| /- |
| Copyright (c) 2026 Humanize/Codex. All rights reserved. |
| Released under Apache 2.0 license as described in the file LICENSE. |
| Authors: Humanize/Codex |
| -/ |
| import Mathlib |
|
|
| /-! |
| Algebraic answer claims for the IPhO 2024 theory problems. The definitions |
| record the official closed-form or numerical answers from the HF JSONL source; |
| the named theorems expose one answer statement for each source row. |
| -/ |
|
|
| namespace Ipho20212023Archon.Problems.Y2024 |
|
|
| noncomputable section |
|
|
| def fourthRoot (x : ℝ) : ℝ := Real.sqrt (Real.sqrt x) |
|
|
| def closedInterval (lo hi x : ℝ) : Prop := lo ≤ x ∧ x ≤ hi |
|
|
| section Problem1 |
|
|
| def solarConstant (sigma TS RS d : ℝ) : ℝ := sigma * TS ^ 4 * (RS / d) ^ 2 |
|
|
| def solarConstantNumerical : ℝ := 1350 |
|
|
| theorem IPhO_2024_1_A_1 (sigma TS RS d : ℝ) : |
| (solarConstant sigma TS RS d, solarConstantNumerical) = |
| (sigma * TS ^ 4 * (RS / d) ^ 2, 1350) := by |
| sorry |
|
|
| def radiativeEquilibriumTemperature (S0 sigma : ℝ) : ℝ := |
| fourthRoot (S0 / (4 * sigma)) |
|
|
| def radiativeEquilibriumTemperatureFromSun (RS d TS : ℝ) : ℝ := |
| Real.sqrt (RS / (2 * d)) * TS |
|
|
| def radiativeEquilibriumTemperatureNumerical : ℝ := 278 |
|
|
| theorem IPhO_2024_1_A_2 (S0 sigma RS d TS : ℝ) : |
| (radiativeEquilibriumTemperature S0 sigma, |
| radiativeEquilibriumTemperatureFromSun RS d TS, |
| radiativeEquilibriumTemperatureNumerical) = |
| (fourthRoot (S0 / (4 * sigma)), Real.sqrt (RS / (2 * d)) * TS, 278) := by |
| sorry |
|
|
| def planckPeakAuxiliary (x : ℝ) : ℝ := 5 * (1 - Real.exp (-x)) - x |
|
|
| theorem IPhO_2024_1_A_3 (x : ℝ) : |
| planckPeakAuxiliary x = 5 * (1 - Real.exp (-x)) - x := by |
| sorry |
|
|
| def wienPeakX : ℝ := 496 / 100 |
|
|
| def wienConstantNmK : ℝ := 2890000 |
|
|
| theorem IPhO_2024_1_A_4 : |
| closedInterval (496 / 100) (497 / 100) wienPeakX ∧ |
| closedInterval (2890000 : ℝ) 2900000 wienConstantNmK := by |
| sorry |
|
|
| def sunPeakWavelengthNm : ℝ := 501 |
|
|
| def earthPeakWavelengthNm : ℝ := 10400 |
|
|
| theorem IPhO_2024_1_A_5 : |
| closedInterval (501 : ℝ) 502 sunPeakWavelengthNm ∧ |
| earthPeakWavelengthNm = 10400 := by |
| sorry |
|
|
| def spectralFluxRatioByTemperature (d RS TE TS : ℝ) : ℝ := |
| (d / RS) ^ 2 * (TE / TS) ^ 5 |
|
|
| def spectralFluxRatioByWavelength (d RS lambdaS lambdaE : ℝ) : ℝ := |
