SherbyNet - /page/wikibook/entry.mu

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<title>Shell theorem</title>

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Shell theorem

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In <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>, the shell theorem gives <a href="Gravitational" class="mw-redirect" title="Gravitational">gravitational</a> simplifications that can be applied to objects inside or outside a spherically <a href="Symmetry" title="Symmetry">symmetrical</a> body. This theorem has particular application to <a href="Astronomy" title="Astronomy">astronomy</a>.

<a href="Isaac_Newton" title="Isaac Newton">Isaac Newton</a> proved the shell theorem<a href="#cite_note-Newton_philo-1">[1]</a> and stated that:

<ol><li>A <a href="Sphere" title="Sphere">spherically</a> symmetric body affects external objects gravitationally as though all of its <a href="Mass" title="Mass">mass</a> were concentrated at a <a href="Point_mass" class="mw-redirect" title="Point mass">point</a> at its center.</li>

<li>If the body is a spherically symmetric shell (i.e., a hollow ball), no net <a href="Gravitational_force" class="mw-redirect" title="Gravitational force">gravitational force</a> is exerted by the shell on any object inside, regardless of the object's location within the shell.</li></ol>

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 <a href="Mass" title="Mass">mass</a>. This can be seen as follows: take a point within such a sphere, at a distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> is proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle r^{3}}</annotation>

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</math><img src="./cd69c630eae852a57d5b776d88626b7afe7bb8e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.103ex; height:2.676ex;" alt="{\displaystyle r^{3}}" loading="lazy"> (because it is based on volume). The gravitational force exerted on a body at radius r will be proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m/r^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle m/r^{2}}</annotation>

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</math><img src="./64d9de9fc5fee152de3a3843364500209696a71b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.306ex; height:3.176ex;" alt="{\displaystyle m/r^{2}}" loading="lazy"> (the <a href="Inverse_square_law" class="mw-redirect" title="Inverse square law">inverse square law</a>), so the overall gravitational effect is proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{3}/r^{2}=r}">

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<annotation encoding="application/x-tex">{\displaystyle r^{3}/r^{2}=r}</annotation>

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</math><img src="./fcc8a50de1b664ccd2354dfee9e2314583ba9ad9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.515ex; height:3.176ex;" alt="{\displaystyle r^{3}/r^{2}=r}" loading="lazy">, so is linear in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy">.

These results were important to Newton's analysis of planetary motion; they are not immediately obvious, but they can be proven with <a href="Calculus" title="Calculus">calculus</a>. (<a href="Gauss's_law_for_gravity" title="Gauss's law for gravity">Gauss's law for gravity</a> offers an alternative way to state the theorem.)

In addition to <a href="Gravity" title="Gravity">gravity</a>, the shell theorem can also be used to describe the <a href="Electric_field" title="Electric field">electric field</a> generated by a static spherically symmetric <a href="Charge_density" title="Charge density">charge density</a>, or similarly for any other phenomenon that follows an <a href="Inverse_square_law" class="mw-redirect" title="Inverse square law">inverse square law</a>. The derivations below focus on gravity, but the results can easily be generalized to the <a href="Electrostatic_force" class="mw-redirect" title="Electrostatic force">electrostatic force</a>.

<div class="mw-heading mw-heading2"><h2 id="Derivation_of_gravitational_field_outside_of_a_solid_sphere">Derivation of gravitational field outside of a solid sphere</h2></div>

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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

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</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> at a position called <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)=(-p,0)}">

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<annotation encoding="application/x-tex">{\displaystyle (x,y)=(-p,0)}</annotation>

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</math><img src="./3a17ff4b82a013c32d7f78aae325a85e18cf9b4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.41ex; height:2.843ex;" alt="{\displaystyle (x,y)=(-p,0)}" loading="lazy"> on the x-axis due to a point of mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> at the origin is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{point}}={\frac {GM}{p^{2}}}}">

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<mtext>point</mtext>

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<annotation encoding="application/x-tex">{\displaystyle E_{\text{point}}={\frac {GM}{p^{2}}}}</annotation>

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Suppose that this mass is moved upwards along the y-axis to the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,R)}">

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<annotation encoding="application/x-tex">{\displaystyle (0,R)}</annotation>

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</math><img src="./9f526dcac0232a0e6aa67fc463c323c46d1c86e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.77ex; height:2.843ex;" alt="{\displaystyle (0,R)}" loading="lazy">. The distance between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> and the point mass is now longer than before; It becomes the hypotenuse of the right triangle with legs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> which is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {p^{2}+R^{2}}}}">

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<annotation encoding="application/x-tex">{\textstyle {\sqrt {p^{2}+R^{2}}}}</annotation>

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</math><img src="./b1e0b53c89a3625b163e0afa853bb49e7e96f727.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.206ex; height:3.509ex;" alt="{\textstyle {\sqrt {p^{2}+R^{2}}}}" loading="lazy">. Hence, the gravitational field of the elevated point is:

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<mtext>elevated point</mtext>

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<annotation encoding="application/x-tex">{\displaystyle E_{\text{elevated point}}={\frac {GM}{p^{2}+R^{2}}}}</annotation>

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The magnitude of the gravitational field that would pull a particle at point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> in the x-direction is the gravitational field multiplied by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\theta )}">

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<annotation encoding="application/x-tex">{\displaystyle \cos(\theta )}</annotation>

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</math><img src="./aaac7b75cda6d5570780075aa2622d27b21117cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.011ex; height:2.843ex;" alt="{\displaystyle \cos(\theta )}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

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<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

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</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> is the angle adjacent to the x-axis. In this case, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\theta )={\frac {p}{\sqrt {p^{2}+R^{2}}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \cos(\theta )={\frac {p}{\sqrt {p^{2}+R^{2}}}}}</annotation>

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</math><img src="./210661d847328610c49b51b59a05326e36ff3d74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.152ex; height:6.176ex;" alt="{\displaystyle \cos(\theta )={\frac {p}{\sqrt {p^{2}+R^{2}}}}}" loading="lazy">. Hence, the magnitude of the gravitational field in the x-direction, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{x}}">

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<annotation encoding="application/x-tex">{\displaystyle E_{x}}</annotation>

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</math><img src="./029e49fbec18ece71cdd1e68bc478444e2c99d30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.888ex; height:2.509ex;" alt="{\displaystyle E_{x}}" loading="lazy"> is:

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<annotation encoding="application/x-tex">{\displaystyle E_{x}={\frac {GM\cos {\theta }}{p^{2}+R^{2}}}}</annotation>

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Substituting in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\theta )}">

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<annotation encoding="application/x-tex">{\displaystyle \cos(\theta )}</annotation>

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</math><img src="./aaac7b75cda6d5570780075aa2622d27b21117cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.011ex; height:2.843ex;" alt="{\displaystyle \cos(\theta )}" loading="lazy"> gives

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<annotation encoding="application/x-tex">{\displaystyle E_{x}={\frac {GMp}{\left(p^{2}+R^{2}\right)^{3/2}}}}</annotation>

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Suppose that this mass is evenly distributed in a ring centered at the origin and facing point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> with the same radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">. 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> due to the ring is the same as a point mass located at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> units above the y-axis:

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<annotation encoding="application/x-tex">{\displaystyle E_{\text{ring}}={\frac {GMp}{\left(p^{2}+R^{2}\right)^{3/2}}}}</annotation>

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To find the gravitational field at point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> due to a disc, an infinite number of infinitely thin rings facing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">, each with a radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">, width of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dy}">

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<annotation encoding="application/x-tex">{\displaystyle dy}</annotation>

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</math><img src="./5c5eda9ec854eb0076d43c147eb8956637a1003f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.371ex; height:2.509ex;" alt="{\displaystyle dy}" loading="lazy">, and mass of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dM}">

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<annotation encoding="application/x-tex">{\displaystyle dM}</annotation>

