---
title: "Follow the gas and the newly made elements"
description: "Build a one-zone mass budget, distinguish recycled material from net yields, and derive why a well-mixed wind need not lower abundance at fixed time."
canonical_url: "https://sandboxphysics.com/stories/galactic-chemical-evolution/follow-the-gas-and-new-elements/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/galactic-chemical-evolution/follow-the-gas-and-new-elements/"
last_modified: "2026-11-05"
---

[Galactic chemical evolution](https://sandboxphysics.com/stories/galactic-chemical-evolution/)

Galactic chemical evolution · L1

# Follow the gas and the newly made elements

Sandbox Physics · 2026-11-05

A chemical-evolution calculation needs two ledgers. The first follows gas, stars and outflows. The second follows each element inside those components. Keeping them separate prevents a common mistake: counting material returned by a star as though it were all newly synthesized. We will build both ledgers, then use them to explain a surprising wind comparison.

## Before you begin

Ratios and multiplication. A derivative below means a rate of change; the product rule is introduced when it is needed. Mass fractions and rates are defined before use.

## A generation returns old material as well as new elements

Suppose a million solar masses of gas form stars. In the instantaneous-recycling approximation used here, a fraction 0.4 is returned immediately and 0.6 remains locked in long-lived stars and remnants. The approximation combines processes with different actual lifetimes into an effective return fraction. It is not a claim that every star instantly ejects forty percent of its mass.

If the initial iron mass fraction was 0.001, that generation inherited a thousand solar masses of iron. Four hundred return with the recycled material and six hundred remain locked. Newly synthesized iron is an additional contribution. The fixed core-collapse net yield is 0.000246 per unit total mass formed, so this generation supplies 246 solar masses of new iron. Multiplying this yield by 0.4 again would undercount the new iron; it is already defined per unit total mass formed.

The effective yields are fixed to the declared VICE 1.3.1 input convention. They assume a particular Kroupa stellar-mass distribution and do not open a new grid of progenitor yields. The delayed magnesium yield is small but nonzero.

Reading: VICE v1.3.1 fixed effective yields and solar reference · Weinberg, Andrews & Freudenburg, Equilibrium and Non-Equilibrium Aspects of Galactic Chemical Evolution, arXiv:1604.07435v1

## Write the total-gas rate first

Let $G$ be gas mass, $\psi$ the total rate at which gas forms stars, $I$ the inflow rate, and $\eta$ the outflow mass per unit mass of stars formed. A wind rate of $\eta$ times $\psi$ has units of mass per time, just like the other terms. With return fraction $R$, only one minus $R$ is permanently removed into stars in this approximation.

$$
\dot G=I-(1-R+\eta)\psi
$$

A dot denotes change per unit time. The three destinations are inflow, net stellar locking, and wind. Here $R$ is fixed at 0.4.

Reading: Independent one-zone chemical-evolution Python materials · Weinberg, Andrews & Freudenburg, Equilibrium and Non-Equilibrium Aspects of Galactic Chemical Evolution, arXiv:1604.07435v1

## Now follow one element through those same flows

Write the element mass in the gas as $G$ times $Z$, where $Z$ is its mass fraction. A well-mixed wind removes gas at the current gas composition; locking material into stars does the same. Let $S$ be the rate of newly synthesized element input and let the inflow have element fraction $Z_{\rm in}$. Both equations use the same gas composition and flow rates.

$$
\frac{d(GZ)}{dt}=S+Z_{\rm in}I-Z(1-R+\eta)\psi
$$

The left side is element mass per time. $S$ includes both the prompt and delayed net sources. In this teaching family the inflow has zero magnesium and iron.

Reading: Independent one-zone chemical-evolution Python materials · Weinberg, Andrews & Freudenburg, Equilibrium and Non-Equilibrium Aspects of Galactic Chemical Evolution, arXiv:1604.07435v1

## A complete numerical step, before any integration

Take ten million solar masses of gas with iron fraction 0.001. Let one solar mass per year form stars, no gas enter, and $\eta$ equal 0.5. Consider only the prompt iron source for this calculation. The gas decreases by 1.1 solar masses per year: 0.6 is locked and 0.5 leaves in the wind.

The prompt source adds 0.000246 solar masses of iron per year, while locking and wind remove 0.0011. The iron inventory therefore decreases by 0.000854 solar masses per year. Yet the iron fraction increases, because the total gas reservoir shrinks as well. An inventory and a fraction answer different questions.

The product rule says that the change in $GZ$ equals $Z$ times the change in $G$ plus $G$ times the change in $Z$. Substituting the two ledgers cancels the composition-matched removal terms. With metal-free inflow the remaining equation is:

$$
G\dot Z=S-IZ
$$

In the example $I$ is zero, so the iron fraction increases at 0.000246 divided by ten million, or 2.46 times ten to the minus eleven per year.

