K^2

300km is pushing it. Not because of Earth's rotation, but because a fall from 300km is going to get you going at nearly 2km/s by the time you enter thick enough atmosphere to start slowing down. But what use is that, anyways?

It is an option. I don't think anyone is saying it is not. You are not going to do worse landing than you are taking off. In fact, you can look at LM as an extreme case. Lithobraking wasn't an option, so they had to burn fuel to land. Same deal with aerobraking on re-entry. The problem, of course, is that it takes an enormous amount of fuel to do powered landing on Earth. As I've mentioned above, I haven't come up with anything that improves dramatically on simply killing (almost) all of your orbital speed. And that's 6-7km/s worth of fuel. When your typical rocket is capable of something on the order of 10km/s total, adding 6-7 more on top of that is insanely challenging. Right now, it's simply cheaper to have craft that are either not re-usable at all, or require servicing of the heat shields. And we can't do better with chemical fuels. We are going to have to find a completely different kind of fuel/propellant for our ships before we start looking at a heat shield as too much hassle. MSMH could do it, but we don't know if that mythical substance even exists. Much less how to produce it in necessary quantities for cheap. At higher energies are the core electrons, which nobody knows how to work with, or nuclear. Fission or fusion are going to be very troublesome for small ships, but maybe we don't have to go small. There is also some promise in nuclear isomers. Since there is no decay involved with these, the only radiation you have to deal with is gamma, making for a much cleaner reactor. Nuclear isomer batteries are also rechargeable, at least in theory, but that might not be worth the trouble. At any rate, there are some options in power generation. The biggest limitation with alternative power sources is getting decent TWR. If you want to do powered landing, you don't need quite as high TWR as you do for liftoff, but anything significantly less than 1 is going to be problematic. The best we've got is NERVA, and that only gives you about a fifth of its own weight in thrust. You might be able to do powered landing on one of these if that's all you are carrying. But you aren't lifting off on that. And once you add everything you need for liftoff, the ship's too heavy for powered landing on NERVA. (Naturally, I assume that multi-stage defeats the very purpose of building something that can be simply refueled.) So in short, yes, powered landing can replace heat shields. Yes, it would be very useful to have that capability. But no, we don't have anything even close to the tech you'd need to make it happen.

Nah. I did the math on that once. It's a good idea on paper, but by the time you have enough lift, you're in too dense an atmosphere moving too fast to survive without a heat shield. You might be able to do something with a powered descent, but all of the "simple" things I've tried result in you burning as much or more fuel than you'd need to simply kill your orbital velocity and drop down.

Sharp's right. If you simply broadcast a signal from 50,000km, there won't be much to receive due to inverse square law. You have to be able to focus it into a beam directed at Earth, or wherever your receiver is, and that means working with UHF or higher. If you want that beam to be steerable, it has to be a phase array. These things are way more advanced than a regular radio. If direction can be fixed, you just need a dish. And like I said, what's your background? How much do you actually know about electronics, because this isn't something you are going to do from a blank slate. If you are serious about it, and you don't have much in terms of starting knowledge, you'll have to start with books and probably some classes. I can point you towards some resources, but I need to know what you are starting with. How much do you know about electronics, general physics, and mathematics.

What's your background? But basically, you need to understand how every part of a superheterodyne works to understand radio communication.

But can it sell me car insurance? There is a lot it can do while staying tethered to a home station, which can be an ordinary rover. But yeah, such a machine would be most useful in places where a rope won't reach. So it'd have to be fully autonomous. Capable of mapping the cave and finding its way back to the surface. We do have systems that allow for that, but it'd be a risk.

As some people have mentioned or alluded to, re-entry without heat shields is entirely possible, depending on maximum altitude attained. Re-entry from orbital flight, however, is not. A drop from 100km is only going to have 1.3km/s worth of energy in it. You don't need heat shields to survive that. Re-entry from LEO, on the other hand, is going to be on the order of 7km/s. That's a whole different game. This does suggest one way to return from orbit without heat shields, however. Just do a retrograde burn to kill all your orbital velocity and drop almost straight down into the atmo. Problem is that the fuel required to do that is going to be heavier than a heat shield.

Not that simple. The problem is that the integration method matters a lot. Euler just isn't going to cut it. In fact, any explicit Runge Kutta method, including Verlet which is so popular for physics simulation, are going to give you very poor results. The 1/R potential of gravity is very tough to integrate over. Any explicit method is going to end up accumulating kinetic energy errors. And because solution is very sensitive to object's velocity, you are going to end up with a very large error in trajectory. Unfortunately, the only way to get a good trajectory computation for a ship affected by many attractors, even if later are assumed to be on rails, is to run an implicit method. Typically, 2nd or 3rd order Gauss-Legendre method is used. I did some tests a while ago when the news of the comet heading for Mars first appeared. I managed to pull NASA data for the comet and all of the important massive bodies in the Solar system to see if I can run my own integration routine for predicting the closest approach. Even using Verlet with 1 second steps, after just a few months, the trajectory was off by enough to make it useless, and velocity was off by a few percent. If you are going to make a video game, it's not such a bit deal. You can go ahead and just Verlet the whole thing. But you'll end up with precession of orbits, overly energetic fly-bys and many other artifacts of poor computation. If there is no need for player to do any super-precise trajectory estimates, that's not too bad. But if you want to have any predictive power at all in your simulation, you have to spend a lot of effort on that integration method.

