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
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
where 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
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
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 [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
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
):
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
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
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]:
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,
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).
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