Is there some combination of a plane that flies high enough, with a railgun small enough to fit into the fuselage, and a satellite small and durable enough, to make this feasible?
That depends upon what would satisfy your needs as an "orbit". I think its unlikely to do what you are hoping.
A good way of thinking about the problem to make it more intuitive is to say that there is a rule that applies to all satellite orbits that, aside from the effects of the atmosphere, from the point that a manoeuvre ends the satellite will conduct a revolution in "some ellipse" and must pass through the same point in space on the second orbit, no matter how high it goes in the mean time. Atmosphere and other perturbations aside it will continue to do that indefinitely.
Even without an atmosphere this idealisation isn't right because the Earth isn't a point mass, but it will do for the thought experiment.
In our case the end of the manoeuvre is the point that the satellite leaves the railgun. This means that however high the aircraft and however big the railgun the satellite will return to the same altitude after one orbit. If we include the atmosphere, as its necessary for the aircraft, then we have to accept that the first orbit will probably be the last for the satellite due to atmospheric drag.
There are two ways of getting past this constraint.
- The railgun gives the satellite such an almighty kick that it takes up a parabolic trajectory at escape velocity and will never be seen again.
- The satellite has some means of performing a second big manoeuvre, when it has gained a lot more altitude.
In case 2 this has the effect of "raising the perigee", causing the satellite not to dip back into the atmosphere.
Going back to our thought experiment rule about orbits now causes the satellite to return to the engine cut-off point of the second manoeuvre and it is that point that now defines where the satellite must now return on subsequent orbits.
In summary, an orbital delivery system has to impart velocity increments at at least two points in the trajectory to go from rest at the Earth's surface to a stable orbit. Just in case you are wondering, I've described an idealised impulsive system, if you had a very long thrust duration, with an arc extending most of an orbit, then you'd find a way to do it also.
Is it possible? Yes, but not with current technology (sorry not sorry for that pun). Right now railguns can't shoot projectiles fast enough to launch picosats from altitude. At least that's what my back of the envelope calculations say, maybe someone will check my math.
Escape velocity on the surface of the earth is 11km/s. The 32 MJ can launch a projectile at about 2.5km/s (5500mph). That's not going to do the trick so let's load the whole rig into the back of a Lockheed Martin c130-30J. The pulsed power system that runs it can fit in a 20 foot shipping container. So the gun itself and the power system can fit in the 55 foot compartment.
According to wikipedia the c130j can get up to 12km in the air. The wikipedia article on escape velocity says that at 9km is you need 7.1km/sec to get escape velocity so this won't quite work out. You'll either need to get higher or build a more powerful railgun.
You might be able to do it in a hybrid airship like the one lockheed martin is making. By my calculations you would need to get up to about 22km to be able to use the railgun.
Of interest, NASA researched a program to do this but in reverse. The idea was to use a 2 mile long railgun at Kennedy Space Center that would launch a scramjet that would then deploy a rocket stage with the payload. You can read more about it here https://www.nasa.gov/topics/technology/features/horizontallaunch.html
Another problem that would rear it's ugly head if you were to try this: Wake turbulence.
Your satellite will be leaving the barrel at a stupendous velocity. It's going to slam into the atmosphere in front of the plane as it goes. Your launching aircraft is going to slam into that turbulence. I would be very surprised if that wasn't followed by pieces of the aircraft slamming into the ground.
And that's assuming you've already managed to overcome:
Puffin's point that you need a circularization motor.
ViennaCodex's point that the railguns of today are nowhere near powerful enough.
The fact that drag is going to be brutal. Drag goes at at least the square of velocity and the smaller the craft the higher the drag/mass ratio and thus the faster it's brought to a stop. Realistically, you need something pretty big and heavy to hope to push it through the atmosphere.
There's also a little problem of recoil. Using the same plane as ViennaCodex I find a max weight of 67,000kg. Lets figure your satellite launcher weighs 100kg and you're kicking it out at 7,000m/s. (I'm ignoring drag here, reality is much worse) You just jolted your plane 10 m/s. So what, you say? The cargo compartment is 12.5m long. To boost it you need 200,000g (note that this is beyond even gun-rated stuff!), applied for 3.5ms. While the gun is boosting your plane is being pushed backwards at 290g. The railgun rips the plane apart. You need a humongous recoil buffer to survive this--and where are you going to put it since the railgun is already as long as the plane?
Now, to power this. You need 2.45GJ assuming 100% efficiency. The best capacitors available are somewhat under 10J/cc. I'll assume 10 as I'm having to pull a number off a graph. You need 245 m^3 of capacitor to power this. That's more than 3x the total cargo capacity of our C130j. Those beefy capacitors didn't come with weight information but looking elsewhere I'm getting around 2g/cc. Your capacitors are something like 490,000kg--about 15x the total payload of the plane.