Stellar evolution and chemical enrichment

化学演化是宇宙学流体动力学模拟中星系演化的自然结果。化学演化模型已纳入宇宙结构形成的宇宙学模拟中，以恰当地研究化学增丰过程：星际介质（ISM）和星系周介质（CGM）中金属的分布记录了关于过去恒星形成和反馈历史的宝贵信息，是流体动力学模拟的一项关键特征（例如 [173] ,[174] ）。与明确分辨单颗恒星的观测不同，宇宙学模拟通过星粒子对星族进行粗略采样。大多数最先进的宇宙学流体动力学模拟（涵盖大宇宙体积和单个星系）中的星粒子质量，取决于分辨率，通常介于 $10^8$ 到 $10^3$ M$_{\odot}$ 之间（例如 [36] ,[40] ,[39] ,[175] ,[41] ,[176] ,[139] ,[6] ,[177] ,[178] ,[179] ,[180] ,[155] ,[135] ,[181] ）。在这些模拟中，星粒子是分辨率元素：每个星粒子代表一个简单星族（SSP），即一组具有相同初始金属丰度的同时代恒星的集合。每个星粒子的初始化学组分与其母气体元素相同，并由初始质量函数（IMF）表征。

Chemical evolution is a natural outcome of galaxy evolution in cosmological hydrodynamical simulations. Models of chemical evolution have been included in cosmological simulations of cosmic structure formation to properly address the study of the chemical enrichment process: the distribution of metals in the ISM and in the circum-galactic medium (CGM) encodes valuable information of the past history of star formation and feedback, and is a crucial feature of hydrodynamical simulations (e.g. [173] ,[174] ). At variance with observations that explicitly resolve individual stars, cosmological simulations provide a coarse sampling of stellar populations by means of star particles. The majority of state-of-the-art cosmological hydrodynamical simulations of both large cosmological volumes and individual galaxies have star particles whose mass typically ranges between $10^8$ and $10^3$ M$_{\odot}$ according to resolution (e.g. [36] ,[40] ,[39] ,[175] ,[41] ,[176] ,[139] ,[6] ,[177] ,[178] ,[179] ,[180] ,[155] ,[135] ,[181] ). In these simulations, star particles are resolution elements: each of them represents a simple stellar population (SSP), i.e. an ensemble of coeval stars that share the same initial metallicity. Every stellar particle initially shares the chemical composition of the gas element from which it has been originated, and is characterized by an initial mass function (IMF).

IMF $\phi (m)$ 决定了单位质量区间内的恒星数目：

The IMF $\phi (m)$ determines the number of stars per unit mass interval: $$ \phi (m) = \beta m^{- \alpha} \,\,\,,

该式定义在给定质量范围 $[M_{inf}, M_{sup}]$ 内。系数 $\alpha$ 设定了幂律在整个质量范围（或其内各质量区间）的斜率。对每个质量区间，归一化常数 $\beta$ 由以下条件确定：在整个质量范围上 $\int m , \phi (m) , dm = 1$，并且在各质量区间的边界处保持连续。

$$ within a given mass range $[M_{inf}, M_{sup}]$. The coefficient $\alpha$ sets the slope of the power law over the mass range or in the different mass intervals within the mass range. For each mass interval, a normalization constant $\beta$ is computed by imposing that $\int m \, \phi (m) \, dm = 1$ over the global mass range and continuity at the edges of subsequent mass intervals.

常用的 IMF 包括单斜率（[182] ,[183] ）或多斜率（[184] ,[185] ,[186] ）幂律，但也存在更复杂的形状（例如 [187] ）。读者可参阅 [188] ,[189] ,[190] ,[191] ,[192] ,[193] ,[164] 以了解不同 IMF 的详细信息，并理解采用不同 IMF 如何影响宇宙学模拟中的金属分布。

Commonly assumed IMFs consist of single- [182] ,[183] or multiple-slope [184] ,[185] ,[186] power law, but more sophisticated shapes have been proposed (e.g. [187] ). We refer the reader to [188] ,[189] ,[190] ,[191] ,[192] ,[193] ,[164] for details about different IMFs and to appreciate how adopting different IMFs can have an impact on metal distribution on cosmological simulations.

IMF 直接决定了不同初始质量恒星之间的相对比例，进而影响不同类型恒星贡献的元素相对丰度。恒星演化理论预言，所有质量大于 $M_{up}=8$ M$_\odot$ 的恒星将以核坍缩超新星或 II 型超新星（SNe II）的形式结束其生命（参见 [194] 及其参考文献）。在此假设下，可计算每个星粒子能向周围气体释放的总能量（通常为每次超新星 $10^{51}$ erg）。爆发为 SNII 的大质量恒星，其典型寿命一般不超过模拟的典型时间步长；在此近似下，死亡恒星在所谓的"瞬时循环近似"中注入反馈能量和金属，即在同一时间步长内完成。另一方面，Ia 型超新星（SNe Ia）普遍认为源自白矮星的热核爆炸（参见 [195] 及其参考文献）。这类爆炸相对于星粒子创建时间有显著延迟，因为双星系统中的白矮星需要从伴星吸积物质并达到触发热核燃烧的质量阈值。渐近巨星支（AGB）恒星经历显著的恒星质量损失，并对重元素的核合成做出重要贡献（参见 [196] 及其参考文献）。

The IMF directly regulates the relative ratio between stars of different initial mass, thus affecting the relative abundance of elements contributed by different types of stars. Stellar evolution predicts that all stars with masses larger than $M_{up}=8$ M$_\odot$ end their life as core collapse or type-II supernovae (SNe II) (see [194] and references therein). Under this assumption, the total amount of energy (typically $10^{51}$ erg per SN) that each star particle can release in the surrounding gas can be calculated. Within the approximation that the typical lifetime of massive stars which explode as SNII does not exceed the typical time step of the simulation, feedback energy and metals are injected by the dying star in the so-called ``instantaneous recycling approximation'', i.e. in the same time step. On the other hand, type-Ia SNe (SNe Ia) are believed to arise from the thermonuclear explosion of white dwarfs (see [195] and references therein). They lead to an explosion which is significantly delayed with respect to the time of creation of the star particle, as the white dwarf in a binary system has to accrete matter from the companion and reach the mass threshold for the onset of thermonuclear burning. Stars in the asymptotic giant branch (AGB) experience significant stellar mass loss and contribute significantly to the nucleosynthesis of heavy elements (see [196] and references therein).

