The Roche Limit Is Not One Distance: It Differs by a Factor of Two Depending on What Holds the Satellite Together ── in units of the planet's radius times the cube root of the density ratio, the limit is 1.259921 for a solid resisting tides alone and 2.455 for a fluid, a factor of 1.948535 apart ── the separator is whether a satellite holds its shape by fluid self-gravity, solid self-gravity or material strength ── [Paper 784]

Abstract

A satellite that comes too close to its planet is torn apart by tides. That distance is called the Roche limit and is spoken of as one value. How many distances exist under the same name, by differences in what holds a satellite together, is counted. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──setting the breakup distance by balancing tides against self-gravity (Roche), the coefficient 2.455 for fluid satellites (Chandrasekhar), the solid coefficients, and small bodies holding their shape by material strength are all known (Chandrasekhar 1969, Murray and Dermott 1999). The fluid coefficient is not derived──2.455 is used as input, and equilibrium figures are not computed. The origin of the rings is not treated──how the rings formed or why they lie where they do is not stated. Values are representative──densities of Earth 5514, Moon 3344, Saturn 687, ice 917 and porous particles 500 kg/m^3, ice strength 10^6 Pa, and Pan's radius 14.1 km and density 400 kg/m^3 are for comparison; measurements are not claimed. Strength is compared by central pressure only──fracture modes and internal structure are not treated. Relation to earlier papers: Paper 505 showed that the "sphere of influence" is three spheres, and dividing the mass ratio by 1000 shrinks them differently──its scope note stated that nothing beyond the three was counted and that the Roche limit and gravitational capture radius were not included. There the sphere of influence split into three by what the gravity is balanced against. Here the same "Roche limit" is counted as splitting into three by what holds the satellite together. What is added is setting the three coefficients side by side and showing fluid and solid differ by 1.948535, solving the exact balance by bisection and confirming it returns to the closed formula for small satellites and departs for large ones, placing Saturn's rings and small moons against the three limits, and placing the separator on what holds the shape. First, even for solids there are two limits──1.259921 for a solid resisting tides alone and 1.442250 for a synchronously rotating solid (in units of planet radius times the cube root of the density ratio) (Section 2). Second, and this is the core. For a fluid, the limit is about twice as far──the fluid's 2.455 is 1.948535 times that of the tides-only solid (Section 2). Third, Saturn's rings sit at the edge of the fluid limit──the fluid limit for ice (density 917) is 2.229704 Saturn radii, the outer edge of the A ring 2.269446, and for porous particles (500) the limit is 2.729274 (Section 3). Fourth, this limit does not apply to small moons──the self-gravitational pressure of Pan (2.216500), inside the rings, is 4446.543499 Pa, 4.446543x10^-3 of the strength of ice (Section 3). Fifth, the closed formula is an approximation for small satellites──a relative difference of 5.273x10^-5 for a satellite of 1 km radius, but 1.633126 times for one as large as the Moon (formula 1.488482) (Section 2). Sixth, the separator is whether a satellite holds its shape by fluid self-gravity, solid self-gravity or material strength──"inside