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Galactic chemical evolution

Galactic chemical evolution · L2

Calculate a delayed enrichment track

Sandbox Physics ·

The source entering the gas today need not come from stars born today. To calculate that memory, we need a distribution of delays and the earlier formation history. A simple prompt-only solution gives us a reference before we add the delayed source. It also tells us which wind comparisons are physically comparable.

A closed box gives an independent control

Start with no inflow, no wind and only a prompt net yield. Gas decreases as a fixed fraction of each new generation remains locked. Dividing the element-fraction rate by the gas-mass rate eliminates the formation rate. Integrating from the initial gas mass to the remaining gas gives:

Z−Z0=y1−Rln⁡ ⁣(G0G)Z-Z_0=\frac{y}{1-R}\ln\!\left(\frac{G_0}{G}\right)
Here ln⁡\ln is the natural logarithm. The initial composition Z0Z_0 remains present; setting it to zero would be another initial condition.

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

Work through a half-depleted reservoir

With return fraction 0.4, prompt iron yield 0.000246 and initial iron fraction one millionth, half the initial gas remaining gives an iron fraction of approximately 0.000285190. The factor multiplying the natural logarithm is the yield divided by 0.6. The logarithm is ln⁡2\ln 2, not log⁡102\log_{10}2.

For a well-mixed wind, replace 0.6 in that factor by 0.6 plus η\eta. At the same remaining gas fraction, stronger loss means less new iron enrichment. At the same elapsed time under a prescribed formation history, however, it can leave less gas and a larger iron fraction. Our prompt-only control gives 0.00008236 without wind and 0.00010154 with η\eta one after the same time. Neither statement contradicts the other.

These controls isolate the prompt-only mathematical limit; the full calculation retains its original prompt and delayed effective yields.

Prompt-only analytic iron fractions with wind loading zero and one at the same time
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

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

Normalize a distribution of waiting times

A delay-time distribution tells us how a population releases its integrated delayed yield over time. In this package the dimensionless delay uu is the actual delay divided by one gigayear. The kernel is a power law between 0.15 and 15 gigayears and zero elsewhere. Its coefficient makes its integral over that entire interval equal to one.

K(τ)=AGyru−1.1,u=τGyr,A=0.10.15−0.1−15−0.1K(\tau)=\frac{A}{\mathrm{Gyr}}u^{-1.1},\quad u=\frac{\tau}{\mathrm{Gyr}},\quad A=\frac{0.1}{0.15^{-0.1}-15^{-0.1}}
This kernel has units of inverse time. Multiplying by the integrated delayed net yield gives element release per unit stellar mass formed per unit delay.

Reading: Independent one-zone chemical-evolution Python materials · VICE v1.3.1 fixed effective yields and solar reference

Do not renormalize an unfinished history

Let FF be the integral of the kernel from its minimum delay up to a specified delay. At eight gigayears, FF is about 0.889076. Nearly eleven percent of the integrated delayed yield of an ideal population born at time zero still lies beyond that age. Stopping the simulation at eight gigayears does not make it legitimate to rescale the completed fraction to one.

For example, a million solar masses formed at once eventually contribute 2580 solar masses of new delayed iron under the fixed input. By eight gigayears they have delivered about 2293.82 solar masses. The small delayed magnesium yield similarly delivers about 7.85054 solar masses. These are cumulative new inputs before later locking or wind removal; they are not the gas inventories at eight gigayears.

Reading: Independent one-zone chemical-evolution Python materials · VICE v1.3.1 fixed effective yields and solar reference

Add all the earlier generations causally

A generation born at time bb has delay t−bt-b when the clock reaches tt. Multiply its formation rate by the kernel at that delay, then add every eligible birth time. This is a causal convolution: no future star formation contributes to a present source.

C(t)=∫0tψ(b)K(t−b) db,Si(t)=yiCCψ(t)+yiIaC(t)C(t)=\int_0^t\psi(b)K(t-b)\,db,\qquad S_i(t)=y_i^{\rm CC}\psi(t)+y_i^{\rm Ia}C(t)
CC has the same units as the star-formation rate. The integrated yield coefficients convert it into the new-element source rate for each element ii.
Prompt and delayed new iron rates from the same renewed-gas record
Original renewed-gas history. Both sources use the saved formation and delayed-response rates and the fixed net iron yields. Source rates are not the iron inventory remaining in the gas. Open the full figure

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

Compute the source and the reservoir together

The independent Python calculation uses adaptive quadrature for the continuous convolution and a high-order differential-equation solver for the inventories. It integrates separately on each side of the first allowed delay, where the source switches on. Independent positive quadrature and a changed order of time integration check the source.

The reservoir starts slightly enriched: its iron fraction is one millionth, and its initial magnesium-to-iron ratio matches the fixed prompt yield ratio. No older stellar population is silently added to produce delayed events before the simulation begins. Inflow, the prescribed formation history and winds then act on the same gas record.

A knee in the stellar track depends on the competition among prompt sources, accumulated delayed sources, dilution and the times when stars actually form. It is not an exact marker of the first delayed event. The time integration, source quadrature and birth bins involve separate approximations. The reference checks the quantities affected by each of them.

After the gas trajectory is checked, the stellar population still requires accurate birth integration. The next chapter connects those weighted births to a measurement density, whose time integration receives a separate refinement test.

Reading: Independent one-zone chemical-evolution Python materials

Exercises and explained answers

  1. At eight gigayears, should the kernel be renormalized so that its completed integral is one?

    Read the explanation

    No. It was normalized on the declared 0.15-to-15-gigayear interval. An eight-gigayear record contains a genuinely incomplete delayed contribution. Renormalizing it changes the model and releases future material too early.

  2. Why does the convolution contain the formation rate at bb and the kernel at t−bt-b?

    Read the explanation

    The first quantity describes how much stellar mass was born then. The second describes how that population releases its delayed yield after the elapsed waiting time. Using the rate at tt for every delay would erase the earlier formation history.

  3. If two tracks have the same iron abundance, must they represent the same elapsed time?

    Read the explanation

    No. Different gas supplies, formation rates, outflows and delayed contributions can reach the same abundance at different times. Even one track can revisit an abundance after dilution.

Try the prediction

Compare histories in one gas reservoir

  1. Compare the actual prompt and delayed iron source histories. Locate the first allowed delayed contribution and distinguish it from the bend in the stellar abundance track.
  2. Read the cumulative delayed fraction at the final time and explain why the kernel still has support beyond the displayed record.

Sources and calculation scope

  1. Weinberg, Andrews & Freudenburg, Equilibrium and Non-Equilibrium Aspects of Galactic Chemical Evolution, arXiv: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

    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

    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.

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