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First of all, a brief description of the two-body problem:

The general solution of the two-body problem is:

Consider a system of two bodies of mass m and M. There are no internal forces other than that of gravity, and there are no external forces acting on the system.

The position vectors of each body are, respectively, [tex]\vec{r}_m[/tex] and [tex]\vec{r}_M[/tex], measured from an arbitrary inertial frame of reference.

The vector [tex]\vec{r}=\vec{r}_m-\vec{r}_M[/tex] is the position of m with respect to M (a vector from M to m, pointing to m).

The gravitational accelerations of m and M are, respectively:

[tex]\ddot{\vec{r}}_m=-G\frac{M}{r^2}\hat{r}[/tex], where [tex]r=|\vec{r}|[/tex] and [tex]\hat{r}=\frac{\vec{r}}{r}[/tex] is a unit vector with the direction of [tex]\vec{r}[/tex], and the two dots above represent the second derivative with respect to time.

[tex]\ddot{\vec{r}}_M=G\frac{m}{r^2}\hat{r}[/tex]

Subtracting the latter from the former:

[tex]\ddot{\vec{r}}_m-\ddot{\vec{r}}_M=-G\frac{M}{r^2}\hat{r}-G\frac{m}{r^2}\hat{r}[/tex]

Therefore:

[tex]\ddot{\vec{r}}=-G\frac{(M+m)}{r^2}\hat{r}[/tex],

which is the equation of relative motion of m with respect to M.

This leads to:

[tex]r=\frac{p}{1+e\cos{\nu}}[/tex], which describes a conic section (for example, an ellipse), where r is the distance from body m to body M - which is at one focus -, p is the semi-latus rectum, e is the eccentricity and [tex]\nu[/tex] is the angle measured from the periapsis to the radius vector of body m.

Now, my question:

The equations above describe the trajectory of body m, but relative to body M (that is, a focus of the conic section). Here, M is the origin of this coordinate system.

If I were to describe the trajectory of body m with respect to thecenter of massof the system, it would make no difference if I used the equations above, since, generally, the central body has a mass M much greater than m, therefore the motion of M with respect to the center of mass is negligible.

But what if m is comparable to M (and the center of mass is located outside of M)? In this case, the motion of M with respect to the center of mass wouldn't be negligible. How could I describe, then, the motion of m and M with respect to the center of mass? Can I still use the equation above, somehow?

Thank you in advance.

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# Question about two-body problem

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