.. _enr: Enrichment ========== VICE takes a general approach in modeling nucleosynthesis. All elements are treated equally; there are no special considerations for any element. In this documentation we derive the analytic form of :ref:`the enrichment equation ` for an arbitrary element :math:`x` with arbitrary nucleosynthetic yields for arbitrary evolutionary histories. This is an integro-differential equation of the element's mass as a function of time, which VICE solves as an initial-value problem by imposing the boundary condition that its abundance at time zero is given by the primordial abundance from big bang nucleosynthesis. In this version of VICE, helium is the only element for which this value is nonzero. .. _enr_eq: The Enrichment Equation ----------------------- The enrichment equation quantifies the rate of change of an element's total mass present in the interstellar medium (ISM). At its core, it is a simple sum of source and sink terms. .. math:: \dot{M}_x = \dot{M}_x^\text{CC} + \dot{M}_x^\text{Ia} + \dot{M}_x^\text{AGB} - \frac{M_x}{M_g}\left[ \dot{M}_\star + \xi_\text{enh}\dot{M}_\text{out} \right] + \dot{M}_x^\text{r} + Z_{x,\text{in}}\dot{M}_\text{in} where :math:`M_x` is the mass of the element :math:`x` in the interstellar medium, :math:`\dot{M}_x` its time-derivative, and :math:`M_g` the mass of the ISM gas. :math:`\dot{M}_x^\text{CC}`, :math:`\dot{M}_x^\text{Ia}`, and :math:`\dot{M}_x^\text{AGB}` quantify the rate of production from core-collapse supernovae (CCSNe), type Ia supernovae (SNe Ia), and asymptotic giant branch (AGB) stars, respectively. We detail each term individually here. .. _enr_ccsne: .. include:: ccsne.rst .. _enr_sneia: .. include:: sneia.rst .. _enr_agb: .. include:: agb.rst Subsequent Terms ---------------- The remaining terms in the enrichment equation make simple statements about remaining source and sink terms. VICE retains the assumption that stars are born at the same metallicity as the ISM from which they form. This motivates the sink term .. math:: -\left(\frac{M_x}{M_g}\right)\dot{M}_\star where the mass of the element :math:`x` is depleted at the metallicity of the ISM :math:`Z_x = M_x/M_g` in proportion with the star formation rate :math:`\dot{M}_\star`. Many galactic chemical evolution models to date have assumed that outflows from galaxies occur at the same metallicity of the ISM. This would suggest that :math:`\dot{M}_x^\text{out} \approx (M_x/M_g)\dot{M}_\text{out}`. However, recent work in the astronomical literature from both simulations (e.g. Christensen et al. (2018) [5]_) and observations (e.g. Chisholm, Trimonti & Leitherer (2018) [6]_) suggest that this may not be the case. Therefore, VICE allows outflows to occur at some multiplicative factor :math:`\xi_\text{enh}` above or below the ISM metallicity, which may vary with time. This motivates the sink term .. math:: -\left(\frac{M_x}{M_g}\right)\xi_\text{enh}\dot{M}_\text{out} Because :ref:`VICE works with net rather than absolute yields `, simulations must quantify the rate at which stars return mass to the ISM at their birth metallicity. This is mathematically similar to the rate of total gas recycling, but weighted by the metallicities of the stars recycling. Since stars are assumed to form at the metallicity of the ISM, .. math:: \dot{M}_x^\text{r} = \int_0^t \dot{M}_\star(t') Z_{x,\text{ISM}}(t') \dot{r}(t - t') dt where :math:`r(\tau)` is the :ref:`cumulative return fraction ` from a single stellar population of age :math:`\tau`. This is approximated numerically as .. math:: \dot{M}_x^\text{r} \approx \sum_i \dot{M}_\star(i\Delta t) Z_{x,\text{ISM}}(i\Delta t) \left[r((i + 1)\Delta t) - r(i\Delta t)\right] where the summation is taken over all previous timesteps. The need to differentiate :math:`r` with time is eliminated in the numerical approximation by allowing each stellar population to be weighted by :math:`\Delta r` between the current timestep and the next, made possible by the quantization of timesteps. In the event that the user has specified instantaneous recycling: .. math:: \dot{M}_x^\text{r} = r_\text{inst}\dot{M}_\star Z_{x,\text{ISM}} At any given timestep, there is gas infall onto the simulated galaxy of a given metallicity :math:`Z`. In most cases this term is negligibly small, but in some interesting cases it may not be (e.g. a major merger event). This necessitates the final term :math:`Z_{x,\text{in}}\dot{M}_\text{in}`. Relevant Source Code: - ``vice/src/singlezone/recycling.c`` - ``vice/src/singlezone/element.c`` - ``vice/src/singlezone/ism.c`` Extension to Multizone Models ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The only subsequent term of the enrichment equation modified in multizone simulations is that quantifying the rate of recycling of an element :math:`x`. The migration of star particles into and out of zones can affect the recycling rate in a given zone. In a singlezone simulation it is exactly as expected for the star formation history, but in a multizone model, it is coupled to the star formation histories in other zones. Because VICE knows the zone each star particle occupies at all times in simulation, the rate of recycling of some element :math:`x` should be expressed not as an integral over the star formation history, but as a summation over the stellar populations in the zone: .. math:: \dot{M}_x^\text{r} \approx \sum_i M_i Z_{x,i} [r(\tau_i + \Delta t) - r(\tau_i)] where :math:`M_i`, :math:`Z_i`, and :math:`\tau_i` are the mass, metallicity, and age, respectively, of the :math:`i`'th star particle in a given zone at a given time. Relevant Source Code: - ``vice/src/multizone/recycling.c`` - ``vice/src/multizone/element.c`` - ``vice/src/multizone/ism.c`` Sanity Checks ------------- At all timesteps VICE forces the mass of every element to be non-negative. If the mass is found to be below zero at any given time, it is assumed to not be present in the interstellar medium and is assigned a mass of exactly zero. Absent this, the mass of each element reported by VICE is merely the numerically estimated solution to the enrichment equation. Relevant source code: - ``vice/src/singlezone/element.c`` .. [5] Christensen et al. (2018), ApJ, 867, 142 .. [6] Chisholm, Trimonti & Leitherer (2018), MNRAS, 481, 1690