  Recent VLA and CARMA observations have shown proto-stars in binaries with
unprecedented resolution. Specifically, the proto-stellar masses of systems
such as CB230 IRS1 and L1165-SMM1 have been detected in the range of $0.1-0.25
\, M_{\odot}$. These are much more massive than the masses generally obtained
by numerical simulations of binary formation, around $0.01 \, M_{\odot}$.
Motivated by these discrepancies in mass, in this paper we study the formation
mass of a binary system as a function of the total mass of its parent core. To
achieve this objective, we present a set of numerical simulations of the
gravitational collapse of a uniform and rotating core, in which azimuthal
symmetric mass seeds are initially implemented in order to favor the formation
of a dense filament, out of which a binary system may be formed by direct
fragmentation. We first observed that this binary formation process is
diminished when the total mass of the parent core $M_0$ is increased; then we
increased the level of the ratio of kinetic energy to the gravitational energy,
denoted by $\beta$, initially supplied to the rotating core, in order to
achieve the desired direct fragmentation of the filament. Next, we found that
the mass of the binary fragment increases with the mass of the parent core, as
expected. In this paper we confirm this expectation and also measure the binary
mass $M_f$ obtained from an initial $M_0$. We then show a schematic diagram
$M_0$ vs $\beta$, where the desired binary configurations are located while the
ratio of the thermal energy to the gravitational energy $\alpha$ is kept fixed.
We also report some basic physical data of the proto-stellar fragments that
form the binary system, including the formation mass $M_f$, and its
corresponding $\alpha_f$ and $\beta_f$.
