Energy of Binaries
Within a binary star system, there are four types of energy that comprise
the total energy of the system. Each star has an individual kinetic
energy (K_{1} and K_{2}) as they orbit the system's center
of mass. There is also the gravitational potential energy between the two
stars (V_{bin}) and the kinetic energy of the binary system as a
whole (K_{bin}). By determining all of these components we can
learn about the total energy of a binary star system.


However, the kinetic energy of each star can be difficult to determine.
So, in order to easily study the energy of binary, we use the Virial
Theorem which states that the kinetic energy of each individual star plus
the potential energy of the system is equal to half the potential energy,
as shown below. This greatly simplifies the energy of a binary system and
makes it much more easy to calculate.


Binary energy becomes important when a binary system interacts with other
stars. If a single star gravitationally interacts with a binary system
and there is no mass transfer, the energy of the system before and after
must be equal, as shown below. However, energy could have been
transferred within the system, from potential energy to kinetic energy, or
the other way around. Therefore, we can use the binary system's energy to
learn more about this energy transfer.


If we apply the Virial Theorem to the equation for the initial and final
binary and single star system from above, we are left with the equation
below. However, unless the original binary and single star are traveling at
extremely high velocities, we don't have to consider the initial kinetic
energies, which are crossed off in the equation. This further simplifies
the issue of determining the binary system energy.


After simplifying the above equation and isolating the potential energies,
we are left with the equation below. This shows the relationship between
the change in potential energies to the final kinetic energies of the
system.


What this equation tells us is that in the case of a single star
encounter, the difference between the initial and final binary energies
goes directly into the final kinetic energies of the binary system and the
single star. So, any significant difference between the initial and final
binary energies will cause higher final velocities. For this reason,
dynamical encounters within an open cluster seem to add energy into
the cluster. This is very interesting in itself and it is reason enough to
further study binaries in open clusters.