| (d / RS) ^ 2 * (lambdaS / lambdaE) ^ 5 |
|
|
| def spectralFluxRatioNumerical : ℝ := 12 / 1000 |
|
|
| theorem IPhO_2024_1_A_6 (d RS TE TS lambdaS lambdaE : ℝ) : |
| spectralFluxRatioByTemperature d RS TE TS = |
| (d / RS) ^ 2 * (TE / TS) ^ 5 ∧ |
| spectralFluxRatioByWavelength d RS lambdaS lambdaE = |
| (d / RS) ^ 2 * (lambdaS / lambdaE) ^ 5 ∧ |
| closedInterval (120 / 10000) (121 / 10000) spectralFluxRatioNumerical := by |
| sorry |
|
|
| def oneLayerSurfaceTemperature (rA S0 sigma : ℝ) : ℝ := |
| fourthRoot (((1 - rA) * (S0 / 2)) / sigma) |
|
|
| def oneLayerAtmosphereTemperature (rA S0 sigma : ℝ) : ℝ := |
| fourthRoot (((1 - rA) * (S0 / 4)) / sigma) |
|
|
| def oneLayerSurfaceTemperatureNumerical : ℝ := 307 |
|
|
| def oneLayerAtmosphereTemperatureNumerical : ℝ := 258 |
|
|
| theorem IPhO_2024_1_B_1 (rA S0 sigma : ℝ) : |
| (oneLayerSurfaceTemperature rA S0 sigma, |
| oneLayerAtmosphereTemperature rA S0 sigma, |
| oneLayerSurfaceTemperatureNumerical, |
| oneLayerAtmosphereTemperatureNumerical) = |
| (fourthRoot (((1 - rA) * (S0 / 2)) / sigma), |
| fourthRoot (((1 - rA) * (S0 / 4)) / sigma), 307, 258) := by |
| sorry |
|
|
| def effectiveAlbedo (rA rE : ℝ) : ℝ := |
| rA + ((1 - rA) ^ 2 * rE) / (1 - rA * rE) |
|
|
| def effectiveAlbedoNumerical : ℝ := 313 / 1000 |
|
|
| theorem IPhO_2024_1_B_2 (rA rE : ℝ) : |
| effectiveAlbedo rA rE = |
| rA + ((1 - rA) ^ 2 * rE) / (1 - rA * rE) ∧ |
| effectiveAlbedoNumerical = 313 / 1000 := by |
| sorry |
|
|
| def greySurfaceTemperature (alpha epsilon S0 sigma : ℝ) : ℝ := |
| fourthRoot (((1 - alpha) * S0) / (2 * sigma * (2 - epsilon))) |
|
|
| def greyEmissivityNumerical : ℝ := 81 / 100 |
|
|
| theorem IPhO_2024_1_B_3 (alpha epsilon S0 sigma : ℝ) : |
| greySurfaceTemperature alpha epsilon S0 sigma = |
| fourthRoot (((1 - alpha) * S0) / (2 * sigma * (2 - epsilon))) ∧ |
| closedInterval (807 / 1000) (811 / 1000) greyEmissivityNumerical := by |
| sorry |
|
|
| def greySurfaceTemperatureDerivative (alpha epsilon S0 sigma : ℝ) : ℝ := |
| (1 / 4) * fourthRoot (((1 - alpha) * S0) / (2 * sigma * (2 - epsilon))) * |
| (1 / (2 - epsilon)) |
|
|
| def onePercentGreyTemperatureIncrease : ℝ := 49 / 100 |
|
|
| theorem IPhO_2024_1_B_4 (alpha epsilon S0 sigma : ℝ) : |
| greySurfaceTemperatureDerivative alpha epsilon S0 sigma = |
| (1 / 4) * fourthRoot (((1 - alpha) * S0) / (2 * sigma * (2 - epsilon))) * |
| (1 / (2 - epsilon)) ∧ |