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</math><img src="./7b8515decc1231e7e511ab7178f8959e2f7fe94c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.658ex; height:2.176ex;" alt="{\displaystyle dM}" loading="lazy"> may be placed inside one another to form a disc. The mass of any one of the rings <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dM}">

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<annotation encoding="application/x-tex">{\displaystyle dM}</annotation>

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</math><img src="./7b8515decc1231e7e511ab7178f8959e2f7fe94c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.658ex; height:2.176ex;" alt="{\displaystyle dM}" loading="lazy"> is the mass of the disc multiplied by the ratio of the area of the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi y\,dy}">

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<annotation encoding="application/x-tex">{\displaystyle 2\pi y\,dy}</annotation>

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</math><img src="./c3161be9059f5c8af7b5e97c2d3a451b25835260.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.408ex; height:2.509ex;" alt="{\displaystyle 2\pi y\,dy}" loading="lazy"> to the total area of the disc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi R^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \pi R^{2}}</annotation>

</semantics>

</math><img src="./db4bdad3187c4001ed7079d18509f3b52d18479c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.15ex; height:2.676ex;" alt="{\displaystyle \pi R^{2}}" loading="lazy">. So, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle dM={\frac {M\cdot 2y\,dy}{R^{2}}}}">

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle dM={\frac {M\cdot 2y\,dy}{R^{2}}}}</annotation>

</semantics>

</math><img src="./dc724c424e9f0b2214ca85206db86ee1be387c0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.48ex; height:4.509ex;" alt="{\textstyle dM={\frac {M\cdot 2y\,dy}{R^{2}}}}" loading="lazy">. Hence, a small change in the gravitational field, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> is:

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dE={\frac {Gp\,dM}{(p^{2}+y^{2})^{3/2}}}}</annotation>

</semantics>

</math>

Substituting in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dM}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dM}</annotation>

</semantics>

</math><img src="./7b8515decc1231e7e511ab7178f8959e2f7fe94c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.658ex; height:2.176ex;" alt="{\displaystyle dM}" loading="lazy"> and integrating both sides gives the gravitational field of the disk:

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=\int {\frac {GMp\cdot {\frac {2y\,dy}{R^{2}}}}{(p^{2}+y^{2})^{3/2}}}}</annotation>

</semantics>

</math>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=0}</annotation>

</semantics>

</math><img src="./094f824655138f6b11d96a0da32e7f0716ba6959.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=0}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=R}</annotation>

</semantics>

</math><img src="./fc53039ce1aa45c1ad3db0f1eccc4eaa5fd1d75a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.018ex; height:2.509ex;" alt="{\displaystyle y=R}" loading="lazy">, resulting in:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\text{disc}}={\frac {2GM}{R^{2}}}\left(1-{\frac {p}{\sqrt {p^{2}+R^{2}}}}\right)}</annotation>

</semantics>

</math>

To find the gravitational field at point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> due to a sphere centered at the origin, an infinite amount of infinitely thin discs facing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">, each with a radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">, width of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dx}</annotation>

</semantics>

</math><img src="./845c817e348381a13f3fad5184169ce0e021c685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.546ex; height:2.176ex;" alt="{\displaystyle dx}" loading="lazy">, and mass of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dM}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dM}</annotation>

</semantics>

</math><img src="./7b8515decc1231e7e511ab7178f8959e2f7fe94c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.658ex; height:2.176ex;" alt="{\displaystyle dM}" loading="lazy"> may be placed together.

These discs' radii <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> follow the height of the cross section of a sphere (with constant radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">) which is an equation of a semi-circle: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle R={\sqrt {a^{2}-x^{2}}}}">

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle R={\sqrt {a^{2}-x^{2}}}}</annotation>

</semantics>

</math><img src="./5b5d3b16ab103c30abc376fe209d7de6ea9e3744.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.695ex; height:3.509ex;" alt="{\textstyle R={\sqrt {a^{2}-x^{2}}}}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> varies from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -a}</annotation>

</semantics>

</math><img src="./6e0982b5868a66be1ed3ad7ef4bcd3d3db20f982.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.176ex;" alt="{\displaystyle -a}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">.

The mass of any of the discs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dM}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dM}</annotation>

</semantics>

</math><img src="./7b8515decc1231e7e511ab7178f8959e2f7fe94c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.658ex; height:2.176ex;" alt="{\displaystyle dM}" loading="lazy"> is the mass of the sphere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> multiplied by the ratio of the volume of an infinitely thin disc divided by the volume of a sphere (with constant radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">). The volume of an infinitely thin disc is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi R^{2}\,dx}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi R^{2}\,dx}</annotation>

</semantics>

</math><img src="./c965abc5e9e731aa3385159d3703b76413a3ff8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.083ex; height:2.676ex;" alt="{\displaystyle \pi R^{2}\,dx}" loading="lazy">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \pi \left(a^{2}-x^{2}\right)dx}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \pi \left(a^{2}-x^{2}\right)dx}</annotation>

</semantics>

</math><img src="./7e0795023ea30071efe4bdf421b0763bd2ef1579.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.29ex; height:3.176ex;" alt="{\textstyle \pi \left(a^{2}-x^{2}\right)dx}" loading="lazy">. So, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle dM={\frac {\pi M(a^{2}-x^{2})\,dx}{{\frac {4}{3}}\pi a^{3}}}}">

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle dM={\frac {\pi M(a^{2}-x^{2})\,dx}{{\frac {4}{3}}\pi a^{3}}}}</annotation>

</semantics>

</math><img src="./070fdb70ae4d3897c541cf7ce66fb58fb94d3853.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:18.479ex; height:6.009ex;" alt="{\textstyle dM={\frac {\pi M(a^{2}-x^{2})\,dx}{{\frac {4}{3}}\pi a^{3}}}}" loading="lazy">. Simplifying gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle dM={\frac {3M(a^{2}-x^{2})\,dx}{4a^{3}}}}">

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle dM={\frac {3M(a^{2}-x^{2})\,dx}{4a^{3}}}}</annotation>

</semantics>

</math><img src="./e01bb3fb07aa47b4514c77ea7af5b65a5d5aae04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:18.36ex; height:4.676ex;" alt="{\textstyle dM={\frac {3M(a^{2}-x^{2})\,dx}{4a^{3}}}}" loading="lazy">.

Each discs' position away from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> will vary with its position within the 'sphere' made of the discs, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> must be replaced with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p+x}</annotation>

</semantics>

</math><img src="./cb30572cb498c5a41261792cb81d0f33c13397e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.429ex; height:2.343ex;" alt="{\displaystyle p+x}" loading="lazy">.

Replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dM}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dM}</annotation>

</semantics>

</math><img src="./7b8515decc1231e7e511ab7178f8959e2f7fe94c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.658ex; height:2.176ex;" alt="{\displaystyle dM}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a^{2}-x^{2}}}}">

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {a^{2}-x^{2}}}}</annotation>

</semantics>

</math><img src="./4ac5221f5b3e35f1befb0da8c656b5b5081a0e6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.832ex; height:3.509ex;" alt="{\displaystyle {\sqrt {a^{2}-x^{2}}}}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p+x}</annotation>

</semantics>

</math><img src="./cb30572cb498c5a41261792cb81d0f33c13397e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.429ex; height:2.343ex;" alt="{\displaystyle p+x}" loading="lazy"> in the 'disc' equation yields:

<mfrac>

<mrow>

<mfrac>

<mrow>

<mrow>

<mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dE={\frac {\left({\frac {2G\left[3M\left(a^{2}-x^{2}\right)\right]}{4a^{3}}}\right)}{{\sqrt {a^{2}-x^{2}}}^{2}}}\cdot \left(1-{\frac {p+x}{\sqrt {(p+x)^{2}+{\sqrt {a^{2}-x^{2}}}^{2}}}}\right)\,dx}</annotation>

</semantics>

</math>

Simplifying,

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int dE=\int _{-a}^{a}{\frac {3GM}{2a^{3}}}\left(1-{\frac {p+x}{\sqrt {p^{2}+a^{2}+2px}}}\right)dx}</annotation>

</semantics>

</math>

Integrating the gravitational field of each thin disc from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=-a}</annotation>

</semantics>

</math><img src="./9f726a9f4cff586cefe8198e5c628a55d70b5571.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.466ex; height:2.176ex;" alt="{\displaystyle x=-a}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=+a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=+a}</annotation>

</semantics>

</math><img src="./8814caa6e38d62bcfb2abd0536056ce084338746.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.466ex; height:2.176ex;" alt="{\displaystyle x=+a}" loading="lazy"> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">, and doing some careful algebra, yields Newton's shell theorem:

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E={\frac {GM}{p^{2}}}}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is the distance between the center of the spherical mass and an arbitrary point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">. 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.

