Stellar Lifetimes¶
The simplest description of the relationship between a star’s mass and its lifetime (i.e. the mass-lifetime relationship, hereafter MLR) is a single power law:
where \(\tau_\odot\) is the sun’s main sequence lifetime and \(\alpha\)
is the power law index of the MLR.
The constants SOLAR_LIFETIME and MASS_LIFETIME_PLAW_INDEX, both
declared in vice/src/ssp.h, assign the values of \(\tau_\odot\) = 10
Gyr and \(\alpha\) = 3.5, respectively.
Motivated by a popular exercise in undergradaute astronomy courses, this form
follows from an assumed single-power law relating the mass and luminosity of
stars: \(L \sim M^{1 + \alpha}\).
Because the lifetime of a star scales with its luminosity per unit mass, the
power law then follows trivially: \(\tau \sim M/L \sim M/M^{1 + \alpha}
\sim M^{-\alpha}\).
VICE generalizes this form to describe the total lifetime of a star of mass \(m\) by simply amplifying the main sequence lifetime by a factor of \(1 + p_\text{MS}\):
where \(p_\text{MS}\) is the ratio of a star post main sequence lifetime to its main sequence lifetime (e.g. if the post main sequence lifetime is 10% of the main sequence lifetime, then \(p_\text{MS}\) = 0.1). As a consequence, this quantity describes the time interval between a star’s formation and when it produces a remnant.
By interpreting \(\tau_\text{total}\) as lookback time, VICE solves for the mass of “remnant-producing” stars from a stellar population of known age:
Since these are the stars that are at the ends of the lifetimes, VICE adopts \(m_\text{postMS}\) as written here as the mass of AGB stars enriching the interstellar medium at a given timestep. This equation also allows the solution of the main sequence turnoff mass by simply taking \(p_\text{MS}\) = 0.
The scaling of \(\tau_\text{MS} \sim m^{-3.5}\) fails for higher mass stars (\(\gtrsim 4 M_\odot\)). However, these stars generally have lifetimes that are short compared to the relevant timescales of galactic chemical evolution (\(\lesssim\) 100 Myr compared to \(\sim\) few Gyr). Regardless of its accuracy for low mass stars (\(\lesssim 0.5 M_\odot\)), their lifetimes are considerably longer than the age of the universe anyway, and VICE does not support simulations on timescales longer than 15 Gyr. Consequently, this approximation generally suffices for galactic chemical evolution models.
The important exception to this rule is that an accurate MLR for \(4 \lesssim M/M_\odot \lesssim 8\) stars is necessary when considering elements with significant nucleosynthetic yields from these stars (e.g. nitrogen, Johnson et al. 2021 [1]). Prior to version 1.3.0, VICE implemented this single power law relationship only. In subsequent versions, a handful of additional forms are available to fill this need:
Larson (1974) [2] (default in versions >= 1.3.0)
This form is a metallicity-independent parabola in \(\log\tau-\log m\) space describing the main sequence lifetimes only:
\[\log_{10}\tau = \alpha + \beta\log_{10}m + \gamma (\log_{10}m)^2\]where the original values of \(\alpha\) = 1.02, \(\beta\) = -3.57, and \(\gamma\) = 0.90 were derived from a fit to the compilation of evolutionary lifetimes presented in Tinsley (1972) [3]. VICE however adopts the updated values of \(\alpha\) = 1.0, \(\beta\) = -3.42, and \(\gamma\) = 0.88 from Kobayashi (2004) [4] and David, Forman & Jones (1990) [5]. In detail, the value of \(\alpha\) directly quantifies the log of the main sequence lifetime of the sun \(\log_{10}\tau_\odot\) in whatever units \(\tau\) is in (\(\alpha\) = 1.0 for \(\tau_\odot\) = 10 Gyr; \(\alpha\) = 10.0 for \(\tau_\odot = 10^{10}\) yr).
