The temperature of 4 stars is listed below. Which star will burn up its hydrogen and move off the main sequence first?

Question

BlackTom AI · Accepted Answer

Higher surface temperature corresponds to higher stellar mass and much higher luminosity; more massive, hotter main-sequence stars burn their hydrogen fuel far faster and therefore leave the main sequence sooner.

1) Relevant concepts
- Main-sequence lifetime ≈ available fuel (∝ mass M) divided by luminosity L: t_MS ∝ M / L.
- For main-sequence stars, luminosity scales strongly with mass, roughly L ∝ M^3.5 (approximate mass–luminosity relation for intermediate-to-high mass stars).
- Combining these gives t_MS ∝ M / M^3.5 = M^-2.5, so lifetime decreases steeply with increasing mass (and thus with increasing surface temperature).

2) Step-by-step reasoning
- The listed temperatures (3,500 K, 7,000 K, 15,000 K, 30,000 K) correspond qualitatively to spectral types M, F, B, and O respectively. An O-type star (30,000 K) is the most massive and most luminous.
- Because L grows much faster than M, the hottest (most massive) star consumes hydrogen fastest and has the shortest main-sequence lifetime.

3) Example calculation (illustrative approximate numbers)
- Assign approximate masses: 30,000 K → 40 M☉; 15,000 K → 8 M☉; 7,000 K → 1.3 M☉; 3,500 K → 0.4 M☉.
- Use t ∝ M^-2.5. Compute M^2.5:
  - 40^2.5 = 40^2 × 40^0.5 = 1600 × 6.3249 ≈ 10,120 → t ∝ 1/10,120 ≈ 9.9×10^-5 (relative units)
  - 8^2.5 = 64 × 2.828 ≈ 181 → t ∝ 1/181 ≈ 5.5×10^-3
  - 1.3^2.5 ≈ 1.926 → t ∝ 1/1.926 ≈ 0.52
  - 0.4^2.5 ≈ 0.1012 → t ∝ 1/0.1012 ≈ 9.88
- These relative lifetimes show the 40 M☉ (30,000 K) star has by far the shortest lifetime.

4) Why the correct answer is correct
- The 30,000 K star is hottest, therefore most massive and most luminous, so it exhausts hydrogen fastest and moves off the main sequence first.

5) Why the other choices are incorrect
- 15,000 K (B-type) is hot and massive but less so than 30,000 K, so it lasts longer than the 30,000 K star.
- 7,000 K (F-type) and 3,500 K (M-type) are progressively cooler, lower-mass, and much longer-lived; the 3,500 K star has the longest lifetime.

Conclusion: The 30,000 K star will burn its hydrogen and leave the main sequence first, matching the provided correct answer: 30,000 K.

The temperature of 4 stars is listed below. Which star will burn up its hydrogen and move off the main sequence first?单项选择题

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