<div class="mw-heading mw-heading2"><h2 id="Outside_a_shell">Outside a shell</h2></div>

A solid, <a href="Sphere" title="Sphere">spherically</a> <a href="Symmetry" title="Symmetry">symmetric</a> body can be modeled as an infinite number of <a href="Concentric" class="mw-redirect" title="Concentric">concentric</a>, 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\theta }</annotation>

</semantics>

</math><img src="./75ae6ca1248d081ef3fcfdd3e17ba0e3f6c02ee9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.306ex; height:2.176ex;" alt="{\displaystyle d\theta }" loading="lazy"> in the diagram refers to the small angle, not the <a href="Arclength" class="mw-redirect" title="Arclength">arc length</a>. The arc length is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle R\,d\theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle R\,d\theta }</annotation>

</semantics>

</math><img src="./878b871964c4fd5395b281005c580f4afedb7d57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.457ex; height:2.176ex;" alt="{\textstyle R\,d\theta }" loading="lazy">.)

Applying <a href="Newton's_Universal_Law_of_Gravitation" class="mw-redirect" title="Newton's Universal Law of Gravitation">Newton's Universal Law of Gravitation</a>, the sum of the forces due to the mass elements in the shaded band is

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dF={\frac {Gm}{s^{2}}}dM.}</annotation>

</semantics>

</math><img src="./40d290ef082fdfe9f2a3314fbab9c0006db0ff78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.063ex; height:5.676ex;" alt="{\displaystyle dF={\frac {Gm}{s^{2}}}dM.}" loading="lazy"></dd></dl>

However, since there is partial cancellation due to the <a href="Euclidean_vector" title="Euclidean vector">vector</a> nature of the force in conjunction with the circular band's symmetry, the leftover <a href="Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">component</a> (in the direction pointing towards <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">) is given by

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dF_{r}={\frac {Gm}{s^{2}}}\cos(\varphi )\,dM}</annotation>

</semantics>

</math><img src="./1ecd1757b10e54bebac35dc8f5ca22236cf4b6a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:22.359ex; height:5.676ex;" alt="{\displaystyle dF_{r}={\frac {Gm}{s^{2}}}\cos(\varphi )\,dM}" loading="lazy"></dd></dl>

The total force on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">, 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:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{r}=\int dF_{r}}</annotation>

</semantics>

</math><img src="./1663d0f2bdd313c00f67da2ae53f8600023aba68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.832ex; height:5.676ex;" alt="{\displaystyle F_{r}=\int dF_{r}}" loading="lazy"></dd></dl>

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> are constants, they may be taken out of the integral:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{r}=Gm\int {\frac {\cos(\varphi )}{s^{2}}}\,dM.}</annotation>

</semantics>

</math><img src="./1e50c37dbe8ddd6221457409a7fb10c4b7467896.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.37ex; height:6.176ex;" alt="{\displaystyle F_{r}=Gm\int {\frac {\cos(\varphi )}{s^{2}}}\,dM.}" loading="lazy"></dd></dl>

To evaluate this integral, one must first express <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dM}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dM}</annotation>

</semantics>

</math><img src="./7b8515decc1231e7e511ab7178f8959e2f7fe94c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.658ex; height:2.176ex;" alt="{\displaystyle dM}" loading="lazy"> as a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\theta }</annotation>

</semantics>

</math><img src="./75ae6ca1248d081ef3fcfdd3e17ba0e3f6c02ee9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.306ex; height:2.176ex;" alt="{\displaystyle d\theta }" loading="lazy">

The total surface of a spherical shell is

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4\pi R^{2}}</annotation>

</semantics>

</math><img src="./8ebf047836476dc57dd8c3758e0fb2aeb950ef8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.313ex; height:2.676ex;" alt="{\displaystyle 4\pi R^{2}}" loading="lazy"></dd></dl>

while the surface area of the thin slice between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta +d\theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta +d\theta }</annotation>

</semantics>

</math><img src="./2fdc46aa00ae4b2451dee3430ee8df381fef6914.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.237ex; height:2.343ex;" alt="{\displaystyle \theta +d\theta }" loading="lazy"> is

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi R\sin(\theta )R\,d\theta =2\pi R^{2}\sin(\theta )\,d\theta }</annotation>

</semantics>

</math><img src="./2f56c0e260450f7aa75efc77424cd675b5a6f64b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.105ex; height:3.176ex;" alt="{\displaystyle 2\pi R\sin(\theta )R\,d\theta =2\pi R^{2}\sin(\theta )\,d\theta }" loading="lazy"></dd></dl>

If the mass of the shell is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, one therefore has that

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dM={\frac {2\pi R^{2}\sin(\theta )}{4\pi R^{2}}}M\,d\theta ={\frac {1}{2}}M\sin(\theta )\,d\theta }</annotation>

</semantics>

</math><img src="./acdc350ea5120fef546ebe5390cbe067c5d6b3ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:40.559ex; height:6.176ex;" alt="{\displaystyle dM={\frac {2\pi R^{2}\sin(\theta )}{4\pi R^{2}}}M\,d\theta ={\frac {1}{2}}M\sin(\theta )\,d\theta }" loading="lazy"></dd></dl>

and

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{r}={\frac {GMm}{2}}\int {\frac {\sin(\theta )\cos(\varphi )}{s^{2}}}\,d\theta }</annotation>

</semantics>

</math><img src="./f4c5d698e18494f78ce2054b163147397c52e207.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.793ex; height:6.176ex;" alt="{\displaystyle F_{r}={\frac {GMm}{2}}\int {\frac {\sin(\theta )\cos(\varphi )}{s^{2}}}\,d\theta }" loading="lazy"></dd></dl>

By the <a href="Law_of_cosines" title="Law of cosines">law of cosines</a>,

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(\varphi )={\frac {r^{2}+s^{2}-R^{2}}{2rs}}}</annotation>

</semantics>

</math><img src="./a46f5fdbc2669600a3cef7595bda702846ed95fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.122ex; height:5.676ex;" alt="{\displaystyle \cos(\varphi )={\frac {r^{2}+s^{2}-R^{2}}{2rs}}}" loading="lazy"></dd></dl>

and

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(\theta )={\frac {r^{2}+R^{2}-s^{2}}{2rR}}.}</annotation>

</semantics>

</math><img src="./51ae9b5722af8023d5329e07e5b09c57f2bf04f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.339ex; height:5.843ex;" alt="{\displaystyle \cos(\theta )={\frac {r^{2}+R^{2}-s^{2}}{2rR}}.}" loading="lazy"></dd></dl>

These two relations link the three parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> that appear in the integral together. As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> increases from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

</semantics>

</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> radians, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> varies from the initial value 0 to a maximal value before finally returning to zero at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta =\pi }</annotation>

</semantics>

</math><img src="./ab4db588619489e27efb50a1d0d5aa016c49ce15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.521ex; height:2.176ex;" alt="{\displaystyle \theta =\pi }" loading="lazy">. At the same time, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> increases from the initial value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r-R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r-R}</annotation>

</semantics>

</math><img src="./38ecf5b8f06f327c77acb677282c3d4839afc6dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.653ex; height:2.343ex;" alt="{\displaystyle r-R}" loading="lazy"> to the final value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r+R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r+R}</annotation>

</semantics>

</math><img src="./3f189b74de1af8d63589582e17d39ffbe8ef2649.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.653ex; height:2.343ex;" alt="{\displaystyle r+R}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> increases from 0 to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

</semantics>

</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> radians. This is illustrated in the following animation:

(Note: As viewed from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">, the shaded blue band appears as a thin <a href="Annulus_(mathematics)" title="Annulus (mathematics)">annulus</a> whose inner and outer radii converge to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\sin(\theta )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\sin(\theta )}</annotation>

</semantics>

</math><img src="./1df4f0ce7510fb17bdb12139b910a148db018224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.906ex; height:2.843ex;" alt="{\displaystyle R\sin(\theta )}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\theta }</annotation>

</semantics>

</math><img src="./75ae6ca1248d081ef3fcfdd3e17ba0e3f6c02ee9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.306ex; height:2.176ex;" alt="{\displaystyle d\theta }" loading="lazy"> vanishes.)

