K^2

Of course. That's the way I would compute the above.

The equation for that funnel is going to be h = 1/r - exactly the same as equation for gravitational potential energy.

You can generalize rule to an arbitrary number of dimensions, but we are talking about a closed surface, yes. Euler Characteristic for a hemisphere is same as that of a disk, 1. Which means you can build a hemisphere out of triangles with every interior vertex having 6 neighbors, so long as on the boundary, for example, most vertices have 4 neighbors, with exactly 6 vertices having 3 neighbors. In fact, lets generalize this a little bit. Consider an interior vertex with n neighbors. It accounts for n/2 edges and n/3 faces. So its contribution to Euler Characteristic is 1 - n/2 + n/3 = 1 - n/6. So for n = 6, this is exactly 0. For n = 5, this is +1/6. So in order to get +2 of the sphere, where all vertices are going to be interior vertices, you need to have 12 vertices with 5 neighbors, each adding 1/6 to Euler Characteristic giving you total of +2. If you have a vertex with 7 neighbors, it contributes -1/6, so you must balance it with another 5-neighbor vertex. Example: Consider an octahedron. It is built out of 8 triangles having 6 vertices. At each vertex you have 4 triangles, so each vertex has 4 neighbors and. Therefore, each one contributes 1 - 4/6 = +1/3 to Euler Characteristic. That gives you a total of +2, which is same as for a sphere. And indeed, all regular polyhedra are topologically equivalent to spheres. On the boundary, however, each vertex is going to have 1 more edge than face. So if it has n neighbors, it contributes 1 - n/2 + (n-1)/3 = 2/3 - n/6 to Euler Characteristic. So now, you get 0 contribution with n = 4. While n = 3 neighbored vertices are going to add +1/6. So if all of the interior vertices have 6 neighbors, for a total Euler Characteristic of 0, then on the edge you need to accumulate +1 of the disk, which can be done with 6 vertices of 3 neighbors each. Example: Consider a regular pentagon broken up into 5 triangles. The interior vertex has five neighbors, so it contributes +1/6 to the total. There are 5 vertices on the boundary having 3 neighbors each, providing +1/6 each for a total of +5/6. The total for this object is +1, which is equivalent to that of the disk.

I've actually put my hand into the container with liquid nitrogen. If done briefly enough, vapor shield protects your skin. Very weird experience. On topic of the video, this is definitely a simulation of central potential interaction, but not quite gravitational. Gravity is a -1/r potential, and what you have looks like it's roughly r² potential. That's a harmonic potential which does result in closed, elliptic orbits, but while gravity-induced orbits have the center of potential at the focus of the ellipse, here the geometrical center of the ellipse is going to coincide with center of the potential well.

So? It just means we have energy source and losses to heat. We know exactly how much energy is released per unit of fuel burned and how much of that energy becomes kinetic energy of the exhaust. You don't have to work with a closed system to make use of the conservation laws. You just need to know how much is being added and how much is being lost. If we choose an arbitrary coordinate system, then we don't know how much is being added by time-dependent interactions without solving equations of motion, which are going to be quite complex in a general coordinate system.

Yeah, that's definitely everything I would need. So we have 3 burns in planned bi-elliptic. 295 by 295 at 28.5° to 295 by 80k at 20.75° 295 by 80k at 20.75° to 35,786 by 80k at 0° 35,786 by 80k at 0° to 35,786 by 35,786 at 0° I'll run these.

Do they have to be regular hexagons? Do they have to be all of the same size?

Bi-elliptic is 2,802 + 926 + 490 = 4,218m/s. Alternative: 2,802 + 375 + 1,467 = 4,644m/s. Neither of these include inclination change burn, but that's going to be identical, making bi-elliptic transfer significantly cheaper. Do you know how much inclination change is required? I want to see how much bi-elliptic saves compared to doing inclination change in LEO.

No, the LEO to GTO burn is about 2.5km/s. GEO to LTO is, indeed, about 3.1km/s. For sake of completion, GTO to GEO is another 1.5km/s. Could you have been thinking of total LEO to GEO, which is about 3.9km/s?

Actually, there sort of is when you are dealing with central potentials. Energy is conserved in a system where Hamiltonian is time-independent. If you choose a coordinate system in which source of gravity is moving, Hamiltonian is time-dependent, and energy is not conserved. So when we are talking about a ship in orbit of a planet, choice of coordinate system in which planet is at rest is a "better" coordinate system. Naturally, we can still describe the problem from any other coordinate system, but then we lose conservation of energy, and math becomes way more complicated.

