K^2

Precisely at the keys, the value will always be exactly what you've specified in the key. I suggest you review splines if it is not clear why it is so.

atan2 will give you the angle of the tangent. You don't need that. You just need the actual slope at each point. Then you construct a 3rd degree polynomial that matches the two end points and the two slopes at either end of the interval between the two keys.

I don't know if you can turn off atmosphere, but you can turn off drag for individual parts of the ship. If you go and find the config file for the parts, it has drag coefficients for them. Set these to 0, and atmosphere won't affect your ship. Of course, this only makes sense for a small ship with small number of parts. Also, be careful with parachutes. You should be able to set both deployed and stowed drag. If you set both to zero, parachutes won't help you land. If you keep deployed drag as it is, you should still be able to use parachutes without worrying about drag while testing things out.

Yeah, it's all sorts of wrong for a suborbital prototype. Not that a lot of valuable know-how needed for suborbitals can't be acquired from such a project. Though, if I had to point a finger at an X-something that most likely serves some of the military's suborbital research interest, I'd go with X-51, the WaveRider. DARPA's main interest in the project is probably as a cruise missile, but if you want a single-stage suborbital, you want hypersonic capabilities, compression lift, and scram jets. Well, air-augmented rocket, to be precise, but challenges in building one are mostly the same. I should really sit down and run some numbers on that. Just to see what it'd take to get a good single-stage suborbital, see how much better that is than an SSTO, and try to guestimate the launch and operation costs from that.

If it's going to be passing by Earth, it might be enough to maneuver it for an Earth-Moon fly-by, boosting it up to a couple km/s, and that gives you great many possible targets for study.

On a short jump like that, you don't need more than 5-6km/s. So it's going to be less than 10 minutes to accelerate and just as much to decelerate at 1 g. That's not where bulk of the flight time is going to go. Though, approach and landing can easily take another 20 minutes or so.

People who can afford sub-orbital aren't the kind of people who wait in lines to check in. They are the kind of people who are going to fly in to NYC or London on a private jet or a heli. Oh, and ICBMs from Russia don't take 2 hours to get to NYC. Sub-orbital to London will be under an hour with the takeoff and landing. Cost is another matter. People who could afford this are also not the sort of people who are going to pack like sardines into tiny capsule like some sort of peasants that have to fly economic. Nor wait for a week for a convenient flight time. If you can't organize at least one flight a day with first class sitting for a reasonable price, it's not going to work. With cryo LH2/LOX, composite tanks, composite structure, and something like an aerospike engine, preferably one that can run as a scram jet to improve efficiency, you might be able to just make it viable. But this is a huge risk on a fairly small and unpredictable market that no private company is ready to take. There is some military interest in a similar vehicles. USMC has payed good money for various studies on the matter. A vehicle that can deliver a small platoon of Marines would be just perfect for such a venture. If they have actually started research on it and they go to building prototypes in secret right around now (this isn't something you can test without it becoming public) they can have one in, optimistically looking, a decade. If they are working with one of the aerospace contractor giants, like Boeing, and part of the agreement is them being able to make use of the tech, (not unprecedented, consider AgustaWestland AW609) we might be looking at private operations of sub-orbital shuttles within two decades. And that's a very optimistic prognosis for it. Another route would be to try and use something like Falcon 9R first stage to get a boost for something with a much simpler build, and that can get you something like 20T sub-orbital, which can fit a small airliner-worth of people to make the whole thing relatively affordable, but we're back to the sardines scenario here, which I don't think people who can afford it would go for.

Hm. That's not so crazy. If they get cost of launch down with Falcon 9R as much as they are planning to, cost of launch per seat could be as low as quarter of a million USD. I don't think that's going to account for multiple launches every week, but this is affordable enough for the rich and famous to make some sort of an orbital establishment a viable venture.

If the difference in inclination of orbits is 12.6°, then the maximum angle between the two orbits is 25.2°. The rest of that figure is probably required for altitude adjustments. That isn't right. Orbiter's mass is 68,585kg. With 65,000lb payload at 1000ft/s delta-V and the 316s ISP of its engines, it would have only 10,000kg of OMS propellant on board. In contrast, de-orbit burn can last up to 1,250s at 26.7kN, which would require over 20,000kg of fuel. And the figure I found actually quoted for this is 21,660kg of fuel in two pods. I suppose, they might have been going for something like 1,000ft/s each, but in either case, Orbiter is capable of significantly higher delta-V. Unfortunately, even taking empty weight and full OMS fuel, I still come up with only 850m/s of delta-V, which would be easily enough sufficient to move to necessary altitude and match orbits if it weren't for inclination change. Even if ascending nodes of the two orbits were drifting fast enough to have them match before life support runs out, that's still almost 1.7km/s burn, as maccollo points out. And inclination change isn't large enough to save any fuel with a bi-elliptic transfer, or any other trick for reducing delta-V required for inclination change. At least, not by any amount that's going to matter. Thanks to everyone who helped clear this up for me.

