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 c. 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 M1 and M2 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 μ moving in an external gravitational potential. The value of the external potential is determined by the total mass of the system

M = M1 + M2. (5 )
The reduced mass μ is
M M μ = --1--2. (6 ) M
The body μ moves in an elliptic orbit with eccentricity e and major semi-axis a. The orbital period P and orbital frequency Ω = 2π ∕P are related to M and a by Kepler’s third law
( 2π )2 GM Ω2 = --- = ---3-. (7 ) P a
This relationship is valid for any eccentricity e.

Individual bodies M1 and M2 move around the barycentre of the system in elliptic orbits with the same eccentricity e. The major semi-axes ai of the two ellipses are inversely proportional to the masses

a1 M2 ---= ---, (8 ) a2 M1
and satisfy the relationship a = a1 + a2. The position vectors of the bodies from the system’s barycentre are ⃗r1 = M2 ⃗r∕(M1 + M2 ) and ⃗r2 = − M1 ⃗r∕(M1 + M2 ), where ⃗r = ⃗r1 − ⃗r2 is the relative position vector. Therefore, the velocities of the bodies with respect to the system’s barycentre are related by
⃗V1 M2 − ---= ---, (9 ) ⃗V2 M1
and the relative velocity is ⃗V = ⃗V1 − ⃗V2.

The total conserved energy of the binary system is

M1 ⃗V12 M2 ⃗V22 GM1M2 μ ⃗V 2 GM1M2 GM1M2 E = ------ + ------ − -------- = -----− -------- = − --------, (10 ) 2 2 r 2 r 2a
where r is the distance between the bodies. The orbital angular momentum vector is perpendicular to the orbital plane and can be written as
⃗J = M ⃗V × ⃗r + M V⃗ × ⃗r = μ ⃗V × ⃗r. (11 ) orb 1 1 1 2 2 2
The absolute value of the orbital angular momentum is
∘ ------------- |⃗Jorb| = μ GM a(1 − e2). (12 )

For circular binaries with e = 0 the distance between orbiting bodies does not depend on time,

r(t,e = 0) = a,

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,

∘ ------- V ≡ |⃗V | = Ωa = GM ∕a, (13 )
and the orbital angular momentum becomes
2 |J⃗orb| = μV a = μΩa . (14 )

3.1.2 Gravitational radiation from a binary

The plane of the orbit is determined by the orbital angular momentum vector ⃗Jorb. The line of sight is defined by a unit vector ⃗n. The binary inclination angle i is defined by the relation ⃗ cos i = (⃗n, Jorb∕Jorb) such that ∘ i = 90 corresponds to a system visible edge-on.

Let us start from two point masses M1 and M2 in a circular orbit. In the quadrupole approximation [211Jump To The Next Citation Point], the two polarization amplitudes of GWs at a distance r from the source are given by

5∕3 h+ = G---1-2(1 + cos2i)(πf M )2∕3μ cos(2πf t), (15 ) c4 r G5∕31- 2∕3 h× = ± c4 r 4cos i(πf M ) μ sin(2πf t). (16 )
Here f = Ω∕ π is the frequency of the emitted GWs (twice the orbital frequency). Note that for a fixed distance r and a given frequency f, the GW amplitudes are fully determined by μM 2∕3 = ℳ5 ∕3, where the combination
ℳ ≡ μ3∕5M 2∕5

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

( ) ( )1 ∕2 5∕3 5∕3 h (f,ℳ, r ) = ⟨h2 ⟩ + ⟨h2⟩ 1∕2 = 32- G----ℳ----(πf )2∕3. (17 ) + × 5 c4 r

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 [211Jump To The Next Citation Point] 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 dE carried by a gravitational wave along its direction of propagation per area dA per time dt is given by

[ ] 3 ( )2 ( )2 -dE---≡ F = --c--- ∂h+- + ∂h×- . (18 ) dA dt 16πG ∂t ∂t
The energy output dE ∕dt from a localized source in all directions is given by the integral
dE ∫ --- = F (𝜃,ϕ)r2 dΩ. (19 ) dt
Replacing
( ∂h+ )2 ( ∂h ×)2 ---- + ---- = 4π2f2h2 (𝜃, ϕ) ∂t ∂t

and introducing

∫ h2 = 1-- h2 (𝜃,ϕ)dΩ, 4π

we write Equation (19View Equation) in the form

dE c3 --- = --(πf )2h2r2. (20 ) dt G

Specifically for a binary system in a circular orbit, one finds the energy loss from the system (sign minus) with the help of Equations (20View Equation) and (17View Equation):

