Shell theorem: Difference between revisions

Revision as of 06:44, 21 August 2024

Revision as of 20:28, 21 November 2024

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{{Short 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.

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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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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>

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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 [[Inscribed_angleInscribed angle|inscribed angle theorem]], the triangles IPH and KPL are similar. The lines KH and IL are rotated about the axis JM to form two cones that intersect the sphere in two 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 two 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 two 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).