Is there a reason why STS-107 could not rendezvous with ISS? Seems like it would have had sufficient fuel, propellant, and time to do any necessary orbit adjustments. They would not have any way to actually dock with the station, but surely, it would have been easier to EVA everyone between the two stations, perhaps, carrying some suits back and forward between Columbia and ISS, then it would have been to organize a shuttle-to-shuttle transfer in such short time.

Completely wrong. Electromagnetic field carries energy and momentum. Even if the field is your reaction mass, there is still reaction mass, which brings you back to limitations of the rocket formula. Best example of that is the photon drive.

Ship will maintain proper acceleration of 1G. As it gets closer to the speed of light, coordinate acceleration is going to diminish. That means the ship needs to accelerate continuously through the flight to keep increasing its proper velocity. You aren't going to get better coordinate time, but you can dramatically improve proper time required to reach destination. In other words, while it will always take 4.37 years to get to Proxima, by ship time, you can reduce it to months or even weeks if you can accelerate sufficiently fast. I recommend reading Wikipedia article on proper acceleration.

Anything interstellar. Of course, warp bubble solves that problem rather nicely, as pointed out in this thread already.

That is not what inertia is or how it works.

It's not that simple. If you take a circular-to-circular transfer, the optimization problem has very simple solution. Infinitely short burn to go to Hohmann, then another infinitely short burn at ap to circularize. Naturally, you can't do infinitely short burns, as that requires infinite thrust, but you do the best you can, and that's your fuel-optimal policy. When you start talking about transfers between general orbits, things are a bit more complicated, but you can still find optimum fairly easily by optimizing energy and angular momentum gains. Former would be accounted for with Obereth effect, yes. Later is optimal at maximum distance from parent body. (And that's the reason why you get the ap/pe burns in circular-to-circular optimization.) But things get way more complicated once you introduce drag. Losses to drag are velocity-dependent. Which means you can't optimize transfer points. You have to optimize the entire trajectory. And that's what this thread has been about. So far, the most successful method has been projection method with collocation points on Pontryagin conditions, it seems, done using a library designed specifically to solve this sort of a problem. The downside is that such a trajectory has to be pre-computed. What would be far more valuable is on-the-fly method. Unfortunately, it doesn't look like a precise solution can be done on-the-fly, but there are approximate methods which lead to a very sensible solution which would be very hard to beat. I've posted a bit of it earlier, for early ascent. I believe, I have a solution that will guide the rocket through the entire ascent at near-optimal fuel consumption. Should be far better than anything Mechjeb does, at least.

There aren't any. Obereth effect only affects the energy gains. Delta-V is always the same and depends only on ISP and mass ratio of consumed fuel.

Actually, Nuke is right that this is going to hurt the ISP. But it's usually not so bad as to make it useless.

That's not true. Rate of burn of fuel in SRB has exponential dependence on pressure. Increasing nozzle size would drop the pressure, reducing the thrust and the burn rate. Unfortunately, you'll also be losing a lot in terms of ISP. So while it's a way to throttle SRB, it's not an efficient one. There is a way to build an SRB whose throttle varies as the fuel burns. It's only going to vary in a predetermined way, but in most situations, that's all you really need. That and an ability to shut SRB down is sufficient to fly a ship powered by solid fuel only all the way to orbit.

Idobox, just being pedantic, perhaps, but moving charges don't radiate, because you can always go into coordinate frame where they are static. It's the accelerating charges that radiate. Of course, motion in any closed orbit requires acceleration, so motion in orbit always results in radiation. And yes, you are absolutely correct that this will also happen for accelerating magnetic dipoles and accelerating mass.

Precisely. That's why it's fusion-boosted, rather than a fusion bomb. Almost all of the energy still comes from fission, but way more of your fuel burns through. Yeah, I probably should have made it clear in my post. As for p-p chain, yes, the branching fraction of He-2 -> D is very low. But it's still dramatically higher than probability of more than two protons fusing together at the same time. And so as slow as it is, it's the dominant process in a lot of stars. As I've mentioned, the other process that may be dominant is the CNO cycle. p-p-p is nowhere even close, and p-p-p-p is just silly.

