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Appendix A: Binary initial mass function and normalization of the simulation

In the initial binary system, the more massive component is called the primary. We use primary-constrained pairing to construct “zero-age” binaries (Kouwenhoven et al. 2008). The primary masses M_primary of the zero-age main sequence binaries are drawn from the stellar initial mass function (IMF) of primaries in massive star clusters that we derive from the results by Kroupa (2001), where M is the stellar mass and 0.08 ≤ M/M_⊙ ≤ 100. The mass ratio 0 < M_secondary/M_primary ≤ 1 of the components is subsequently drawn from a constant distribution (Kraicheva et al. 1989; Hogeveen 1992) − secondary masses lower than 0.08  M_⊙ are accepted. The eccentricity e distribution is proportional to e between 0 and 1, and the semi-major axis a distribution is inversely proportional to a (Popova et al. 1982; Abt 1983), up to 10⁶  R_⊙ (Duquennoy & Mayor 1991) − the lower limit is set by the requirement that the initial stellar radii fit inside the circularized orbit.

The specific binary fraction as a function of M is given by the observationally practical definition (Reipurth & Zinnecker 1993; Kouwenhoven et al. 2009) (A.1)where N_single(M) is the distribution of single stars of mass M, N_binary(M_primary = M) the distribution of binary systems containing a primary of mass M, and IMF(M) the IMF of systems (single stars and multiple systems combined) by Kroupa (2001). Based on observations summarized in Kouwenhoven et al. (2009); Kraus & Hillenbrand (2009); Sana et al. (2012) we approximate (A.2)where we assume all multiple systems to be binaries. Equation (A.1) can be separated as (A.3)

It follows that single stars are more common than binary systems; there are 1.6 single stars for each binary system. The mass per binary system including the corresponding single stars (which can be a fractional number) is given by (A.4)and the average star forming mass for each binary system formed (i.e., including mass from single stars) by (A.5)(the factor 1/2 appears because the average secondary mass is equal to half of the average primary mass for the chosen constant mass ratio distribution). This number is the sum of the average primary mass (0.86  M_⊙)¹¹, the average secondary mass (0.43  M_⊙) and the corresponding average mass in single stars per binary system (0.64  M_⊙). A lower limit of 0.1  M_⊙ increases the average mass per binary by ~12%. Overall two-thirds of the star-forming mass is in binaries. The total number of binaries that forms in the Galactic bulge is normalized using the total number of stars (A.6)Of all primaries, 1.3% have a mass higher than 8  M_⊙. For these masses, the power-law slope of the primary IMF (defined over linear mass intervals), from which we draw primary masses, varies between − 2.15 (for M = 8  M_⊙) and − 2.2 (M = 100  M_⊙), compared to the estimate of − 2.3 by Kroupa (2001) for the combined IMF of single stars and primary components. The IMF of primary components N_binary(M_primary = M) is flatter than the IMF of systems IMF(M) because Eq. (A.2) is an increasing function (most low-mass stars are single whereas massive stars are usually in binaries)¹². The IMF for single stars only is steeper than − 2.3 and steepens towards high mass.

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