Gravity Rotation Vrot Flat Of Linear Menc

Flat rotation from linear mass

In Newtonian gravity, the orbital speed of a star around a central mass depends on how much mass lies inside its orbit. The standard formula, v = sqrt(G M / r), comes from balancing gravitational pull against centripetal acceleration. If the enclosed mass M stays constant as you move outward, the speed falls off as 1/sqrt(r), which is why planets farther from the Sun move slower. But if the enclosed mass grows linearly with radius, so that M(r) = α r for some constant α, then the r cancels: v becomes sqrt(G α), a fixed number independent of distance. That is the flat rotation curve, the signature astronomers observe in spiral galaxies.

The result is not new physics. It is a direct algebraic consequence of Newton's law, and it has been known since the 1970s as the simplest explanation for why galaxy rotation curves stay flat: the mass inside the orbit must keep increasing with radius. In the framework's machine-checked library of formal theorems, this statement is codified in a lemma named vrot_flat_of_linear_Menc, where Menc is the enclosed mass function of a rotation system. The lemma assumes a rotation system with gravitational constant G and a nonnegative enclosed mass, and it proves that if Menc(r) = α r for all positive radii, then the rotation velocity vrot(r) equals sqrt(G α) for every r > 0. The proof is a short algebraic calculation using the square root identity.

The framework also proves a companion statement: under the same linear mass condition, the centripetal acceleration g(r) equals (G α)/r, falling off as 1/r. This is consistent with the flat velocity, since acceleration is v²/r. A separate lemma, labeled Newtonian, adds the explicit assumption that α ≥ 0, which guarantees the square root is well-defined over the reals.

What the declaration does not claim is just as important. It does not say that real galaxies have linearly growing mass; that is an empirical question about dark matter halos, not a theorem. It does not derive the value of α or G from first principles. And it does not explain why the mass distribution is linear, only what the rotation curve would be if it were. The lemma is a conditional statement: if the mass grows linearly, then the curve is flat. It is a tool for checking consistency between a mass model and observed velocities, not a prediction of the mass model itself.

/-- If the enclosed mass grows linearly, `Menc(r) = α r` with `α ≥ 0`, then the rotation curve is flat:
    `vrot(r) = √(G α)` for all `r > 0`. -/
lemma vrot_flat_of_linear_Menc (S : RotSys) (α : ℝ)
  (hlin : ∀ {r : ℝ}, 0 < r → S.Menc r = α * r) :
  ∀ {r : ℝ}, 0 < r → vrot S r = Real.sqrt (S.G * α) := by
  intro r hr
  have hM : S.Menc r = α * r := hlin hr
  have hrne : r ≠ 0 := ne_of_gt hr
  have hfrac : S.G * S.Menc r / r = S.G * α := by
    calc
      S.G * S.Menc r / r = S.G * (α * r) / r := by rw [hM]
      _ = S.G * α * r / r := by ring
      _ = S.G * α := by field_simp [hrne]
  dsimp [vrot]
  rw [hfrac]

/-- Under linear mass growth `Menc(r) = α r`, the centripetal acceleration scales as `g(r) = (G α)/r`. -/
lemma g_of_linear_Menc (S : RotSys) (α : ℝ)
  (hlin : ∀ {r : ℝ}, 0 < r → S.Menc r = α * r) :
  ∀ {r : ℝ}, 0 < r → g S r = (S.G * α) / r := by
  intro r hr
  have hM : S.Menc r = α * r := hlin hr
  have hrne : r ≠ 0 := ne_of_gt hr
  dsimp [g]
  have hvrot_sq : (vrot S r) ^ 2 = S.G * α := by
    have hfrac : S.G * S.Menc r / r = S.G * α := by
      calc
        S.G * S.Menc r / r = S.G * (α * r) / r := by rw [hM]
        _ = S.G * α * r / r := by ring
        _ = S.G * α := by field_simp [hrne]
    dsimp [vrot]
    have hnonneg : 0 ≤ S.G * S.Menc r / r := by
      have hnum_nonneg : 0 ≤ S.G * S.Menc r := by
        have hM : 0 ≤ S.Menc r := S.nonnegM r
        exact mul_nonneg (le_of_lt S.posG) hM
      exact div_nonneg hnum_nonneg (le_of_lt hr)
    calc
      Real.sqrt (S.G * S.Menc r / r) ^ 2 = S.G * S.Menc r / r := by
        rw [Real.sq_sqrt hnonneg]
      _ = S.G * α := by rw [hfrac]
  calc
    g S r = (vrot S r) ^ 2 / r := by dsimp [g]
    _ = (S.G * α) / r := by rw [hvrot_sq]

What this page does not claim

Real galaxies necessarily have linearly growing enclosed mass. The value of G or α is derived from the framework. The lemma explains why a galaxy's mass distribution is linear.

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/Gravity/Rotation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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