To find a <a href="Primitive_function" class="mw-redirect" title="Primitive function">primitive function</a> to the integrand, one has to make <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> the independent integration variable instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy">.

Performing an <a href="Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">implicit differentiation</a> of the second of the "cosine law" expressions above yields

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\sin(\theta )\,d\theta ={\frac {-2s}{2rR}}\,ds}</annotation>

</semantics>

</math><img src="./01cd0f32ed428515f0c163e1617e006f735e7339.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.333ex; height:5.343ex;" alt="{\displaystyle -\sin(\theta )\,d\theta ={\frac {-2s}{2rR}}\,ds}" loading="lazy"></dd></dl>

and thus

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(\theta )\,d\theta ={\frac {s}{rR}}\,ds.}</annotation>

</semantics>

</math><img src="./cd23383c1da2e1e183320f3e05e4547477383a8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.536ex; height:4.843ex;" alt="{\displaystyle \sin(\theta )\,d\theta ={\frac {s}{rR}}\,ds.}" loading="lazy"></dd></dl>

It follows that

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{r}={\frac {GMm}{2}}{\frac {1}{rR}}\int {\frac {s\cos(\varphi )}{s^{2}}}\,ds={\frac {GMm}{2rR}}\int {\frac {\cos(\varphi )}{s}}\,ds}</annotation>

</semantics>

</math><img src="./e31d7725ede7898ed87bd883ffaf8b609ff01856.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:53.958ex; height:6.176ex;" alt="{\displaystyle F_{r}={\frac {GMm}{2}}{\frac {1}{rR}}\int {\frac {s\cos(\varphi )}{s^{2}}}\,ds={\frac {GMm}{2rR}}\int {\frac {\cos(\varphi )}{s}}\,ds}" loading="lazy"></dd></dl>

where the new integration variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> increases from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r-R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r-R}</annotation>

</semantics>

</math><img src="./38ecf5b8f06f327c77acb677282c3d4839afc6dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.653ex; height:2.343ex;" alt="{\displaystyle r-R}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r+R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r+R}</annotation>

</semantics>

</math><img src="./3f189b74de1af8d63589582e17d39ffbe8ef2649.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.653ex; height:2.343ex;" alt="{\displaystyle r+R}" loading="lazy">.

Inserting the expression for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\varphi )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(\varphi )}</annotation>

</semantics>

</math><img src="./72e404f8ab6ec664c4b8ba46f414035c54c2d2ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.441ex; height:2.843ex;" alt="{\displaystyle \cos(\varphi )}" loading="lazy"> using the first of the "cosine law" expressions above, one finally gets that

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{r}={\frac {GMm}{4r^{2}R}}\int \left(1+{\frac {r^{2}-R^{2}}{s^{2}}}\right)\ ds\ .}</annotation>

</semantics>

</math><img src="./0e30d75cb089dcec301d2eaf2affad772445c2a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.203ex; height:6.343ex;" alt="{\displaystyle F_{r}={\frac {GMm}{4r^{2}R}}\int \left(1+{\frac {r^{2}-R^{2}}{s^{2}}}\right)\ ds\ .}" loading="lazy"></dd></dl>

A primitive function to the integrand is

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s-{\frac {r^{2}-R^{2}}{s}}\ ,}</annotation>

</semantics>

</math><img src="./6a7c5267dbbca60116954381fde1880c74892ad6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.756ex; height:5.676ex;" alt="{\displaystyle s-{\frac {r^{2}-R^{2}}{s}}\ ,}" loading="lazy"></dd></dl>

and inserting the bounds <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r-R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r-R}</annotation>

</semantics>

</math><img src="./38ecf5b8f06f327c77acb677282c3d4839afc6dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.653ex; height:2.343ex;" alt="{\displaystyle r-R}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r+R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r+R}</annotation>

</semantics>

</math><img src="./3f189b74de1af8d63589582e17d39ffbe8ef2649.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.653ex; height:2.343ex;" alt="{\displaystyle r+R}" loading="lazy"> for the integration variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> in this primitive function, one gets that

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{r}={\frac {GMm}{r^{2}}},}</annotation>

</semantics>

</math><img src="./11d0b3a40f25c2ab9e47e76894f862440c4211dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.359ex; height:5.676ex;" alt="{\displaystyle F_{r}={\frac {GMm}{r^{2}}},}" loading="lazy"></dd></dl>

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

<div class="mw-heading mw-heading3"><h3 id="Spherical_shell_to_solid_sphere">Spherical shell to solid sphere</h3></div>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dM}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dM}</annotation>

</semantics>

</math><img src="./7b8515decc1231e7e511ab7178f8959e2f7fe94c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.658ex; height:2.176ex;" alt="{\displaystyle dM}" loading="lazy">, and we can obtain the total gravity contribution of a solid ball to the object outside the ball

<msub>

<mtext>total</mtext>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{\text{total}}=\int dF_{r}={\frac {Gm}{r^{2}}}\int dM.}</annotation>

</semantics>

</math><img src="./fd7c1a8764a5bb522ee3ba0bc80bfa9c2e3105ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.546ex; height:5.843ex;" alt="{\displaystyle F_{\text{total}}=\int dF_{r}={\frac {Gm}{r^{2}}}\int dM.}" loading="lazy"></dd></dl>

Uniform density means between the radius of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+dx}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x+dx}</annotation>

</semantics>

</math><img src="./5ace2cb84eca9933bdaa742302f2167f5dd6c705.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.716ex; height:2.343ex;" alt="{\displaystyle x+dx}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dM}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dM}</annotation>

</semantics>

</math><img src="./7b8515decc1231e7e511ab7178f8959e2f7fe94c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.658ex; height:2.176ex;" alt="{\displaystyle dM}" loading="lazy"> can be expressed as a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">, i.e.,

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dM={\frac {4\pi x^{2}dx}{{\frac {4}{3}}\pi R^{3}}}M={\frac {3Mx^{2}dx}{R^{3}}}}</annotation>

</semantics>

</math><img src="./2f56fb3de30109a266b7afd0615841a3e53129b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:29.928ex; height:7.343ex;" alt="{\displaystyle dM={\frac {4\pi x^{2}dx}{{\frac {4}{3}}\pi R^{3}}}M={\frac {3Mx^{2}dx}{R^{3}}}}" loading="lazy"></dd></dl>

Therefore, the total gravity is

<msub>

<mtext>total</mtext>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{\text{total}}={\frac {3GMm}{r^{2}R^{3}}}\int _{0}^{R}x^{2}\,dx={\frac {GMm}{r^{2}}}}</annotation>

</semantics>

</math><img src="./aa7640c7afa24f923c3d157ee929d5e596f0aff0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.792ex; height:6.176ex;" alt="{\displaystyle F_{\text{total}}={\frac {3GMm}{r^{2}R^{3}}}\int _{0}^{R}x^{2}\,dx={\frac {GMm}{r^{2}}}}" loading="lazy"></dd></dl>

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.

