K^2

None. Because buoyancy also decreases with altitude, so you have to make the chamber lighter, and no pressure can the chamber be both light enough to float and strong enough not to collapse.

For space combat, take a look at how Babylon 5: I've Found Her handled it. It's a free game, so just search for it. Go through tutorials. The way they did targeting system is pretty much perfect. Also, save for unrealistic accelerations and fighter dV resource, this is probably the most realistic space combat game I've ever seen.

Right, so I'm not sure why you are calling it B.S. 3-body physics is a big part of why the trajectory is more expensive. And anything you approximate with conics just isn't going to give you that. Your 3/4% increase is just poor planning. With SoI approximation, you can do much better.

That's... Different. You'll have to get transitions between different regimes right, each with its own characteristics. With an airplane, you just aren't ever going to have to model the belly-forward flight correctly. With a re-entering SSTO fighter, you really do. And the problem there is that center of pressure is going to move as you change the AoA. Ok, here is what I would do. I would still have only one form/skin drag computation per ship. I would model each lifting surface with one or more center of lift points. These would generate their own drag, but only lift-related. The fact that you might be re-entering belly-forward, with wings generating much of your shock wave, I would absorb into the body drag. The body drag object is going to be way more complicated now, however. I'd still keep 3 axes, but with each one I'd associate a center of drag and a CD curve. The later should depend on velocity and absorb the effects of wave drag in transonic regions and increasing drag in hypersonic. Keeping in mind that transition regions are going to depend on speed of sound in your atmosphere, which depends primarily on temperature. This is a pretty complicated model, but doable. I was playing around with building an orbital combat game myself, but I wasn't really planning to touch atmosphere, at least initially. I got basic planet rendering going, and was working on ships. Got swamped in building ship computer systems. I wanted the thing to be very customizable, down to the firmware running on each ship.

Then I suggest you do rigorous computations of the trajectory in the 3-body case and compare the two scenarios. Have fun with that.

There we go, then. And yeah, for internet, it makes more sense than satellites due to bandwidth crunch.

I'll see if I can catch a glimpse of it through a small telescope. Naked eye is out of the question from a city, but I should be able to see a mag 6 easy enough if the weather cooperates. Got to be up before 6am, though. Ugh.

I wouldn't say completely useless. It could make for a pretty good replacement for comsats. In fact, if we didn't have satellites, we'd probably have some transmitters hanging on balloons 30km up. Satellites just work out to be more effective, and in the long term, cheaper than balloons.

Nah. Try it in KSP. If you think in terms of SoI, there is almost no difference. Just plot your transfer to touch Mun SoI in direction of one of the poles. Same for the return trip. dV is almost the same. In the real world, however, you are dealing with 3-body physics. You are never influenced by just the Earth or just the Moon. That makes transfer from Earth to a roughly equatorial Lunar orbit to be significantly easier than a transfer to a polar orbit. I don't think there is a simple way to visualize this. You kind of have to work in a rotating frame, where Earth and Moon are almost static with respect to each other due to effective potential of the rotating frame. Then see how the forces on a transfer trajectory work to note that it's easier to organize a transfer in the plane of Earth-Moon system than out of the plane. Basically, the simple answer is 3-body physics, and complicated answer is complicated.

If the limiting factor is mass, then the defending side can make their ships much, much lighter by simply not holding the fuel necessary to do the complicated maneuvers. Defender has huge advantage in either case. There is simply no way to organize a good attack in space, unless you have warp technology. Attack will never be worth it in terms of resources you'd have to spend on it, and we won't see any serious space battles until we have warp.

Doesn't matter. Materials that would allow a vacuum airship to out-perform a helium-filled one are not just beyond current material strengths, but beyond any theoretically possible material strengths. You will always do better by filling your air ship with a light gas than you would by trying to maintain a vacuum.

Yup. And the only reason you need more than one is because difference in drag/lift on wings during turn is actually a significant effect while flying. Say you are in level flight and you decide to yaw left using the pedals. Plane ends up banking left as well. Why? Because your right wing is moving just a little faster than your left wing and generates just a little bit more drag. If you don't take this into account in your flight model, the plane will actually bank right due to torque from the rudder. On the other hand, when performing a roll, you'll also end up experiencing different drag on two wings, since the wing that generates less lift generates less drag as well. So all of these effects work together. In principle, if you have a very simple airplane, you might only need the drag tensor for the body, and for the wings just do FD = FL/Constant. Except, you want to reduce lift past critical AoA only after you computed drag, so drag of stalled wing remains high. But with a more complex aircraft, you probably want to have engines have their own drag tensor, for example. Adding landing gear with its own drag tensor that turns "on" and "off" when gear is deployed or retracted is also a good idea. But I don't know exactly what sort of craft you were planning on modeling. If it's something different from conventional airplanes, it might be way off. By the way, having a bit of flight experience helps a lot. So if you are serious about writing a good flight model, you might want to invest in a few flight lessons. Just enough to get a feel for it.

Yup, and then the defending side has twice as much fuel as you to maneuver, and just going to start doing things you either can't match, or you will match, but end up running of fuel and give them easy target practice.

Unfortunately, Verlet integration is very bad for gravity problems. If you know anything about general Runge Kutta methods and multi-variable optimization algorithms, I strongly suggest you take the time to build an implicit integration scheme. In particular, 6th order Gauss-Legendre method should work for this much better than anything you can compute explicitly.

