  Star-forming regions, characterized by dense environments, experience
frequent encounters that significantly influence binary systems, leading to
their hardening, softening, or ionization. We extend the Hut \& Bahcall
formalism to derive an analytical expression for the ionization cross-section
in extreme mass ratio binary systems, allowing us to investigate the orbital
evolution and population dynamics of binary planets and binary brown dwarfs in
star clusters, while considering ongoing binary system formation. Our findings
reveal that for low-mass soft binaries, the semi-major axis distribution
asymptotes to a universal power law between $\propto a^{-8/3}$ and $\propto
a^{-5/3}$ over the derived ionization timescale. We also discuss the
implications of our results for the candidate Jupiter-mass binary objects
putatively reported in the Trapezium cluster. We demonstrate that if their
existence is verified, they likely form continuously with a spectrum
proportional to $a^{1}$, aligning better with the ejection mechanism than with
the in-situ formation mechanism, which predicts a distribution roughly
proportional to $a^{-1}$. However, this implies an impractically high ejection
formation rate. Alternatively, if these objects are binary brown dwarfs,
continuous in-situ formation ($\propto a^{-1}$) with an initial minimal
semi-major axis around 20 AU and a formation rate of 100 Myr$^{-1}$ plausibly
matches the observed number of single objects, binary number, binary fraction,
and semi-major axis distribution.
