Since the interstellar medium is highly ionised, the physical collecting funnel can be augmented by an electromagnetic field system, which will draw nuclei into the funnel from a much wider area.
Since a ramjet gathers both its fuel and reaction mass as it goes, the standard reaction mass calculations do not apply. Some other relevant equations can be derived, however.
Δ momentum = m (v - u)By the conservation of momentum, this is equal to the change in momentum of the ramjet:
m (v - u) = M Δ Vwhere:
M = the mass of the ramjet shipNote: This equation is an approximation which neglects the small amount of collected mass which is converted into energy by the nuclear fusion reaction. For hydrogen fusion, less than 1% of the mass is lost in this way, so any error is quite small. The acceleration of the ramjet a is then given by:
Δ V = the change in velocity of the ramjet ship.
a = dV / dt = m (v - u) / M dtwhere:
dt = an "infinitesimal change in time" (I am not bothering with strict formalities of calculus here).Now, the change in kinetic energy of the interstellar medium material Δ (m v2) / 2 is equal to the generated engine power P multiplied by the change in time:
P dt = Δ (m v2) / 2But (v + u) / 2 is the average speed V of the ramjet relative to the interstellar medium over the time increment in question. Substituting this, and the acceleration formula above:
= m (v2 - u2) / 2
= m (v - u) (v + u) / 2
P dt = a M dt VNow consider the volume of interstellar medium swept up by the ramjet funnel. If the effective funnel (including any electromagnetic attraction fields) is circular, with a radius r, then in a time dt it sweeps through a volume of:
P = a M V
π r2 V dtIf the density of hydrogen nuclei in the interstellar medium is ρ (in mass per unit volume units), then the mass of hydrogen nuclei swept up in time dt is:
π r2 V ρ dtThis mass is available for conversion into energy, with a nuclear fusion efficiency η (η is 0.753% for hydrogen fusion), so:
E = m c2where:
P dt = π r2 V ρ η c2 dt
c = the speed of light.Substituting the formula for power above and rearranging:
a = π r2 ρ η c2 / M
This means the acceleration of a ramjet is dependent only on the size of the collecting funnel, density of the interstellar medium, efficiency of the nuclear fusion reaction, and mass of the ship, and is a constant over time. In other words, the ship's velocity will increase linearly with time.
The limit to this velocity increase is the speed of light, and close to the speed of light the equation derived above will break down due to the effects of special relativity.
If we assume a threshold mass-collection rate dm/dt (the units are mass per unit time), then the rate of mass collection by the funnel π r2 ρ V needs to be greater than the threshold. This gives a threshold velocity:
Vt > (dm/dt) / ( π r2 ρ )Below this velocity, the ramjet engine will not work.
In order to get up to the threshold velocity, a ramjet may be equipped with a reaction engine with its own power and reaction mass supply. This engine can be switched off once the ramjet begins to work.
Consider a ramjet moving at speed V with respect to the interstellar medium. If matter collected by the funnel is stored in the ship, then in a time increment dt an amount of mass dm is given a change in momentum equal to the change in momentum of the ship, but in the opposite direction:
dm V = - M dVBut the mass collected in this time interval is as given under Acceleration of a Ramjet above, so:
π r2 ρ V2 dt = - M dVIntegrating with respect to V from time to when speed is Vo to time t when speed is V:
dt / dV = - M / ( π r2 ρ V2 )
t = [ M / ( π r2 ρ ) ] (1 / V - 1 / Vo)Rearranging to make speed the subject as a function of time:
V = M Vo / (M + π r2 ρ Vo t )Note that the drag generated on the ship by the incoming interstellar medium does not affect the acceleration calculated above, since only the total change of momentum is relevant (and is how the acceleration was calculated).