Solutions to the inverse function (i.e. mass as a function of lifetime) follow directly from the quadratic formula: \(\log_{10}m = (-\beta \pm \sqrt{\beta^2 - 4\gamma(\alpha - \log_{10}\tau)}) / (2\gamma)\). The correct solution comes when choosing subtraction in the numerator as this corresponds to increasing lifetimes with decreasing stellar mass.
This is the default MLR for VICE version 1.3.0 and later.
Maeder & Meynet (1989) [6]
This form is a metallicity-independent broken power law quantifying only the main sequence lifetimes:
\[\tau = \Bigg \lbrace { 10^{\alpha\log_{10}m + \beta}\ (m < 60 M_\odot) \atop 1.2m^{-1.85} + 3\ \text{Myr}\ (m \geq 60 M_\odot) }\]where the values of \(\alpha\) and \(\beta\) are given by:
Mass Range
\(\alpha\)
\(\beta\)
\(m \leq 1.3\)
-0.6545
1
\(1.3 < m \leq 3\)
-3.7
1.35
\(3 < m \leq 7\)
-2.51
0.77
\(7 < m \leq 15\)
-1.78
0.17
\(15 < m \leq 60\)
-0.86
-0.94
In detail, the values of \(\beta\) shift linearly depending on the choice of units for \(\tau\); those listed here are appropriate for \(\tau\) in Gyr. For a shift to \(\tau\) in yr, all values should increase by 9 (e.g. \(\beta\) = 10.35 for masses between 1.3 and 3 \(M_\odot\)).
Though this form was originally published in Maeder & Meynet (1989), the exact form as written here was taken from Romano et al. (2005) [7]. While analytic solutions to the inverse function (i.e. mass as a function of lifetime) are possible, VICE takes a numerical approach, implementing a recursive version of the bisection algorithm described in chapter 9 of Press, Teukolsky, Vetterling & Flannery (2007) [8].
Padovani & Matteucci (1993) [9]
This form is a metallicity-independent curve describing the main-sequence lifetime only:
\[\log_{10}\tau = \frac{\alpha - \sqrt{\beta - \gamma \left(\eta - \log_{10}m\right)}}{\mu}\]The values of the coefficients are given by:
Coefficient
Value
\(\alpha\)
0.334
\(\beta\)
1.790
\(\gamma\)
0.2232
\(\eta\)
7.764
\(\mu\)
0.1116
These values are appropriate for \(\tau\) in units of Gyr. Solutions to the inverse function (i.e. mass as a function of lifetime) follow analytically. Though this form was originally published in Padovani & Matteucci (1993), the form as written here was taken from Romano et al. (2005).
Kodama & Arimoto (1997) [10]
Using the stellar evolution code presented in Iwamoto & Saio (1999) [11], Kodama & Arimoto (1997) tabulate the total lifetimes (i.e. including post main sequence evolution) of stars as a function of both initial mass and metallicity. VICE stores internal data at 41 initial masses and 9 metallicities, using 2-dimensional linear interpolation to approximate a smooth function based on these discrete points.
Because of the necessary interpolation, solutions to the inverse function (i.e. mass as a function of lifetime and metallicity) follow numerically, for which VICE implements a recursive version of the bisection algorithm described in chapter 9 of Press, Teukolsky, Vetterling & Flannery (2007).
Hurley, Pols & Tout (2000) [12]
This is a metallicity-dependent characterization of the main sequence lifetimes of stars given by:
\[\tau = \text{max}(\mu, x) t_\text{BGB}\]where \(t_\text{BGB}\) is the time required for a star to reach the base of the giant branch on the Hertzsprung-Russell diagram:
\[t_\text{BGB} = \frac{ a_1 + a_2 m^4 + a_3 m^{5.5} + m^7 }{ a_4 m^2 + a_5 m^7 }\]The coefficients \(a_n\) vary with metallicity according to:
\[a_n = \alpha_n + \beta_n \zeta + \gamma_n \zeta^2 + \eta_n\zeta^3\]VICE stores the values of \(\alpha\), \(\beta\), \(\gamma\), and \(\eta\) for the coefficients \(a_n\) as internal data, and the quantity \(\zeta\) is related to the metallicity by mass \(Z\) by \(\zeta = \log_{10}(Z / 0.02)\). The value of 0.02 corresponds to the metallicity of the sun; although there has been some evolution in the accepted value of \(Z_\odot\), VICE takes this value of 0.02 always when calculating lifetimes according to the Hurley, Pols & Tout (2000) parameterization regardless of the user’s setting in a chemical evolution model.
