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I am looking to find out how many stages are needed to reach a certain orbit. I came across equations to compute the height reached by a stage over a certain time; however, I couldn't find material on how to plan a launch. Given a chosen orbit, how long should each stage last, and how many stages should there be to reach that orbit?

I would like to develop a model that would be able to launch a rocket at different orbits. Thus during the launch I'd have accelerations computed for $x$, $y$, and $\phi$, and an integrator to find velocity and position state vectors.

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  • $\begingroup$ Take a look at some of the existing launch vehicles to get an idea of what the scope of the possible are. $\endgroup$ – zeta-band Jun 13 '19 at 16:53
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The number of stages required to reach a given orbit varies with the design of the stages and the specifics of the payload.

For liquid rocket engine stages, it's most typical to see two stages to low Earth orbit, and either two or three to geosynchronous orbit.

Solid rockets have a lower specific impulse (a measure of fuel efficiency) so launchers using solids often need three or even four stages.

The answers to this question describe some implementations of orbital launch simulations, which may be helpful for you.

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  • $\begingroup$ I have seen this stack topic but was hoping on finding a scientific article going through coast time, boost time, spacing between boosts, required thrust force, etc And from this understanding why choosing a certain number of stages maybe ? $\endgroup$ – Emma Vande Wouwer Jun 15 '19 at 8:15
  • $\begingroup$ For liquid fuel two stages are enough right? (and then adjust the size of the stages) One for before exiting the atmosphere, and one with vacuum optimized engine(s) $\endgroup$ – Speedphoenix Feb 26 at 23:07
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This will be highly dependent on the characteristics of your fuel tanks, engines, and payload. A rocket uses stages because it's a good way to drop weight that's no longer needed, most notably, empty fuel tanks. A single-stage rocket approaching the end of its burn is still lugging around an almost-empty fuel tank. You could split that evenly into two stages, allowing you to drop the half the weight of your fuel tank halfway through the burn, but now, you need a second engine (one for the first stage, one for the second), as well as some way to separate the stages.

When adding stages, there's always a tradeoff in fuel savings due to the ability to drop weight, and fuel cost due to the increased weight of additional engines and infrastructure required to safely separate stages. If your engines and infrastructure are very light, more stages are preferable since you'll make up the difference by dropping empty tanks, but if they're very heavy, you'd be better off just carrying the empty tank instead of everything required for an independent stage.

I highly recommend the game Kerbal Space Program as a fun and extremely illustrative way to get a feel for staging considerations in spacecraft.

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  • $\begingroup$ Keep in mind that unless you use the Real Solar System and Real Fuels mod, KSP is giving you a scaled-down version of the solar system where SSTOs are much easier $\endgroup$ – ikrase Feb 27 at 6:53
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There is no answer to that question specifically. Atmosphere aside, reaching orbit requires that you have horizontal orbital velocity at your insertion velocity (e.g.: around 8.7 km/s at 140 km altitude). To do this, your stage or stages must provide you with a total change in velocity (hereafter called delta-V or dV) that is equal to that orbital velocity.

However, because we have an atmosphere, we will lose energy to drag (drag losses). The most efficient way to get to orbital velocity is to go completely sideways, but we have an atmosphere in the way. To minimize drag losses we will have to go up above the atmosphere before we start going very fast, and fighting gravity (going up unnecessarily) incurs gravity losses.

The higher the TWR (thrust to weight ratio) of our rocket, the faster we can get up and stop fighting gravity. However, the higher the TWR, the faster we will be going in the low atmosphere, and the more we'll use to drag. The best balance seems to be an SLT (sea-level TWR) of 1.2 off the pad.

All said, for a typical rocket, it takes about 9.2 km/s dV to get to orbit after you include drag & gravity losses.

Now for staging. The rocket equation is how one calculates the dV imparted by a given rocket stage. Play around with it a bit, and you'll see that with a sufficiently efficient engine and a light enough tank (you need a very high wet-to-dry mass ratio), you can get 9.2 km/s dV out of a single stage. This is called an SSTO.

In practice, TSTO (two stages), 3STO, or TSTO with boosters are much more common. There are several reasons, but one you may not have considered is that because of the decreasing atmospheric density as you ascend, a static nozzle (that's most of them) will be either underexpanded at altitude or overexpanded at sea level. Here's a brief introduction. A TSTO can optimize its first-stage engines for sea level and its upper-stage engines for vacuum.

This is unsourced, but as a rule of thumb, for a TSTO, you want ~4 km/s dV and a ~1.2 SLT on the first stage and ~5 km/s dV and a ~0.8 ignition TWR on the upper stage. For a 3STO, you want ~3 km/s dV per stage with the same SLTs as before, unless you're using a kick stage to push you into orbit at apogee, in which case your ignition TWR should be as high as you can get away with.

As for ascent guidance, it's a very non-trivial problem. Luckily, I know some guys who have developed an excellent freeware ascent guidance calculator based on Primer Vector Guidance and are working on some new super magic ascent calculator that even considers the atmosphere. This is after they wrote one based on Powered Explicit Guidance, used by the shuttle, and before that a much more basic one that's probably similar to what you're interested in. They can help point you in the right direction. Here's their repo: Mechjeb-PVG.

But anyways, to answer your question: There is no number of stages needed to get to orbit. You can use as many or as few as will provide sufficient dV, accounting for drag & gravity losses.

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Single-stage-to-orbit (SSTO). Current and previous SSTO projects include the Japanese Kankoh-maru project, the Skylon, ARCA Haas 2C, and the Indian Avatar spaceplane. https://en.wikipedia.org/wiki/Single-stage-to-orbit

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    $\begingroup$ One important sentence from the wikipedia article you linked: "No Earth-launched SSTO launch vehicles have ever been constructed." So SSTO from Earth is theory only. $\endgroup$ – Uwe Jun 13 '19 at 20:33

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