5.1 Analytical estimates

A rough estimate of the formation rate of double compact binaries can be obtained ignoring many details of binary evolution. To do this, we shall use the observed initial distribution of binary orbital parameters and assume the simplest conservative mass transfer (M1 + M2 = const) without kick velocity imparted to the nascent compact stellar remnants during SN explosions.

Initial binary distributions.   From observations of spectroscopic binaries it is possible to derive the formation rate of binary stars with initial masses M 1, M 2 (with mass ratio q = M ∕M ≤ 1 2 1), orbital semimajor axis A, and eccentricity e. According to [327Jump To The Next Citation Point], the present birth rate of binaries in our Galaxy can be written in factorized form as

( ) −1( ) −2.5 ----dN------- A--- M1-- dA dM1 dqdt ≈ 0.087 R⊙ M ⊙ f (q), (56 )
where f(q) is a poorly constrained distribution over the initial mass ratio of binary components.

One usually assumes a mass ratio distribution law in the form f (q ) ∼ q −αq where αq is a free parameter; another often used form of the q-distribution was suggested by Kuiper [204]:

2 f(q) = 2∕(1 + q) .

The range of A is 10 ≤ A∕R ⊙ ≤ 106. In deriving the above Equation (56View Equation), the authors of [327] took into account selection effects to convert the “observed” distribution of stars into the true one. An almost flat logarithmic distribution of semimajor axes was also found in [4]. Integration of Equation (56View Equation) yields one binary system with M1 ≳ 0.8M ⊙ and 6 10R ⊙ < A < 10 R⊙ per year in the Galaxy, which is in reasonable agreement with the Galactic star formation rate estimated by various methods; the present-day star formation rate is about several M ⊙ per year (see, for example, [257401]).

Constraints from conservative evolution.   To form a NS at the end of thermonuclear evolution, the primary mass should be at least 10 M ⊙. Equation (56View Equation) says that the formation rate of such binaries is about 1 per 50 years. We shall restrict ourselves by considering only close binaries, in which mass transfer onto the secondary is possible. This narrows the binary separation interval to 10 –1000 R ⊙ (see Figure 1View Image); the birth rate of close massive (M1 > 10 M ⊙) binaries is thus 1/50 × 2/5 yr–1 = 1/125 yr–1. The mass ratio q should not be very small to make the formation of the second NS possible. The lower limit for q is derived from the condition that after the first mass transfer stage, during which the mass of the secondary increases, M2 + ΔM ≥ 10 M ⊙. Here ΔM = M1 − MHe and the mass of the helium core left after the first mass transfer is MHe ≈ 0.1(M1 ∕M ⊙)1.4. This yields

1.4 m2 + (m1 − 0.1m 1 ) > 10,

where we used the notation m = M ∕M ⊙, or in terms of q:

q ≥ 10∕m1 + 0.1m0.14− 1. (57 )

An upper limit for the mass ratio is obtained from the requirement that the binary system remains bound after the sudden mass loss in the second supernova explosion5. From Equation (45View Equation) we obtain

0.1[m2-+-(m1-−--0.1m1.41-)]1.4-−-1.4 2.8 < 1,

or in terms of q:

0.4 q ≤ 14.4∕m1 + 0.1m 1 − 1. (58 )

Inserting m1 = 10 in the above two equations yields the appropriate mass ratio range 0.25 < q < 0.69, i.e. 20% of the binaries for Kuiper’s mass ratio distribution. So we conclude that the birth rate of binaries which potentially can produce double NS system is ≲ 0.2 × 1∕125 yr −1 ≃ 1∕600 yr−1.

Of course, this is a very crude upper limit – we have not taken into account the evolution of the binary separation, ignored initial binary eccentricities, non-conservative mass loss, etc. However, it is not easy to treat all these factors without additional knowledge of numerous details and parameters of binary evolution (such as the physical state of the star at the moment of the Roche lobe overflow, the common envelope efficiency, etc.). All these factors should decrease the formation rate of double NS. The coalescence rate of compact binaries (which is ultimately of interest for us) will be even smaller – for the compact binary to merge within the Hubble time, the binary separation after the second supernova explosion should be less than ∼ 100R ⊙ (orbital periods shorter than ∼ 40 d) for arbitrary high orbital eccentricity e (see Figure 3View Image). The model-dependent distribution of NS kick velocities provides another strong complication. We also stress that this upper limit was obtained assuming a constant Galactic star-formation rate and normalization of the binary formation by Equation (56View Equation).

Further (semi-)analytical investigations of the parameter space of binaries leading to the formation of coalescing binary NSs are still possible but technically very difficult, and we shall not reproduce them here. The detailed semi-analytical approach to the problem of formation of NSs in binaries and evolution of compact binaries has been developed by Tutukov and Yungelson [419420Jump To The Next Citation Point].

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