Robert Wagner 

Footnote 18 Honors Project 

Brewer MAT267, Fall 2007 

Rocket Science 

As the world's space programs have progressed, rocket engineers have settled on three-stage rockets as the best compromise between efficiency and cost for getting payloads into orbit.  Single-stage rockets require far too much fuel to propel both their payload and their fuel tanks to the desired final velocity, so multiple stages are required, each of which has smaller fuel tanks that it needs to propel.  An ideal rocket, from a mass efficiency perspective, would have an infinite number of stages, so that as each atom of fuel is used, the structure needed to hold that fuel would drop off.  However, in actual practice, three stages has been decided on as the best trade-off between mass efficiency and cost of design and construction.  In this project, I will derive equations to calculate the minimum masses of each of the three stages need to accelerate a payload to a given velocity, based on the mass of the payload, the exhaust velocity of the rocket, and a structural factor relating the dry mass of a stage to the mass with fuel. 

The equation for the velocity change of a single-stage rocket with a given fuel-to-mass ratio S, a constant exhaust velocity c, a mass , and a payload P, is . 

In order to find the minimum mass for a three-stage rocket, three copies of this equation need to be added to each other, with appropriate values for the payloads and masses.  First, however, in order to simplify later calculations, the equation will be modified as follows: 

First, convert the 1 inside the natural log into a fraction with the same denominator as the other fraction:   

Next, combine the two fractions, and multiply all values through on the top:  

Finally, remove the oppositely-signed terms from the top of the fraction, and raise the interior of the natural log to the -1 power, thus removing the negative sign from the front of the equation: 

If the mass of the first stage is M1, the mass of the second stage is M2, the mass of the third stage is M3, and the mass of the final payload is A, it is possible to substitute these values in for and in equation (1) to get values for the three stages of a three-stage rocket: 

The final velocity of a multistage rocket is the sum of the velocity changes of its stages, so to get the final velocity of a three-stage rocket, vf, the three equations above can be added together, giving the following equation: 

Next, in order to allow easier calculation of the minimum mass of the rocket, three new variables N1, N2, and N3 will be defined such that the above equation can be expressed as .  If we set M to be the mass of the rocket stage in question, P to be the payload of that stage (including any later stages), and N to be the value N1, N2, or N3, as applicable for that stage, .  This equation can be further modified to a point where it can be easily used to find the minima of N1, N2, and N3: 

Multiply both sides by (P + SM), and multiply the N through on the right:  

Subtract SMN from both sides:   

Divide both sides by P:  

Subtract SN from both sides:  

Factor this to get: , or .  

Dividing both sides by (1 - SN), and converting the 1 on the left into P/P, gives equation (2): 

Substituting the appropriate values of M, N, and P into equation (2) gives the following equations for each stage of the rocket: 

If these are multiplied together, this gives: 

Simplifying the expression, and substituting in M for M1+M2+M3 gives this final equation: 

is minimized at the same place as M, for a constant value of A.  This can be shown by it being equal to ln(M + A) - ln(A), which is a simple increasing function plus a constant, the same as in the case of the function M.  Given this, and the equation for  just developed, the minimum for M occurs at the same point as the minima for N1, N2, and N3.  Using Lagrange multipliers, minimum functions for N1, N2, and N3 can be found subject to the constraint , the formula for the velocity of a three-stage rocket developed above. 

First, it is necessary to break up equation (6) using the laws of logarithms in order to make it easy to take its partial derivatives with respect to N1, N2, and N2: 

The Lagrange equations for minimizing this function with respect to the constraint are as follows: 

Multiplying each of the firs three of these equations by the appropriate N value, and subtracting 1 from both sides, gives the following equality: 

By taking any two parts of this equality, it can be shown that all three N values are identical: 

Choose two arbitrary parts:  

Cross-multiply the denominators:  

Multiply everything through:  

The identical terms can be removed, and both sides divided by S, leaving , for the minimal values of N1 and N2.  The same sequence can be used to show that and .  Due to this equality, N1, N2, and N3 will be referred to as N from here on. 

To get a formula for the minimum of N in terms of , it is possible to simply substitute N into the constraint equation , giving the new equation , or . 

To get an expression for the minimum value of M, which is the value we wish to know, it is simply necessary to solve equation (6) for M, and then substitute in the expression for N: 

M is the combined mass of the three stages of the rocket, A is the mass of the payload, S is the ratio between the mass of the rocket without payload and the mass fully loaded and fueled, c is the exhaust speed, and is the desired final velocity.  Furthermore, formulae for the masses of the individual sections can be determined by substituting the value of N into equations (3), (4), and (5), and solving for the desired masses: 

With these equations, it is finally possible to get actual numbers for the masses of rockets.  First calculated will be the mass needed to put an arbitrarily-sized satellite into orbit 100 miles up, which requires a velocity of approximately 17,500 mi/h.  Additional assumptions are that the structural factor S is 0.2, and the exhaust speed is 6000 mi/h. 

The mass of the total rocket is: 

The masses of the individual sections are: 

If the three section masses are summed together, they add up to 89.4, which matches the separately-calculated total mass within rounding error, showing that the four equations relate to each other properly.  Furthermore, the ratio of fuel to payload seems fairly reasonable for a rocket, so the equations were likely derived properly. 

Last, the mass needed to launch a 500-pound probe out of Earth's gravity well will be calculated.  The structural constant and exhaust velocity will be the same as in the previous calculation, and the required final velocity is 24,700 mi/h.  Substituting these numbers into the mass equations for the three stages gives the following calculations: 

Total mass: , or roughly 833 tons.