  We characterize the stellar and gas volume density, potential, and
gravitational field profiles in the central $\sim$ 0.5 pc of the Orion Nebula
Cluster (ONC), the nearest embedded star cluster (or rather, proto-cluster)
hosting massive star formation available for detailed observational scrutiny.
We find that the stellar volume density is well characterized by a Plummer
profile $\rho_{stars}(r) = 5755\,{\rm M}_{\odot}\,{\rm
pc}^{-3}\,(1+(r/a)^2)^{-5/2}$, where $a = 0.36$ pc. The gas density follows a
cylindrical power law $\rho_{gas}(R) = 25.9\,{\rm M}_{\odot}\,{\rm
pc}^{-3}\,(R/{\rm pc})^{-1.775}$. The stellar density profile dominates over
the gas density profile inside $r\,\sim\,1$ pc. The gravitational field is
gas-dominated at all radii, but the contribution to the total field by the
stars is nearly equal to that of the gas at $r\,\sim\,a$. This fact alone
demonstrates that the proto-cluster cannot be considered a gas-free system or a
virialized system dominated by its own gravity. The stellar proto-cluster core
is dynamically young, with an age of $\sim$ 2-3 Myr, a 1D velocity dispersion
of $\sigma_{\rm obs} = 2.6$ km s$^{-1}$, and a crossing time of $\sim$ 0.55
Myr. This timescale is almost identical to the gas filament oscillation
timescale estimated recently by Stutz & Gould (2016). This provides strong
evidence that the proto-cluster structure is regulated by the gas filament. The
proto-cluster structure may be set by tidal forces due to the oscillating
filamentary gas potential. Such forces could naturally suppress low density
stellar structures on scales $\gtrsim\,a$. The analysis presented here leads to
a new suggestion that clusters form by an analog of the "slingshot mechanism"
previously proposed for stars.
