# mass of propellant required + rocket equation

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

• If you are using a staged rocket like the Atlas you must do the equation for each stage. See here for an Atlas example planetary.org/blogs/guest-blogs/2017/… For the first stage, the payload mass is the upper stages including the actual payload. The g in the equation is Earth surface gravity in the units you are using, it's really just a conversion factor. Don't use the gravity for some other planet. Apr 6, 2020 at 0:49
• thank you! problem solved! Apr 6, 2020 at 13:55
• If you care to, you could write an answer showing how you solved it. Answering your own question is perfectly ok. Apr 6, 2020 at 13:59

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.