Shell theorem: Difference between revisions

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{{shortShort description|Statement on the gravitational attraction of spherical bodies.}}

In [[classical mechanics]], the '''shell theorem''' gives [[gravitational]] simplifications that can be applied to objects inside or outside a spherically [[symmetry|symmetrical]] body. This theorem has particular application to [[astronomy]].

[[Isaac Newton]] proved the shell theorem<ref name="Newton philo">{{cite book|last=Newton|first=Isaac|title=Philosophiae Naturalis Principia Mathematica|url=https://archive.org/details/philosophinatur03newtgoog|date=1687|___location=London|pages=193, Theorem XXXI}}</ref> and stated that:

# A [[sphere|spherically]] [[symmetry|symmetric]] body affects external objects gravitationally as though all of its [[mass]] were concentrated at a [[point mass|point]] at its center.

# If the body is a spherically symmetric shell (i.e., a hollow ball), no net [[gravitational force]] is exerted by the shell on any object inside, regardless of the object's ___location within the shell.

A corollary is that inside a solid sphere of constant density, the gravitational force within the object varies linearly with distance from the center, becoming zero by symmetry at the center of [[mass]]. This can be seen as follows: take a point within such a sphere, at a distance <math>r</math> from the center of the sphere. Then you can ignore all of the shells of greater radius, according to the shell theorem (12). But the point can be considered to be external to the remaining sphere of radius r, and according to (21) all of the mass of this sphere can be considered to be concentrated at its centre. The remaining mass <math>m</math> is proportional to <math>r^3</math> (because it is based on volume). The gravitational force exerted on a body at radius r will be proportional to <math display="inline">\frac{m}{/r^2}</math> (the [[inverse square law]]), so the overall gravitational effect is proportional to {{nowrap|<math display="inline">\frac{r^3}{/r^2} =r</math>,}} so is linear in {{nowrap|<math>r</math>.}}

These results were important to Newton's analysis of planetary motion; they are not immediately obvious, but they can be proven with [[calculus]]. ([[Gauss's law for gravity]] offers an alternative way to state the theorem.)

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==Derivation of gravitational field outside of a solid sphere==

There are three steps to proving Newton's shell theorem (1). First, the equation for a gravitational field due to a ring of mass will be derived. Arranging an infinite number of infinitely thin rings to make a disc, this equation involving a ring will be used to find the gravitational field due to a disk. Finally, arranging an infinite number of infinitely thin discs to make a sphere, this equation involving a disc will be used to find the gravitational field due to a sphere.

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>(0,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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Applying [[Newton's Universal Law of Gravitation]], the sum of the forces due to the mass elements in the shaded band is

:<math>dF = \frac{Gm}{s^2} dM.</math>

However, since there is partial cancellation due to the [[Euclidean vector|vector]] nature of the force in conjunction with the circular band's symmetry, the leftover [[Vector (geometry)#Vector componentsDecomposition|component]] (in the direction pointing towards {{nowrap|<math>m</math>)}} is given by

:<math>dF_r = \frac{Gm}{s^2} \cos(\varphi) \, dM</math>

The total force on {{nowrap|<math>m</math>,}} then, is simply the sum of the force exerted by all the bands. By shrinking the width of each band, and increasing the number of bands, the sum becomes an integral expression:

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:<math>F_r = \frac{GMm}{4r^2 R} \int \left( 1 + \frac{r^2 - R^2}{s^2} \right)\ ds\ .</math>

A [[primitive function]] to the integrand is

:<math>s - \frac{r^2 - R^2}{s}\ ,</math>

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saying that the gravitational force is the same as that of a point mass in the center of the shell with the same mass.

=== Spherical shell to solid sphere ===

Finally,It integrateis allpossible 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>

BetweenUniform 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.,

:<math>dM = \frac{4 \pi x^2 dx}{\frac{4}{3} \pi R^3} M = \frac{3Mx^2 dx}{R^3}</math>

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:<math>F_\text{total} = \frac{3GMm}{r^2 R^3} \int_0^R x^2 \, dx = \frac{GMm}{r^2}</math>

whichAs found earlier, this suggests that the gravity of a solid spherical ball to an exterior object can be simplified as that of a point mass in the center of the ball with the same mass.

== Inside a shell ==

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:<math>\int_S {\mathbf g}\cdot \,d{\mathbf {S}} = \int_S {\mathbf g}\cdot {\hat\mathbf{n}}\,dS</math>

is the [[surface integral]] of the [[gravitational field]] '''<math>\mathbf{g'''}</math> over any [[closed surface]] inside which the total mass is ''M'', the [[unit vector]] <math>\hat\mathbf{n}</math> being the outward normal to the surface.

