The Blue of the Sky Returns the Same Number With or Without Molecules, and the Formula Without Them Diverges at the Critical Point ── for air at 550 nm, Rayleigh's formula with independent molecules and Einstein's formula written with density fluctuations alone both return an attenuation coefficient of 1.039926×10⁻⁵ per metre, and the formula without molecules returns their number through kT κ_T; in water the two routes split by a factor 0.096418, and at the critical point the fluctuation formula diverges as 4(1+ε)/(9ε) ── the separator is whether what scatters is counted as molecules or as fluctuations ── [Paper 1023]

Abstract

The sky is blue because air scatters short wavelengths more strongly. Rayleigh (1899) counted this with molecules placed as independent scatterers; Einstein (1910) wrote the same phenomenon without molecules, from density fluctuations and the isothermal compressibility alone. This paper counts that the two formulas return the same number for air, that the formula without molecules returns their number through the compressibility, that the two split in water, and that the formula without molecules diverges at the critical point. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──the inverse-fourth-power law of Rayleigh scattering and the molecular scattering formula (Rayleigh 1899), the scattering formula from density fluctuations and the theory of critical opalescence (Smoluchowski 1908, Einstein 1910), that the isothermal compressibility of an ideal gas is 1/p, the Lorentz-Lorenz relation, and the critical point of the van der Waals equation are all known. Depolarisation is not included──the correction for the anisotropy of air molecules (the King factor) is left out; molecules are counted as isotropic. The real sky is not claimed──aerosols, multiple scattering and the density profile are left out; a scale height of 8 km, a refractive index of 1 + 2.78×10⁻⁴ and the Loschmidt number are representative values. Water is not compared with measurement──the refractive index 1.333, the compressibility 4.5×10⁻¹⁰ per Pa and the number density are representative values, and no agreement with measured scattering coefficients is claimed. Critical exponents are not included──the divergence is counted in mean field (van der Waals) as 1/ε, without the exponents of real fluids; that the divergence is halted once the correlation length exceeds the wavelength (Ornstein and Zernike) is only named. Relation to earlier papers: the scope note of Paper 544 said that it does not reproduce the thirteen routes, does not follow Perrin’s methods one by one, and assembles only three routes that can be computed today; this paper sits in that seat and assembles the route of the blue sky in two forms, with and without molecules, returning the same number (no earlier occupant). Paper 670 showed that the green flash needs scattering as well as dispersion, taking the wavelength dependence of Rayleigh scattering as known──here the coefficient of that inverse-fourth-power law is obtained from two routes. Paper 249 showed that the 4π that catches molecules by diffusion and the 4π that spreads a field share a root──here it is counted that molecular scattering and fluctuation scattering share a root, including the case where the tying equation fails. Paper 300 showed that whether two