<div class="mw-heading mw-heading2"><h2 id="Inside_a_shell">Inside a shell</h2></div>

For a point inside the shell, the difference is that when θ is equal to zero, ϕ takes the value π radians and s the value R − r. When θ increases from 0 to π radians, ϕ decreases from the initial value π radians to zero and s increases from the initial value R − r to the value R + r.

This can all be seen in the following figure

Inserting these bounds into the <a href="Primitive_function" class="mw-redirect" title="Primitive function">primitive function</a>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s-{\frac {r^{2}-R^{2}}{s}}}</annotation>

</semantics>

</math><img src="./1036e5fc1221ce0a7b94173fe8ceea79952c5d5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.529ex; height:5.676ex;" alt="{\displaystyle s-{\frac {r^{2}-R^{2}}{s}}}" loading="lazy"></dd></dl>

one gets that, in this case

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{r}=0,}</annotation>

</semantics>

</math><img src="./a32bf3f996c720c75432a21a14645302a20a4a3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.376ex; height:2.509ex;" alt="{\displaystyle F_{r}=0,}" loading="lazy"></dd></dl>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\frac {k}{r^{p}}}}">

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f={\frac {k}{r^{p}}}}</annotation>

</semantics>

</math><img src="./ffcec102e34040bc358d0f37faafa9f6fc8809db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.321ex; height:5.343ex;" alt="{\displaystyle f={\frac {k}{r^{p}}}}" loading="lazy">, the resultant force inside the shell is:

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(r)={\frac {GMm}{4r^{2}R}}\int _{R-r}^{R+r}\left({\frac {1}{s^{p-2}}}+{\frac {r^{2}-R^{2}}{s^{p}}}\right)\,ds}</annotation>

</semantics>

</math><img src="./eaf8211c8a19e6d7a9691c68fbd61b12b6b7e4e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.607ex; height:6.343ex;" alt="{\displaystyle F(r)={\frac {GMm}{4r^{2}R}}\int _{R-r}^{R+r}\left({\frac {1}{s^{p-2}}}+{\frac {r^{2}-R^{2}}{s^{p}}}\right)\,ds}" loading="lazy"></dd></dl>

The above results into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(r)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(r)}</annotation>

</semantics>

</math><img src="./00c291246c498df91cb717d1c4b7211b2016f7db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.599ex; height:2.843ex;" alt="{\displaystyle F(r)}" loading="lazy"> being identically zero if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p=2}</annotation>

</semantics>

</math><img src="./d62e4100b94c1939c67f2d4b8580d26c78106c44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=2}" loading="lazy">

Outside the shell (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r>R}</annotation>

</semantics>

</math><img src="./8971c9610113faec012a76ec2d47fa6235e16d2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle r>R}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r<-R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r<-R}</annotation>

</semantics>

</math><img src="./eea9050e3789a90ca1b5ff5a477a21909a61546c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.719ex; height:2.343ex;" alt="{\displaystyle r<-R}" loading="lazy">):

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(r)={\frac {GMm}{4r^{2}R}}\int _{r-R}^{r+R}\left({\frac {1}{s^{p-2}}}+{\frac {r^{2}-R^{2}}{s^{p}}}\right)\,ds}</annotation>

</semantics>

</math><img src="./019aca6bafc5ffea3c1436df17d18cc020cdc0e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.607ex; height:6.343ex;" alt="{\displaystyle F(r)={\frac {GMm}{4r^{2}R}}\int _{r-R}^{r+R}\left({\frac {1}{s^{p-2}}}+{\frac {r^{2}-R^{2}}{s^{p}}}\right)\,ds}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Derivation_using_Gauss's_law">Derivation using Gauss's law</h2></div>

The shell theorem is an immediate consequence of <a href="Gauss's_law_for_gravity" title="Gauss's law for gravity">Gauss's law for gravity</a> saying that

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{S}{\mathbf {g} }\cdot \,d{\mathbf {S} }=-4\pi GM}</annotation>

</semantics>

</math><img src="./1c8fee61070a2da7d00b74c449b39fbab5ec393e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.746ex; height:5.676ex;" alt="{\displaystyle \int _{S}{\mathbf {g} }\cdot \,d{\mathbf {S} }=-4\pi GM}" loading="lazy"></dd></dl>

where M is the mass of the part of the spherically symmetric mass distribution that is inside the sphere with radius r and

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{S}{\mathbf {g} }\cdot \,d{\mathbf {S} }=\int _{S}{\mathbf {g} }\cdot {\hat {\mathbf {n} }}\,dS}</annotation>

</semantics>

</math><img src="./7671e4475f4724263856d76047f8530a5841c9be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.75ex; height:5.676ex;" alt="{\displaystyle \int _{S}{\mathbf {g} }\cdot \,d{\mathbf {S} }=\int _{S}{\mathbf {g} }\cdot {\hat {\mathbf {n} }}\,dS}" loading="lazy"></dd></dl>

is the <a href="Surface_integral" title="Surface integral">surface integral</a> of the <a href="Gravitational_field" title="Gravitational field">gravitational field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} }</annotation>

</semantics>

</math><img src="./8cdf843789e9564a867aee3ff184453b72ecbafe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.337ex; height:2.009ex;" alt="{\displaystyle \mathbf {g} }" loading="lazy"> over any <a href="Closed_surface" class="mw-redirect" title="Closed surface">closed surface</a> inside which the total mass is M, the <a href="Unit_vector" title="Unit vector">unit vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {n} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {n} }}}</annotation>

</semantics>

</math><img src="./aae87b164ba005e99b51066c46d1eacc7f56564a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {n} }}}" loading="lazy"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {n} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {n} }}}</annotation>

</semantics>

</math><img src="./aae87b164ba005e99b51066c46d1eacc7f56564a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {n} }}}" loading="lazy"> 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

</mrow>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} =g(r){\hat {\mathbf {n} }}}</annotation>

</semantics>

</math><img src="./c35fd2e7c90ef79bac7041136c4df673962d0790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.894ex; height:2.843ex;" alt="{\displaystyle \mathbf {g} =g(r){\hat {\mathbf {n} }}}" loading="lazy"></dd></dl>

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, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {n} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {n} }}}</annotation>

</semantics>

</math><img src="./aae87b164ba005e99b51066c46d1eacc7f56564a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {n} }}}" loading="lazy">, is precisely the direction pointing away from the point of symmetry of the mass distribution.

One, therefore, has that

</mrow>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} =g(r){\hat {\mathbf {n} }}}</annotation>

</semantics>

</math><img src="./c35fd2e7c90ef79bac7041136c4df673962d0790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.894ex; height:2.843ex;" alt="{\displaystyle \mathbf {g} =g(r){\hat {\mathbf {n} }}}" loading="lazy"></dd></dl>

and

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{S}\mathbf {g} \cdot \,d{\mathbf {S} }=g(r)\int _{S}\,dS=g(r)4\pi r^{2}}</annotation>

</semantics>

</math><img src="./017eeb192215d78fa420b71062b1f39c44fc7d19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.279ex; height:5.676ex;" alt="{\displaystyle \int _{S}\mathbf {g} \cdot \,d{\mathbf {S} }=g(r)\int _{S}\,dS=g(r)4\pi r^{2}}" loading="lazy"></dd></dl>

as the area of the sphere is 4πr2.

