2.1 Equation of motion
One center problem
The Restricted Two-Body Problem is when one of the masses is much smaller than the other \(m\ll M\).Then according to Neton's law of gravitation: \[ \begin{align*} \vec{F} = -G\frac{Mm}{r^3}\vec{r} \end{align*} \] According to Newton's 2nd law, \[ \begin{align*} \vec{F} = m\ddot{\vec{r}} \end{align*} \] Thus, \[ \begin{align*} \ddot{\vec{r}}+\frac{Mm}{r^3}\vec{r} = 0 \end{align*} \]
Two-Body problem: relative motion
\[ \begin{align*} M_1 \ddot{\vec{\rho_1}} &= G\frac{M_1 M_2}{r^3}\vec{r} \\ M_2 \ddot{\vec{\rho_2}} &= -G\frac{M_1 M_2}{r^3}\vec{r} \end{align*} \] Adding these,\(M_1 \ddot{\vec{\rho_1}}+M_2 \ddot{\vec{\rho_2}}\) = 0
integrate,
\(M_1 \dot{\vec{\rho_1}}+M_2 \dot{\vec{\rho_2}} = \vec{A}\)
\(M_1 \vec{\rho_1}+M_2 \vec{\rho_2} = \vec{A}t+\vec{B}\)
Consider the center of mass a.k.a. barycenter of the system: \[ \begin{align*} \vec{R} &= \frac{M_1\vec{\rho_1}+M_2\vec{\rho_2}}{M_1+M_2} \\ \vec{R} &= \frac{\vec{A}}{M_1+M_2}t+\frac{\vec{B}}{M_1+M_2} \\ \dot{\vec{R}} &=\frac{\vec{A}}{M_1+M_2} \end{align*} \] which means that the center of mass moves along a straight line \(\vec{R}\) at a constant speed.
Application: Sirius, the brightest star on the sky. In 1844, Friedrich Wilhelm Bessel analyzed the astrometric data and found the trajectory on Sirius on the sky to be “sine”-like. He interpreted this by a presence of an invisible companion of comparable mass. If such a companion (“Sirius B”) were present, the center of mass of Sirius A + B could be moving along a straight line. Later on, this companion was indeed discovered (by Clark, 1862). It turned out to be the first white dwarf ever discovered. White dwarfs have normal stellar masses, but much lower luminosities, which makes them faint. Sirius B has mag = 8m. This explains why the companion was not observed earlier.
We can now write the equation of motion: \[ \begin{align*} \ddot{\vec{r}} + G\frac{M_1+M_2}{r^3}\vec{r} = 0 \end{align*} \]Barycentric equation of motion
fillt his \[ \begin{align*} \ddot{\vec{r_2}} + G\frac{M_1^3(M_1+M_2)^2}{r_2^3}\vec{r_2} = 0 \end{align*} \]Comparision
Comparing these equaitons, we see that they all take the same form: \[ \begin{align*} \ddot{\vec{r}} + \kappa^2\frac{\vec{r}}{r^3} = 0 \end{align*} \] where \(\kappa\) is a constant. \(\vec{r}\) and \(\kappa\) vary from problem to problem, but mathematically all these versions of the two body problem are the same.These equations of motions are too complicated to solve directly. Instead, we can use 'integrals of motion'. An integral of motion is of the form $$f(t,\vec{r},\dot{\vec{r}}) = \text{const.}$$