Shell theorem

In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy. E p o i n t = G M p 2 {displaystyle E_{point}={frac {GM}{p^{2}}}} E e l e v a t e d p o i n t = G M ( p 2 + R 2 ) {displaystyle E_{elevatedpoint}={frac {GM}{(p^{2}+R^{2})}}} E x = G M c o s ( θ ) s q r t ( p 2 + R 2 ) 2 {displaystyle E_{x}={frac {GMcos( heta )}{sqrt(p^{2}+R^{2})^{2}}}} E x = G M p ( p 2 + R 2 ) 3 / 2 {displaystyle E_{x}={frac {GMp}{(p^{2}+R^{2})^{3/2}}}} E r i n g = G M p ( p 2 + R 2 ) 3 / 2 {displaystyle E_{ring}={frac {GMp}{(p^{2}+R^{2})^{3/2}}}} d E = G p ∗ d M ( p 2 + y 2 ) 3 / 2 {displaystyle dE={frac {Gp*dM}{(p^{2}+y^{2})^{3/2}}}} ∫ d E = ∫ 2 G M R 2 ∗ p ∗ y ∗ d y ( p 2 + y 2 ) ( 3 / 2 ) {displaystyle int dE=int {frac {{frac {2GM}{R^{2}}}*p*y*dy}{(p^{2}+y^{2})^{(}3/2)}}} E d i s c = 2 G M R 2 ∗ ( 1 − p s q r t ( p 2 + R 2 ) ) {displaystyle E_{disc}={frac {2GM}{R^{2}}}*(1-{frac {p}{sqrt(p^{2}+R^{2})}})} d E = 2 G ∗ ( 3 M ∗ ( a 2 − x 2 ) ) 4 a 3 s q r t ( a 2 − x 2 ) 2 ∗ ( 1 − p + x s q r t ( ( p + x ) 2 + s q r t ( a 2 − x 2 ) 2 ) ) d x {displaystyle dE={frac {frac {2G*(3M*(a^{2}-x^{2}))}{4a^{3}}}{sqrt(a^{2}-x^{2})^{2}}}*(1-{frac {p+x}{sqrt((p+x)^{2}+sqrt(a^{2}-x^{2})^{2}))}}dx} ∫ d E = ∫ − a a 3 G M 2 a 3 ∗ ( 1 − p + x s q r t ( p 2 + a 2 + 2 p x ) ) d x {displaystyle int dE=int _{-a}^{a}{frac {3GM}{2a^{3}}}*(1-{frac {p+x}{sqrt(p^{2}+a^{2}+2px)}})dx} In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy. Isaac Newton proved the shell theorem and stated that: A corollary is that inside a solid sphere of constant density, the gravitational force within the object varies linearly with distance from the centre, becoming zero by symmetry at the centre of mass. This can be seen as follows: take a point within such a sphere, at a distance r {displaystyle r} from the centre of the sphere. Then you can ignore all the shells of greater radius, according to the shell theorem. So, the remaining mass m {displaystyle m} is proportional to r 3 {displaystyle r^{3}} (because it is based on volume), and the gravitational force exerted on it is proportional to m / r 2 {displaystyle m/r^{2}} (the inverse square law), so the overall gravitational effect is proportional to r 3 / r 2 = r {displaystyle r^{3}/r^{2}=r} , so is linear in r {displaystyle r} . These results were important to Newton's analysis of planetary motion; they are not immediately obvious, but they can be proven with calculus. (Alternatively, Gauss's law for gravity offers a much simpler way to prove the same results.) In addition to gravity, the shell theorem can also be used to describe the electric field generated by a static spherically symmetric charge density, or similarly for any other phenomenon that follows an inverse square law. The derivations below focus on gravity, but the results can easily be generalized to the electrostatic force. Moreover, the results can be generalized to the case of general ellipsoidal bodies.