You are thinking in 1D. In 2D, rope would only kill radial momentum. Any angular momentum they've had relative to the point where the ropes are attached would remain, so while they are no longer drifting away from the station, they are going to be drifting across, generating some tension in the ropes. It's not going to be much, but it really wouldn't take much given the situation. In contrast, once Kowalski disconnects, that very tension is going to accelerate her towards the station. Again, perfectly accurate.

Hm? Why? I can easily compute the proper acceleration vector for an object at rest in the moving Schwarzschild metric using Christoffel symbols.

First of all, you have to be careful when you say that mass in creases. Inertial, or relativistic mass increases. Invariant, or rest mass is going to remain exactly the same. In modern physics, the word "mass" typically means the later. Now, gravitational mass is, indeed, identical to the relativistic mass. But it's still a little more complicated than that. Source of gravity isn't just mass, but the stress-energy tensor. For a point object, it depends not only on object's mass, but also on its momentum. An object moving really close to speed of light is going to have a different gravitational field around it, but not in such a trivial way. Unfortunately, this is way outside of what you can describe with classical physics. This really has to be treated from perspective of GR, where the answer is going to be governed by a Lorentz-boosted Scwarzschild metric. Qualitative effect, however, is going to be a lot like magnetism. You will experience stronger pull if you are moving in direction opposite to the object, but a lesser pull if you are traveling in the same direction.

Ok, fair enough. But it's still somewhat limited. Harpoon's warhead is "only" 221kg. Yeah, I know that's still a lot, but compare it to a typical missile cutter anti-ship which is typically in 400kg ranges. The LRAM's warhead is also expected to be in 1,000lb ranges. The fact that these things are being built, and indeed, still developed, suggests that the 200kg warheads of some anti-ship missiles just isn't getting the job done. What surprised me was that I was unable to find any current anti-ship missile in U.S. service with a warhead heavier than Harpoon's. Tomahawk TASM seemed to be the weapon of choice, but it's been withdrawn in the 90's with no replacement. And indeed, looking at Zumwalt's armament, there is absolutely nothing there for fighting larger ships. It carries some anti-sub missiles, the ESSMs which are good against FACs, and tactical Tomahawks. It doesn't look like U.S. plans to deploy their stealth destroyers against any serious navy. So any comparison to a sub is a flawed one to begin with. Zumwalt's mission is pretty clear. It's designed to engage military land installations using Tomahawk missiles. The rest of its armament is designed to protect it from a 3rd World navy. Though, the fact that LRAM is being developed suggests that they are keeping other threats in mind, and it will be easy enough to retrofit ships like Zumwalt class with these.

Operate a passive radar, receive communications via sat*, carry ship-to-ship missiles, not have to worry about being detected while coming up for air, and still be cheaper to build and operate for the same offensive capability. There are a lot of advantages. Of course, they can't completely replace the subs either. * As a license diver I can tell you first hand that you can't even get cellphone coverage just 30 feet bellow.

And the result will be nothing like you'd expect from Special Relativity. SR deals specifically with a Minkowski metric. Near the event horizon, the metric is going to look very different. The line element in polar coordinates in SR looks like ds² = dt² - dr² - r²(dθ² + sin²θ dƲ). In the exterior of the BH, it's (1-rs/r)dt² - 1/(1-rs/r)dr² - r²(dθ² + sin²θ dƲ). When you are far from the event horizon, r >> rs, the two are very similar and you can talk about effects of Special Relativity on a ship in orbit. But for r only slightly above rs you have to discard all of your SR notions and deal with General Relativity directly. Kepler's laws break down way before you reach event horizon. In fact, there are no stable orbits bellow 3rs. So you can be a full diameter away from event horizon, and Kepler's Laws are already completely useless.

H2O is not the best propellant. You are much better off with H2. Basically, the lighter the gas, the faster the molecules are at the same temperature. So you get better ISP. The other problem is that an electric arc is a horribly inefficient way to heat something up. You'll be getting more heat deposited in your wiring than propellant. You should be able to do much, much better with a microwave. Set up a cavity to heat up the propellant with RF radiation and use the heated gas for propulsion. You won't beat an ion drive in efficiency this way, but you might be able to get better TWR without huge losses. Well, these theories lead to NERVA with the same modification. Use H2 gas instead of H2O for better ISP.

Well, sure. I don't think a ball of electric tape would survive a re-entry. But all of the wiring is held together with electric tape, and I wouldn't be surprised if some of the structural connections are also held with tape. If you ever have to fly a Soviet/Russian space craft, have a roll of tape and an army knife with you. Might save your life.

Lots of electrical tape. Lots. They didn't have duct tape in Soviet Union, but they did have electrical tape and they weren't afraid to use it. Go to Smithsonian. Look at the Soyuz that hangs there. You'll know it's true.