具有精确恒星演化和化学增丰模型的宇宙学流体动力学模拟，可计算逐渐老化并最终爆发为超新星的恒星数量，以及污染周围 ISM 的金属量。除设定 IMF 外，常用的模型（例如 [172] ）还需恒星寿命、延迟函数和恒星产额表，以计算主要由 SNe Ia、SNe II 和 AGB 星留下的化学-能量印记。

Cosmological hydrodynamical simulations which feature an accurate model for stellar evolution and chemical enrichment can evaluate the number of stars aging and eventually exploding as SNe, as well as the amount of metals polluting the surrounding ISM. Besides assuming an IMF, commonly adopted models (e.g. [172] ) include stellar lifetimes, delay functions and table of stellar yields to calculate the chemo-energetic imprint mainly left by SNe Ia, SNe II, and AGB stars.

化学演化模型采用依赖于质量的寿命函数（即质量为 $m$ 的恒星死亡时的年龄函数，例如 [197] ,[198] ,[199] ）来考虑不同质量恒星的演化时标。初始质量高于 $M_{up}$ 且低于 $M_{sup, SNII}=40$ $M_\odot$ 的恒星，假定以核坍缩超新星的形式结束生命；而质量超过 $M_{sup, SNII}$ 的恒星，则假定直接坍缩为黑洞，因此不会对进一步的化学增丰或恒星反馈能量做出贡献。

Chemical evolution models account for the evolutionary timescales of stars with different masses by adopting mass-dependent lifetime functions (i.e. functions that describe the age at which a star of mass $m$ dies, e.g. [197] ,[198] ,[199] ). Stars with an initial mass larger than $M_{up}$ and lower than $M_{sup, SNII}=40$ $M_\odot$ are assumed to end their life exploding as core-collapse SNe, while stars that are more massive than $M_{sup, SNII}$ are assumed to implode in BHs directly, and thus do not contribute to further chemical enrichment or stellar feedback energy.

假定在整个质量范围中，一定比例（通常为常数）的恒星处于双星系统内，并作为 SNe Ia 的前身。采用 SNe Ia 的恒星演化情景（例如 [200] 的模型），结合寿命函数和延迟时间分布函数，可计算 SN Ia 的爆发率。

A fraction (usually constant) of stars relative to the whole mass range is assumed to be located in binary systems that are progenitors of SNe Ia. Assuming a stellar evolution scenario for SNe Ia (e.g. according to the model by [200] ) and adopting lifetime functions and delay time distribution functions, the rate of SN Ia explosions can be computed.

采用恒星产额追踪演化并最终爆发的恒星所产生的不同重金属。产额表示质量为 $m$、初始金属丰度为 $Z$ 的恒星产生的不同金属元素 $i$ 的抛出质量，即 $p_{Z_{i}}(m,Z)$。通常，至少需要三个主要过程的预测：AGB 恒星的持续质量损失、SNe II 和 SNe Ia。依赖于质量和金属丰度的恒星产额表已纳入最先进的宇宙学流体动力学模拟的化学演化模型中。恒星产额预测仍存在不确定性，主要原因是恒星演化模型中星风质量损失过程尚未充分理解。常用的恒星产额集包括：SNe Ia 的 [201] ；AGB 星的依赖于质量和金属丰度的产额 [202] ,[203] ,[204] ；SNe II 的依赖于质量和金属丰度的产额 [205] ,[206] ,[207] ,[208] 。读者可参阅（例如 [209] ,[210] ）获取解析形式的全面综述。

The production of different heavy metals by stars that evolve and eventually explode is followed by adopting stellar yields. They represent the ejected mass of different metal species $i$ produced by a star of mass $m$ and initial metallicity $Z$, i.e. $p_{Z_{i}}(m,Z)$. In general, predictions for at least three main processes are needed: the continuous mass loss of AGB stars, SNe II and SNe Ia. Tables of mass- and metallicity-dependent stellar yields are incorporated in the chemical evolution models of state-of-the-art cosmological hydrodynamical simulations. Predictions for stellar yield are still affected by uncertainties, mainly due to the poorly understood process of mass loss through stellar winds in stellar evolution models. Commonly adopted sets of stellar yields include: [201] for SNe Ia, mass- and metallicity-dependent yields by [202] ,[203] ,[204] for AGB stars, mass- and metallicity-dependent yields by [205] ,[206] ,[207] ,[208] for SNe II. We refer the reader to (e.g. [209] ,[210] ) for a comprehensive review of the analytic formalism.

最后需指出，将模拟结果与单颗恒星观测进行比较时，通常基于以下假设：模拟中星粒子的金属含量在统计上再现了该星粒子所采样 SSP 的平均金属丰度。更精确的技术见 [211] ,[212] 的介绍。

As a final caveat, we note that comparisons of results from simulations to observations of single stars are usually performed under the assumption that e.g., the metal content of a star particle in simulations statistically reproduces the mean metallicity of the SSP that the star particle samples. More accurate techniques are introduced in [211] ,[212] .

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