| closedInterval (487 / 1000) (492 / 1000) onePercentGreyTemperatureIncrease := by |
| sorry |
|
|
| def coupledEmissivity (TE TA sigma alpha S0 : ℝ) : ℝ := |
| (sigma * TE ^ 4 - (1 - alpha) * (S0 / 4)) / |
| (sigma * (TE ^ 4 - TA ^ 4)) |
|
|
| def conductiveCoefficient (TE TA sigma alpha S0 : ℝ) : ℝ := |
| ((2 * TA ^ 4 - TE ^ 4) * (sigma * TE ^ 4 - (1 - alpha) * (S0 / 4))) / |
| ((TE ^ 4 - TA ^ 4) * (TE - TA)) |
|
|
| def coupledEmissivityNumerical : ℝ := 85 / 100 |
|
|
| def conductiveCoefficientNumerical : ℝ := 36 / 100 |
|
|
| theorem IPhO_2024_1_B_5 (TE TA sigma alpha S0 : ℝ) : |
| coupledEmissivity TE TA sigma alpha S0 = |
| (sigma * TE ^ 4 - (1 - alpha) * (S0 / 4)) / |
| (sigma * (TE ^ 4 - TA ^ 4)) ∧ |
| conductiveCoefficient TE TA sigma alpha S0 = |
| ((2 * TA ^ 4 - TE ^ 4) * (sigma * TE ^ 4 - (1 - alpha) * (S0 / 4))) / |
| ((TE ^ 4 - TA ^ 4) * (TE - TA)) ∧ |
| closedInterval (847 / 1000) (852 / 1000) coupledEmissivityNumerical ∧ |
| closedInterval (357 / 1000) (366 / 1000) conductiveCoefficientNumerical := by |
| sorry |
|
|
| def coupledSensitivityEquationOne (epsilon TE TA dTE dTA : ℝ) : Prop := |
| epsilon * (1 / (TE - TA) + 4 * TE ^ 3 / (2 * TA ^ 4 - TE ^ 4)) * dTE = |
| 1 + epsilon * (8 * TA ^ 3 / (2 * TA ^ 4 - TE ^ 4) + 1 / (TE - TA)) * dTA |
|
|
| def coupledSensitivityEquationTwo |
| (epsilon TE TA sigma alpha S0 dTE dTA : ℝ) : Prop := |
| 1 + epsilon * |
| (4 * TE ^ 3 / (TE ^ 4 - TA ^ 4) - |
| 4 * sigma * TE ^ 3 / (sigma * TE ^ 4 - (1 - alpha) * (S0 / 4))) * dTE = |
| (4 * TA ^ 3 / (TE ^ 4 - TA ^ 4)) * epsilon * dTA |
|
|
| def coupledOnePercentTemperatureIncrease : ℝ := 525 / 1000 |
|
|
| theorem IPhO_2024_1_B_6 (epsilon TE TA sigma alpha S0 dTE dTA : ℝ) : |
| (coupledSensitivityEquationOne epsilon TE TA dTE dTA ↔ |
| epsilon * (1 / (TE - TA) + 4 * TE ^ 3 / (2 * TA ^ 4 - TE ^ 4)) * dTE = |
| 1 + epsilon * |
| (8 * TA ^ 3 / (2 * TA ^ 4 - TE ^ 4) + 1 / (TE - TA)) * dTA) ∧ |
| (coupledSensitivityEquationTwo epsilon TE TA sigma alpha S0 dTE dTA ↔ |
| 1 + epsilon * |
| (4 * TE ^ 3 / (TE ^ 4 - TA ^ 4) - |
| 4 * sigma * TE ^ 3 / (sigma * TE ^ 4 - (1 - alpha) * (S0 / 4))) * |
| dTE = |
| (4 * TA ^ 3 / (TE ^ 4 - TA ^ 4)) * epsilon * dTA) ∧ |
| closedInterval (521 / 1000) (528 / 1000) coupledOnePercentTemperatureIncrease := by |
| sorry |
|
|
| end Problem1 |
|
|
| section Problem2 |
|
|
| def ringElectricField (lambda eps0 R x y z : ℝ) : ℝ × ℝ × ℝ := |
