Cosmology · Conditional futures of the Universe

Declining star formation and long-lived populations

Follow gas recycling through a finite population ledger, keeping surviving stars, substellar objects and compact remnants distinct.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

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After reading, you can

Identify how gas supply and recycling affect later stellar births.
Distinguish surviving stars and brown dwarfs from compact remnants.
Interpret a finite population ledger without turning its supplied fractions into a universal future timeline.

A remote population is not one universal sequence

Future star formation depends on available gas, stellar evolution and interactions. Recycling can return some material to gas, while gas loss, infall and mergers can alter a system’s supply.

Surviving stars are different from compact remnants. White dwarfs, neutron stars and black holes represent different stellar outcomes; the abundance of each depends on the progenitors and their evolution.

Brown dwarfs form as substellar objects rather than as extinguished main-sequence stars. Isolated objects without sustained hydrogen-burning balance can cool; temporary light-element burning does not make every brown dwarf a long-lived hydrogen-burning star.

Use a finite supplied inventory to follow these distinctions. It will model allocations and recycling, not actual stellar lifetimes, a mass function, a cooling curve or a cosmic schedule.

Begin with gas and a specified update rule

Start with 100 mass-proxy units of gas and zero in six other categories: short-lived stars S, long-lived stars L, brown dwarfs BD, white dwarfs WD, neutron stars NS and black holes BH. Each proxy unit represents an arbitrary mass M_*.

A stage is an arbitrary proper-time interval Δt. All of the previous stage’s short-lived cohort is processed first: half its proxy returns to gas, one quarter goes to WD, and one eighth each to NS and BH.

Let A = G+S/2 be the gas available after that return. Allocate one quarter of A to births, split into 3/5 long-lived stars, 3/10 short-lived stars and 1/10 brown dwarfs. The other three quarters remains gas.

Long-lived stars and all non-gas categories other than S persist through the displayed eight stages. Long-lived means surviving this finite window; no category is declared immune to all future physical processes.

Recycling slows the depletion without ending it

At the first stage, A = 100. The 25 birth-proxy units contain 15 long-lived stars, 7.5 short-lived stars and 2.5 brown dwarfs, leaving 75 gas. No compact remnant has yet formed in this update order.

At the next stage, 3.75 from the old short-lived cohort returns to gas, giving A = 78.75. Its old remnant allocations are 1.875 to WD and 0.9375 each to NS and BH.

The new birth allocation is 19.6875, of which 17.71875 goes to stars and 1.96875 to brown dwarfs. Gas after these births is 59.0625. New stellar births exclude the separately counted brown-dwarf category.

The seven reservoirs sum to 100 at every stage. This is exact conservation of the supplied proxy ledger; nuclear binding changes and radiated energy are omitted, so it is not literal relativistic mass-energy bookkeeping.

Selected stages of the closed supplied proxy inventory; all numbers are mass-proxy allocations, not object counts
Stage	Gas	Short-lived stars	Surviving long-lived stars	Brown dwarfs	White dwarfs	Neutron stars	Black holes
0	100	0	0	0	0	0	0
1	75	7.5	15	2.5	0	0	0
2	59.0625	5.90625	26.8125	4.46875	1.875	0.9375	0.9375
4	36.627979	3.662798	43.440439	7.240073	4.514355	2.257178	2.257178
8	14.086893	1.408689	60.147362	10.024560	7.166248	3.583124	3.583124

Optional mathematical path: the available-gas recurrence

The supplied update gives G_next = 3A/4 and S_next = 3A/40. Therefore the next stage’s available gas is A_next = G_next+S_next/2 = 63A/80.

The fraction 63/80 is less than one. Hence available gas and each subsequent birth allocation decline geometrically under these fixed rules, even though some material is recycled.

New stellar proxy per stage is 9A/40, since nine tenths of the total A/4 births are stars. It decreases from 22.5 in the first step to about 3.328028 for the step following stage eight.

