Lifetimes, Luminosity and the Giant Phase

The BPT law

BPT Part 2.9 gives the Main-Sequence Lifetime, τ⊛ = τ☉·(M/M☉)^(−2.5): "The heavier the star, the faster it burns: lifetime falls as mass to the −2.5", with τ☉ ≈ 10¹⁰ years. "A star's fuel grows with its mass but its brightness grows much faster, about as M^3.5, so the heaviest stars die first … Its first deaths, and its first Stellar Null Wells, come within a few million years of its first stars; its white dwarfs accumulate for billions of years; its red dwarfs outlive everything else in it." BPT Part 6.10 states the Mass–Luminosity Relation, L⊛ = L☉·(M/M☉)^3.5, "a star's brightness rises steeply with its mass", 3.5 being "the average over the stars that matter most for the light of a galaxy".

BPT Part 2.9's Giant Phase, t⋆ᵍ = 0.1·t⋆: "Through about nine tenths of its life it burns hydrogen in its core on the main sequence; in the last tenth it changes shape and colour." A star like the Sun swells into a red giant; one of more than about eight Suns into a supergiant. Red dwarfs, in the book's table, fade and are never giants. BPT Part 6.10's Game's Reading sets the clock: the lifetime M^(−2.5) with "the game's Sun living a set share of its universe's life".

The established physics

Mass–luminosity and lifetime power laws, and a giant phase of about a tenth of the main sequence (Kippenhahn, Weigert & Weiss 2012; Iben 1967; cited in BPT Part 2.9). Below 0.075 M☉ there is no hydrogen burning.

The game's math

lifetime(m, τ☉) = ∞                 if m < 0.075          stars.js:43
                = τ☉ · m^(−2.5)     otherwise              (Tempos)
luminosity(m)   = 10⁻⁴              if m < 0.075          stars.js:45
                = m^3.5             otherwise              (Suns)

τ☉ = matter.tauSun = max(8, lifeEndTempo − universePulses) at the burst (app.js:1246): the Sun lives the whole life from the Physical Nova to the final well, 2^Lm Tempos. The fallback in stepMatterPost() is max(8, rupturePulses) (app.js:665).

A lot becomes a giant once (s.giant) when it is alive, not a remnant, at least 0.3 Suns, and now − born ≥ 0.9·life (stars.js:207). It is counted under giants while alive.

luminosity() does not move any mass. It weights each star's light in the galaxy readout (app.js:2977) and in the drawing (scene3d).

Worked case, desktop (τ☉ = 64 Tempos, one sub-step = 1/8 Tempo):

So massive stars live and die within a sub-step or three of their birth, as in BPT Part 2.9 ("its first deaths come within a few million years of its first stars").

In the game

The ordering in time is what the player sees. The large blue-white points of O and B stars appear and are gone in moments, each ending in a supernova flash (Part 5.9); F and G stars, the Sun's kind, shine through most of the run and the heavier ones swell into orange giants late in it; K and M dwarfs shine to the end. The Stars line's giants count rises in the last part of the life, and white dwarfs pile up behind them.

Luminosity decides what is drawn. A body's drawn points take its stars brightest first (scene3d.js:761), so a clump holding one O star shows it before its hundreds of red dwarfs; a lone point counts as a star only if it holds one as bright as the Sun, else it shows its gas (scene3d.js:764). A star's drawn size grows with the logarithm of 1 + L (scene3d.js:669, 672), and the starlight on the torus's skin and a galaxy's color are weighted by n·L (scene3d.js:916; app.js:2977). A giant is drawn swollen: a clear orange at a size of 0.06, a supergiant redder at 0.1, against 0.014 for a red dwarf (scene3d.js:668, 671).

The scale model

The lifetimes are shares of the scale model's life: a star of mass m lives m^(−2.5) of the life wherever it is drawn. The true life is 2^L Tempos; the shares are the same.

Where the game and the theory differ

• The game lets stars from 0.3 Suns become giants (stars.js:207); BPT Part 2.9's table makes giants of stars from 0.5 Suns and says red dwarfs never are. With τ☉ the whole life, a star under 0.5 Suns never reaches nine tenths of its life in the run, so no red dwarf is ever drawn as a giant.
• BPT Part 2.9's Sun lives about 10¹⁰ years in a universe of 11.0 × 10⁹ years to its final well; the game's Sun lives the whole life from the Physical Nova to the final well. This is the "set share" of BPT Part 6.10's Game's Reading, taken as one.

Fixed numbers