# How to convert bi-propellant fuel into Delta-v (for ex., for JWST)

The following news gave the details on the quantity of fuel embarked on JWST: NASA gives green light to fuel the James Webb Telescope.

The Webb telescope’s spacecraft bus, built by Northrop Grumman, will be filled with 42 gallons (159 liters) of hydrazine and 21 gallons (79.5 liters) of dinitrogen tetroxide, a mix of storable fuel and oxidizer to feed the mission’s 20 rocket thrusters, according to Mark Voyton, NASA’s launch site manager for Webb.

How can we convert this into weight and Delta-v?

Let's start the dependency chain. For delta-v, we need the rocket equation

$$\Delta v = v_e \ln\left(\frac{m_0}{m_1}\right)$$

Which means we need the exhaust velocity of this propellant combo $$v_e$$, the dry mass of the telescope $$m_0$$ and the gross mass of the telescope $$m_1$$

$$m_0$$ is a quick lookup, wikipedia says 6500kg launch mass

$$m_1 = m_0 - m_{propellant}$$

So we need the propellant mass, also requested in the question.

$$m_{propellant} = \rho_{fuel} v_{fuel} + \rho_{oxidiser} v_{oxidiser}$$

So what we really need is basically all the properties of these two chemicals. Luckily, Encyclopedia Astronautica has just that. $$v_e = 3325m/s$$, $$\rho_{N_2O_4} = 1450 kg/m^3$$ and $$\rho_{N_2H_4} = 1008 kg/m^3$$

So, 159kg of hydrazine and 115kg of N2O4.

Encyclopedia Astronautica also lists the oxidiser to fuel ratio as 1.34, and we're pretty close here at 1.38, so it looks like everything is alright so far, with a total of 274kg of propellant.

Plugging in all the numbers in the initial equation yields ~143m/s of delta-v.

That's the gist of "how to convert". You might get more accurate by finding a more exact launch mass for the telescope (the wiki number is from 2015), and the exact performance of the thrusters used, instead of just the general performance of the propellant combination. But the approach is the same.

• Great! For the avoidance of confusion, can you detail the conversion from specific impulse to exhaust velocity? (The Encyclopedia does not provide the latter). Dec 15, 2021 at 17:51
• @NgPh ISP = V/EarthNormalGravity of 9.81. The exact same conversion as from Newton to Kilogram. Dec 15, 2021 at 18:06
• Doing everything at the same time can get confusing too, so I'll drop a link here that covers that in more detail: space.stackexchange.com/questions/20817/… Dec 15, 2021 at 18:07
• Perfect. 339 x 9.81 =3325. Dec 15, 2021 at 18:40
• The actual mass of JWST is 6200 Kg -> Delta-v=150 m/s (as expected). NASA update Dec 15, 2021 at 18:51