It is, indeed, DT fusion in fusion bombs, but modern devices don't store them as deuterium and tritium, because tritium is unstable. Instead, the fusion payload (as well as the fusion booster for the primary) are made out of lithium-6 deuteride which can be stored for a very long time without decaying into something inert. During the explosion, the fast neutrons strike lithium-6, taking it to lithium-7*, which has high odds of decaying into helium-4 + tritium. That tritium, consequently, fuses with deuterium to produce another helium and a fast neutron. So there is actually one more fission stage in a modern fusion bomb. The fact that lithium deuteride absorbs neutrons overall is the main reason why a fissile core is inserted into the secondary of a modern TU device. By the way, most implosion type fission bombs are also fusion-boosted. The yield of a pure fission bomb is only about 20kT, which isn't enough to light up a fusion bomb. A fusion-boosted bomb has lithium deuteride core sandwiched between the plutonium hemispheres. That takes the primary to about 200kT. This can be used either in its own warhead, such a smaller tactical nuke, or used to ignite the secondary of the TU device. In the later case, you get your .5 - 2MT yield of a typical ICBM warhead. Or more, if you needed, but warheads larger than 1-2MT are not cost-efficient. Fusing four protons at the same time is impossible even in an actual star. That's not how proton-proton fusion works. p-p fusion follows a p-p chain. And even in some stars, that's not the dominant cycle. See also the CNO cycle.

Probably, but how close do you need to get, anyhow? Are you not you while experiencing a strong emotion, like anger or excitement? How about when you have a flu? Or if you had a bit to drink? The brain is clearly capable of operating under quite a range of not quite perfect parameters. And no simulation will ever get it perfectly, but we just don't have to have it perfect. We can adjust the parameters we have control over until the person simulated feels "normal". Even if that "normal" isn't identical to real neutral mood behavior, we are still simulating that person's brain. Might be that person's brain when it's operating slightly off peak, but if it's not noticeable to the person being simulated, what else can we ask for?

That's actually a complicated question, and that knitting needle video is a big part of the answer. You can't have magnets orbiting each other the same way you do with planets. The reason is that there is no such thing as magnetic monopoles. All magnets have magnetic dipole, but no net magnetic charge. And dipoles, instead of the normal inverse square law follow the inverse cube law. And to have stable orbits, you do need inverse square law. But that's where that knitting needle video comes in. The electrostatic attraction there is also a dipole attraction. There is some net charge on the needle, but not on the droplet. However, the small net charge of the needle polarizes the droplet, and droplet starts to work like an electrostatic equivalent of a magnet. No net charge, but lots of dipoles. So why does it stay in orbit around the needle? The trick here is that it is a needle, and not a sphere. The fact that it's a long cylinder makes it work like a 2D problem. Only distance from nearest point on the needle matters. And in 2D, dipoles follow an inverse square law. So the droplet orbits the needle in the plane perpendicular to the needle, but mostly free to bounce around along the length of the needle. (There are some interesting edge effects that prevent it from "sliding off" the needle during all of this. So can this work with magnets? In principle, yes. You need one magnet that is long and cylindrical. But it's not just a bar magnet. Its North and South poles shouldn't be on opposite ends, but rather on opposite faces of the cylinder. This is important for stability. Now, you should be able to send a small magnet orbiting around it. I'd take a little puck-shaped magnet, and give it a bit of a spin to gyro-stabilize it, so that it's north and south poles are always aligned to attract to the cylinder magnet. Give the small magnet a bit of a push, and it should stay in a circular or elliptic orbit around the long one.

No, it is not stable. It is enough to go into a rotating frame of reference to see that the effective potential has a local maximum at the center. That means that the Kerbin's location in this system is an equilibrium, but it is not a stable one. It will start falling towards one of the Jools, and either end up its moon or get knocked out of the system. The too Jools will continue orbiting each other. If you put a ship in the center, instead of the planet. It can remain there by using very small amounts of fuel for station-keeping. Because as I've mentioned above, while it's not stable, it is an equilibrium, so you don't need a lot of force to keep something there.

And that's bad how? If you were to copy human brain exactly, molecule for molecule, and run original and copy side by side, they wouldn't behave exactly the same way either. Because that's what chaos means. So why should you demand more from simulation than the original?