The gravitational field of a spherically symmetric mass distribution like a mass point, a spherical shell or a homogeneous sphere must also be spherically symmetric. If <math>\hat\mathbf{n}</math> is a unit vector in the direction from the point of symmetry to another point the gravitational field at this other point must therefore be

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== Converses and generalizations ==

It is natural to ask whether the [[Theorem#Converse|converse]] of the shell theorem is true, namely whether the result of the theorem implies the law of universal gravitation, or if there is some more general force law for which the theorem holds. MoreIf specificallywe require only that the force outside of a spherical shell is the same as for an equal point mass at its center, then there is one mayadditional askdegree of freedom for force laws.<ref name=Gurzadyan>{{cite journal| last=Gurzadyan |first=Vahe |authorlink=vahe Gurzadyan|title=The cosmological constant in McCrea-Milne cosmological scheme|journal=The Observatory|date= 1985|volume=105|pages=42–43|bibcode=1985Obs...105...42G}} https://adsabs.harvard.edu/full/1985Obs...105...42G&lang=en</ref><ref name=Arens>{{cite journal| last=Arens| first=Richard| authorlink=Richard Friederich Arens|title=Newton's observations about the questionfield of a uniform thin spherical shell|journal=Note di Matematica|date=January 1, 1990|volume=X|issue=Suppl. n. 1|pages=39–45}}</ref> The most general force, as given by the [[Gurzadyan theorem]], is:<ref name="Gurzadyan"/>

:<math> F(r) = -\frac{G M m}{r^2} -+ \frac{\Lambda M m c^2 r}{3} </math>

{{block indent|Suppose there is a force <math>F</math> between masses ''M'' and ''m'', separated by a distance ''r'' of the form <math>F = M m f(r)</math> such that any spherically symmetric body affects external bodies as if its mass were concentrated at its center. Then what form can the function <math>f</math> take?}}

where <math>G</math> and <math>\Lambda</math> can be constants taking any value. The first term is the familiar law of universal gravitation; the second is an additional force, analogous to the [[cosmological constant]] term in [[general relativity]]. However, the inverse-square potential is the only potential such that the net force inside the shell is also zero.<ref name=Gurzadyan/>

In fact, this allows exactly one more class of force than the (Newtonian) inverse square.<ref name=Gurzadyan>{{cite journal| last=Gurzadyan |first=Vahe |authorlink=vahe Gurzadyan|title=The cosmological constant in McCrea-Milne cosmological scheme|journal=The Observatory|date= 1985|volume=105|pages=42–43|bibcode=1985Obs...105...42G}} http://adsabs.harvard.edu/full/1985Obs...105...42G</ref><ref name=Arens>{{cite journal| last=Arens| first=Richard| authorlink=Richard Friederich Arens|title=Newton's observations about the field of a uniform thin spherical shell|journal=Note di Matematica|date=January 1, 1990|volume=X|issue=Suppl. n. 1|pages=39–45}}</ref> The most general force as derived by [[Vahe Gurzadyan]] in <ref name="Gurzadyan"/> (Gurzadyan

theorem) is:"

The force described by the [[Yukawa potential]]

:<math> F = -\frac{G M m}{r^2} - \frac{\Lambda M m r}{3} </math>

:<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>

where <math>G</math> and <math>\Lambda</math> can be constants taking any value. The first term is the familiar law of universal gravitation; the second is an additional force, analogous to the [[cosmological constant]] term in [[general relativity]].

If we further constrain the force by requiring that the second part of the theorem also holds, namely that there is no force inside a hollow ball, we exclude the possibility of the additional term, and the inverse square law is indeed the unique force law satisfying the theorem.

On the other hand, if we relax the conditions, and require only that the field everywhere outside a spherically symmetric body is the same as the field from some point mass at the center (of any mass), we allow a new class of solutions given by the [[Yukawa potential]], of which the inverse square law is a special case.

Another generalization can be made for a disc by observing that

:<math>dM=\frac{R^2}{2} \frac{d\theta \, \sin^2(\theta)}{\pi R^2}M=\frac{ \sin^2(\theta)}{2 \pi}M \, d\theta </math>

so:

:<math>F_r = \frac{GMm}{2 \pi} \int \frac{ \sin^2 (\theta) \cos(\varphi)} {s^2} \, d\theta,</math>

where {{nowrap|<math>M=\pi R^2 \rho</math>,}} and <math>\rho</math> is the density of the body.

Doing all the intermediate calculations we get:

:<math>F(r) = \frac{G m \rho}{8r^3} \int_{R-r}^{R+r} { \frac{\left(r^2 + s^2 - R^2\right)\sqrt{2\left(r^2 R^2 + r^2 s^2 + R^2 s^2\right) - s^4 - r^4 - R^4} }{s^2} } \, ds</math>

== 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.

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[[File:Attraction Interior Sphere.png|Attraction interior sphere]]

Fig. 2 is a cross-section of the hollow sphere through the center, S and an arbitrary point, P, inside the sphere. Through P draw two lines IL and HK such that the angle KPL is very small. JM is the line through P that bisects that angle. From the geometry[[Inscribed ofangle|inscribed circlesangle theorem]], the triangles IPH and KPL are similar. The lines KH and IL are rotated about the axis JM to form 2two cones that intersect the sphere in 2two closed curves. In Fig. 1 the sphere is seen from a distance along the line PE and is assumed transparent so both curves can be seen.