A useful thing to remember is that icosahedron is a dual of the dodecahedron. If the later has 12 faces and 20 vertices, the former has 20 faces and 12 vertices. Makes it easy to remember. In general, every platonic solid is a dual of a platonic solid, with tetrahedron being its own dual. It's probably worth clarifying to Idobox or anyone else not familiar with theory, that this is true for anything that's topologically a sphere. If we are still talking about a triangular mesh (or pure simplical 2-complex as a topologist might call it) then roughly speaking, it is equivalent to a sphere if you can drag all of its vertices and re-position them, without overlapping, to make something resembling a sphere. So a 3D model of a sofa is probably going to be equivalent to a sphere. While a common example of a surface equivalent to a torus is a surface of a coffee cup. A torus has Euler Characteristic of 0, so it can be built out of all vertices having exactly 6 neighbors. But more realistically, since you'll encounter positive and negative curvature, you'll be looking at vertices with 5 and 7 neighbors, but there has to be the same amount of both. Actually, if anyone has interest in geometry or 3D modeling, I would recommend reading up on Euler Characteristic. Wikipedia article is pretty easy to follow, and it can be useful in many situations. I've made great use of the concept in some simple simulations of deforming objects. If an Euler Characteristic has changed, it likely signals that the body you are simulating has fragmented. Made for some good optimization. @ZetaX: For some reason, I never thought about it in terms of conserved total curvature. That's a good way to look at it. Thanks.

First question. Do you actually understand all of the equations involved in computing drag and what to do with these? If so, read on. Otherwise, you need to brush up on basic mechanics. Start with a capsule first. Besides orbital velocity, it only has one parameter to vary. Basically, re-entry angle. There might be more efficient ways to go about it, but with modern computers, just go for it. Take your ship, write equations for forces acting on it based on current velocity and altitude, and then use a numerical differential equation solver to compute trajectory. Compute loading factors and heating along trajectory, adjust re-entry angle and repeat. Do this for a range of possible entry angles, and use the data you get to establish the corridor. For the numerical solver, I'd use Mathematica. It has a lot of nifty symbolic algebra features that makes this kind of computation a lot easier. Results probably won't be as reliable as you'd want if your life is going to depend on it, but you can definitely compute a plausible corridor for a simulation.

Do all faces have to be hexagons? That won't be very neat. A full sphere has a non-zero Euler Characteristic, so you can't build it out of hexagons. Half a sphere is possible, but it wouldn't be very even or pretty. Simplest solution for a sphere is a truncated icosahedron. Basically, a soccer ball shape. Half of that should work for you pretty well, but you'll need a few pentagonal faces. In practice, when large hemispherical domes are built, all faces are triangular. That gives the most stable construction. Each vertex of the frame is connected to 5 or 6 other vertices. In a full sphere, you'll have exactly 12 vertices with 5 neighbors. Rest will have 6. So for a hemisphere, you'd build the most stable structure with 6 such 5-neighbored vertices. Surface area will depend on the choice of structure, of course. Truncated icosahedron is relatively easy to compute, if that's what you are leaning towards.

You are right, but this isn't quite what we are talking about. If you could vary velocity of the exhaust, launching it at the speed equal to that of the rocket maximizes energy benefit in the chosen frame. (There is some question of frame choice*, but they are easy to resolve for a 2-body problem.) But energy is not the only thing you have to worry about. Consider a rocket traveling at 6km/s in low Earth orbit. Energy-optimal exhaust velocity is going to be the same 6km/s in the other direction. If you expel 1kg of mass, you'll gain 18MJ of energy. Now, suppose, the same rocket is traveling near Geo-stationary orbit at just 3km/s. If it can achieve exhaust of 3km/s, it's going to have the same energy-efficiency as before. But now to get the same 18MJ difference, it has to expel 4kg of mass. Clearly, lower orbit is more efficient in terms of propellant. In case of a realistic chemical rocket, you are dealing with an engine that produces constant exhaust velocity. Here, it is easy to convince yourself that once again, lower orbit allows you to gain most energy for a given quantity of fuel burned. Hence the Oberth effect. * As a toy problem, consider a rocket in nearly-empty space traveling at velocity v. As discussed above, you get most energy gain if you launch exhaust at velocity -v. Now, suppose there in alien on a nearby asteroid, and that asteroid is buzzing by fast enough to observe the rocket moving by at velocity 2v. So from perspective of the alien, rocket should be accelerating exhaust to 2v. Who's right? Well, it doesn't matter. Trying to optimize energy only makes sense in problems with gravity. Otherwise, you are optimizing delta-V, and that's a different problem.

Completely different effects that happen to have similar appearance and potential outcomes purely by coincidence. POGO effect is due to variations in pressure of fuel and combustion chamber. Since KSP does not model either of these, it cannot replicate POGO effect. On the other hand, instead of a semi-rigid rocket body, you have perfectly rigid sections connected by spring-like constraints. The constraints can oscillate a little. If some driving force hits a resonance of one of these, your entire ship starts shaking itself apart, which is visually very similar to what happens due to POGO. So yeah, not on purpose, but somewhat realistic nonetheless.

Hm. That might actually be an interesting question. Suppose, you have a point mass. Well, a black hole. Obviously, you still don't want to dip bellow the event horizon, so the "don't crash," rule is still in effect. But could the optimal periapsis for boosting past a black hole actually be at some specific point above the event horizon? Once we include relativity, I'm not sure that the answer would be the trivial, "as low as possible."