From Gauss's law it then follows that

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(r)4\pi r^{2}=-4\pi GM,}</annotation>

</semantics>

</math><img src="./43bca57205e012fb36bd08456d17520d25479614.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.888ex; height:3.176ex;" alt="{\displaystyle g(r)4\pi r^{2}=-4\pi GM,}" loading="lazy"></dd></dl>

or,

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(r)=-{\frac {GM}{r^{2}}}.}</annotation>

</semantics>

</math><img src="./39dc6574fe454d96b3b60ae23b11e5a075e62e42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.632ex; height:5.676ex;" alt="{\displaystyle g(r)=-{\frac {GM}{r^{2}}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Converses_and_generalizations">Converses and generalizations</h2></div>

It is natural to ask whether the <a href="Theorem#Converse" title="Theorem">converse</a> 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.<a href="#cite_note-Gurzadyan-2">[2]</a><a href="#cite_note-Arens-3">[3]</a> The most general force, as given by the <a href="Gurzadyan_theorem" title="Gurzadyan theorem">Gurzadyan theorem</a>, is:<a href="#cite_note-Gurzadyan-2">[2]</a>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(r)=-{\frac {GMm}{r^{2}}}+{\frac {\Lambda mc^{2}r}{3}}}</annotation>

</semantics>

</math><img src="./22951920360b1e0eda37bf6665e7dd2975bef1d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.09ex; height:6.009ex;" alt="{\displaystyle F(r)=-{\frac {GMm}{r^{2}}}+{\frac {\Lambda mc^{2}r}{3}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation>

</semantics>

</math><img src="./0ac0a4a98a414e3480335f9ba652d12571ec6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" loading="lazy"> 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 <a href="Cosmological_constant" title="Cosmological constant">cosmological constant</a> term in <a href="General_relativity" title="General relativity">general relativity</a>. However, the inverse-square potential is the only potential such that the net force inside the shell is also zero.<a href="#cite_note-Gurzadyan-2">[2]</a>

The force described by the <a href="Yukawa_potential" title="Yukawa potential">Yukawa potential</a>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U(r)=-{\frac {GMm}{r}}e^{-\lambda r}}</annotation>

</semantics>

</math><img src="./49853819f5506ba67e3e85a4076bc7710d725c6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.987ex; height:5.343ex;" alt="{\displaystyle U(r)=-{\frac {GMm}{r}}e^{-\lambda r}}" loading="lazy"></dd></dl>

has the property that the force outside of a spherical shell is also a Yukawa potential with the same range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/\lambda }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/\lambda }</annotation>

</semantics>

</math><img src="./1add931eabb9aec89275acb2f50d50428010b0fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.68ex; height:2.843ex;" alt="{\displaystyle 1/\lambda }" loading="lazy"> and centered at the shell's center, but for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda >0}</annotation>

</semantics>

</math><img src="./eea25afc0351140f919cf791c49c1964b8b081de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda >0}" loading="lazy"> the equivalent point mass is not the same as the mass of the shell.<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a> For a shell of radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> and mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, the equivalent point mass is:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{\text{eff}}=M{\frac {\sinh \lambda r}{\lambda r}}}</annotation>

</semantics>

</math><img src="./326da2250cffa2e52b5f97d8feae62975db7fd0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.539ex; height:5.509ex;" alt="{\displaystyle M_{\text{eff}}=M{\frac {\sinh \lambda r}{\lambda r}}}" loading="lazy">.</dd></dl>

For an <a href="Ellipsoid" title="Ellipsoid">ellipsoidal</a> shell, the two halves of the shell theorem are generalized by different types of shells. The shell bound by two <a href="Concentric" class="mw-redirect" title="Concentric">concentric</a>, <a href="Similarity_(geometry)" title="Similarity (geometry)">similar</a>, and aligned ellipsoids (a <a href="Homoeoid" class="mw-redirect" title="Homoeoid">homoeoid</a>) exters no gravitational force on a point inside of it.<a href="#cite_note-7">[7]</a> Meanwhile, the shell bound by two concentric, <a href="Confocal" title="Confocal">confocal</a> ellipsoids (a <a href="Focaloid" class="mw-redirect" title="Focaloid">focaloid</a>) has the property that the gravitational force outside of two concentric, confocal focaloids is the same.<a href="#cite_note-rodrigues-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="Newton's_proofs">Newton's proofs</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Introduction">Introduction</h3></div>

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.

The proof of Proposition 71 is more historically significant. It forms the first part of his proof that the gravitational force of a solid sphere acting on a particle outside it is inversely proportional to the square of its distance from the center of the sphere, provided the density at any point inside the sphere is a function only of its distance from the center of the sphere.

Although the following are completely faithful to Newton's proofs, very minor changes have been made to attempt to make them clearer.

<div class="mw-heading mw-heading3"><h3 id="Force_on_a_point_inside_a_hollow_sphere">Force on a point inside a hollow sphere</h3></div>

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 <a href="Inscribed_angle" title="Inscribed angle">inscribed angle theorem</a>, 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle PJI=\angle PMK}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \angle PJI=\angle PMK}</annotation>

</semantics>

</math><img src="./859e8e4a7f0c58d0db0ed196efb6e45dba82c55c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.097ex; height:2.176ex;" alt="{\displaystyle \angle PJI=\angle PMK}" loading="lazy">.

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {IH}{KL}}={\frac {PJ}{PM}}}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {IH}{KL}}={\frac {PJ}{PM}}}</annotation>

</semantics>

</math><img src="./657f24e66806014f9cd70b03fb7ff186dfad9c06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.607ex; height:5.176ex;" alt="{\displaystyle {\frac {IH}{KL}}={\frac {PJ}{PM}}}" loading="lazy">.

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \propto {\frac {PJ^{2}}{PM^{2}}}}">

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \propto {\frac {PJ^{2}}{PM^{2}}}}</annotation>

</semantics>

</math><img src="./a2856179cad8b6b1275b7c91be6d56abba1ded0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.588ex; height:5.843ex;" alt="{\displaystyle \propto {\frac {PJ^{2}}{PM^{2}}}}" loading="lazy">.

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.

Note: Newton simply describes the arcs IH and KL as 'minimally small' and the areas traced out by the lines IL and HK can be any shape, not necessarily elliptic, but they will always be similar.

<div class="mw-heading mw-heading3"><h3 id="Force_on_a_point_outside_a_hollow_sphere">Force on a point outside a hollow sphere</h3></div>

Fig. 1 is a cross-section of the hollow sphere through the center, S with an arbitrary point, P, outside the sphere. PT is the tangent to the circle at T which passes through P. HI is a small arc on the surface such that PH is less than PT. Extend PI to intersect the sphere at L and draw SF to the point F that bisects IL. Extend PH to intersect the sphere at K and draw SE to the point E that bisects HK, and extend SF to intersect HK at D. Drop a perpendicular IQ on to the line PS joining P to the center S. Let the radius of the sphere be a and the distance PS be D.

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle IH\cdot \zeta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle IH\cdot \zeta }</annotation>

</semantics>

</math><img src="./d4b019a4a622ccd187f32e74a1466cf9b75a6601.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.01ex; height:2.509ex;" alt="{\displaystyle IH\cdot \zeta }" loading="lazy">, and its mass is proportional to this product.

The force due to this mass on the particle at P <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \propto {\frac {IH\cdot \zeta }{PI^{2}}}}">

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \propto {\frac {IH\cdot \zeta }{PI^{2}}}}</annotation>

</semantics>

</math><img src="./1a6fbeb13009d00447b5cdcf55ae7d5ae234d806.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.299ex; height:5.509ex;" alt="{\displaystyle \propto {\frac {IH\cdot \zeta }{PI^{2}}}}" loading="lazy"> and is along the line PI.

The component of this force towards the center <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \propto {\frac {IH\cdot PQ\cdot \zeta }{PI^{3}}}}">

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \propto {\frac {IH\cdot PQ\cdot \zeta }{PI^{3}}}}</annotation>

</semantics>

</math><img src="./c1c80a7967622bc2a5b937e3bc2df761421b543e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.562ex; height:5.509ex;" alt="{\displaystyle \propto {\frac {IH\cdot PQ\cdot \zeta }{PI^{3}}}}" loading="lazy">.

