The first thing I did was to rewrite the program so that it is a true orbital simulation, not ballistic. That is to say, an object travelling at orbital velocity will follow a curved path.
The preceeding work identified that of the three engine choices the most likely was the
Olympus fitted with a military afterburner system.
The flight path has to be modified to take into account the performance limitations of a gas turbine. I set limits
of Mach 3 and 20 km for a gas turbine. Then I designed a second stage that would use this as a starting point to
get the same mass into orbit as Apollo did. According to this simulation Apollo could get 122 tonnes into orbit,
which is a little high compared with the real result (117 tonnes), but does not seem unreasonable, given that this
sim has no limitations so far as aerodynamics or variations of specific impulse with atmospheric pressure.
The results is that the second stage would need to be 4 times larger than Apollo's, and so it would have 20 J5
engines, and would need 1800 tonnes of fuel. That is, it is roughly the size of the first stage of Saturn V...
rather rendering the whole process pointless.
A gas turbine first stage for that would need 150 engines, and 400 tonnes of fuel.
The turquoise line in the speed plot is the vertical speed, times 10. The first thing the second stage does is to rotate back to vertical, to gain some vertical speed. The rather alarming zigzags in the orientation plot at t>700 could be eliminated with a better control strategy, but since they are tending to damp out I left them. They do use a bit more fuel than a carefully optimised path.