Yes, but only slightly, and only in the interior of the Solar system. In the outer Solar system, trajectory will become elliptic. This is due to interaction with other planets in the solar system, and actually pretty common for comets from Oort cloud. The simplest way to think about it is that it has enough energy to escape Sun's mass, but not the combined Solar System mass.

How is your background on classical mechanics? I would recommend to start with a mechanics text, since they would usually have a good section on central potential motion. I just looked in Goldstein, and it covers central potential in general, Kepler problem, and goes a bit into 3-body motion. If you understand that, you will understand absolutely everything there is to understand about types of orbits KSP considers. But real orbital mechanics goes way beyond that. Real objects are not perfect spheres, and you never have a pure two-body problem to deal with.

@ ZetaX. They couldn't put something on a collision course without a mid-flight correction if they tried. Odds of actually hitting anything in space are virtually zero. But if the navs and propulsion work, it should be completely safe to fly by within a kilometer of the station. If something fails, then the thing will just miss by significantly more. I don't think anyone would have a problem with that. Speaking of which, it would be nice to have the launch on a default sub-orbital trajectory that results in good re-entry angle. So if propulsion or maneuvering fail, there is no danger of the crew getting stuck up there.

I'm sure it'd be possible to write up papers so that only property/equipment is insured, with the pilot signing whatever needs to be signed to wave it. There is simply no way to impose regulations preventing this by law in United States. Of course, if SpaceX flat out refuses to launch a manned mission without certification, that's their right. Can we maybe talk to the Russians about launching with their rocket? A launch from Baikonur can put the bus on the rendezvous orbit with ISS. I don't think they'd let it get anywhere close to docking, but it'd be fun just to fly-by and flash the lights at them. I'd also honk the horn. They wouldn't be able to hear it, but I'd do it anyways.

All that says to me is that if NASA was launching this bus, or if it was launched for, on behalf of, or in any other way in connection with NASA, it would require this certification. If you are a private company asking another private company to do the launch, NASA's regulations are absolutely irrelevant. To comply with US law, all you need is to slap an FAA Experimental sticker on the thing, and launch it in such a way as to be compliant with FAR on experimental aircraft.

Permit from whom? Except for re-entry, making the thing habitable is the easy part. As brought up a few times in this thread already, pilot would wear a simple space suit tied by umbilical to a simple life support system. Compared to propulsion and navs, this is peanuts. In terms of returning pilot to Earth, I have a really crazy idea that might be simple enough to work. Basically, I propose wake-boarding a heat shield down. The most extreme extreme sport ever. The bus would be put into a correct re-entry trajectory, and the pilot/crew would just have to bail out of the back doors with the heat shield and hold on for dear life until reaching terminal velocity in the thicker layers of atmosphere. At that point, the pilot would drop the shield and become a skydiver. A small oxygen tank, an inflatable one man raft, and a parachute is all the equipment the pilot needs on this ride. The suit would have to have valves designed to let air in once pressure outside becomes greater than inside so that the pilot doesn't get crushed once at low enough altitude. The complicated bit is making sure the heat shield allows for a stable ride into the atmosphere. This would require some testing. Everything else is tried and tested, though, and it works. I'd go for it.

KSP is totally all about taking a bus and welding on it a bunch of thrusters and scientific equipment. This might not be totally random enough for you, but just launching a bus would be boring. You wouldn't be able to see what happens to it. You have to put at least tracking, video, and communication equipment on it. But then you also want to drop it somewhere safely, so you do need engines. And at that point, you might as well put a pilot in and make it a proper mission.

The main purpose of the propulsion system is de-orbiting. Unfortunately, that puts strict limits on TWR that can only be fulfilled by a chemical rocket.

Absolutely none of the predictions of supersymmetry or string theory, these that are distinct from standard model, have been observed. As such, we have no means of gauging at what energy levels these effects are meant to be found. Pure supersymmetry would require identical masses for superpartners. That's obviously not the case, so if such a symmetry exists, it is a broken symmetry. How badly broken? There is no way of telling. But the more broken it needs to be to fit data, the less plausible it becomes. And regardless of what somebody might have expected, LHC experiments are pushing these boundaries further back. Like I said, I don't know much about string theory. What I can tell you is that the last few conferences I have been to on particle and nuclear physics, there have not been any talks focused purely on string theory, and very few on supersymmetry. People are interested in string theory and things like AdS/QCD as tools of studying QCD, but not as independent theories. As for GUT, I have not seen any attempts at building a completely stand-alone theory. Which makes sense. We have Yang-Mills, and we know how to build a unified action. The key problem is that this does not, directly, lead to a renormalizable QFT. So most of the progress there stems from either trying to come up with renormalizable approximation or in learning to work with non-renormalizable QFT. There is progress in both areas, but it's outside of my expertise.