Nuclear Timescale & Stellar Lifetimes

Nuclear Timescale & Stellar Lifetimes

Stars shine by nuclear fusion — converting hydrogen to helium in their cores. The energy released per unit mass is determined by Einstein's $E = mc^2$: hydrogen fusion converts 0.7% of the rest-mass energy into radiation.

Nuclear Timescale

The total nuclear burning lifetime is:

$t_{\text{nuc}} = \frac{\varepsilon M c^2}{L}$

where:

• $\varepsilon = 0.007$ is the hydrogen fusion efficiency (0.7% of rest mass)
• $M$ is the stellar mass in kg
• $c = 2.998 \times 10^8$ m/s
• $L$ is the luminosity in watts

For the Sun: $t_{\text{nuc}} \approx 104$ Gyr for complete H→He conversion, or ~10 Gyr for core hydrogen (roughly 10% of the total mass fuses before the star leaves the main sequence).

Mass-Luminosity Relation

On the main sequence, luminosity scales steeply with mass:

$L \propto M^4$

Combining this with the nuclear timescale:

$t_{\text{nuc}} \propto \frac{M}{L} \propto \frac{M}{M^4} = M^{-3}$

Massive stars burn out dramatically faster. A 10 $M_\odot$ star lives about 1000 times shorter than the Sun. A 0.1 $M_\odot$ red dwarf could shine for trillions of years.

Your Task

Implement three functions. All constants must be defined inside each function.

• nuclear_timescale_s(M_kg, L_W) — returns $t_{\text{nuc}}$ in seconds
• nuclear_timescale_Gyr(M_solar, L_solar) — returns $t_{\text{nuc}}$ in gigayears
• main_sequence_lifetime_Gyr(M_solar) — uses $L \propto M^4$ to return lifetime in Gyr

Use $c = 2.998 \times 10^8$ m/s, $M_\odot = 1.989 \times 10^{30}$ kg, $L_\odot = 3.828 \times 10^{26}$ W, 1 Gyr $= 3.1558 \times 10^{16}$ s.