Dividing those proxy masses by Δt would define this model’s new stellar mass rate. Neither the interval nor the allocation fractions were calibrated to a real star-formation history.

The supplied closed inventory starts with 100 mass-proxy units of gas and zero in six other categories. Each unit is arbitrary mass M_*, not an object count or literal relativistic energy budget. Stage is an arbitrary proper-time interval. Panel A shows selected stages 0, 1, 2, 4 and 8 on a zero-based inventory axis from 0 to 100. All actual bar segments carry the matching legend hatch and outline. Bottom-to-top order is explicitly numbered: 1 gas G, 2 short-lived stars S, 3 surviving long-lived stars L, 4 brown dwarfs BD, 5 white dwarfs WD, 6 neutron stars NS, 7 black holes BH. The native table gives rounded rows (stage,G,S,L,BD,WD,NS,BH): (1,75,7.5,15,2.5,0,0,0), (2,59.063,5.906,26.813,4.469,1.875,0.938,0.938), and (8,14.087,1.409,60.147,10.025,7.166,3.583,3.583). Before new births, the old S cohort returns half to gas, one quarter to WD and one eighth each to NS and BH. Available A = G+S/2 then puts A/4 into births, split 3/5 L, 3/10 S and 1/10 BD; 3A/4 remains gas. Panel B separately shows the NEXT step's new stellar proxy, excluding brown dwarfs, on a zero-based axis from 0 to 24. Integer-stage markers follow 22.5(63/80)^n; the connecting curve is a guide, not continuous star formation. It falls from 22.5 to 3.328028 after stage eight. Seven full-precision allocations sum to 100. Survival is stipulated only for this finite window. No real IMF, cooling curve, stellar lifetime or universal remnant path is predicted. Nuclear binding and radiation are omitted from the proxy ledger. Original Ark teaching model.

Supply additional physics before predicting a real population

An actual stellar calculation needs a distribution of stellar masses, composition, lifetimes, mass return and remnant outcomes. A fixed aggregate short-lived cohort does not correspond to one progenitor mass with several arbitrary fates.

The model does not calculate brown-dwarf luminosity or cooling. Isolated cooling, irradiation and possible interactions require thermal and dynamical calculations beyond a conserved category label.

Gas infall, expulsion, stellar encounters and mergers could change the ledger. In some conditional settings, interactions between substellar objects can enable new stellar formation; that is a separate mechanism rather than a guaranteed final stage.

The useful lesson is the dependence of a population on resources and pathways. Surviving stars, cooling substellar objects and compact remnants can coexist, with their proportions and persistence determined by additional assumptions.

Worked reasoning

Allocate the first birth cohort

What forms during the first stage, starting with 100 gas-proxy units?

One quarter of the available gas becomes births: 25 units.
The supplied shares give 15 long-lived stars, 7.5 short-lived stars and 2.5 brown dwarfs.
The other 75 stays gas; no earlier short-lived cohort existed to make remnants.

The first stage contains gas, two surviving stellar cohorts and a distinct brown-dwarf category, with no compact remnants yet.

Try the reasoning

How much gas is available before the second birth allocation?
Show the answer and reasoning
78.75 proxy units.
The remaining 75 receives half of the previous 7.5 short-lived cohort.
Are the model’s brown dwarfs white-dwarf remnants?
Show the answer and reasoning
No.
They occupy a separate substellar birth category, while white dwarfs receive an allocation from an older stellar cohort.
Does the growing long-lived category imply that those stars survive indefinitely?
Show the answer and reasoning
No.
Their survival is stipulated only over the finite eight-stage window; no physical lifetime was assigned.

What to carry forward

Gas availability and recycling jointly affect subsequent stellar births.
Surviving stars, substellar objects and compact remnants require distinct categories.
A supplied finite ledger illustrates pathways without establishing real fractions or a universal timeline.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.