The coefficients \(\mu\) and \(x\) are given by:
\[\mu = \text{max}\left(0.5, 1.0 - 0.01 \text{max}\left( \frac{a_6}{m^{a_7}}, a_8 + \frac{a_9}{m^{a_{10}}} \right) \right)\]\[x = \text{max}\left(0.95, \text{min}\left[ 0.95 - 0.03\left(\zeta + 0.30103\right) \right] \right)\]Solutions to the inverse function (i.e. mass as a function of lifetime and metallicity) are numerical, for which VICE implements a recursive version of the bisection algorithm described in chapter 9 of Press, Teukolsky, Vetterling & Flannery (2007).
Vincenzo et al. (2016) [13]
This form characterizes the total lifetimes of stars (i.e. including the post main sequence evolution) as a function of stellar mass and metallicity according to:
\[\tau = A \exp(B m^{-C})\]where the coefficients \(A\), \(B\), and \(C\) depend on metallicity. VICE stores their values sampled at 299 values of the metallicity \(Z\) as internal data, interpolating linearly between them to approximate smooth functions out of the discrete points. With their values known at a given metallicity, the inverse function (i.e. mass as a function of lifetime) follows analytically from the above equation.
Vincenzo et al. (2016) determined the values of these coefficients by using isochrones computed using the PARSEC stellar evolution code (Bressan et al. 2012 [14]; Tang et al. 2014 [15]; Chen et al. 2015 [16]) in combination with a one-zone chemical evolution model parameterized to reproduce the color-magnitude diagram of the Sculptor dwarf galaxy.
Here we plot stellar lifetime as a function of progenitor mass
according to each of these forms along with the single power law described
above; its failure at high masses compared to the other, more sophisticated
parameterizations is quite clear.
VICE affords users the ability to evaluate these functions using the
vice.mlr module (e.g. vice.mlr.hpt2000 correspond to the Hurley, Pols
& Tout (2000) form, and vice.mlr.ka1997 to the Kodama & Arimoto (1997)
form).
The form to be adopted in all chemical evolution models and single stellar
population calculations is assigned via a global setting stored at
vice.mlr.setting.
Of these parameterizations of the MLR, the following take into account the metallicity dependence:
Vincenzo et al. (2016)
Hurley, Pols & Tout (2000)
Kodama & Arimoto (1997)
The following require numerical solutions for the inverse function (i.e. stellar mass as a function of lifetime):
Hurley, Pols & Tout (2000)
Kodama & Arimoto (1997)
Maeder & Meynet (1989)
The following quantify the total lifetimes a priori, making calculations of purely main sequence lifetimes unavailable:
Vincenzo et al. (2016)
Kodama & Arimoto (1997)
Except where measurements of the total lifetime is available, VICE always implements the simplest assumption of allowing the user to specify the parameter \(p_\text{MS}\) (see above), and the total lifetime then follows trivially via:
Relevant source code:
vice/core/mlr.py
vice/src/ssp.h
vice/src/ssp/mlr.c
vice/src/ssp/mlr/powerlaw.c
vice/src/ssp/mlr/vincenzo2016.c
vice/src/ssp/mlr/hpt2000.c
vice/src/ssp/mlr/ka1997.c
vice/src/ssp/mlr/pm1993.c
vice/src/ssp/mlr/mm1989.c
vice/src/ssp/mlr/larson1974.c
vice/src/ssp/mlr/root.c
