  The recently observed population of 540 free-floating Jupiter-mass objects,
including 40 dynamically soft pairs in the Trapezium cluster have raised
interesting questions on their formation and evolution. We test various
scenarios for the origin and survivability of these free floating Jupiter-mass
objects and Jupiter-mass Binary Objects (JuMBOs) in the Trapezium cluster. The
numerical calculations are performed by direct N-body integration of the stars
and planets in the Trapezium cluster starting with a wide variety of planets in
various configurations. We discuss four models: SPP, in which selected stars
have two outer orbiting Jupiter-mass planets; SPM, where selected stars are
orbited by Jupiter-mass planet-moon pairs; ISF in which JuMBOs form in situ
with the stars, and FFC, where we introduce a population of free-floating
single Jupiter-mass objects, but no initialized binaries. Models FFC and SPP
fail to produce enough JuMBOs. Models SPM can produce sufficient JuMBOs, but
requires unusually wide orbits for the planet-moon system around the star. The
observed JuMBOs and free-floating Jupiter-mass objects in the Trapezium cluster
are best reproduced if they formed in pairs and as free-floaters together with
the other stars in a smooth (Plummer) density profile with a virial radius of
0.5pc. A fractal stellar distribution also works, but requires relatively
recent formations (>0.2Myr after the other stars formed) or a high (50%)
initial binary fraction. This would make the primordial binary fraction of
JuMBOs even higher than the already large observation fraction of 8%. The
fraction of JuMBOs will continue to drop with time, and the lack of JuMBOs in
Upper Scorpius could then result in its higher age, causing more JuMBOs to be
ionized. We then also predict that the interstellar density of Jupiter-mass
objects (mostly singles with 2% lucky surviving binaries) is 0.05/pc$^{3}$.