( ) 7∕3 dE- 32- G---- 10∕3 dt = − 5 c5 (ℳ πf ) . (21 )
This expression is exactly the same one that can be obtained directly from the quadrupole formula [211],
4 2 2 dE- = − 32-G--M-1M-2-M-, (22 ) dt 5 c5 a5
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 Ω. The GW frequency f = Ω ∕π and the GW amplitude h are also increasing functions of time. The rate of the frequency change is4
( ) ˙ 96- G5-∕3- 8∕3 5∕3 11∕3 f = 5 c5 π ℳ f . (23 )

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

dE c3πf 2 (|| ||2 || ||2) c3 πf2 ------= ------ |&tidle;h(f )+ | + |&tidle;h(f)× | ≡ -- ----S2h(f), (24 ) dA df G 2 G 2
where we have introduced the spectral density S2 (f ) h of the gravitational wave field h. In the case of a binary system, the quantity S h is calculable from Equations (15View Equation, 16View Equation):
G5 ∕3 π ℳ5 ∕3 1 Sh = ---3-------2------7∕3. (25 ) c 12 r (πf )

3.1.4 Binary coalescence time

A binary system in a circular orbit loses energy according to Equation (21View Equation). For orbits with non-zero eccentricity e, the right-hand-side of this formula should be multiplied by the factor

( ) f(e) = 1 + 73-e2 + 37e4 (1 − e2)−7∕2 24 96

(see [309Jump To The Next Citation Point]). The initial binary separation a0 decreases and, assuming Equation (22View Equation) is always valid, it should vanish in a time

c5 5a4 5c5(P ∕2 π)8∕3 ( P )8∕3( ℳ ) −5∕3 t0 = --------0---= -------0-------≈ (9.8 × 106 yr) --0- ---- . (26 ) G3 256M 2μ 256 (G ℳ )5∕3 1 h M ⊙
As we noted above, gravitational radiation from the binary depends on the chirp mass ℳ, which can also be written as 3∕5 ℳ ≡ M η, where η is the dimensionless ratio η = μ ∕M. Since η ≤ 1∕4, one has ℳ ≲ 0.435M. For example, for two NS with equal masses M1 = M2 = 1.4 M ⊙, the chirp mass is ℳ ≈ 1.22M ⊙. This explains the choice of normalization in Equation (26View Equation).
View Image

Figure 3: The maximum initial orbital period (in hours) of two point masses which will coalesce due to gravitational wave emission in a time interval shorter than 1010 yr, as a function of the initial eccentricity e0. The lines are calculated for 10 M ⊙ + 10 M ⊙ (BH + BH), 10 M ⊙ + 1.4M ⊙ (BH + NS), and 1.4 M ⊙ + 1.4M ⊙ (NS + NS).

The coalescence time for an initially eccentric orbit with e0 ⁄= 0 and separation a0 is shorter than the coalescence time for a circular orbit with the same initial separation a0 [309]:

tc(e0) = t0f (e0), (27 )
where the correction factor f (e0) is
2 4 ∫ e ( 121 2)1181∕2299 48--------(1-−-e0)-------- 0-1-+--304e---------- 29∕19 f(e0) = 19e48∕19(1 + 121e2)3480∕2299 0 (1 − e2)3∕2 e de. (28 ) 0 304 0
To merge in a time interval shorter than 10 Gyr the binary should have a small enough initial orbital period P0 ≤ Pcr(e0,ℳ ) and, accordingly, a small enough initial semimajor axis a0 ≤ acr(e0,ℳ ). The critical orbital period is plotted as a function of the initial eccentricity e0 in Figure 3View Image. The lines are plotted for three typical sets of masses: two neutron stars with equal masses (1.4M ⊙ + 1.4 M ⊙), a black hole and a neutron star (10M ⊙ + 1.4 M ⊙), and two black holes with equal masses (10 M ⊙ + 10M ⊙). Note that in order to get a significantly shorter coalescence time, the initial binary eccentricity should be e ≥ 0.6 0.

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 0.3 –1.5M ⊙. 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 [435Jump To The Next Citation Point] 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] V ∝ t−1∕2 (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,

˙ 4 2 JMSW--∼ − Ro-GM---, (29 ) Jorb Mc a5
where R o is the radius of the optical companion and M c is the mass of the compact star.

Radii of stars filling their Roche lobes should be proportional to binary separations, Ro ∝ a, which means that the characteristic time of orbital angular momentum removal by an MSW is τMSW ≡ (J˙MSW ∕Jorb)− 1 ∝ a. This should be compared with AML by GWs with τGW ∝ a4. 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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