mass of propellant required + rocket equation

Question

I'm doing some research for a class project and I have a questions about the use of the rocket equation to get the propellant mass required. If $m_1$ is the final mass, and $m_0$ is the initial mass, after a particular $\Delta v$

$m_1 = m_0\, \exp(-\frac{\Delta v}{I*g})$,

where $I$ is the specific impulse and g is the gravity acceleration (which I guess it depends on where I'm applying the $\Delta v$, it could be the gravity of the Earth or the object where I'm traveling to, depending on the case).

I'm considering to use an Atlas V 401 (http://spaceflight101.com/spacerockets/atlas-v-401/) to go from the Earth to X object by doing a few $\Delta v$. Also, I'm assuming that each $\Delta v$ will be possible by using the Centaur Upper Stage of the vehicle.

My question is about the initial mass in the equation. If I'm carrying a payload with some mass $M_p$, I'm note exactly sure which mass contributes to the initial mass in the equation. I know the centaur has an inert mass of about 2000 kg, and the maximum amount of propellant it can carry is 20830 kg. For a correct calculation, should I consider $m_0 = M_p + 2000 + 20830$, and then subtract successively $m_i$ resulting from a $\Delta v_i$ maneuver to recompute a new $m_0$? I'm not sure whether this is the correct way to do it. In particular, I'm wondering if I could be over-estimating the amount of propellant required if I'm traveling with the tank at its full capacity.

Thank you

Answer 1

Here is what I did.

The Centaur second stage of an atlas V401 will perform 4 maneuvers such that $\Delta v_{\rm tot} = \Delta v_1 + \Delta v_2 + \Delta v_3 + \Delta v_4 =$ 6.3 km/s.

For the Centaur of an atlas V401, the propellant mass is about 20830 kg, the inert mass is 2243 kg, and the exhaust velocity is $v_e = 4420$ m/s. Further, the payload mass is about 1141 kg. Therefore, the initial total mass $m_0 = 24214$ kg.

Applying the rocket equation

$m_{\rm final} = m_0 \exp\left(-\Delta v_{\rm tot}/v_e\right) \approx 5821.32$ kg

Therefore, $\Delta m_{\rm req} = m_0 - m_{\rm final} \approx$ 18392 kg

I got the same result applying the rocket equation at each $\Delta v_i$ and writing $m_{0,i}$ accordingly.