| (-(lambda * x) / (4 * eps0 * R ^ 2), |
| -(lambda * y) / (4 * eps0 * R ^ 2), |
| lambda * z / (2 * eps0 * R ^ 2)) |
|
|
| def ringOscillationFrequency (Q lambda eps0 R m : ℝ) : ℝ := |
| Real.sqrt (Q * lambda / (4 * eps0 * R ^ 2 * m)) |
|
|
| theorem IPhO_2024_2_A_1 (lambda eps0 R x y z Q m : ℝ) : |
| (ringElectricField lambda eps0 R x y z, |
| ringOscillationFrequency Q lambda eps0 R m, |
| ringOscillationFrequency Q lambda eps0 R m) = |
| ((-(lambda * x) / (4 * eps0 * R ^ 2), |
| -(lambda * y) / (4 * eps0 * R ^ 2), |
| lambda * z / (2 * eps0 * R ^ 2)), |
| Real.sqrt (Q * lambda / (4 * eps0 * R ^ 2 * m)), |
| Real.sqrt (Q * lambda / (4 * eps0 * R ^ 2 * m))) := by |
| sorry |
|
|
| def paulTrapK (Q lambda0 eps0 R m : ℝ) : ℝ := |
| Real.sqrt (Q * lambda0 / (2 * eps0 * R ^ 2 * m)) |
|
|
| def paulTrapA (Q u eps0 R m Omega : ℝ) : ℝ := |
| Q * u / (2 * eps0 * R ^ 2 * m * Omega ^ 2) |
|
|
| theorem IPhO_2024_2_A_2 (Q lambda0 eps0 R m u Omega : ℝ) : |
| (paulTrapK Q lambda0 eps0 R m, paulTrapA Q u eps0 R m Omega) = |
| (Real.sqrt (Q * lambda0 / (2 * eps0 * R ^ 2 * m)), |
| Q * u / (2 * eps0 * R ^ 2 * m * Omega ^ 2)) := by |
| sorry |
|
|
| def rapidCoordinateAcceleration (p a Omega t : ℝ) : ℝ := |
| p * a * Omega ^ 2 * Real.cos (Omega * t) |
|
|
| def rapidCoordinateSolution (p a Omega t : ℝ) : ℝ := |
| -p * a * Real.cos (Omega * t) |
|
|
| theorem IPhO_2024_2_A_3 (p a Omega t : ℝ) : |
| (rapidCoordinateAcceleration p a Omega t, rapidCoordinateSolution p a Omega t) = |
| (p * a * Omega ^ 2 * Real.cos (Omega * t), -p * a * Real.cos (Omega * t)) := by |
| sorry |
|
|
| def slowCoordinateAcceleration (k a Omega p : ℝ) : ℝ := |
| (k ^ 2 - a ^ 2 * Omega ^ 2 / 2) * p |
|
|
| def paulTrapStable (Omega k a : ℝ) : Prop := Omega > Real.sqrt 2 * k / a |
|
|
| theorem IPhO_2024_2_A_4 (k a Omega p : ℝ) : |
| slowCoordinateAcceleration k a Omega p = |
| (k ^ 2 - a ^ 2 * Omega ^ 2 / 2) * p ∧ |
| (paulTrapStable Omega k a ↔ Omega > Real.sqrt 2 * k / a) := by |
| sorry |
|
|
| def paulTrapKNumerical : ℝ := 200000 |
|
|
| def paulTrapOmegaMinNumerical : ℝ := 7000000 |
|
|
| theorem IPhO_2024_2_A_5 : |
| (paulTrapKNumerical, paulTrapOmegaMinNumerical) = (200000, 7000000) := by |
| sorry |
|
|
| def naturalLinewidth (tau : ℝ) : ℝ := 1 / tau |
|
|
| theorem IPhO_2024_2_B_1 (tau : ℝ) : |
| naturalLinewidth tau = 1 / tau := by |
| sorry |
