Consider Sun’s total gravitational potential energy as its power source to maintain its current luminosity. How long would it take to radiate all of it away? (Hint: t = Energy/Luminosity; 1 W = 1 J/s, 1 J = 1 kg 1 m2 s-2)

Question

BlackTom AI · Accepted Answer

Let's work through the problem step by step and evaluate each answer option.

Option a: 30 million years.
- We are asked to estimate the time it would take for the Sun to radiate away its total gravitational potential energy, using t = E/L. The gravitational potential energy of a star like the Sun is roughly Epot ~ GM^2/R. Plugging in M ~ 2×10^30 kg, R ~ 7×10^8 m, and G ~ 6.7×10^-11 m^3 kg^-1 s^-2 gives Epot on the order of  GM^2/R ≈ (6.7×10^-11)(4×10^60)/(7×10^8) ≈ a few ×10^41 J. The Sun’s luminosity is L ≈ 3.8×10^26 W (J/s). Therefore t ≈ E/L ≈ (a few×10^41 J)/(3.8×10^26 W) ≈ a few×10^15 s. Converting seconds to years (1 year ≈ 3.15×10^7 s) yields t ≈ a few×10^7 years, i.e., on the order of 30 million years. This closely matches option a, making it a plausible estimate.

Option b: 10 billion years.
- 10^10 years is about the Sun’s main-sequence lifetime with respect to nuclear fusion, not the gravitational energy drain. The gravitational energy reservoir is far smaller than the energy produced by fusion over that timescale, so this overestimates t for E/L and is not consistent with the simple energy budget t = E/L for gravitational energy. Thus this is not correct for the question as posed.

Option c: 2 billion years.
- While 2×10^9 years is a common timescale in stellar evolution contexts, it is still much shorter than the simple estimate from E/L with Epot ≈ GM^2/R for the Sun. The computed t from E/L is more like 10^7–10^8 years range, not billions of years, so this option is inconsistent with the gravitational-energy-only calculation here.

Option d: Less than 1 year.
- This would require Epot/L < 1 year. Since Epot is ~10^41 J and L is ~10^26 W, E/L is ~10^15 s, which is about 3×10^7 years, far longer than a year. Therefore this option grossly underestimates the timescale and is incorrect for the given model.

Summary: Evaluating the energy budget with Epot ~ GM^2/R and L ~ 3.8×10^26 W yields t on the order of 10^15 seconds, i.e., tens of millions of years, which aligns with option a. The other options either reflect nuclear-timescale interpretations or are inconsistent with the straightforward gravitational-energy-to-luminosity calculation.

ASTR 350 ONL FA25: Big Bang,Black Holes,Universe (Liu, X)

Consider Sun’s total gravitational potential energy as its power source to maintain its current luminosity. How long would it take to radiate all of it away? (Hint: t = Energy/Luminosity; 1 W = 1 J/s, 1 J = 1 kg 1 m2 s-2)

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