Sun's life expectancy (a) Estimate the average kinetic...

Sun's life expectancy (a) Estimate the average kinetic energy of the particles in the Sun. Assume that it is made of atomic hydrogen and that its average temperature is $100,000 \mathrm{K} .$ The mass of the Sun is $2 \times 10^{30} \mathrm{kg}$ (b) For how long would the Sun shine using this energy if it radiates $4 \times 10^{26} \mathrm{W} / \mathrm{s} ?$ Is your answer reasonable? Explain.

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Key Concepts

Equipartition Theorem

This theorem states that for a system in thermal equilibrium, each degree of freedom contributes (1/2)k_B T to the energy, leading to an average kinetic energy of (3/2)k_B T for a monoatomic gas particle. This concept is used to relate the temperature of a gas to the energy of its particles.

Mass-to-Particle Number Conversion

In astrophysics, to determine the total energy content of a star made predominantly of hydrogen, one converts the star’s mass into the number of hydrogen atoms by dividing the total mass by the mass of a single hydrogen atom. This conversion is essential for estimating the aggregate kinetic (thermal) energy available.

Luminosity and Energy Output

Luminosity, defined as the energy radiated per unit time, is a key measure in astrophysics. By dividing a star's total available energy by its luminosity, one can estimate the time over which this energy would be radiated; this forms the basis for calculating the star’s lifetime based solely on its thermal energy content.

Order-of-Magnitude Estimation

This concept involves making approximate calculations using powers of ten to gauge whether an estimate is reasonable. In the context of the problem, it helps in assessing the plausibility of the computed energy values and lifetimes, serving as a cross-check against known astrophysical timescales.

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