Formula for Determining Time to Perform Delta-V Burn Given Propulsion System Parameters

Question

Question is as stated. Determining how much delta-V one needs to change orbits is pretty straight forward with the rocket equation. However, I’d like to see if there’s a way to calculate delta-V from a propulsion system’s power, to compare the trade offs between maneuver time, spacecraft mass, and exhaust velocity.

I initially thought I could just go from $P = F \cdot v_{ex}$ to determine this.

$P = m \cdot a \cdot v_{ex}$

$a = \frac{dV}{dt}$

$dV = P \cdot \frac{dt}{m \cdot v_{ex}}$

As you can see, increasing exhaust velocity seems to decrease delta-V, which doesn’t make any sense when compared to what exhaust velocity does to delta-V in the rocket equation. Probably misconstruing a variable in this derivation, any insights are appreciated.

Answer 1

Let's assume you already know the thrust and specific impulse of the propulsion system, in addition to the initial mass of the spacecraft (prior to the burn) and the delta-v for the burn.

We can start by evaluating the mass flow rate of the propulsion system i.e., how much propellant is burned per unit time $[\mathrm{kg/s}]$. This can be determined from the thrust and specific impulse via the following expression:

$$\dot{m} = \frac{F}{I_{sp}\cdot g_{0}}$$

In the above $F$ is the thrust $[\mathrm{N}]$, $I_{sp}$ is the specific impulse $[\mathrm{s}]$ and $g_{0}$ is the standard gravity ($9.80665 \ [\mathrm{m/s^2}]$). To simplify we will assume the mass flow rate is constant throughout the burn.

We can then evaluate the propellant mass required to perform the manoeuvre via the rocket equation:

$$m_{prop} = m_0 - m_f = m_0 \cdot \left(1 - e^{\frac{-\Delta v}{I_{sp}\cdot g_{0}}} \right)$$

Knowing that the mass flow rate can be simply approximated as the mass of propellant divided by the burn time we can rearrange the above equations to get a single expression to give the approximate burn time to provide the given delta-v:

$$t_{burn} = \frac{m_{prop}}{\dot{m}} = \frac{m_0 \cdot \left(1 - e^{\frac{-\Delta v}{I_{sp}\cdot g_{0}}} \right)}{\left(\frac{F}{I_{sp}\cdot g_{0}}\right)}$$