If now the arc HI is rotated completely about the line PS to form a ring of width HI and radius IQ, the length of the ring is 2π·IQ and its area is 2π·IQ·IH. The component of the force due to this ring on the particle at P in the direction PS becomes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \propto {\frac {IH\cdot IQ\cdot PQ}{PI^{3}}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \propto {\frac {IH\cdot IQ\cdot PQ}{PI^{3}}}}</annotation>

</semantics>

</math><img src="./fc7c72b867a2f7f5469061b0595b9b8023877b67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.477ex; height:5.509ex;" alt="{\displaystyle \propto {\frac {IH\cdot IQ\cdot PQ}{PI^{3}}}}" loading="lazy">.

The perpendicular components of the force directed towards PS cancel out since the mass in the ring is distributed symmetrically about PS. Therefore, the component in the direction PS is the total force on P due to the ring formed by rotating arc HI about PS.

From similar triangles: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {IQ}{PI}}={\frac {FS}{D}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {IQ}{PI}}={\frac {FS}{D}}}</annotation>

</semantics>

</math><img src="./917c599c891ae7d605db5d1be2e4f6c7fa6b52b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.021ex; height:5.343ex;" alt="{\displaystyle {\frac {IQ}{PI}}={\frac {FS}{D}}}" loading="lazy">; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {PQ}{PI}}={\frac {PF}{D}}}">

<mfrac>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {PQ}{PI}}={\frac {PF}{D}}}</annotation>

</semantics>

</math><img src="./02b90c3d79f7526658f292452194d62528ce9fc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.841ex; height:5.343ex;" alt="{\displaystyle {\frac {PQ}{PI}}={\frac {PF}{D}}}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {RI}{PI}}={\frac {DF}{PF}}}">

<mfrac>

<mrow>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {RI}{PI}}={\frac {DF}{PF}}}</annotation>

</semantics>

</math><img src="./6b7dcae14d89edf66f6b6a6745bfa25462d2e4d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.372ex; height:5.176ex;" alt="{\displaystyle {\frac {RI}{PI}}={\frac {DF}{PF}}}" loading="lazy">.

If HI is sufficiently small that it can be taken as a straight line, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle SIH}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \angle SIH}</annotation>

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</math><img src="./d882128c02998f4d5b1ae41f679b92a37422d92b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.413ex; height:2.176ex;" alt="{\displaystyle \angle SIH}" loading="lazy"> is a right angle, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle RIH=\angle FIS}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \angle RIH=\angle FIS}</annotation>

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</math><img src="./18cc5ad7e1483d07d072f4b969c22ed6c64b0933.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.866ex; height:2.176ex;" alt="{\displaystyle \angle RIH=\angle FIS}" loading="lazy">, so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {HI}{RI}}={\frac {a}{IF}}}">

<mfrac>

<mrow>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {HI}{RI}}={\frac {a}{IF}}}</annotation>

</semantics>

</math><img src="./23ab91c5c8e064917d385ae6cf11beabecca9405.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.919ex; height:5.343ex;" alt="{\displaystyle {\frac {HI}{RI}}={\frac {a}{IF}}}" loading="lazy">.

Hence the force on P due to the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \propto {\frac {IH\cdot IQ\cdot PQ}{PI^{3}}}={\frac {a\cdot DF\cdot FS\cdot PF}{IF\cdot PF\cdot D\cdot D}}={\frac {a\cdot DF\cdot FS}{IF\cdot D^{2}}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \propto {\frac {IH\cdot IQ\cdot PQ}{PI^{3}}}={\frac {a\cdot DF\cdot FS\cdot PF}{IF\cdot PF\cdot D\cdot D}}={\frac {a\cdot DF\cdot FS}{IF\cdot D^{2}}}}</annotation>

</semantics>

</math><img src="./2542e00e05e62bb83b6abe5061c0a78d59bb7aeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:52.497ex; height:5.509ex;" alt="{\displaystyle \propto {\frac {IH\cdot IQ\cdot PQ}{PI^{3}}}={\frac {a\cdot DF\cdot FS\cdot PF}{IF\cdot PF\cdot D\cdot D}}={\frac {a\cdot DF\cdot FS}{IF\cdot D^{2}}}}" loading="lazy">.

Assume now in Fig. 2 that another particle is outside the sphere at a point p, a different distance d from the center of the sphere, with corresponding points lettered in lower case. For easy comparison, the construction of P in Fig. 1 is also shown in Fig. 2. As before, ph is less than pt.

Generate a ring with width ih and radius iq by making angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fiS=FIS}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle fiS=FIS}</annotation>

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</math><img src="./2a05fcdc0e920becc38349b7764e1f831722a958.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.09ex; height:2.509ex;" alt="{\displaystyle fiS=FIS}" loading="lazy"> and the slightly larger angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dhS=DHS}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dhS=DHS}</annotation>

</semantics>

</math><img src="./212402026dbbb19232638a3302d7caa52735d9be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.64ex; height:2.176ex;" alt="{\displaystyle dhS=DHS}" loading="lazy">, 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

<mfrac>

<mrow>

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<mrow>

<msup>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \propto {\frac {ih\cdot iq\cdot pq}{pi^{3}}}={\frac {a\cdot df\cdot fS}{if\cdot d^{2}}}}</annotation>

</semantics>

</math><img src="./97405e5daaa1a970b58c20ea5da58dcc1eecc55e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.695ex; height:6.009ex;" alt="{\displaystyle \propto {\frac {ih\cdot iq\cdot pq}{pi^{3}}}={\frac {a\cdot df\cdot fS}{if\cdot d^{2}}}}" loading="lazy"></dd></dl>

Clearly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fS=FS}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle fS=FS}</annotation>

</semantics>

</math><img src="./93db82611a0bff566f2bdb5da556332bda5924ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.116ex; height:2.509ex;" alt="{\displaystyle fS=FS}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle if=IF}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle if=IF}</annotation>

</semantics>

</math><img src="./7e115cfd0f44f8998deacdcc2a6b685f05c24d93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.092ex; height:2.509ex;" alt="{\displaystyle if=IF}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle eS=ES}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle eS=ES}</annotation>

</semantics>

</math><img src="./38fdf67bd15bfaa2398f86d1f3f2a14ec4cee65c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.956ex; height:2.176ex;" alt="{\displaystyle eS=ES}" loading="lazy">.

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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {DF}{PF}}={\frac {ED}{ES}}}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {DF}{PF}}={\frac {ED}{ES}}}</annotation>

</semantics>

</math><img src="./b9c93384497c3a3b6c10642514d207e6603285db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.979ex; height:3.676ex;" alt="{\textstyle {\frac {DF}{PF}}={\frac {ED}{ES}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ED^{2}=(DF+FS)^{2}-ES^{2}}">

<msup>

</mrow>

</msup>

<msup>

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</msup>

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle ED^{2}=(DF+FS)^{2}-ES^{2}}</annotation>

</semantics>

</math><img src="./04af0620687ff11181d2af11c04f95ec58cac786.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.653ex; height:3.176ex;" alt="{\displaystyle ED^{2}=(DF+FS)^{2}-ES^{2}}" loading="lazy">, giving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\left(PF^{2}-ES^{2}\right)DF^{2}}{PF^{2}}}+2\cdot FS\cdot DF+FS^{2}-ES^{2}=0}">

<mfrac>

<mrow>

<mrow>

<msup>

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</msup>

<msup>

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</msup>

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<msup>

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</msup>

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<msup>

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<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\left(PF^{2}-ES^{2}\right)DF^{2}}{PF^{2}}}+2\cdot FS\cdot DF+FS^{2}-ES^{2}=0}</annotation>

</semantics>

</math><img src="./397b94b2385aebac498720d976e7e54ff72f1c2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:52.829ex; height:6.176ex;" alt="{\displaystyle {\frac {\left(PF^{2}-ES^{2}\right)DF^{2}}{PF^{2}}}+2\cdot FS\cdot DF+FS^{2}-ES^{2}=0}" loading="lazy">. Solving the quadratic for DF, in the limit as ES approaches FS, the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DF=ES-FS}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle DF=ES-FS}</annotation>

