Dead weights are for the second stage plus satellite, the red line is the orbital threshold, required Isp for each dead weight is read off by looking where each blue line crosses the red line
I’m sure that anyone following ACW’s discussion of the Safir-Omid will have picked up that a number of us find it hard to accept that a two-stage Safir could put the Omid satellite into orbit. I thought I’d try to explain why, if it really only used two stages, this is so surprising/worrying. Fortunately, this can be done rather simply without worrying about the details of the trajectory if a couple of simplifying assumptions are made. Both of these make it seem easier for a two-stage Safir to get into orbit so if it still seems surprising then you know it is really surprising! These assumptions are
1) The second stage doesn’t lose any energy due to atmospheric drag
2) The total second stage burn time, regardless of how many times it is turned off and on or coasts before being lit, is short compared to the flight time to its orbital altitude. (Gravity is, of course, included during all of the “coast” period.)
The second assumption means we are ignoring the effects of gravity during the burn time of the second stage; this effectively adds an unknown Delta V (change in velocity) to the second stage. But, as I said, I am loading up the assumptions on the side of making the 2-stage hypothesis easier to accept because it will end up still being so surprising! My model for the Safir gives a first stage burn out at about 30 km altitude and a speed of 1.4 km/s. Thirty km altitude is pretty high but there should still be some aerodynamic drag. Again, however, I am giving this “bonus” to the two-stage hypothesis.
In order to put the Omid satellite into its observed orbit, the second stage needs to lift it to an 243.5 km altitude and give it a speed (relative to the Earth) of 7.54 km/s. Converting that change in potential energy into a change in speed and adding in the change in speed needed to boost it up to orbital speed means that the second stage needs to supply a total of 8.16 km/s. This is the threshold and is shown in red in the plot above.
That plot shows the change in speed (Delta V) supplied by various engine/propellant combinations that vary in strength ( specific impulse or Isp ) for several different dead weight fractions. This dead weight not only includes the engine masses and the propellant tank masses but also the navigation and control units and the satellite mass. Notice that even a 5% dead weight, which would be an amazing improvement in the stage’s structure, requires an effective Isp of 278 s while an optimistic SCUD-type rocket might have an Isp of 240! More “realistic” dead weights require an even greater improvement in engine/propellant power. This is why some of us have a hard time accepting a two-stage Safir. It is also why a two-stage Safir, which all the experimental data seems to point to, would indicate why the rocket would have to be so much more sophisticated and worrying than what we expected.