How Stars End

The BPT law

BPT Part 2.9 (How the Stars End): "A star dies when its core runs out of fuel it can burn. What it leaves depends on what can still hold the core up." Electron pressure holds a core of up to about 1.4 Suns, the Chandrasekhar Limit, a white dwarf; neutron pressure up to about 2.3 Suns, the neutron-star limit (Tolman; Oppenheimer & Volkoff), a neutron star; above that "the core falls inside its own horizon". Its summary: "Birth mass decides the remnant: a white dwarf from 0.075 to 8 M☉, a neutron star from 8 to 25 M☉, a Stellar Null Well from 25 to 130 M☉; in metal-poor gas, nothing from 130 to 250 M☉ and direct collapse above." The table gives the events: red giants and planetary nebulae for stars of 0.5–8 Suns, core-collapse supernovae and neutron stars of 1.2–2.3 Suns for 8–25, black holes of 5–50 Suns for 25–130, "pair-instability supernova: the whole star blown apart" for metal-poor 130–250, and "direct collapse, no explosion", "black hole of 100 M☉ or more", above 250. The Last Tenth adds that a Sun-like star "at the end blows off its outer layers as a glowing shell, a planetary nebula".

BPT Part 6.10 gives the formal pieces: the White Dwarf Initial–Final Mass Relation, M⊘f ≈ 0.109·M + 0.394 M☉ for birth masses 0.8–8 M☉ (Kalirai et al. 2008); the Chandrasekhar Limit, M⊘ = 3.098·m_P³/(μₑ·m_H)², 1.43 M☉ for carbon–oxygen; and the Stellar Null Well Threshold, M▣ > M_TOV ≈ 2.3 M☉.

The established physics

Kalirai et al. (2008), the initial–final mass relation; Chandrasekhar (1931); Tolman (1939), Oppenheimer & Volkoff (1939); Woosley, Heger & Weaver (2002); Heger et al. (2003); Heger & Woosley (2002) for pair instability.

The game's math

fate(m, metalFree) stars.js:48:

m < 0.5                       none (outlives the universe)
m < 8                         white dwarf, rem = min(1.4, 0.109·m + 0.394)
m < 25                        neutron star, rem = 1.4, supernova
m < 130                       Stellar Null Well, rem = min(47, 5 + 0.4·(m − 25)), supernova
m < 250 and metal-free        pair-instability supernova, rem = 0
otherwise                     direct collapse, rem = 0.9·m, no supernova

In stepStars() stars.js:210–237, for each lot whose life is over:

ejecta = m − rem;   ej = n · ejecta · S.sun   cells returned to the body's gas
white dwarf: class WD, m ← rem;  a planetary nebula flash (up to 24 a step)
neutron star: class NS, m ← rem
pair instability: class gone, m ← 0
well: class gone; the body loses n·rem·S.sun cells; up to min(n, 4) wells are reported,
      each of mass (n / min(n, 4))·rem·S.sun

Remnant white dwarfs and neutron stars stay in the body (its mass includes them). The tallies S.sn, S.wd, S.ns, S.pisn, S.bh add n each.

Worked case: a lot of 3 stars of 40 Suns dies: rem = 5 + 0.4 × 15 = 11 Suns each; ejecta 29 Suns each; with S.sun = 3.8 cells (desktop, level 28), 3 × 29 × 3.8 ≈ 330 cells go back to gas and 3 wells of 11 Suns (42 cells) each are reported.

The code's linear remnant law gives 5–47 Suns over 25–130, inside the book's 5–50.

In the game

Every death with a supernova flashes where it happened: a shell that starts white-hot and turns blue-violet as it spreads, growing as the square root of its age and fading as the square of what is left, over a flash life of 16, twice a merger's 8 (stepFlashes() scene3d.js:1049–1062, FLASH_LIFE scene3d.js:18). A white dwarf's birth glows as a soft teal planetary nebula, flash life 10, round the new remnant (scene3d.js:1060). The dead star's point becomes a small pale white dwarf or a tiny neutron star, one in five of which blinks as a pulsar (Part 5.4); a pair-instability giant or a collapsed core leaves no point at all.

Each kind of death is announced once, by starToasts() app.js:731–738:

• First supernova: "A star of N Suns has exploded, returning its metals to the gas"
• First white dwarf: "A Sun-like star has shed its outer layers and left a white dwarf"
• First neutron star: "A supernova has left a neutron star behind"
• Pair-instability supernova: "A metal-free giant has blown itself apart, leaving nothing"

When the universe is nine tenths grown the banner "the web stands" adds "· N supernovae so far" (app.js:673). The Stars line keeps the running counts of white dwarfs, neutron stars, supernovae and stellar wells, and the end card's Stars born box lists "… supernovae · … white dwarfs · … neutron stars · … stellar wells" (app.js:2783). The wells themselves are Part 5.13.

The scale model

Masses are in the level's Suns; each drawn remnant stands for starMult real ones.

Where the game and the theory differ

• A metal-rich star of 130–150 Suns (possible under the Kroupa top of 150) is not metal-free, so it takes the last branch, direct collapse to 0.9·m, a case BPT Part 2.9's table does not cover.
• BPT Part 2.9 gives direct collapse above 250 Suns "a black hole of 100 M☉ or more"; the game keeps nine tenths of the star, 0.9·m.
• Every neutron star is 1.4 Suns; BPT Part 2.9 gives a range of 1.2–2.3.
• The initial–final mass relation, which BPT Part 6.10 states for birth masses of 0.8–8 Suns, is applied from 0.5.

Fixed numbers