|
|
| def photonCollisionRatePlus (sL alpha omegaL v c : ℝ) : ℝ := |
| sL + alpha * omegaL * (v / c) |
|
|
| def photonCollisionRateMinus (sL alpha omegaL v c : ℝ) : ℝ := |
| sL - alpha * omegaL * (v / c) |
|
|
| def momentumAbsorptionPlus (sPlus hbar kPlus : ℝ) : ℝ := sPlus * (-hbar * kPlus) |
|
|
| def momentumAbsorptionMinus (sMinus hbar kMinus : ℝ) : ℝ := sMinus * (hbar * kMinus) |
|
|
| def opticalMolassesForce (alpha hbar kL v : ℝ) : ℝ := -(2 * alpha * hbar * kL ^ 2) * v |
|
|
| theorem IPhO_2024_2_B_2 (sL alpha omegaL v c sPlus sMinus hbar kPlus kMinus kL : ℝ) : |
| (photonCollisionRatePlus sL alpha omegaL v c, |
| photonCollisionRateMinus sL alpha omegaL v c, |
| momentumAbsorptionPlus sPlus hbar kPlus, |
| momentumAbsorptionMinus sMinus hbar kMinus, |
| opticalMolassesForce alpha hbar kL v) = |
| (sL + alpha * omegaL * (v / c), |
| sL - alpha * omegaL * (v / c), |
| sPlus * (-hbar * kPlus), |
| sMinus * (hbar * kMinus), |
| -(2 * alpha * hbar * kL ^ 2) * v) := by |
| sorry |
|
|
| def recoilHeatingPower (hbar kL m tau : ℝ) : ℝ := hbar ^ 2 * kL ^ 2 / (m * tau) |
|
|
| theorem IPhO_2024_2_B_3 (hbar kL m tau : ℝ) : |
| recoilHeatingPower hbar kL m tau = hbar ^ 2 * kL ^ 2 / (m * tau) := by |
| sorry |
|
|
| def opticalCoolingPower (alpha hbar kL v : ℝ) : ℝ := -2 * alpha * hbar * kL ^ 2 * v ^ 2 |
|
|
| def equilibriumMeanSquareVelocity (hbar Gamma alpha m : ℝ) : ℝ := |
| hbar * Gamma / (2 * alpha * m) |
|
|
| def dopplerTemperature (hbar Gamma alpha kB : ℝ) : ℝ := |
| hbar * Gamma / (2 * alpha * kB) |
|
|
| theorem IPhO_2024_2_B_4 (alpha hbar kL v Gamma m kB : ℝ) : |
| (opticalCoolingPower alpha hbar kL v, |
| equilibriumMeanSquareVelocity hbar Gamma alpha m, |
| dopplerTemperature hbar Gamma alpha kB) = |
| (-2 * alpha * hbar * kL ^ 2 * v ^ 2, |
| hbar * Gamma / (2 * alpha * m), |
| hbar * Gamma / (2 * alpha * kB)) := by |
| sorry |
|
|
| def dopplerTemperatureNumerical : ℝ := 2 / 10000 |
|
|
| theorem IPhO_2024_2_B_5 : |
| dopplerTemperatureNumerical = 2 / 10000 := by |
| sorry |
|
|
| end Problem2 |
|
|
| section Problem3 |
|
|
| def binaryEffectivePotential (M1 M2 G a x y : ℝ) : ℝ := |
| -G * M1 / Real.sqrt ((x + M2 / (M1 + M2) * a) ^ 2 + y ^ 2) - |
| G * M2 / Real.sqrt ((x - M1 / (M1 + M2) * a) ^ 2 + y ^ 2) - |
| (1 / 2) * (G * (M1 + M2) / a ^ 3) * (x ^ 2 + y ^ 2) |
|
|
| theorem IPhO_2024_3_A_1 (M1 M2 G a x y : ℝ) : |
