### 3.1 Keplerian binary system and radiation back reaction

We start with some basic facts about Keplerian motion in a binary system and the simplest case of evolution of two point masses due to gravitational radiation losses. The stars are highly condensed objects, so their treatment as point masses is usually adequate for the description of their interaction in the binary. Furthermore, Newtonian gravitation theory is sufficient for this purpose as long as the orbital velocities are small in comparison with the speed of light . The systematic change of the orbit caused by the emission of gravitational waves will be considered in a separate paragraph below.

#### 3.1.1 Keplerian motion

Let us consider two point masses and orbiting each other under the force of gravity. It is well known (see [210]) 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

The reduced mass is
The body moves in an elliptic orbit with eccentricity and major semi-axis . The orbital period and orbital frequency are related to and by Kepler’s third law
This relationship is valid for any eccentricity .

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

and satisfy the relationship . The position vectors of the bodies from the system’s barycentre are and , where is the relative position vector. Therefore, the velocities of the bodies with respect to the system’s barycentre are related by
and the relative velocity is .

The total conserved energy of the binary system is

where is the distance between the bodies. The orbital angular momentum vector is perpendicular to the orbital plane and can be written as
The absolute value of the orbital angular momentum 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,

and the orbital angular momentum becomes

#### 3.1.2 Gravitational radiation from a binary

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 [211], the two polarization amplitudes of GWs at a distance from the source are given by

Here is the frequency of the emitted GWs (twice the orbital frequency). Note that for a fixed distance and a given frequency , the GW amplitudes are fully determined by , where the combination

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

#### 3.1.3 Energy and angular momentum loss

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 [211] 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 [11]). The energy carried by a gravitational wave along its direction of propagation per area per time is given by

The energy output from a localized source in all directions is given by the integral
Replacing

and introducing

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):

This expression is exactly the same one that can be obtained directly from the quadrupole formula [211],
rewritten using the definition of the chirp mass and Kepler’s law. Since energy and angular momentum are continuously carried away by gravitational radiation, the two masses in orbit spiral towards each other, thus increasing their orbital frequency . The GW frequency and the GW amplitude are also increasing functions of time. The rate of the frequency change is

In spectral representation, the flux of energy per unit area per unit frequency interval is given by the right-hand-side of the expression

where we have introduced the spectral density of the gravitational wave field . In the case of a binary system, the quantity is calculable from Equations (15, 16):

#### 3.1.4 Binary coalescence time

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 [309]). The initial binary separation decreases and, assuming Equation (22) is always valid, it should vanish in a time

As we noted above, gravitational radiation from the binary depends on the chirp mass , which can also be written as , where is the dimensionless ratio . Since , one has . For example, for two NS with equal masses , the chirp mass is . This explains the choice of normalization in Equation (26).

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  [309]:

where the correction factor is
To merge in a time interval shorter than 10 Gyr the binary should have a small enough initial orbital period and, accordingly, a small enough initial semimajor axis . The critical orbital period is plotted as a function of the initial eccentricity in Figure 3. The lines are plotted for three typical sets of masses: two neutron stars with equal masses (), a black hole and a neutron star (), and two black holes with equal masses (). Note that in order to get a significantly shorter coalescence time, the initial binary eccentricity should be .

#### 3.1.5 Magnetic stellar wind

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 [371]. 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 [435] 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 [379] (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,

where is the radius of the optical companion and is the mass of the compact star.

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., [324] for recent discussion and references).