Reading: Independent one-zone chemical-evolution Python materials

## Why the wind can raise abundance at a fixed time

The cancellation does not make the wind irrelevant. Changing the wind loading changes $G$, the mass of gas into which a given source is mixed. In this family the star-formation history is prescribed independently of the remaining gas, provided the reservoir stays positive. A stronger wind can therefore leave less gas to receive the same new element mass and raise its fraction at a specified time.

A separate prompt-only control starts with unit gas, iron fraction one millionth, and a constant formation rate of 0.3 per time unit. After one time unit, $\eta$ zero leaves gas 0.82 and iron fraction about 0.00008236. A wind loading of $\eta=1$ leaves gas 0.52 and iron fraction about 0.00010154. Both are valid budgets; the latter has less gas and a higher iron fraction.

At the same remaining gas fraction, the conclusion changes: a stronger well-mixed wind reduces the effective yield per logarithmic gas depletion. Always state what is held fixed. Tying the formation rate to gas mass requires solving that relation too, and can change the resulting history.

![Prompt-only analytic iron fractions with wind loading zero and one at the same time](https://sandboxphysics.com/_astro/wind.L7SmWEPX.svg)

Separate prompt-only mathematical control: identical prescribed constant formation, no inflow, same initial composition. The stronger well-mixed wind leaves less gas and raises its iron fraction at the same time. A same-gas-fraction comparison answers a different question. [Open the full figure](https://sandboxphysics.com/_astro/wind.L7SmWEPX.svg)

Reading: Independent one-zone chemical-evolution Python materials · Weinberg, Andrews & Freudenburg, Equilibrium and Non-Equilibrium Aspects of Galactic Chemical Evolution, arXiv:1604.07435v1

## What this first ledger leaves out

The gas is one mixed reservoir. Outflows carry the current reservoir composition; a preferentially metal-rich wind would require a different element-loss term. The fixed return approximation, trace-element treatment and fixed hydrogen fraction do not solve a full stellar mass-return calculation or gas thermodynamics. Histories that exhaust the permitted gas reservoir are refused before integration.

A saved record carries the same gas and element inventories into its abundance track and stellar birth groups. That common record is what makes later comparisons testable. The next chapter supplies the delayed source that enters $S$.

Reading: Independent one-zone chemical-evolution Python materials

## Exercises and explained answers

1. Two million solar masses form stars with $R$ equal to 0.4. How much is returned and locked?

   Read the explanation

   Eight hundred thousand solar masses return and 1.2 million remain locked. The amount of a newly made element still requires its separate net yield per total mass formed.

2. In the worked rate example, how can iron inventory decrease while iron fraction increases?

   Read the explanation

   The gas inventory falls even faster relative to its size. Using the product rule removes the composition-matched losses and leaves $G$ times the iron-fraction rate equal to the positive prompt source.

3. Would the same cancellation hold for a metal-enhanced wind with composition different from the gas?

   Read the explanation

   No. The element-loss term would no longer be the gas fraction multiplied by the total wind rate. An additional composition-dependent term would remain; that case lies outside the current model.

## Try the prediction

### [Compare histories in one gas reservoir](https://sandboxphysics.com/labs/galactic-chemical-evolution/)

1. Compare histories with and without a wind while following both gas mass and gas iron fraction. State whether you compare the same time or the same remaining gas fraction.
2. Export the gas, locked and outflow element inventories and reconstruct their sum including newly produced material.

[Previous: Why magnesium and iron remember star formation](https://sandboxphysics.com/stories/galactic-chemical-evolution/why-magnesium-and-iron-remember-star-formation/) [Next: Calculate a delayed enrichment track](https://sandboxphysics.com/stories/galactic-chemical-evolution/calculate-a-delayed-enrichment-track/)

## Sources and calculation scope

1. [Weinberg, Andrews & Freudenburg, Equilibrium and Non-Equilibrium Aspects of Galactic Chemical Evolution, arXiv:1604.07435v1](https://arxiv.org/abs/1604.07435v1)

   Complete fixed 30-page paper read again, including both appendices. Net yields, recycled material and analytic controls follow sections 2–3; delay approximations and birth weighting follow section 5. Its oxygen and exponential-delay examples are not silently relabeled as our fixed magnesium and finite power-law kernel. Its gas-linked formation law differs from our prescribed histories.

2. [VICE v1.3.1 fixed effective yields and solar reference](https://github.com/giganano/VICE/tree/v1.3.1)

   Previously frozen native defaults: Kroupa convention, effective net core-collapse and Type Ia magnesium/iron yields, Asplund 2009 reference and the declared hydrogen conversion. Public materials contain scalar values and identities, not a new yield grid or VICE executable. The independent Python does not execute VICE.

3. [Independent one-zone chemical-evolution Python materials](https://sandboxphysics.com/stories/galactic-chemical-evolution/materials/chemical-evolution-science-v1.zip)

   Original continuous convolution/DOP853 producers, saved fourteen-field inventories and birth populations, finite refinement, normalized conditional mixtures and mathematical limits. Public inputs and figures are original synthetic data. Default is offline; optional fixed Hill source is fetched and analyzed on the reader’s computer. No App/Engine, original paper, observed rows or measured-derived plot is included.