| binaryEffectivePotential M1 M2 G a x y = |
| -G * M1 / Real.sqrt ((x + M2 / (M1 + M2) * a) ^ 2 + y ^ 2) - |
| G * M2 / Real.sqrt ((x - M1 / (M1 + M2) * a) ^ 2 + y ^ 2) - |
| (1 / 2) * (G * (M1 + M2) / a ^ 3) * (x ^ 2 + y ^ 2) := by |
| sorry |
|
|
| def rocheOverflowRatio : ℝ := 36 / 100 |
|
|
| theorem IPhO_2024_3_A_3 : |
| rocheOverflowRatio = 36 / 100 := by |
| sorry |
|
|
| def binarySeparationRate (beta a M1 M2 : ℝ) : ℝ := |
| -2 * beta * a * (1 / M1 - 1 / M2) |
|
|
| def binaryPeriodRate (beta a G M M1 M2 : ℝ) : ℝ := |
| -6 * Real.pi * Real.sqrt (a ^ 3 / (G * M)) * beta * (1 / M1 - 1 / M2) |
|
|
| theorem IPhO_2024_3_A_4 (beta a G M M1 M2 : ℝ) : |
| (binarySeparationRate beta a M1 M2, binaryPeriodRate beta a G M M1 M2) = |
| (-2 * beta * a * (1 / M1 - 1 / M2), |
| -6 * Real.pi * Real.sqrt (a ^ 3 / (G * M)) * beta * (1 / M1 - 1 / M2)) := by |
| sorry |
|
|
| def accretionDiscTemperature (G M1 beta sigma r : ℝ) : ℝ := |
| fourthRoot (G * M1 * beta / (8 * Real.pi * sigma * r ^ 3)) |
|
|
| theorem IPhO_2024_3_A_5 (G M1 beta sigma r : ℝ) : |
| accretionDiscTemperature G M1 beta sigma r = |
| fourthRoot (G * M1 * beta / (8 * Real.pi * sigma * r ^ 3)) := by |
| sorry |
|
|
| def accretionDiscTemperatureNumerical : ℝ := 9000 |
|
|
| theorem IPhO_2024_3_A_6 : |
| accretionDiscTemperatureNumerical = 9000 := by |
| sorry |
|
|
| def maximumPostExplosionVelocity (G M1p M2 a : ℝ) : ℝ := |
| Real.sqrt (2 * G * (M1p + M2) / a) |
|
|
| def minimumPostExplosionMass (M1 M2 : ℝ) : ℝ := (M1 - M2) / 2 |
|
|
| theorem IPhO_2024_3_A_7 (G M1p M2 a M1 : ℝ) : |
| (maximumPostExplosionVelocity G M1p M2 a, minimumPostExplosionMass M1 M2) = |
| (Real.sqrt (2 * G * (M1p + M2) / a), (M1 - M2) / 2) := by |
| sorry |
|
|
| def centralGravity (G rhoC r : ℝ) : ℝ := -(4 * Real.pi * G * rhoC * r) / 3 |
|
|
| theorem IPhO_2024_3_B_1 (G rhoC r : ℝ) : |
| centralGravity G rhoC r = -(4 * Real.pi * G * rhoC * r) / 3 := by |
| sorry |
|
|
| def stellarH1 (rho r gamma : ℝ) : ℝ := r ^ 2 * Real.rpow rho (gamma - 2) |
|
|
| def stellarH2 (G K gamma r : ℝ) : ℝ := 4 * Real.pi * G * r ^ 2 / (K * gamma) |
|
|
| theorem IPhO_2024_3_B_2 (rho r gamma G K : ℝ) : |
| (stellarH1 rho r gamma, stellarH2 G K gamma r) = |
| (r ^ 2 * Real.rpow rho (gamma - 2), 4 * Real.pi * G * r ^ 2 / (K * gamma)) := by |
| sorry |
|
|
| def stellarLengthScale (G pc rhoC : ℝ) : ℝ := Real.sqrt pc / (Real.sqrt G * rhoC) |
|
|