the Roche limit a satellite breaks up" has no truth value until one says what holds the satellite together (Section 4). The Roche limit is not one distance: it differs by a factor of two depending on what holds the satellite together. Taking planet radius times the cube root of the density ratio as the unit, the limit is 1.259921 for a tides-only solid, 1.442250 for a synchronous solid and 2.455 for a fluid, so fluid and solid differ by 1.948535──the closed formula approximates small satellites; for one as large as the Moon the exact balance gives 1.633126. Saturn's fluid limit for ice is 2.229704 Saturn radii, and the A ring's outer edge at 2.269446 lies just outside it, inside for porous particles. Pan, within the rings at 2.216500, has a self-gravitational central pressure of 4446.543499 Pa, only 4.446543x10^-3 of the strength of ice, and moons smaller than 92.235583 km hold their shape by strength──the limit bites only on satellites held together by self-gravity. The separator is whether a satellite holds its shape by fluid self-gravity, solid self-gravity or material strength. Placed among the earlier papers──Paper 505 counted the sphere of influence splitting into three by what it is balanced against; this counts the Roche limit splitting into three by how the satellite holds together. To be honest──the fluid coefficient is an input, and neither the rings' origin nor the strength of real moons is treated. 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: Roche limit, tidal force, self-gravity, Saturn's rings, material strength, satellites. ----- 惑星に近づきすぎた衛星は潮汐で引き裂かれる。その距離はロシュ限界と呼ばれ、一つの値のように語られる。同じ名前のもとに、衛星が形を保つ仕組みの違いでいくつの距離があるかを数える。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──潮汐と自己重力の釣り合いで衛星が壊れる距離を決めること(ロシュ)、流体の衛星の係数 2.455(チャンドラセカール)、固体の係数、小さな天体が材料の強さで形を保つことは、いずれも既知である(チャンドラセカール 1969、マレー&ダーモット 1999)。流体の係数を導かない──2.455 は入力として使い、平衡形の計算には立ち入らない。環の起源を扱わない──環がどうできたか、なぜその位置にあるかは述べない。値は代表値である──地球 5514・月 3344・土星 687・氷 917・すかすかの粒 500 kg/m^3、氷の強さ 10^6 Pa、パンの半径 14.1 km と密度 400 kg/m^3 は比べるための値で、実測は主張しない。強さの見積もりは中心の圧力だけで比べる──割れ方や内部の構造は扱わない。既刊との関係:論文505 は、「勢力圏」が三つあり、質量比を 1000 分の 1 にすると別々に縮むことを示した──その射程注記は、三つ以外を数えておらず、ロッシュ限界や重力捕獲の半径は入れていないと明記していた。そこでは何と釣り合わせるかで勢力圏が三つに分かれた。ここでは同じ「ロシュ限界」が、衛星が何で形を保つかで三つに分かれることを数える。加えたのは、三つの係数を並べ、流体と固体が 1.948535 倍ひらくことを示したこと、正確な釣り合いを二分法で解き、小さな衛星で閉じた式に戻り、大きな衛星でずれることを確かめたこと、土星の環と小衛星の位置を三つの限界と並べたこと、分離子を「何で形を保つか」に置いたことである。 第一に、固体でも、限界は二つある──潮汐だけに耐える固体で 1.259921、同期回転する固体で 1.442250(惑星半径×密度比の三乗根を単位に)(第2節)。 第二に、これが本稿の芯である。流体なら、限界は二倍ほど遠い──流体の 2.455 は潮汐だけの固体の 1.948535 倍(第2節)。 第三に、土星の環は、流体の限界の縁にある──氷(密度 917)の流体の限界は 2.229704 土星半径、A 環の外縁は 2.269446、すかすかの粒(500)なら限界は 2.729274(第3節)。 第四に、小さな衛星には、この限界が当てはまらない──環の中のパン(2.216500)の自己重力の圧力は 4446.543499 Pa で、氷の強さの 4.446543x10^-3 倍(第3節)。 第五に、閉じた式は、衛星が小さいときの近似である──半径 1 km の衛星では相対差 5.273x10^-5、月ほど大きいと 1.633126 倍(式は 1.488482)(第2節)。 第六に、分離子は、衛星が形を保つのが流体の自己重力か、固体の自己重力か、材料の強さかである──「ロシュ限界の内側では衛星は壊れる」は、衛星が何で形を保つかを言うまで真偽が決まらない(第4節)。 ロシュ限界は一つの距離ではなく、衛星が何で形を保つかで二倍ちがう。惑星の半径に密度の比の三乗根を掛けた長さを単位にすると、潮汐だけに耐える固体の限界は 1.259921、同期回転する固体は 1.442250、流体は 2.455 で、流体と固体は 1.948535 倍ひらく──閉じた式は小さな衛星の近似で、月ほど大きいと正確な釣り合いは 1.633126 倍になる。土星の氷の流体の限界は 2.229704 土星半径で、A 環の外縁 2.269446 はそのわずかに外、すかすかの粒なら内になる。環の中の小衛星パン 2.216500 の自己重力の中心の圧力は 4446.543499 Pa で氷の強さの 4.446543x10^-3 倍しかなく、半径 92.235583 km より小さな衛星は強さで形を保つ──限界が効くのは、自己重力で形を保つ衛星だけである。分離子は、衛星が形を保つのが流体の自己重力か、固体の自己重力か、材料の強さかである。既刊との位置──論文505 は勢力圏が釣り合わせる相手で三つに分かれることを数えた。ここはロシュ限界が衛星の保ち方で三つに分かれることを数えた。正直に言えば──流体の係数は入力で、環の起源も実在の衛星の強さも扱っていない。 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:ロシュ限界、潮汐力、自己重力、土星の環、材料の強さ、衛星。