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Arc coupled nonlinear LCR circuit equations were solved simultaneously with the Newtonian arc motion to describe the under-critically damped high arc current and resulting _J × _B propulsion in a vacuum-arc rail gun. Heating of plasma owing to the direct coupling by arc through magneto-hydrodynamic, ion-acoustic, Coulomb, and neutral interactions is formulated in a three-component electron continuity regime including major loss by radiation. The plasma density (ne) so deduced in arc was around 1022-1023m-3, temperature (Te) over hundreds of electronvolts, and attained final velocity ∼2×106cm/s. The launched plasma on emerging from the gun muzzle, rapidly equilibrated through radiative re-combinations and three-body collisions has ne∼5×1021m-3 and Te∼80-90 eV at 50 cm away from the breech as shown through experiments.