# Delta V/Fuel Cost for Satellite Rotation

How would you figure out the fuel cost or delta V for spinning up a S/C to match the orbital period of 27.32 days (moon orbit)? Given that the S/C is initially not rotating about any axes.

For simplicity say the S/C is in an equatorial orbit and only needs to rotate about one axis. If the max torque is 40 N-cm, max thrust is 3N, total mass is 2kgs and ISP is 200s. Just the general equations are sufficient.

• You need to know the moment of inertia of your spacecraft May 18, 2020 at 16:22
• if its 0.00045m^4? still not sure how to relate that back? Thanks May 18, 2020 at 16:36
• Those units don't work. Moment of inertia is in units of $kg\, m^2$, May 18, 2020 at 18:00
• Sorry you're right I got confused. It is 0.00045 kgm² May 18, 2020 at 19:52

$$\dot\omega = T/I = 0.4/0.00045 = 1000 s^{-2}$$
Now the angular velocity you want is $$2\pi/(27.32*86400) = 2.6\times 10^{-6} s^{-1}$$ So full thrust for about $$2.6$$ nanoseconds would be needed to spin it up. Now a thruster with an $$I_{sp}$$ of 200s and a thrust of $$3N$$ consumes $$3/200g kg$$ of fuel per second (where $$g$$ is the acceleration of Earth's gravity at the surface) or about $$1.5gs^{-1}$$. So your spacecraft would need roughly 4 nanograms of fuel to achieve the desired rotation.