Propulsion

Theory

Propulsion equation

This is the equation that links the power, the propulsive force and the ISP.

P = F x Isp x 9,81 / 2ɳ or P= F x V_e /2ɳ

where:

P = Power (Watts)

F = Engine force (Newtons)

Isp = Specific impulse (Seconds)

ɳ = efficiency of engine, for an ion thruster this is between 0,2 and 0,8, for a chemical rocket, or nuclear thermal rocket, essentially 1.

9,81 conversion factor, Earth gravity (m/s²)

V_e = Exhaust Velocity (m/s)

Specific impulse

Isp = V_e / 9,81

Isp = Specific impulse (Seconds)

V_e = Exhaust velocity (m/s)

Fuel consumption

ṁ = F/V_e

ṁ = Fuel consumption (kg/s)

F = Force (Newtons)

V_e = Exhaust velocity (m/s)

Velocity (Rocket equation)

V_f = V_o + V_e x ln(M_o/M_f)

V_e = Exhaust velocity (m/s)

V_f = ship final velocity (m/s)

V_o = Initial velocity (m/s)

ln = natural logarithm

M_o = Initial mass of the ship (kg)

M_f = Final mass of the ship (kg)

The propulsive force is the force the engines produce to push the ship. This is factor of the available energy, the engine efficiency and the propulsive system ISP and ejection velocity.

The energy may come from a chemical reaction, a nuclear reaction or external sources (solar or kinetic).

From Newton's third law of motion, the force exerted on a body by a second body is equal and inverse to the force exerted by the second body on the first body. F₁=-F₂

For a rocket, the first body is the exhaust, a "body" of gas, and the second body is the rocket nozzle and by extension the entire rocket. Since the force is directly proportional to the energy in the exhaust, then the energy added must be equal to exactly twice the energy in the rocket exhaust. Half of the energy added to the engine will go the the exhaust as kinetic energy, and the other half will go to the vehicle as kinetic energy.