Ah, yes, that does make sense. It still makes things easier on the nose as well, even if it wasn't the main factor. You'll notice that re-entry capsules have a blunt shape despite not having any wings to protect. Normal shock in front of a blunt nose does provide extra thermal protection. Same goes for the leading edge on wings. I wouldn't put it quite that way, though, it's not actually wrong. Just paints a picture that might be off. The oblique shock wave as it spreads from the ship is going to have the same angle. That depends only on velocity, temperature, and to a lesser degree density of the gas. So you don't have much control over it. What the flat nose does is create a region of normal shock which is equivalent to shifting the focal point of the oblique shock forward. Of course, this does increase the amount of space inside the shock, so you are absolutely correct about it making a difference to the wings. The cost, of course, is significantly increased drag. Normal shock is much more "expensive" in terms of resistance than an oblique shock. It's a good thing if you are building a re-entry capsule. But it will cost you on the way up if you are building a space plane. That's interesting. I haven't really thought much about what all of this means to an SSTO. Basically, you trade a reduction in drag during later stages of ascent for added complexity in terms of materials and cooling. I would not have guessed it to be worthwhile, but I'm sure Reaction Engines actually done their homework on it. If they think they can make it work, more power to them.

I don't know. It's pretty obvious to anyone who does orbital mechanics regularly. The moment I saw your post about two spheres I knew what was going on. Maybe whoever made the key just didn't think of it as something requiring explanation. But yeah, I can see how this can be confusing if you aren't used to it. When you have just two objects, -GmM/R is the TOTAL gravitational energy. You don't get -GmM/R per object, but rather for the whole system. There is a way to derive that, of course, but it requires a bit of algebra. So GMM/a - GMM/b is the amount of energy that will be shared between the two neutron stars in this problem right before impact. That means the kinetic energy of each object in CoM frame is going to be M(GM/a - GM/b)/2, and this gives velocity as sqrt(GM/a - GM/b). The factor of two goes away. You should also keep in mind the fact that the equations you are using are from classical mechanics. The answer you get is almost a tenth of the speed of light. That should be a warning sign that classical equations are probably going to be wrong. And indeed, when computing velocities for collision with neutron star, especially if the thing colliding into it is another neutron star, you have to use equations from General Relativity. To be honest, I'm not sure I'd know where to start on solving this problem correct. I know how to solve for a small object, like an asteroid, striking a neutron star, and I know the equations I'd use would alredy give me a different answer.

That's not how physics works. Nibb31's complaint was absolutley valid. There HAS to be a force. It could have been an inertial force from a rotating frame, however, as I pointed out.

Reduced mass. This is because both objects are moving. You can either think of it as each object being attracted by a baricenter at reduced mass, or you can think of it as gravitational potential energy ending up being split between the two objects. Either way, you end up with velocity less by sqrt(2). And that's the correct answer.

Then, I guess, the only plausible explanation is that not everyone made it. We've only seen a small section of the ISS. Perhaps some modules lost pressure and have been sealed off by the crew during evacuation. Edit: Or they've only had one pilot, so they decided to take a chance on riding in one Soyuz despite the lack of seats. Re-entry should be surivivable with an overcrowded RM.

Yup. You forgot the reduced mass.

Wherever you got that 28'874'613.94 number, it is not the correct answer for the formula you provided. Even doing this by hand you can see that the answer is about 4x10^7. The 1/b term is absolutely irrelevant, because it is much, much smaller than 1/a. So the whole thing becomes Sqrt(2 * G * M / a) = Sqrt(2 * 6.67x10^-11 * 2.5x10^30 / 2x10^5) = Sqrt(6.67*2.5x10^14) and that's roughly Sqrt(16x10^14) = 4x10^7. Don't over-rely on calculators. Learn to do estimates without them. So either the "correct answer" is wrong, you have the wrong formula, or one of the parameters is wrong. I'd suspect a, because the formula is obviously for the orbit around a star comparable to Sun, and nothing's getting within 200,000m of its center. (If this is for orbit around black hole or neutron star, the formula is wrong to begin with.) Edit: If this is for two objects orbiting a barycenter, have you taken reduced mass into account? It'll make a difference by root(2) in this case, which exactly accounts for the difference.

They explained that in dialog. Both stations had two ships parked to be used in case of emergencies. That does seem to be a typical configuration. Even during the Shuttle age, I see a lot of pictures with two Soyuz ships parked at the ISS. (for example edit: Hm. Or it might be a Progress. But there are at least four places where Soyuz/Progress can dock, so I don't think it's implausible to have two Soyuz there at any rate.) According to the dialog, both stations were evacuated using only one of the ships, leaving the beat up Soyuz at ISS and a working Shenzhou at Tiangong. I was thinking about that, and it's not as implausible as it seems at first. This could have been due a tiniest amount of force. It really wouldn't take much. So if there was even a bit of angular velocity, he'd be slipping away the way they've shown it, creating just enough tension in the ropes to do what happens next. The way they've shown it is a bit strange, but the situation itself is entirely plausible.