Stellar Lifetimes¶
In VICE we adopt the following functional form for the lifetime of a star on the main sequence:
where \(\tau_\odot\) is the sun’s main sequence lifetime, \(\alpha\)
is the power-law index of the mass-lifetime relationship. The constant
SOLAR_LIFETIME declares \(\tau_\odot\) = 10 Gyr, and
MASS_LIFETIME_PLAW_INDEX delcares \(\alpha\) = 3.5. Both constants are
declared in vice/src/ssp.h.
The scaling of \(\tau_\text{MS} \sim m^{-3.5}\) fails for high mass stars (\(\gtrsim 8 M_\odot\)), but these stars have lifetimes that are very short compared to the relevant timescales of galactic chemical evolution (\(\sim\)few` Gyr). This approximation fails for low mass stars as well (\(\lesssim 0.5 M_\odot\)), but these stars have very long lifetimes that are considerably longer than the age of the universe. Because VICE does not support simulations on this long of timescales, this approximation suffices for all timescales of interest.
This is motivated by a conventional power-law relationship between mass and luminosity \(L \sim M^{+\beta}\). The lifetime then scales as \(\tau \sim M/L \sim M^{1 - \beta}\). \(\alpha\) = 3.5 corresponds to \(L \sim M^{4.5}\) in the mass range of interest.
This equation can be generalized to find the the total lifetime of a star of mass \(m\): the time until it produces a remnant by simply amplifying the lifetime by a factor \(1 + p_\text{MS}\):
where \(p_\text{MS}\) is an adopted lifetime ratio of the post main sequence to main sequence phases of stellar evolution.
By interpreting \(\tau_\text{total}\) as lookback time, we can solve for the mass of remnant producing stars under this model.
This equation allows the solution of both the main sequence turnoff mass and the mass of stars at the end of their post main sequence lifetimes by whether or not \(p_\text{MS}\) = 0.
Relevant source code:
vice/src/ssp.h
vice/src/ssp/mlr.c
