Astrophysics Protoplanetary Disk Mass From Jcost

Disk mass from cost

A protoplanetary disk is a rotating reservoir of gas and dust around a young star, the raw material from which planets assemble. Astronomers measure its mass relative to the star, a fraction that observations put near one percent: a star like the Sun typically carries a disk with about a hundredth of its own mass. That number matters because it bounds how much solid material can gather into planets, and it is the quantity the framework's module addresses.

The classical route to disk mass runs through radiation. Dust in the disk absorbs starlight and re-emits it at millimeter wavelengths, so telescopes measure that emission and convert it to a mass using an assumed dust temperature and opacity. The conversion carries large uncertainty, often a factor of a few, because the dust properties and the disk's optical depth are not directly known. A complementary approach uses the disk's rotation curve to infer the gravitational potential, but that measures the star's mass far more reliably than the disk's. The one-percent rule of thumb has held up across surveys, though individual disks range from a tenth of a percent to several percent.

In Recognition Science, the framework models the disk mass fraction as a cost: the recognition cost J(x) = (x + 1/x)/2 - 1 applied to the ratio of disk mass to stellar mass. The framework's library, a machine-checked collection of formal theorems, establishes three general facts about this cost: it vanishes when the ratio equals one, it is never negative for positive masses, and the golden-ratio threshold φ - 3/2 is positive. Those are properties of the cost function itself, not of protoplanetary disks. The module defines the disk cost as J(m/e) without any definition of m or e in astrophysical terms, so it establishes nothing specific to disks.

The research note attached to the module sketches where the idea was meant to go: taking the cost at the golden ratio, J(φ) ≈ 0.118, raising it to the one-tenth power gives about 0.81, and multiplying by a scale factor yields the observed one percent. That chain is a proposal, not a derivation. The formal content establishes only the three general cost facts, and the step from those facts to a disk mass prediction requires defining the masses in the disk's own terms, which the module does not do.

What the module does establish, in plain language, is a template: any physical system whose mass ratio can be expressed as a positive real number has a well-behaved cost with a unique zero and a threshold tied to the golden ratio. The disk application remains open until a definition connects the abstract ratio to measurable disk and stellar masses. The one-percent figure is an empirical anchor, not a theorem, and the framework's contribution is the cost function that could one day reproduce it.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

noncomputable def cert : ProtoplanDiskCert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

What this page does not claim

The module does not derive the one-percent disk mass fraction from the cost function. The golden-ratio threshold φ - 3/2 is not shown to select a physical disk mass. No statement in the module connects the abstract cost to any specific protoplanetary disk observation.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Astrophysics/ProtoplanetaryDiskMassFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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