things share a root is decidable──here that criterion is applied to the two formulas for the blue sky. What is added is matching the attenuation coefficients of the two routes at three wavelengths, inverting the fluctuation route to recover the number density, giving the ratio by which the two routes split in water, and matching the compressibility ratio on the critical isochore between a closed form and numerics. First, and this is the core. The two routes return the same number for air──at 550 nm and standard conditions, the attenuation coefficient by the molecular route is 1.039926x10^-5 per metre and by the fluctuation route 1.039926x10^-5, with ratio 1.000000000000. The scattering length is 96.160661 km, the optical depth of an atmosphere of scale height 8 km is 0.083194, and the transmittance 0.920173 (Section 2). Second, the formula without molecules returns their number──the only material input to the fluctuation formula is kT κ_T, which for an ideal gas equals 1/N (3.721900x10^-26 m³). Inverting the fluctuation-route coefficient gives back the number density 2.686800x10^25 per m³, that is 6.022185x10^23 per mole. The blue of the sky counts molecules without assuming them (Section 2). Third, in water the two routes split──with independent molecules the scattering length is 61.243631 m, by the fluctuation route 635.188804 m, a ratio of 0.096418. The kT κ_T of water is only 0.060832 of the ideal-gas 1/N, and the refractive index enters differently (Lorentz-Lorenz 0.978075 against 0.776889). When molecules are not independent, the fluctuation side is the right one (Section 3). Fourth, at the critical point the fluctuation formula diverges──on the critical isochore of the van der Waals gas the isothermal compressibility is 4(1+ε)/(9ε) times that of the ideal gas (ε the fractional distance from the critical temperature): 4.888889, 44.888889 and 444.888889 at ε = 0.1, 0.01 and 0.001, and 451.007407 at 0.3 K above the critical temperature of carbon dioxide. The critical opalescence that whitens a liquid is the side of the same formula on which the denominator approaches zero (Section 4). Fifth, the two routes share a root by the criterion of Paper 300──not only do they return the same number: one equation, kT κ_T = 1/N, ties them together, and where that equation fails, in water, the numbers split by 0.096418 (Section 5). Sixth, the separator is whether what scatters is counted as molecules or as fluctuations──in air either will do, in water counting molecules is off by a factor of ten, and at the critical point the molecular formula has no place for a divergence (Section 5). The blue of the sky returns the same number with or without molecules, and the formula without them diverges at the critical point. For air at 550 nm, Rayleigh’s formula with independent molecules and Einstein’s formula with fluctuations alone both return an attenuation of 1.039926×10⁻⁵ per metre, and the formula without molecules returns the number density 2.686800×10²⁵ per m³ through kT κ_T = 1/N. In water the two routes split by 0.096418, and at the critical point the fluctuation formula diverges as 4(1+ε)/(9ε)──since both the tying equation and the place where it breaks exist, the two share a root. The separator is whether what scatters is counted as molecules or as fluctuations. Placed among the earlier papers──Paper 670 showed that the green flash needs the wavelength dependence of Rayleigh scattering, Paper 249 that the 4π of diffusion and the 4π of a field share a root, and Paper 300 that sharing a root is decidable. Here the route of the blue sky that Paper 544 did not assemble was assembled in two forms and returned to the same number. To be honest──the real sky, real water and a real critical fluid have been counted only with representative values. