.5 ≈ 11 times greater than the Sun — far more than twice. Since it has only twice the fuel but burns at 11 times the rate, its lifetime is t ∝ M/L ∝ 2/11 ≈ 1/5.7 times the Sun's. So a 2-solar-mass star lives roughly 1.8 billion years versus the Sun's 10 billion — less than one-fifth, not one-half. The nonlinear exponent (3.5) is what makes massive stars so much shorter-lived than naive intuition suggests.

Question 4

The most common stars in the galaxy are also among the brightest, since stars are most numerous at the high-mass end of the initial mass function.

• T. True
• F. False

Answer: False

This reverses the actual situation. The initial mass function strongly favors low-mass stars: roughly 75% of all stars in the galaxy are red dwarfs (M-type stars, less than ~0.5 solar masses). These are by far the most common stars but are so faint — owing to L ∝ M^3.5 — that not a single one is visible to the naked eye from Earth. The bright stars dominating our night sky (Rigel, Sirius, Betelgeuse) are rare, massive, short-lived stars that are spectacularly luminous but cosmically uncommon. Rarity and visibility run in opposite directions for stars.

Question 5

Explain why the main sequence turnoff point of a star cluster can be used to determine the cluster's age, and describe the physical process that causes the turnoff.

Model answer: Stars in a cluster all formed at roughly the same time from the same gas cloud, so they began on the main sequence simultaneously. The most massive stars, burning fuel at enormous rates due to L ∝ M^3.5, exhaust their hydrogen first and evolve off the main sequence to become red giants. As the cluster ages, successively less massive (and less luminous) stars reach the end of their main-sequence lives. The turnoff point — where the cluster's main sequence ends and stars begin evolving redward — marks the mass of stars currently finishing main-sequence life. Applying the lifetime relation t ∝ M^(−2.5) to that mass gives the cluster's age. Astronomers can thus read a cluster's age from a single HR diagram snapshot without observing it over billions of years.

This technique, called isochrone fitting, is one of the most powerful tools in observational astronomy for determining ages. The turnoff is a clock: higher-luminosity turnoff = younger cluster; lower-luminosity (more solar-like) turnoff = older cluster. The mass-luminosity-lifetime chain connects what observers can measure (luminosity at turnoff) to the quantity they want (cluster age) through well-understood stellar physics.