Propellant and Spin Up/Down
In our spacecraft two thrusters, one on each end of the craft, are placed in opposite directions in order to spin up and down the spacecraft to attain the angular velocity needed to generate 1G. Their thrusts have the same value, so the following calculations deal with only one thrust for simplicity.
If T1 is one thrust, J is the moment of inertia (a constant that measures resistance to inertia), and w is the angular speed, then:
T1 * Rc = J * w (d/dt) (6)
For a collection of discrete bodies, J is defined as:
Using (7) and substituting in (6),
T1 * Rc = (Mc * w * Rc2) * (d/dt) (8)
T1 = Mc * Rc (dw/dt) (9)
In (9), Mc and Rc are not affected as time passes because they are constants. In this model, the tether is rigid and of a constant length. Though we have included propellant in the Mc mass, propellant consumption is negligible in terms of the change in mass and does not affect the equation.
From (9) we can separate variables.
T1 * dt = Mc * Rc * dw (10)
Initial angular speed (w0 ) is 0, and Mc and Rc are constants,
I (T1 * dt) = Mc * Rc * w (11)
By definition, the specific impulse (Isp ) is the thrust achieved per flow of propellant rate. It is defined by the equation:
Isp = T/[g * (dm/dt)] (12)
T * dt = Isp * dm * g (13)
Isp and g are constants, and the initial mass of propellant (m0 ) is zero, so after integrating :
T * dt = Isp * m * g (14)
Combining equations (11) and (14),
m = (Mc * Rc * w)/(Isp * g) (16)
As stated before, m is only the mass of propellant for one thruster, and there are two thrusters that must spin-up and down twice (for the trip to Mars, and then back to Earth). Therefore, our final equation is:
mtotal propellant = (8 * Mc * Rc * w)/(Isp * g) (17)
which determines the total mass of propellant needed for the spin up and down cycles as a function of the specific impulse of the propulsive system chosen, the mass of the crew compartment and the radius of the tether.