# Orbit of detached mass

I was wondering what precisely would happen to parts detached by objects in orbit. To specify:

Let a large object of mass $m$ is in a stable orbit around a planet with mass $M$ at a distance $r$ from the planet's center. A part of mass $m$ is detached from the mass. splitting the object into two parts $m_1$ and $m_2$, where $m_2<<m_1$. While detaching, mass $m_1$ applies a force $\vec F$ to $m_2$ within an infinitesimally small time. What can be said about the distance to the planet's center $r_2$ of mass $m_2$? Will mass $m_2$ reach a new stable orbit?

I would also like to know how the change in $r_2$ relate to the magnitude and direction of $\vec F$

• Not clear. What is the significance of detached with a force $F$ within an infinitesimally small time? Are you saying that the two masses fly apart with high relative velocity? Or that the force is just enough to break the bonds holding them together? ... I presume that the initial orbit is circular, defined by a radius $r$. In specifying that the new orbits are defined by radii $r_1, r_2$ are you assuming that they are also circular? Jul 14, 2018 at 12:12
• Possible duplicate of What happen to a spoon which is detached from the satellite? Jul 14, 2018 at 12:20
• @sammygerbil The force described is a vector representing a force on mass $m_2$ by $m_1$ while seperating. By Newton's 3rd law this force is applied to both objects but because $m_1$ >> $m_2$ the effect on $m_1$ will be much smaller. Then, because of the infinitesimally small time it is like an impulse upon $m_2$. The radii $r_1$ and $r_2$ Are the distances for the masses $m_1$ and $m_2$ from the planet's center. They may be time dependent, but may also move to a new equilibrium. That is part of my question. The question you linked is indeed similar, but does not involve the force $F$ . Jul 14, 2018 at 14:03
• @sammygerbil Feel free to correct me if my reasoning is flawed anywhere ) Jul 14, 2018 at 14:03
• @sammygerbil I have changed my question accordingly Jul 14, 2018 at 14:09

Gravitational bound orbits are generally elliptical. The two fragments will follow separate elliptical orbits with the planet at one focus. The velocities of the fragments (magnitude $v$ and direction angle $\alpha$) can be determined from conservation of momentum and the impulse between them. This is the easy part of the calculation, using familiar collision dynamics.
The new orbits of the fragments can be calculated from their velocities ($v, \alpha$) and radius $r$ at the point of separation. From ($r, v, \alpha)$ you can calculate the orbital parameters eccentricity $e$, semi-major axis $a$ and its inclination angle $\theta$ with a reference axis. However, the calculation of ($e, a, \theta$) from ($r, v, \alpha$) is not an easy task. For details see answers to Projectile/orbital motion over very long distance which is the same problem except that the launch point is at the surface of the planet.