The key assumptions were 5% gravity loss, and 5% drag loss for the first stage, and 5% gravity loss only for the second stage, on a fast ascent trajectory, plus 4.5% stage inert weights, stage payload included in that accounting. I assumed stage 1 burnout outside the “sensible atmosphere”, at 3.05 km/s achieved velocity. LEO was assumed to be 7.79 km/s achieved velocity. I completely ignored the “boost” effect of the Earth’s eastward rotation.
I did some crude engine ballistics based on characteristic velocity c* at 1000 psia chamber pressures (c* = 5900 ft/s for LOX-RP1, from the vintage 1970 version of the old Pratt & Whitney “Vest-Pocket Handbook”), and a bell divergence-corrected thrust coefficient CF chart computed for specific heat ratio 1.20. The divergence thrust correction factor is 0.983, pretty much an average 15-degree half angle, and equivalent to most modern curved expansion bells.
The first stage is expanded “perfectly” to 14.7 psia (101.3 Kpa, 1013 mbar, 760 mm Hg) backpressure. The chart says CF = 1.57 at expansion ratio 9.00 when read at pressure ratio 68. There is a simple relationship among CF, c*, and Isp, leading to Isp = 287.9 s for the first stage. A very slightly-oversimplified model then estimates exhaust velocity as 2.835 km/s.
For the second stage, the ambient backpressure is zero (out in the vacuum). I looked at the chart for pressure ratio set to an arbitrary 1000:1, and got CF = 1.826 at expansion ratio 65:1. For that same 1000 psia LOX-RP1, c* = 5900 ft/s, I got Isp = 334.8 s, and exhaust velocity 3.284 km/s.
5% gravity loss plus 5% drag loss is a total 10% loss for the first stage, making the effective required velocity not 3.05 but 3.355 km/s. That corresponds to a mass ratio of 3.2820, and a propellant fraction of 0.6953. The corresponding stage payload fraction is 0.2597. For a nominal liftoff weight of 500 metric tons, the stage 1 payload (stage 2 plus the “real payload”) is 129.85 tons. That ignores any interstage weights, which might be around a ton, for a second stage ignition weight of 128.85 tons.
5% gravity loss on the second stage velocity increment of 4.74 km/s results in an effective velocity requirement on the second stage of about 4.977 km/s. At the higher second stage Isp, the mass ratio is 4.55183, and the propellant fraction for that stage is 0.7803. For the same stage 4.5% inert fraction, that leaves 0.1747 for the stage 2 “real payload” fraction. That’s near 22.51 metric tons for a stage 2 ignition weight of 128.85 tons.
That payload has to ride inside some sort of protective, aerodynamic shroud. For the sake of argument, assume that shroud also weighs about 1 ton. Therefore, the real delivered payload is nearer 21 metric tons for a vehicle massing 500 tons at launch.
The real Falcon-9 is a little over 500 tons at ignition, and is rated to deliver 13 metric tons to LEO. My 21 tons is in the ballpark, clearly, but still quite a ways “off” for purposes of “exact” estimates. All in all, I’d say these ballpark estimating techniques are actually quite good, especially for relative-comparison calculations. But it takes a better model than this, to really “pin things down”.
For example, use the 5%-5% loss factors on the entire trajectory to LEO, with 4.5% inerts, and the lower-performing first stage engine performance, but as a single-stage to orbit vehicle. It really does calculate as technically feasible, just at only 1.48 ton payload, and with propellant tanks “stretched” by 37% (volume) to maintain a 500 ton ignition weight.
That 1.48 tons gets compared to the 21 tons for the 500 ton two-stage bird. For one stage versus two, the launch cost could only be factor 2 lower for the one-stage bird, at the very most. On a per-unit-delivered payload basis, the one stage version will always deliver less mass for a higher cost. That basic effect is why staging was invented, over 6 decades ago.