4. The Gas Supply

4.1. Inflows, Star Formation, and Efficiency

Like the enrichment equation, the time derivative of the mass of the gas in the interstellar medium (ISM) \(M_g\) is a simple sum of source and sink terms. For an infall rate (IFR) \(\dot{M}_\text{in}\), star formation rate (SFR) \(\dot{M}_\star\), and outflow rate (OFR) \(\dot{M}_\text{out}\):

\[\dot{M}_g = \dot{M}_\text{in} - \dot{M}_\star - \dot{M}_\text{out} + \dot{M}_\text{r}\]

where \(\dot{M}_\text{r}\) is the rate of recycling from stars producing remnants and return gas to the ISM at their birth metallicity. Because VICE is implemented with a Forward Euler solution, this equation is evaluated via:

\[\Delta M_g \approx \dot{M}_g\Delta t = \dot{M}_\text{in}\Delta t - \dot{M}_\star \Delta t - \dot{M}_\text{out} \Delta t + \dot{M}_\text{r}\Delta t\]

By construction, VICE operates such that the user specifies either an infall history (\(\dot{M}_\text{in}\) as a function of time), a star formation history (\(\dot{M}_\star\) as a function of time), or the gas history (\(\dot{M}_\text{gas}\) as a function of time). The user also specifies a star formation efficiency timescale 1:

\[\tau_\star \equiv \frac{M_g}{\dot{M}_\star}\]

Users may specify an arbitrary function of time in Gyr to describe \(\tau_\star\), whose units are assumed to be Gyr. With one of either \(\dot{M}_\text{in}\), \(\dot{M}_\star\), or \(\dot{M}_g\) specified by the user, \(\tau_\star\), and the implementation of \(\dot{M}_\text{out}\) and \(\dot{M}_\text{r}\) discussed in this section, the solution to \(M_g\) as a function of time is unique.

VICE also allows users to adopt a formulation of \(\tau_\star\) that depends on the gas supply; this is an application of the Kennicutt-Schmidt relation to the single-zone approximation. This is implemented as a power-law:

\[\tau_\star^{-1} = \tau_{\star,\text{spec}}^{-1} \left(\frac{M_g}{M_{g,\text{Schmidt}}}\right)^\alpha\]

where \(M_{g,\text{Schmidt}}\) is a normalizing gas supply and \(\tau_{\star,\text{spec}}\) is the user-specified \(\tau_\star\). The singlezone object will employ this scaling when the attribute schmidt = True.

Relevant Source Code:

• vice/src/singlezone/ism.c

1

In the astronomical literature, this quantity is often referred to as the “depletion time” rather than star formation efficiency. In the chemical evolution literature, it quantifies the fractioanl rate at which gas is converted into stars, and is thus referred to as star formation efficiency. This is the

4.2. Outflows

In the astronomical literature, the strength/efficiency of outflows are typically quantified according to a dimensionless parameter referred to as the mass loading factor, defined as the ratio of the mass outflow rate to the star formation rate: \(\eta \equiv \dot{M}_\text{out}/\dot{M}_\star\). Johnson & Weinberg (2020) introduced a new parameter to generalize this, dubbed the “outflow smoothing time.” This is the timescale on which the star-formation rate is averaged (i.e. “smoothed”) to determine the outflow rate:

\[\dot{M}_\text{out} = \eta(t)\langle\dot{M}_\star\rangle_{\tau_\text{s}} = \frac{\eta(t)}{\tau_\text{s}}\int_{t - \tau_\text{s}}^t \dot{M}_\star(t') dt'\]

At early times when \(0 \leq t \leq \tau_\text{s}\), this average is taken over only the time interval from 0 to \(t\). This equation is approximated numerically according to:

\[\dot{M}_\text{out} \approx \eta(t) \frac{\Delta t}{\tau_\text{s}} \sum_{i = 0}^{\tau_\text{s}/\Delta t} \dot{M}_\star(t - i\Delta t)\]

Put simply, at each timestep VICE looks backs at the number of timesteps corresponding to the smoothing time, and determines the arithmetic mean of the star formation rate at those timesteps, then multiplies this number by \(\eta(t)\), which may be a user-specified function of time in Gyr. An advantage of this formulation is that when \(\tau_\text{s} < \Delta t\), VICE automatically recovers the traditional relation of \(\dot{M}_\text{out} = \eta(t)\dot{M}_\star(t)\).

Note

It is only the star formation rate which is time averaged. The mass loading factor is not time-averaged in any way.

Relevant Source Code:

• vice/src/singlezone/ism.c

4.3. Recycling

As stars produce remnants, the mass that does not end up in the remnant is returned to the interstellar medium (ISM). The net effect of this from all previous episodes of star formation quantifies the rate of recycling:

\[\dot{M}_\text{r} = \int_0^t \dot{M}_\star(t - t') \dot{r}(t') dt'\]

where \(r(\tau)\) is the cumulative return fraction from a single stellar population of age \(\tau\). This is approximated numerically according to

\[\dot{M}_\text{r} \approx \sum_i \dot{M}_\star(t - i\Delta t) \left[r((i + 1)\Delta t) - r(i\Delta t)\right]\]

This is an instance where the quantization of star forming episodes due to the Forward Euler solution simplifies the implementation; the stars that form in previous timesteps contribute \(\Delta r\) of their mass back to the ISM.

In the case of instantaneous recycling, this simplifies further to

\[\dot{M}_\text{r} \approx r_\text{inst}\dot{M}_\star\]

Weinberg, Andrews & Freudenburg (2017) 2 demonstrate that \(r_\text{inst}\) = 0.4 (0.2) for a Kroupa 3 (Salpeter 4) IMF are good approximations.

Note

Instantaneous recycling refers only previously produced nucleosynthetic products. While this term has been used to refer to instantaneous production of new heavy nuclei in the astronomical literature in the past, VICE retains this approximation only for enrichment from core collapse supernovae.

Relevant Source Code:

• vice/src/singlezone/recycling.c

2

Weinberg, Andrews & Freudenburg (2017), ApJ, 837, 183

3

Kroupa (2001), MNRAS, 322, 231

4

Salpeter (1955), ApJ, 121, 161