Shell theorem: Difference between revisions

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The gravitational field <math>E</math> at a position called <math>P</math> at <math>(x,y) = (-p,0)</math> on the ''x''-axis due to a point of mass <math>M</math> at the origin is <math display="block">E_\text{point} = \frac{GM}{p^2}</math>

[[File:Point2.png|frameless|300x300px]]

Suppose that this mass is moved upwards along the ''y''-axis to the point {{nowrap|<math>(-p0,R)</math>.}} The distance between <math>P</math> and the point mass is now longer than before; It becomes the [[hypotenuse]] of the right triangle with legs <math>p</math> and <math>R</math> which is {{nowrap|<math display="inline">\sqrt{p^2 + R^2}</math>.}} Hence, the gravitational field of the elevated point is:

<math display="block">E_\text{elevated point} = \frac{GM}{p^2+R^2}</math>

[[File:Pointy2.png|frameless|270x270px]]

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It is possible to use this spherical shell result to re-derive the solid sphere result from earlier. This is done by integrating an infinitesimally thin spherical shell with mass of {{nowrap|<math>dM</math>,}} and we can obtain the total gravity contribution of a solid ball to the object outside the ball

:<math>F_\text{total} = \int dF_r = \frac{Gm}{r^2} \int dM.</math>

Uniform density means between the radius of <math>x</math> to {{nowrap|<math>x+dx</math>,}} <math>dM</math> can be expressed as a function of {{nowrap|<math>x</math>,}} i.e.,

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The force described by the [[Yukawa potential]]

:<math> U(r) = -\frac{G M m}{r} e^{-\lambda r}</math>

has the property that the force outside of a spherical shell is also a Yukawa potential with the same range <math>1/\lambda</math> and centered at the shell's center, but for <math>\lambda > 0</math> the equivalent point mass is not the same as the mass of the shell.<ref>{{cite web |last=Kuhn |first=Paulo |title=Debye-Hückel interaction, or Yukawa potential, in different geometries |url=https://wp.ufpel.edu.br/pskuhn/files/2024/09/debye-huckel-interaction-or-yukawa-in-different-geometries.pdf |access-date=14 February 2025}}</ref><ref>{{cite web |last=McDonald | first=Kirk |orig-date=April 17, 1984 |date=December 20, 2021 |title=A Naïve Estimate of the Coupling Constant in Yukawa Theory |url=http://kirkmcd.princeton.edu/examples/yukawa.pdf |access-date=14 February 2025}}</ref><ref>{{cite web |title=Shell theorem for a general potential |url=https://math.stackexchange.com/questions/296180/shell-theorem-for-a-general-potential |website=Mathematics Stack Exchange |access-date=14 February 2025 |language=en}}</ref> For a shell of radius <math>R</math> and mass <math>M</math>, the equivalent point mass is:

:<math>M_\text{eff} = M \frac{\sinh \lambda r}{\lambda r}</math>.

For an [[ellipsoid]]al shell, the two halves of the shell theorem are generalized by different types of shells. The shell bound by two [[concentric]], [[similarity (geometry)|similar]], and aligned ellipsoids (a [[homoeoid]]) exters no gravitational force on a point inside of it.<ref>[[Michel Chasles]], [http://sites.mathdoc.fr/JMPA/PDF/JMPA_1840_1_5_A41_0.pdf ''Solution nouvelle du problème de l’attraction d’un ellipsoïde hétérogène sur un point exterieur''], Jour. Liouville 5, 465–488 (1840)</ref> Meanwhile, the shell bound by two concentric, [[confocal]] ellipsoids (a [[focaloid]]) has the property that the gravitational force outside of two concentric, confocal focaloids is the same.<ref name="rodrigues">{{cite journal |last1=Rodrigues |first1=Hilário |title=On determining the kinetic content of ellipsoidal configurations |journal=Monthly Notices of the Royal Astronomical Society |date=11 May 2014 |volume=440 |issue=2 |pages=1519–1526 |doi=10.1093/mnras/stu353|doi-access=free |arxiv=1402.6541 }}</ref>

== Newton's proofs ==

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=== Introduction ===

Propositions 70 and 71 consider the force acting on a particle from a hollow sphere with an infinitesimally thin surface, whose mass density is constant over the surface. The force on the particle from a small area of the surface of the sphere is proportional to the mass of the area and inversely as the square of its distance from the particle. The first proposition considers the case when the particle is inside the sphere, the second when it is outside. The use of infinitesimals and limiting processes in geometrical constructions are simple and elegant and avoid the need for any integrations. They well illustrate [[Newton's method]] of proving many of the propositions in the ''Principia''.

His proof of Propositions 70 is trivial. In the following, it is considered in slightly greater detail than Newton provides.