Since we've been fielding a lot of questions on this subject lately, I just want to chime in with some of the rules-of-thumb that seem very natural to me. Unlike rockets, a gun sees the maximum density of the atmosphere at its fastest point. So provided you understand the necessity to circularize an orbit, and you understand that high accelerations need hardened equipment, the viability of many proposals can be easily ruled out envelope calculations for the drag.
Here's a helpful metric to look at the atmosphere with: the mass-thickness of the atmosphere is the mass per unit area, looking straight up into the sky at sea-level.
$$ \frac{ \text{Mass of Atmosphere} }{ \text{Area} } := \mu_{atm} \approx 10 \frac{ \text{tons} }{ m^2 } $$
In a naive sense, look at a bullet straight-on. Divide its mass by the area (same area it presents to the atmosphere). A more accurate approach would be to do some manipulation of the drag equation. We're not interested strictly on the force on the projectile. Alternatively, I'll consider the velocity lost due to its trip through the atmosphere, $\Delta v$ here. I obtained the following from the drag equation, under the assumption that the total velocity is notably greater than the loss in the atmosphere (if not, it's nonviable anyway).
$$ \frac{ \Delta v }{ v } = \frac{1}{2} \frac{C_d \epsilon \mu_{atm}}{ \rho D f \sin{( \theta)} } $$
Due to practical considerations, if this ratio is about 1, you don't have a chance. It will burn up in the atmosphere, and even if not, you can't produce those speeds. Speeds near orbital have never been demonstrated. So if you can't push this ratio well below 1, the idea is out the window. I'll go over all the terms below, dividing them into more-or-less into things that are completely impossible to push beyond a certain window, and things that have not inherent limitation.
Hard limits:
- C_d the drag coefficient for bullets are in the neighborhood of 0.2 for well engineered projectiles. You can easily push it a little bit higher or a little lower, but there is no where close to an order of magnitude of wiggle room. The high mach numbers make it an even tighter range for different shapes.
- epsilon, is the correction for a thinner atmosphere at higher altitudes. For Mt. Everest the factor is something like 0.3. I would believe a factor of 0.5 for placing it on a high mountain. Obviously sea launches face the full factor of 1.
- rho, the average density of the projectile is constrained by your propellant if you're going to circularize its orbit. This is very low, probably around 1.0 specific gravity. If you're doing some other scheme (like a Rotovator), you could pack it full of steel payload, getting to 7.0 or 8.0 at theoretical maximum. Unless you're sending Uranium or something like that. I'll give it 2.0 for a practical payload.
- f, the length to diameter ratio of the projectile is limited by aerodynamic considerations. Rockets tend to be very slender, but they have active control systems. Fins can help push the envelope here. But even with that, I'm calling a "hard" limit of about 10.
- sin(theta), this factor works against you, and the best case scenario is 1. This would apply for suborbital vertical launches. For orbital launches, you risk defeating the point of a gun in the first place if this angle isn't low enough. You could shoot something straight up and then burn the necessary 7.8 km/s, but this would make a very poor mass ratio, and probably can't be engineered to withstand the gun acceleration. For Quichlaunch, I would give this a value of 0.5.
Soft limits:
- D, is the diameter, which serves as a metric for the overall size of the payload. How large can it be? How large of a gun can you construct?
So we have to use our (only) soft limit in order to engineer around the other hard limits. I'll take a velocity loss ratio of 0.2 for now. You could stretch this, but not by much. You actually come out with a set of parameters like:
$$ D \approx 4.5 cm \\
M \approx 11.7 kg $$
This doesn't sound shockingly huge. But if you change it to a sea-level launch, the required mass goes up by a factor of 8, due to the fact that mass scales with D^3. I was also extremely generous with the diameter to length factor. A bullet shape that has a factor of 5 would be, again, 8 times the mass. So if we're looking t something like a practical sea launch to ultimately reach orbit, the minimum mass is closer to the scale of 750 kg.
You can see here how the minimum size depends strongly on the assumptions of the type of system you're using. Nonetheless, 10-750 kg is still a starting range for the minimum bullet mass needed for the system to ever work.
But the bullet size and speed also set a limit on the minimum size of the gun itself. If you can get pressures on the order of 50,000 psi, then the 750 kg case (Quicklaunch) at 8 km/s would need a volume of about 110 m^3. That's big.
Compare, a nuclear reactor pressure vessel is probably about 2-3 times the volume of that. It's also about 1/20th the pressure. And it costs upwards of $100 million.
Straightforward material requirements for a pressure vessel are proportional to the (pressure)x(volume) product. So things are not looking good for our space gun, although the economics of heavy forgings are far from simple. On the other hand, a minimalist space gun mounted on the slope of a tall mountain might have a hard cost minimum below the $10 million scale.
Or you could just buy a Falcon 9 flight, for somewhere around $50 million. Of course you could get better economics from the space gun if you used it enough times to amortize the cost sufficiently. However, the total yearly demand for payload into LEO is only about 240 tons. The Falcon 9 can carry 10 tons in one shot. Only a small fraction of the payloads could be substituted by the space gun (hardened equipment). So maybe there would be enough demand for the space gun to replace 1 or 2 rocket launch equivalents. On top of that, the risks are huge. Global flight frequency would have to be much larger for people to seriously put the necessary capital into this.