For a binary system with stars of masses \(m_{1}\) and \(m_{2},\) in circular orbits, with a total separation \(r,\) find an expression for the ratios of the kinetic energies of the two stars.

In astronomy, many celestial bodies follow circular or nearly circular paths around a common center. A circular orbit means that the distance between the two objects in question remains constant. For binary star systems, each star orbits their common center of mass while staying at a certain fixed distance from each other.
These stars continuously experience a gravitational pull that keeps them in this circular orbit. In such systems, the motion can be analyzed by breaking down the forces acting on each star and understanding the speed at which each star must move to maintain that orbit.
This analysis helps us determine relationships like kinetic energy ratios, which depend on velocities and masses of the stars.

The center of mass in a binary star system is the point around which both stars orbit. It’s a balance point determined by the masses of the two stars. For stars of masses \(m_1\) and \(m_2\) separated by a distance \(r\), the center of mass is closer to the more massive star.
Mathematically, the distance to the center of mass from each star (\(r_1\) and \(r_2\)) can be given as:
\[ r_1 = \frac{m_2 r}{m_1 + m_2} \]
\[ r_2 = \frac{m_1 r}{m_1 + m_2} \]
These distances are crucial because they impact the velocities of the stars, which in turn influence their kinetic energies.

Kinetic energy is the energy an object has due to its motion. For celestial bodies in circular motion, like stars in a binary system, the kinetic energy can be expressed as:
\[ KE = \frac{1}{2}mv^2 \]
Where \(m\) is the mass of the star and \(v\) is its velocity. In our case,
\[ KE_1 = \frac{1}{2} m_1 v_1^2 \] and \[ KE_2 = \frac{1}{2} m_2 v_2^2 \]
The ratio of these kinetic energies gives crucial insights into the dynamics of the system. By analyzing this ratio, we can determine how the distribution of mass between the two stars affects their movement and the energy each star spends remaining in orbit.

Gravitational force is the attractive force between two masses. In a binary star system, it is the force keeping the stars in orbit. For two stars with masses \(m_1\) and \(m_2\), the force can be described by Newton's law of gravitation:
\[ F = G \frac{m_1 m_2}{r^2} \]
Where \(G\) is the gravitational constant, and \(r\) is the separation between the stars. This force provides the necessary centripetal force for the stars’ circular motion. The velocities of the stars can thus be related to the gravitational force acting on them:
\[ m_1 v_1^2 / r_1 = G \frac{m_1 m_2}{r^2} \]\br> and
\[ m_2 v_2^2 / r_2 = G \frac{m_1 m_2}{r^2} \]
By manipulating these equations, we can find the velocities and thus calculate the kinetic energies of the stars, revealing how gravitational forces shape their dynamic behaviors.