Let us consider two point masses and orbiting each other under the force of gravity. It is well known (see ) that this problem is equivalent to the problem of a single body with mass moving in an external gravitational potential. The value of the external potential is determined by the total mass of the system
Individual bodies and move around the barycentre of the system in elliptic orbits with the same eccentricity . The major semi-axes of the two ellipses are inversely proportional to the masses
The total conserved energy of the binary system is
For circular binaries with the distance between orbiting bodies does not depend on time,
and is usually referred to as orbital separation. In this case, the velocities of the bodies, as well as their relative velocity, are also time-independent,
The plane of the orbit is determined by the orbital angular momentum vector . The line of sight is defined by a unit vector . The binary inclination angle is defined by the relation such that corresponds to a system visible edge-on.
Let us start from two point masses and in a circular orbit. In the quadrupole approximation , the two polarization amplitudes of GWs at a distance from the source are given by
is called the “chirp mass” of the binary. After averaging over the orbital period (so that the squares of periodic functions are replaced by 1/2) and the orientations of the binary orbital plane, one arrives at the averaged (characteristic) GW amplitude
In the approximation and under the choice of coordinates that we are working with, it is sufficient to use the Landau–Lifshitz gravitational pseudo-tensor  when calculating the gravitational waves energy and flux. (This calculation can be justified with the help of a fully satisfactory gravitational energy-momentum tensor that can be derived in the field theory formulation of general relativity ). The energy carried by a gravitational wave along its direction of propagation per area per time is given by
we write Equation (19) in the form
Specifically for a binary system in a circular orbit, one finds the energy loss from the system (sign minus) with the help of Equations (20) and (17):, 4
In spectral representation, the flux of energy per unit area per unit frequency interval is given by the right-hand-side of the expression
A binary system in a circular orbit loses energy according to Equation (21). For orbits with non-zero eccentricity , the right-hand-side of this formula should be multiplied by the factor
(see ). The initial binary separation decreases and, assuming Equation (22) is always valid, it should vanish in a time
The coalescence time for an initially eccentric orbit with and separation is shorter than the coalescence time for a circular orbit with the same initial separation :
In the case of low-mass binary evolution, there is another important physical mechanism responsible for the removal of orbital angular momentum, in addition to GW emission discussed above. This is the magnetic stellar wind (MSW), or magnetic braking, which is thought to be effective for main-sequence G-M dwarfs with convective envelopes, i.e. in the mass interval . The upper mass limit corresponds to the disappearance of a deep convective zone, while the lower mass limit stands for fully convective stars. In both cases a dynamo mechanism, responsible for enhanced magnetic activity, is thought to be ineffective. The idea behind angular momentum loss (AML) by magnetically coupled stellar wind is that the stellar wind is compelled by magnetic field to corotate with the star to rather large distances where it carries away large specific angular momentum . Thus, it appears possible to take away substantial angular momentum without evolutionary significant mass-loss in the wind. The concept of an MSW was introduced into analyses of the evolution of compact binaries by Verbunt and Zwaan  when it became evident that momentum loss by GWs alone is unable to explain the observed mass-transfer rates in cataclysmic variables. The latter authors based their reasoning on observations of the spin-down of single G-dwarfs in stellar clusters with age  (the Skumanich law). Applying this to a binary component and assuming tidal locking between the star’s axial rotation and orbital revolution, one arrives at the rate of angular momentum loss via an MSW,
Radii of stars filling their Roche lobes should be proportional to binary separations, , which means that the characteristic time of orbital angular momentum removal by an MSW is . This should be compared with AML by GWs with . Clearly, the MSW (if it operates) is more effective in removing angular momentum from binary system at larger separations (orbital periods), and at small orbital periods GWs always dominate. Magnetic braking is especially important in CVs and in LMXBs with orbital periods exceeding several hours and is the driving mechanis for mass accretion onto the compact component.
We should note that the above prescription for an MSW is still debatable, since it is based on extrapolation of stellar rotation rates over several orders of magnitude – from slowly rotating field stars to rapidly spinning components of close binaries. There are strong indications that actual magnetic braking for compact binaries may be much weaker than predictions based on the Skumanich law (see, e.g.,  for recent discussion and references).
© Max Planck Society and the author(s)