In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetric body. This theorem has particular application to astronomy.
Isaac Newton proved the shell theorem and stated that: A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a 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 from the center of the sphere. Then you can ignore all of the shells of greater radius, according to the shell theorem (2). But the point can be considered to be external to the remaining sphere of radius r, and according to (1) all of the mass of this sphere can be considered to be concentrated at its centre. The remaining mass is proportional to (because it is based on volume). The gravitational force exerted on a body at radius r will be proportional to (the inverse square law), so the overall gravitational effect is proportional to so is linear in
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.)
In addition to gravity, the shell theorem can also be used to describe the electric field generated by a static spherically symmetric charge density, or similarly for any other interaction whose force follows an inverse square law. The derivations below focus on gravity, but the results can easily be generalized to the electrostatic force.
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 at a position called at on the x-axis due to a point of mass at the origin is
Suppose that this mass is moved upwards along the y-axis to the point The distance between and the point mass is now longer than before; It becomes the hypotenuse of the right triangle with legs and which is Hence, the gravitational field of the elevated point is:
The magnitude of the gravitational field that would pull a particle at point in the x-direction is the gravitational field multiplied by where is the angle adjacent to the x-axis. In this case, Hence, the magnitude of the gravitational field in the x-direction, is:
Substituting in gives
Suppose that this mass is evenly distributed in a ring centered at the origin and facing point with the same radius Because all of the mass is located at the same angle with respect to the x-axis, and the distance between the points on the ring is the same distance as before, the gravitational field in the x-direction at point due to the ring is the same as a point mass located at a point units above the x-axis:
To find the gravitational field at point due to a disc, an infinite number of infinitely thin rings facing each with a radius width of and mass of may be placed inside one another to form a disc. The mass of any one of the rings is the mass of the disc multiplied by the ratio of the area of the ring to the total area of the disc So, Hence, a small change in the gravitational field, is:
Substituting in and integrating both sides gives the gravitational field of the disk:
Adding up the contribution to the gravitational field from each of these rings will yield the expression for the gravitational field due to a disc. This is equivalent to integrating this above expression from to resulting in:
To find the gravitational field at point due to a sphere centered at the origin, an infinite amount of infinitely thin discs facing each with a radius width of and mass of may be placed together.
These discs' radii follow the height of the cross section of a sphere (with constant radius ) which is an equation of a semi-circle: varies from to
The mass of any of the discs is the mass of the sphere multiplied by the ratio of the volume of an infinitely thin disc divided by the volume of a sphere (with constant radius The volume of an infinitely thin disc is or So, Simplifying gives
Each discs' position away from will vary with its position within the 'sphere' made of the discs, so must be replaced with
Replacing with with and with in the 'disc' equation yields:
Simplifying,
Integrating the gravitational field of each thin disc from to with respect to and doing some careful algebra, yields Newton's shell theorem:
where is the distance between the center of the spherical mass and an arbitrary point The gravitational field of a spherical mass may be calculated by treating all the mass as a point particle at the center of the sphere.
Outside a shell A solid, spherically symmetric body can be modeled as an infinite number of concentric, infinitesimally thin spherical shells. If one of these shells can be treated as a point mass, then a system of shells (i.e. the sphere) can also be treated as a point mass. Consider one such shell (the diagram shows a cross-section):
(Note: the in the diagram refers to the small angle, not the arc length. The arc length is
Applying Newton's law of universal gravitation, the sum of the forces due to the mass elements in the shaded band is
However, since there is partial cancellation due to the vector nature of the force in conjunction with the circular band's symmetry, the leftover component (in the direction pointing towards is given by
The total force on 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:
Since and are constants, they may be taken out of the integral:
To evaluate this integral, one must first express as a function of
The total surface of a spherical shell is
while the surface area of the thin slice between and is
If the mass of the shell is one therefore has that
and
By the law of cosines,
and
These two relations link the three parameters and that appear in the integral together. As increases from to radians, varies from the initial value 0 to a maximal value before finally returning to zero At the same time, increases from the initial value to the final value as increases from 0 to radians. This is illustrated in the following animation:
(Note: As viewed from the shaded blue band appears as a thin annulus whose inner and outer radii converge to as vanishes.)
To find a primitive function to the integrand, one has to make the independent integration variable instead of
Performing an implicit differentiation of the second of the "cosine law" expressions above yields
and thus
It follows that
where the new integration variable increases from
Inserting the expression for using the first of the "cosine law" expressions above, one finally gets that
A primitive function to the integrand is
and inserting the bounds and for the integration variable in this primitive function, one gets that
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 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 and we can obtain the total gravity contribution of a solid ball to the object outside the ball
Uniform density means between the radius of to can be expressed as a function of i.e.,
Therefore, the total gravity is
As 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 For a point inside the shell, the difference is that when θ is equal to zero, ϕ takes the value radians and s the value . When θ increases from 0 to radians, ϕ decreases from the initial value radians to zero and s increases from the initial value to the value .
This can all be seen in the following figure
Inserting these bounds into the primitive function
one gets that, in this case
saying that the net gravitational forces acting on the point mass from the mass elements of the shell, outside the measurement point, cancel out.
Generalization: If the resultant force inside the shell is:
The above results into being identically zero if and only if
Outside the shell (i.e. or ):
Derivation using Gauss's law The shell theorem is an immediate consequence of Gauss's law for gravity saying that
where M is the mass of the part of the spherically symmetric mass distribution that is inside the sphere with radius r and
is the surface integral of the gravitational field over any closed surface inside which the total mass is M, the unit vector 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 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
where g(r) only depends on the distance r to the point of symmetry
Selecting the closed surface as a sphere with radius r with center at the point of symmetry the outward normal to a point on the surface, is precisely the direction pointing away from the point of symmetry of the mass distribution.
One, therefore, has that
and
as the area of the sphere is 4r2.
From Gauss's law it then follows that
or,
Converses and generalizations
It is natural to ask whether the 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. If we 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 additional degree of freedom for force laws. https://adsabs.harvard.edu/full/1985Obs...105...42G&lang=en The most general force, as given by the Gurzadyan theorem, is:
where and 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.
The force described by the Yukawa potential
has the property that the force outside of a spherical shell is also a Yukawa potential with the same range and centered at the shell's center, but for the equivalent point mass is not the same as the mass of the shell. For a shell of radius and mass , the equivalent point mass is: .