That's basically what I did, except that CD can also be different in different directions, so I absorbed the (1/2)ACD into a single value, and built a drag tensor D out of it. Then the total drag was just ÃÂ(vDv). This probably isn't accurate enough for a real flight simulation, but for a game it works great. But no, unfortunately, you can't use a bunch of these, one for each part, to figure out the total. Not in general. For an airplane, having drag center for fuselage and one for each lifting surface works well enough. I've built a pretty nifty simulator based on this idea. But if, say, your fuselage is built out of parts, you can't just look at how much of each "blob" is exposed. The total quadratic drag is due to a combination of a form drag and skin drag. For a long narrow body, nearly all of the form drag comes from the end points, but nearly all of the skin drag comes from the length of the object. And you can't really tell how much of your CD is due to form and how much is due to skin.

Unfortunately, the vacuum balloon problem does not scale at all. Suppose, you have a vacuum balloon. You want to build one 2x larger. (Or 2x smaller.) This increases (decreases) buoyancy by a factor of 8, because volume scales as cube of diameter. This also, however, increases (decreases) the pressure it has to withstand by a factor of 4, all the while the cross-section has increased (decreased) by a factor of 2 only. Former is cross-section area of the entire sphere, which scales with square of the diameter, and later is cross-section of the shell only, which scales linearly with diameter. So now the shell has to be 2 times thicker (thinner). And because you already increased (decreased) total surface area by a factor of 4 already, the total weight of the shell has also increased (decreased) by a factor of 8. So it doesn't matter how big or small you make that shell. The requirements of strength/weight of material from which you want to build it are the same. And it's impossible to build either way. Similar problem occurs if you consider how this changes with atmospheric pressure. If you reduce pressure, you can make the shell lighter, but you get less buoyancy by the same fraction. So again, the problem does not scale. And like I said, materials with sufficient strength/weight to build a uniform shell that can maintain a vacuum and still be light enough to float do not exist.

It would. Less than 1.1km/s to Moon, almost 1.5km/s to skim the Earth's atmosphere. If we start having problems with graveyard orbit, somebody needs to propose this.

That's because of the safety margins. Military operations allow for a much higher risk factor. And if you throw landing into a fully automatic mode, a computer can plot a course and land a ship within at most a revolution from the target, or about 90 minutes. It can also plot intercepts that are very hard to counter. Point is, if you actually get into close combat situation, things can be interesting. Problem is getting close. If anything you might have to worry about would have to approach on similar orbit, you'll have many hours of advanced warning.

You are absolutely wrong. The buoyancy is simply equal to displaced weight. Vacuum is, in fact, the perfect filler for a balloon. It gives you 100% buoyancy. The reason vacuum chambers don't float is because they are heavy. Really, really heavy. They have to be in order to withstand the pressure. And that's the reason we don't build balloons filled with vacuum. If you fill a balloon with, say, helium, the pressure inside is equal to one atmosphere, same as on the outside. Yet, helium can still be lighter than air at the same pressure. So the balloon itself can be very thin and light. Vacuum, on the other hand, has to be contained within a thick, strong shell, or it won't be able to withstand the outside pressure. There is no material in existence that would be strong enough yet light enough to make a vacuum balloon. That said, it's not a complete impossibility. It is possible to build a honeycomb structure with something like Mylar skin that can, in principle be buoyant due to contained vacuum. I don't know if anyone has ever actually built one, but I've seen some papers showing that it is possible. At the end of the day, however, the advantage of vacuum being lighter than any gas is completely nullified by the weight of the container required to maintain said vacuum. It is always more efficient to use a light gas, such as hydrogen or helium. And with these you can, indeed, float to the very edge of space.

It would. More than 1km/s of dV.

Which isn't a problem either, since they would have almost zero relative velocity even if they collide. And won't that surprise some feudal lord when it falls on his castle.

Real atmospheric drag can be modeled extremely well using Fd = Cd Pdyn A. Here, Pdyn = (1/2) ÃÂv² is dynamic pressure of the air flow of density àand velocity v. Cd is the drag coefficient. KSP uses a fairly small constant value for Cd specified for each part. Furthermore, real Cd depends a bit on velocity. It's going to be fairly constant at subsonic speeds, spike through the transonic region, settle on a new nearly-constant value in the supersonic, and start climbing again in hypersonic. Finally, A is the cross-section area as per the article linked above. The way you'd typically use this is bunch together some of these constants, like k = (1/2) Cd A. That way, drag is simply Fd = kÃÂv². You can then experimentally measure k by measuring terminal velocity. Since at terminal velocity drag cancels weight, mg = Fd = kÃÂvt², or k = mg/(ÃÂvt²). The reason this works is that most imperfections of the model are absorbed into the value of Cd. If you were to try and compute it from the shape of the object, you are likely to end up being way off. But if you establish it experimentally, the rest of the equation works extremely well, and you can apply it towards anything from a bullet to a rocket.

I wouldn't worry about it too much. GEO is only useful with an equatorial orbit, so while we might end up putting enough junk there to eventually cascade, it won't create a death zone around the entire planet. All you'd have to do to get around it is launch along an inclined orbit.

We don't actually know that. What we know is that physics we know breaks down at these time/length scales. Whether it's actually a unit of time/length or if we simply have completely different physics governing shorter time/length scales is currently unknown.

Yup. It's the drag model, as a few people mentioned. Specifically, in KSP drag coefficient is mass times a constant. For a real rocket, drag coefficient is independent of mass, but grows with speed. So as a fraction of rocket's mass, it starts out really small, and increases dramatically as the rocket picks up speed. Because of that, it's far more advantageous to do a gravity turn really early.