The surface of the sphere that the cones intersect can be considered to be flat, and {{nowrap|<math> \angle PJI = \angle PMK </math>.}}

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Since the intersection of a cone with a plane is an ellipse, in this case the intersections form two ellipses with major axes IH and KL, where {{nowrap|<math> \frac{IH}{KL} = \frac{PJ}{PM} </math>.}}

By a similar argument, the minor axes are in the same ratio. This is clear if the sphere is viewed from above. Therefore, the two ellipses are similar, so their areas are as the squares of their major axes. As the mass of any section of the surface is proportional to the area of that section, for the 2two elliptical areas the ratios of their masses {{nowrap|<math> \propto \frac{PJ^2}{PM^2} </math>.}}

Since the force of attraction on P in the direction JM from either of the elliptic areas, is direct as the mass of the area and inversely as the square of its distance from P, it is independent of the distance of P from the sphere. Hence, the forces on P from the 2two infinitesimal elliptical areas are equal and opposite and there is no net force in the direction JM.

As the position of P and the direction of JM are both arbitrary, it follows that any particle inside a hollow sphere experiences no net force from the mass of the sphere.

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Let arc IH be extended perpendicularly out of the plane of the diagram, by a small distance ζ. The area of the figure generated is {{nowrap|<math> IH\cdot \zeta </math>,}} and its mass is proportional to this product.

The force due to this mass on the particle at P <math> \propto \frac{IH\cdot \zeta}{PI^2} </math> and is along the line PI.

The component of this force towards the center <math> \propto \frac{IH\cdot PQ\cdot \zeta}{PI^3} </math>.

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Generate a ring with width ih and radius iq by making angle <math> fiS = FIS </math> and the slightly larger angle {{nowrap|<math> dhS = DHS </math>,}} so that the distance PS is subtended by the same angle at I as is pS at i. The same holds for H and h, respectively.

The total force on p due to this ring is

:<math> \propto \frac{ih\cdot iq\cdot pq}{pi^3} = \frac{a\cdot df\cdot fS}{if\cdot d^2} </math>

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Newton claims that DF and df can be taken as equal in the limit as the angles DPF and dpf 'vanish together'. Note that angles DPF and dpf are not equal. Although DS and dS become equal in the limit, this does not imply that the ratio of DF to df becomes equal to unity, when DF and df both approach zero. In the finite case DF depends on D, and df on d, so they are not equal.

Since the ratio of DF to df in the limit is crucial, more detailed analysis is required. From the similar right triangles, <math display="inline"> \frac {DF}{PF} = \frac{ED}{ES}</math> and {{nowrap|<math> ED^2 = (DF + FS)^2 - ES^2 </math>,}} giving {{nowrap|<math> \frac {\left(PF^2 - ES^2\right)DF^2}{PF^2} + 2\cdot FS\cdot DF + FS^2 - ES^2 = 0 </math>.}} Solving the quadratic for DF, in the limit as ES approaches FS, the smaller root, {{nowrap|<math> DF = ES - FS </math>.}} More simply, as DF approaches zero, in the limit the <math> DF^2 </math> term can be ignored: <math> 2\cdot FS\cdot DF + FS^2 - ES^2 = 0 </math> leading to the same result. Clearly df has the same limit, justifying Newton’sNewton's claim.

Comparing the force from the ring HI rotated about PS to the ring hi about pS, the ratio of these 2 forces equals {{nowrap|<math display="inline"> \frac{d^2}{D^2} </math>.}}

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Spherical symmetry implies that the metric has time-independent Schwarzschild geometry, even if a central mass is undergoing gravitational collapse (Misner et al. 1973; see [[Birkhoff's theorem (relativity)|Birkhoff's theorem]]). The [[Metric tensor (general relativity)|metric]] thus has form

:<math>ds^2 = - (1-2M/r)\, dt^2 + (1-2M/r)^{-1} \, dr^2 + r^2 \, d\Omega^2</math>

(using [[Geometrized unit system|geometrized units]], where {{nowrap|<math>G=c=1</math>).}} For <math>r>R>0</math> (where <math>R</math> is the radius of some mass shell), mass acts as a [[delta function]] at the origin. For {{nowrap|<math>r < R</math>,}} shells of mass may exist externally, but for the metric to be [[Singularity (mathematics)|non-singular]] at the origin, <math>M</math> must be zero in the metric. This reduces the metric to flat [[Minkowski space]]; thus external shells have no gravitational effect.

This result illuminates the [[gravitational collapse]] leading to a black hole and its effect on the motion of light-rays and particles outside and inside the event horizon (Hartle 2003, chapter 12).

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==See also==

{{commons category}}

* [[Chasles' theorem (gravitation)]]

* [[Scale height]]

*[[Chasles' theorem (gravitation)]]

==References==

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[[Category:Physics theorems]]

[[Category:Mathematical theorems]]

[[Category:Electrostatics]]

[[Category:Potential theory]]