Page History

{composition-setup}{composition-setup}
{table:border=1|cellpadding=8|cellspacing=0|frame=void|rules=cols}
{tr:valign=top}
{td:width=250|bgcolor=#F2F2F2}
{live-template:Left Column}
{td}
{td}

h1. Gravitation

{excerpt}An interaction between two massive particles resulting in an attractive force exerted on each by the other.  The force is proportional to the gravitational constant G={{{*}6.674 28(67) x 10{*}{*}^\-11{^}* *m{*}{*}{^}3{^}{*}* kg{*}{*}^\-1{^}{*}* s{*}{*}^\-2{^}{*}}}, and the masses of the bodies, and inversely proportional to the square of the distance between them.{excerpt}

h3. Motivation for Concept

Newton's Law of Universal Gravitation provides an effective description of the movement of objects from submillimeter distances to galactic sizes, and is the dominant force on most (macroscopic) astronomical bodies.

h3. Gravitational Force (Newton's Law of Universal Gravitation)


h4. Statement of the Law for Point Masses

Between any two point masses (masses _m{_}{~}1~ and _m{_}{~}2~, respectively) there will exist an attractive force along the line joining the masses.  The force on body 1 due to body 2 will have the form:
{latex}\begin{large}\[ \vec{F}_{12} = - G \frac{m_{1}m_{2}}{r_{12}^{2}} \hat{r}_{12} \]\end{large}{latex}

where, _G_ is the gravitational constant equal to:
{latex}\begin{large}\[ G = \mbox{6.67}\times\mbox{10}^{-11}\mbox{ N}\frac{\mbox{m}^{2}}{\mbox{kg}^{2}} \]\end{large}{latex}

_r{_}{~}12~ is the distance between the two objects and is obtained as the magnitude of the vector difference between the position vector of object 1 and the position vector of object 2. The vector difference is expressed as:
{latex}\begin{large}\[\vec{r}_{12} = \vec{r}_{1}-\vec{r}_{2}\]\end{large}{latex}


h4. Compatibility with Newton's Laws of Motion

Note that the Universal Law of Gravition is consistent with Newton's [Third Law of Motion|Newton's Third Law]:
{latex}\begin{large}\[ \vec{F}_{21} = -G\frac{m_{2}m_{1}}{r_{21}^{2}} \hat{r}_{21}\]\end{large}{latex}
Noting that the differences of the position vectors _r{_}{~}12~ and _r{_}{~}21~ will certainly satisfy:
{latex}\begin{large}\[ \vec{r}_{12} = - \vec{r}_{21}\]\end{large}{latex}
which implies:
{latex}\begin{large}\[ \vec{F}_{12} = - \vec{F}_{21}\]\end{large}{latex}

h4. The Case of Spherical Symmetry

Although the form of the Law of Universal Gravitation is strictly valid only for [point particles|point particle], it is possible to show that for extended objects with a spherically symmetric mass distribution, the Law will hold in the form stated above *provided that the positions of the spherical objects are specified by their centers*.

h3. Gravitational Potential Energy

h4. Form of the Potential Energy

For two spherically symmetric objects (objects 1 and 2), it is customary to analyze the energy of the gravitational interaction by constructing spherical coordinates with one of the objects at the origin (if one of the objects dominates the mass of the system, its position is typically used as the origin).  Newton's Law of Universal Gravitation then takes the form:

{latex}\begin{large}\[ \vec{F} = - G\frac{m_{1}m_{2}}{r^{2}} \hat{r}\]\end{large}{latex}

where _r_ is the position of the object that is not placed at the origin.

It is also customary to make the assignment that the potential energy of the system goes to zero as the separation goes to infinty:

{latex}\begin{large}\[ \lim_{r \rightarrow \infty} U(r) = 0 \]\end{large}{latex}

Thus, we can define the potential for any separation _r_ as:

{latex}\begin{large}\[ U(r) = U(\infty) - \lim_{r_{0}\rightarrow \infty}\int_{r_{0}}^{r} \left(-G\frac{m_{1}m_{2}}{r^{2}}\right) \;dr 
= - Gm_{1}m_{2} \left(\frac{1}{r}-\lim_{r_{0}\rightarrow \infty}\frac{1}{r_{0}}\right)\]
\[U(r) = -G\frac{m_{1}m_{2}}{r} \]\end{large}{latex}

h4. Potential Energy Curve

If the two objects are isolated from other influences, their [potential energy] curve] is then:

!uvsr.gif!

This potential energy curve is somewhat misleading, since the potential is spherically symmetric.  Thus, although in spherical coordinates, _r_ cannot go negative, if we define a one-dimensional coordinate system by following a radial line through the origin (suppose, for instance, we chose to follow the _z_ axis where _z_ = _r{_}cosθ) we would generate a curve:

!uvsz.gif!

which indicates the possibility of stable equilibrium when the objects' separation goes to zero.  Of course, this is technically impossible for objects of finite size.

h4. Gravitational Potential Energy of a System

In a system composed of many spherically symmetric objects, the total gravitational potential energy can be found by adding up the contribution from each distinct _interaction_.  

It is very important to note that any pair of the bodies experiences only one interaction between them.  Take, for example, a system composed of four objects labeled 1, 2, 3 and 4.  There are six distinct interactions among these bodies, each of which has an associated potential energy:

{latex}\begin{large}\[ 1 \leftrightarrow 2 \mbox{ gives rise to }U_{12}\]
\[ 1 \leftrightarrow 3 \mbox{ gives rise to }U_{13}\]
\[ 1 \leftrightarrow 4 \mbox{ gives rise to }U_{14}\]
\[ 2 \leftrightarrow 3 \mbox{ gives rise to }U_{23}\]
\[ 2 \leftrightarrow 4 \mbox{ gives rise to }U_{24}\]
\[ 3 \leftrightarrow 4 \mbox{ gives rise to }U_{34}\]\end{large}{latex}

The total potential energy would then be given by:

{latex}\begin{large}\[ U_{\rm sys} = U_{12}+U_{13}+U_{14}+U_{23}+U_{24}+U_{34}\]\end{large}{latex}

{warning}It is important to beware of the temptation to double-count.  The potential energy _U_~12~ is associated with the interaction _between_ objects 1 and 2, it is not associated separately with object 1 and with object 2.{warning}

{td}
{tr}
{table}
{live-template:RELATE license}