Metals

The BPT law

BPT Part 3.12 (Ripples, Spin, and the Whirling Disks), in Where the Metals Come From: "every metal in the universe was made in a star … So the metals are not the universe's but each galaxy's: a galaxy's gas holds the metals its own stars have returned, and a galaxy that has made few stars cools as the first clumps did." Core-collapse supernovae make most of them; "a metal-free giant of 140 to 260 Suns dies by pair instability: its helium core, 64 to 133 Suns, burns almost wholly to oxygen, silicon and iron".

Its Metals Returned law is M_Z = M_ej·(Z★ + p_Z): "A dying star returns its birth metals and the metals it made", M_ej its birth mass less its remnant, Z★ the metal share of the gas it was born from, p_Z the share of its ejecta that is new metals, "none for a star that dies quietly, about half for a pair-instability giant"; the ejecta "mix first through the gas of the star's own cloud". Its Net Metal Yield, y_Z = M_Z,new/(M★·(1 − R)), is about 0.05 with R ≈ 0.43 for the Kroupa function, "so new metals are about 0.029 of the mass of stars formed". Its game reading sets the game's share: "new metals from a core-collapse supernova at p_Z = 0.17, the one share that makes the game's own population return the yield above". The ruling quoted at stars.js:110 (2026-10-28): "they need their own metals for cooling, and the metals should be generated from supernova".

The established physics

Nomoto, Kobayashi & Tominaga (2013) yields; Vincenzo et al. (2016) net yield; de Avillez & Mac Low (2002) mixing through the cloud; Heger & Woosley (2002) pair-instability metals, 64–133 Suns of metals in stars of 140–260. Critical metallicity Z_crit = 10^(−3.5) Z☉ (Bromm et al. 2001), with Z☉ = 0.019 (Anders & Grevesse 1989).

The game's math

• Mixing. Each step, each cloud's metals are spread evenly through its gas: Z = Σzm / Σgas, then every body's zm = gas·Z (stars.js:155).
• Birth. A lot is born with its cloud's Z; it is metal-free if Z < Z_crit (stars.js:176).
• Death. b.zm += ej·Z★ + made, made = ej·newMetals(m, f) (stars.js:215–216):

newMetals = min(1, clamp(64 + (m − 140)·69/120, 64, 133) / m)    pair instability
          = P_Z_CC = 0.17                                         any other supernova
          = 0                                                      no supernova

P_Z_CC is the share of core-collapse ejecta that is new metals, chosen so that this Kroupa population returns y_Z(1 − R) = 0.05 × 0.576 = 0.029 of the mass of stars formed, its supernovae throwing off 0.168 of it (stars.js:113–117).

• Takes. takeGas() and the winds remove metals in proportion.
• Record. S.metals is all new metals made; S.Z the mean of all gas, for the record only; each galaxy cools at its own (Chapter 4, Part 4.14).

Worked case: a 15-Sun star with Z★ = 0.001 dies as a neutron star: ejecta 13.6 Suns; metals returned 13.6 × (0.001 + 0.17) ≈ 2.3 Suns. A metal-free 200-Sun star: 64 + 60 × 69/120 = 98.5 Suns of new metals, 49 % of it.

In the game

Metals are not drawn; they are read. The first supernova's banner says the star "has exploded, returning its metals to the gas" (app.js:735). A tap in a galaxy gives the metals of its own gas against the Sun's, "metals X of the Sun's", or "metals none yet" below a hundredth of the Sun's (app.js:2986, 2994), next to its heat mode, which reads 'cold' once its metals pass the critical metallicity, 'H₂' below it while the gas can still cool through molecules, and 'too warm to cool' otherwise (heat.js:73–74). The run record keeps the mean metals of all gas as stars.metals (app.js:2211).

What the player sees of metals is their effect: the first generation, from metal-free clouds, is all massive blue points (Part 5.5); once a cloud's supernovae have enriched it past Z_crit, its next stars come from the Kroupa function and the sky fills with small red points; and a galaxy whose gas holds metals cools faster and keeps forming stars (Chapter 4, Part 4.14).

The scale model

Exact: no scale model. Metals are mass shares.

Where the game and the theory differ

• Only supernovae make new metals. BPT Part 3.12 names carbon and nitrogen from the winds of dying stars of one to eight Suns and iron from Type Ia supernovae, "a few hundredths of the metals", and its game reading says they "are not yet made"; in the game a white dwarf's progenitor returns its birth metals only (p_Z = 0).

Fixed numbers