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: Rayleigh scattering, density fluctuations, isothermal compressibility, critical opalescence, Avogadro number, blue of the sky. ----- 空が青いのは、空気が短い波長の光をよく散らすからである。レイリー(1899)は分子を独立な散乱体として置いてこれを数え、アインシュタイン(1910)は分子を置かず、密度のゆらぎと等温圧縮率だけで同じ現象を書いた。本稿は、二つの式が空気で同じ数を返すこと、分子を置かない式が圧縮率を通して分子の数を返すこと、水では二つが分かれること、臨界点で置かない式が発散することを数える。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──レイリー散乱の波長の 4 乗の逆数則と分子による散乱の式(レイリー 1899)、密度のゆらぎによる散乱の式と臨界乳光の理論(スモルコフスキー 1908、アインシュタイン 1910)、理想気体の等温圧縮率が 1/p であること、ローレンツ=ローレンツの式、ファン・デル・ワールスの状態方程式の臨界点は、いずれも既知である。偏光解消を入れない──空気の分子の非等方性による補正(キング因子)は入れず、等方な分子として数える。実在の空を主張しない──エアロゾル、多重散乱、大気の密度の分布は入れず、尺度高 8 km と屈折率 1+2.78×10⁻⁴、ロシュミット数は代表値である。水の測定値と比べない──水の屈折率 1.333・等温圧縮率 4.5×10⁻¹⁰ /Pa・数密度は代表値で、測られた散乱係数との一致を主張しない。臨界指数を入れない──発散は平均場(ファン・デル・ワールス)の 1/ε で数え、実在の流体の指数は入れない。発散が相関長が波長を超えると止まること(オルンシュタインとゼルニケ)は名指すにとどめる。既刊との関係:論文544 の射程注記は、十三通りを再現しない、ペランの用いた方法を一つずつ辿ってはおらず、いま計算できる三つの道だけを組んだと書いた──本稿はその席に座り、空の青の道を、分子を置く形と置かない形の二通りで組んで同じ数に戻す(先客なし)。論文670 は、緑閃光に分散だけでなく散乱も要ることを、レイリー散乱の波長依存を既知として示した──ここでは、その波長の 4 乗の逆数則の係数を二つの道から出す。論文249 は、拡散が捕らえる 4π と場が広がる 4π が同じ根であることを示した──ここでは、分子の散乱とゆらぎの散乱が同じ根であることを、等式が破れる場合まで含めて数える。論文300 は、同根か別根かが判定できることを示した──ここでは、その基準を空の青の二つの式に当てる。加えたのは、二つの道の減衰係数を三つの波長で一致させたこと、ゆらぎの道から数密度を逆算して戻したこと、水で二つの道が分かれる比を出したこと、臨界等容線の圧縮率の倍率を閉じた式と数値で一致させたことである。 第一に、これが本稿の芯である。二つの道は空気で同じ数を返す──550 nm・標準状態で、分子を置く道の減衰係数は 1 メートルあたり 1.039926x10^-5、ゆらぎの道も 1.039926x10^-5 で、比は 1.000000000000 だった。散乱の長さは 96.160661 km、尺度高 8 km の大気の光学的厚さは 0.083194、透過率は 0.920173 である(第2節)。 第二に、分子を置かない式が、分子の数を返す──ゆらぎの式に入るのは kT κ_T だけで、理想気体ではこれが 1/N(3.721900x10^-26 m³)に等しい。ゆらぎの道の減衰係数から数密度を逆算すると 2.686800x10^25 /m³ に戻り、アボガドロ数に直せば 6.022185x10^23 である。空の青は、分子を仮定しなくても分子を数える(第2節)。 第三に、水では二つの道が分かれる──分子を独立に置くと散乱の長さは 61.243631 m、ゆらぎの道では 635.188804 m で、比は 0.096418 である。水の kT κ_T は理想気体の 1/N の 0.060832 倍しかなく、屈折率の入り方(ローレンツ=ローレンツの 0.978075 対 0.776889)も違う。分子が独立でないとき、正しいのはゆらぎの側である(第3節)。 第四に、臨界点ではゆらぎの式が発散する──ファン・デル・ワールスの臨界等容線で、等温圧縮率は理想気体の 4(1+ε)/(9ε) 倍(ε は臨界温度からのずれの割合)で、ε=0.1・0.01・0.001 で 4.888889・44.888889・444.888889 倍、二酸化炭素の臨界温度の 0.3 K 上で 451.007407 倍になる。液体が白く濁る臨界乳光は、空の青と同じ式の分母が 0 に近づく側である(第4節)。 第五に、二つの道は論文300 の基準で同根である──同じ数を返すだけでなく、kT κ_T=1/N という一つの等式が両方を結び、その等式が破れる水では数も 0.096418 倍に分かれる(第5節)。 第六に、分離子は、散らすものを分子と数えるか、ゆらぎと数えるか──空気ではどちらでも同じ、水では分子と数えると 10 倍外れ、臨界点では分子と数える式に発散の出る場所が無い(第5節)。 空の青は分子を置いても置かなくても同じ数を返し、置かない式は臨界点で発散する。550 nm の空気で、分子を独立に置くレイリーの式とゆらぎだけで書くアインシュタインの式はどちらも 1 メートルあたり 1.039926×10⁻⁵ の減衰係数を返し、置かない式は kT κ_T=1/N を通して数密度 2.686800×10²⁵ /m³ を返す。水では二つの道が 0.096418 倍に分かれ、臨界点ではゆらぎの式が 4(1+ε)/(9ε) 倍に発散する──結ぶ等式と破れる場所の両方が在るので、二つは同根である。分離子は、散らすものを分子と数えるか、ゆらぎと数えるかである。既刊との位置──論文670 は緑閃光にレイリー散乱の波長依存が要ることを、論文249 は拡散の 4π と場の 4π が同じ根であることを、論文300 は同根か別根かが判定できることを示した。ここでは、論文544 が組まなかった空の青の道を二通りで組み、同じ数に戻した。正直に言えば──実在の空も水も臨界点の流体も、代表値でしか数えていない。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:レイリー散乱、密度のゆらぎ、等温圧縮率、臨界乳光、アボガドロ数、空の青。