| theorem IPhO_2024_3_B_3 (G pc rhoC : ℝ) : |
| stellarLengthScale G pc rhoC = Real.sqrt pc / (Real.sqrt G * rhoC) := by |
| sorry |
|
|
| def dimensionlessA1 (u x gamma : ℝ) : ℝ := x ^ 2 * Real.rpow u (gamma - 2) |
|
|
| def dimensionlessA2 (x gamma : ℝ) : ℝ := 4 * Real.pi * x ^ 2 / gamma |
|
|
| theorem IPhO_2024_3_B_4 (u x gamma : ℝ) : |
| (dimensionlessA1 u x gamma, dimensionlessA2 x gamma) = |
| (x ^ 2 * Real.rpow u (gamma - 2), 4 * Real.pi * x ^ 2 / gamma) := by |
| sorry |
|
|
| def gammaTwoProfileF (x : ℝ) : ℝ := |
| Real.sin (Real.sqrt (2 * Real.pi) * x) / Real.sqrt (2 * Real.pi) |
|
|
| theorem IPhO_2024_3_B_5 (x : ℝ) : |
| gammaTwoProfileF x = |
| Real.sin (Real.sqrt (2 * Real.pi) * x) / Real.sqrt (2 * Real.pi) := by |
| sorry |
|
|
| def stellarGammaNumerical : ℝ := 167 / 100 |
|
|
| theorem IPhO_2024_3_B_6 : |
| closedInterval (164 / 100) (170 / 100) stellarGammaNumerical := by |
| sorry |
|
|
| def perturbedDensity (rho epsilon : ℝ) : ℝ := rho * (1 - 3 * epsilon) |
|
|
| def perturbedGravity (g epsilon : ℝ) : ℝ := g * (1 - 2 * epsilon) |
|
|
| theorem IPhO_2024_3_B_7 (rho g epsilon : ℝ) : |
| (perturbedDensity rho epsilon, perturbedGravity g epsilon) = |
| (rho * (1 - 3 * epsilon), g * (1 - 2 * epsilon)) := by |
| sorry |
|
|
| def shellAcceleration (gTilde K gamma rhoTilde densityGradient : ℝ) : ℝ := |
| gTilde - K * gamma * Real.rpow rhoTilde (gamma - 2) * densityGradient |
|
|
| theorem IPhO_2024_3_B_8 (gTilde K gamma rhoTilde densityGradient : ℝ) : |
| shellAcceleration gTilde K gamma rhoTilde densityGradient = |
| gTilde - K * gamma * Real.rpow rhoTilde (gamma - 2) * densityGradient := by |
| sorry |
|
|
| def radialPerturbationAcceleration (G rhoC gamma epsilon : ℝ) : ℝ := |
| -(4 * Real.pi * G * rhoC / 3) * (3 * gamma - 4) * epsilon |
|
|
| def stableGammaMinimum : ℝ := 4 / 3 |
|
|
| def stellarOscillationFrequency (G rhoC gamma : ℝ) : ℝ := |
| Real.sqrt ((4 * Real.pi * G * rhoC / 3) * (3 * gamma - 4)) |
|
|
| theorem IPhO_2024_3_B_9 (G rhoC gamma epsilon : ℝ) : |
| (radialPerturbationAcceleration G rhoC gamma epsilon, |
| stableGammaMinimum, |
| stellarOscillationFrequency G rhoC gamma) = |
| (-(4 * Real.pi * G * rhoC / 3) * (3 * gamma - 4) * epsilon, |
| 4 / 3, |
| Real.sqrt ((4 * Real.pi * G * rhoC / 3) * (3 * gamma - 4))) := by |
| sorry |
|
|
| end Problem3 |
|
|
| end |
|
|
| end Ipho20212023Archon.Problems.Y2024 |
|
|