</semantics>

</math><img src="./42506251c7325778038790ab4b7393c33e3e0c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.119ex; height:2.343ex;" alt="{\displaystyle DF=ES-FS}" loading="lazy">. More simply, as DF approaches zero, in the limit the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DF^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle DF^{2}}</annotation>

</semantics>

</math><img src="./a8d52a6f1622ee3b0fa289045669dc1becf3c539.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.793ex; height:2.676ex;" alt="{\displaystyle DF^{2}}" loading="lazy"> term can be ignored: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cdot FS\cdot DF+FS^{2}-ES^{2}=0}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\cdot FS\cdot DF+FS^{2}-ES^{2}=0}</annotation>

</semantics>

</math><img src="./287ed84f4b5096478782a26821558ac0899a792b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:30.036ex; height:2.843ex;" alt="{\displaystyle 2\cdot FS\cdot DF+FS^{2}-ES^{2}=0}" loading="lazy"> leading to the same result. Clearly df has the same limit, justifying Newton'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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {d^{2}}{D^{2}}}}">

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {d^{2}}{D^{2}}}}</annotation>

</semantics>

</math><img src="./91c9c8c52b8c36d900b7b32d5968681e52957668.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:3.028ex; height:4.509ex;" alt="{\textstyle {\frac {d^{2}}{D^{2}}}}" loading="lazy">.

By dividing up the arcs AT and Bt into corresponding infinitesimal rings, it follows that the ratio of the force due to the arc AT rotated about PS to that of Bt rotated about pS is in the same ratio, and similarly, the ratio of the forces due to arc TB to that of tA both rotated are in the same ratio.

Therefore, the force on a particle any distance D from the center of the hollow sphere is inversely proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D^{2}}</annotation>

</semantics>

</math><img src="./7a1e7d554748b105413b259fdfb8c9587acb9f1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.979ex; height:2.676ex;" alt="{\displaystyle D^{2}}" loading="lazy">, which proves the proposition.

<div class="mw-heading mw-heading2"><h2 id="Shell_theorem_in_general_relativity">Shell theorem in general relativity</h2></div>

An analogue for shell theorem exists in <a href="General_relativity" title="General relativity">general relativity</a> (GR).

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 <a href="Birkhoff's_theorem_(relativity)" title="Birkhoff's theorem (relativity)">Birkhoff's theorem</a>). The <a href="Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">metric</a> thus has form

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

<msup>

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</msup>

<msup>

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</msup>

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle ds^{2}=-(1-2M/r)\,dt^{2}+(1-2M/r)^{-1}\,dr^{2}+r^{2}\,d\Omega ^{2}}</annotation>

</semantics>

</math><img src="./81f0256f3069c00a188effeb2f409bba9ef31c92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.177ex; height:3.176ex;" alt="{\displaystyle ds^{2}=-(1-2M/r)\,dt^{2}+(1-2M/r)^{-1}\,dr^{2}+r^{2}\,d\Omega ^{2}}" loading="lazy"></dd></dl>

(using <a href="Geometrized_unit_system" title="Geometrized unit system">geometrized units</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=c=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=c=1}</annotation>

</semantics>

</math><img src="./e39d46bcab0fb0fdacbe443e08e4ff3af308035f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.193ex; height:2.176ex;" alt="{\displaystyle G=c=1}" loading="lazy">). For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>R>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r>R>0}</annotation>

</semantics>

</math><img src="./c446e15a485f0b9c492d08087f0aa20f10e15fe7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.172ex; height:2.176ex;" alt="{\displaystyle r>R>0}" loading="lazy"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> is the radius of some mass shell), mass acts as a <a href="Delta_function" class="mw-redirect" title="Delta function">delta function</a> at the origin. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r<R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r<R}</annotation>

</semantics>

</math><img src="./bda3f5af44b388094e37b9ddeb9524a3433ef30c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle r<R}" loading="lazy">, shells of mass may exist externally, but for the metric to be <a href="Singularity_(mathematics)" title="Singularity (mathematics)">non-singular</a> at the origin, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> must be zero in the metric. This reduces the metric to flat <a href="Minkowski_space" title="Minkowski space">Minkowski space</a>; thus external shells have no gravitational effect.

This result illuminates the <a href="Gravitational_collapse" title="Gravitational collapse">gravitational collapse</a> 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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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Shell_theorem" class="extiw external" title="commons:Category:Shell theorem">Shell theorem</a>.</div></div>

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<ul><li><a href="Chasles'_theorem_(gravitation)" class="mw-redirect" title="Chasles' theorem (gravitation)">Chasles' theorem (gravitation)</a></li>

<li><a href="Scale_height" title="Scale height">Scale height</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFNewton1687" class="citation book cs1">Newton, Isaac (1687). <a rel="nofollow" class="external text" href="https://archive.org/details/philosophinatur03newtgoog">Philosophiae Naturalis Principia Mathematica</a>. London. pp. 193, Theorem XXXI.</cite>

</li>

<li id="cite_note-Gurzadyan-2">^ <a href="#cite_ref-Gurzadyan_2-0">a</a> <a href="#cite_ref-Gurzadyan_2-1">b</a> <a href="#cite_ref-Gurzadyan_2-2">c</a> <cite id="CITEREFGurzadyan1985" class="citation journal cs1"><a href="Vahe_Gurzadyan" title="Vahe Gurzadyan">Gurzadyan, Vahe</a> (1985). "The cosmological constant in McCrea-Milne cosmological scheme". The Observatory. 105: 42–43. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1985Obs...105...42G">1985Obs...105...42G</a>.</cite> <a rel="nofollow" class="external free" href="https://adsabs.harvard.edu/full/1985Obs...105...42G&lang=en">https://adsabs.harvard.edu/full/1985Obs...105...42G&lang=en</a>

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<li id="cite_note-Arens-3"><a href="#cite_ref-Arens_3-0">^</a> <cite id="CITEREFArens1990" class="citation journal cs1"><a href="Richard_Friederich_Arens" title="Richard Friederich Arens">Arens, Richard</a> (January 1, 1990). "Newton's observations about the field of a uniform thin spherical shell". Note di Matematica. X (Suppl. n. 1): 39–45.</cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFKuhn" class="citation web cs1">Kuhn, Paulo. <a rel="nofollow" class="external text" href="https://wp.ufpel.edu.br/pskuhn/files/2024/09/debye-huckel-interaction-or-yukawa-in-different-geometries.pdf">"Debye-Hückel interaction, or Yukawa potential, in different geometries"</a> (PDF). Retrieved 14 February 2025.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFMcDonald2021" class="citation web cs1">McDonald, Kirk (December 20, 2021) [April 17, 1984]. <a rel="nofollow" class="external text" href="http://kirkmcd.princeton.edu/examples/yukawa.pdf">"A Naïve Estimate of the Coupling Constant in Yukawa Theory"</a> (PDF). Retrieved 14 February 2025.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/questions/296180/shell-theorem-for-a-general-potential">"Shell theorem for a general potential"</a>. Mathematics Stack Exchange. Retrieved 14 February 2025.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="Michel_Chasles" title="Michel Chasles">Michel Chasles</a>, <a rel="nofollow" class="external text" href="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</a>, Jour. Liouville 5, 465–488 (1840)

</li>

<li id="cite_note-rodrigues-8"><a href="#cite_ref-rodrigues_8-0">^</a> <cite id="CITEREFRodrigues2014" class="citation journal cs1">Rodrigues, Hilário (11 May 2014). <a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fmnras%2Fstu353">"On determining the kinetic content of ellipsoidal configurations"</a>. Monthly Notices of the Royal Astronomical Society. 440 (2): 1519–1526. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1402.6541">1402.6541</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fmnras%2Fstu353">10.1093/mnras/stu353</a>.</cite>

</li>

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