# What is the minimum rocket mass that can achieve GEO

A tiny model rocket can't get up very high. Massive rockets can make it to GEO.
There must be some theoretical mass threshold where the rocket is "big enough" to make it to GEO - what is that mass?

To put some parameters around the question, assume:

• no atmosphere (simple physics, no air resistance, like launching from moon)
• negligible payload mass
• some solid rocket boosters (with currently available propellant and well designed nozzle) and liquid propellant main engine (like shuttle but without most/all of the vehicle)
• stationary launch (no rail guns, cannons, trebuchets, slingshots, etc)
• "near" the equator, say ±30° latitude
• rocket motor is the only source of lifting thrust
• Also assuming stationary launch, right? So if there's a railgun to give initial velocity, the weight of the railgun counts as part of the vehicle. – Peter Cordes May 29 '16 at 3:17
• @Peter good point. stationary launch. "near" equator - say ±30° latitude. no rail gun (but no atmosphere either). just rocket motor. I've updated the Q to cover these criteria – Bohemian May 29 '16 at 4:41

It depends entirely on what assumptions you make about the fixed mass portions of the launcher.

The needed velocity increment, or delta-v, to GEO, depending on where you launch from, is about 13800 m/s (atmospheric drag makes only about a 1% difference to this value; the exact acceleration profile of the launcher makes a much larger difference). This is the only "theoretical threshold" figure involved; everything else is a practical limit.

The rocket equation relates delta-v to rocket exhaust velocity (or specific impulse) and the ratio of fueled mass to empty mass. To maximize that ratio, and thus delta-v, minimize the empty mass of each stage -- this is structural weight, weight of empty fuel tanks, weight of guidance and control systems, and weight of the rocket engines. These are engineering problems, not theoretical ones.

With off-the-shelf engines using hypergolic propellants, it seems feasible to make a rocket of around 3 tons capable of reaching GEO with no payload*. If you had unlimited budget to design all-new, ultra-small engines specifically for this purpose, it might be possible to get a lot smaller.

At very small sizes, as discussed in another question, the effect of atmospheric drag eventually comes to dominate, so there's a fairly hard practical minimum limit somewhere.

note: I am not a real rocket scientist, so take my estimates with a big grain of salt.

* I based this estimate on a back-of-the-envelope design using an R4D thruster as the upper stage engine with a few smaller ones for steering; I figured anything smaller was impractical because you'd need some fairly robust guidance and control hardware on it.

The thrust of the R4D determined the overall mass limit of the upper stage. I assumed 5% tank mass to fuel mass ratio (which might not be appropriate for 55kg of pressurized MMH/NTO). The mass ratio and engine specific impulse gives the ∆v contribution of the upper stage.

I then iterated that process for two more stages, choosing bigger engines for each -- Aerojet Rocketdyne R40 on the middle stage.

I had assumed that pressure-fed engines would be the only way to go for a very small rocket, turbopump mass being prohibitive, but I ran across the interesting Rutherford from Rocket Lab -- a tiny kerosene engine that uses battery power to drive a pump, giving decent specific impulse, which might work for the bottom stage. I had to guess at the engine mass, though, as Rocket Lab doesn't publish that figure. If it's not a feasible choice due to mass, I'd choose something like a pair of Aestus engines (pressure-fed, hypergolic-fueled) with their nozzles shortened for sea level optimization. In any case, it took three stages totaling almost 3 tons to reach the required ∆v. I've got a spreadsheet I use for this kind of napkin-sketch design, so it came out like this:

• So... with all knobs dialed to 11, about a tonne. – Bohemian May 30 '16 at 8:17
• I had the wrong delta-v figure, so my 2 ton estimate wasn't realistic. – Russell Borogove May 30 '16 at 14:15