Stellar Ages and Main-Sequence Lifetimes

Stellar Ages and Main-Sequence Lifetimes

A star's lifetime on the main sequence is set by two competing factors: how much hydrogen fuel it has ($\propto M$) and how fast it burns it ($L \propto M^4$). The result is a steep mass dependence:

$t_{\text{MS}} \approx t_\odot \left(\frac{M}{M_\odot}\right)^{-2.5}$

where $t_\odot \approx 10$ Gyr is the Sun's main-sequence lifetime.

Main-Sequence Turnoff

The main-sequence turnoff is the key to aging stellar clusters. Stars more massive than the turnoff mass have already evolved off the main sequence; the turnoff mass equals the mass whose lifetime equals the cluster age:

$M_{\text{to}} = M_\odot \left(\frac{t_{\text{age}}}{t_\odot}\right)^{-0.4}$

Turnoff Temperature

Using the mass-temperature scaling $T \propto M^{0.6}$:

$T_{\text{to}} = T_\odot \left(\frac{t_{\text{age}}}{10\text{ Gyr}}\right)^{-0.4 \times 0.6}$

At 1 Gyr, the turnoff temperature is about 10,000 K — an A-type star.

Your Task

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

• main_sequence_lifetime_Gyr(M_solar) — returns $t_{\text{MS}} = 10 \times M^{-2.5}$ in Gyr
• turnoff_mass_solar(age_Gyr) — returns $M_{\text{to}} = (t/10)^{-0.4}$ in $M_\odot$
• turnoff_temperature_K(age_Gyr) — returns $T_{\text{to}} = 5778 \times (t/10)^{-0.24}$ in K

All three functions take a single numeric argument.