K^2

To build something that will take generations a colossal effort? There have always been a reason. Sometimes it's a stupid religious one, as is the case with pyramids, but unless you're proposing that we'll build space elevator because the church decides it's the thing to do, that sort of logic doesn't apply. And for pretty much everything else built on the same scale, nothing short of fear of extinction has ever been enough. As I've said above, launch loop is more feasible, and if we really get to the point where it's that or total annihilation, we'll build that. Space elevator will never be built. I'm pretty confident saying that. Not on Earth, at any rate.

Arbitrarily long is a bad word. You don't need them thousands of kilometers long. A few dozen meters is enough to make a sufficiently durable compound material. But it doesn't matter how strong your materials are. That's not even the most significant limiting factor. Space elevator requires impressive infrastructure. You need a sufficiently massive counterweight, so that Coriolis forces of ascending capsules don't deorbit it, and then you still need to transfer momentum to it. That means you need an angled cable. Whether it's the main cable or an auxiliary one is a separate question. And that brings in a whole new level of challenge. People picture some sort of an idealized picture, with a perfectly vertical column/cable and a moderate size station as anchor. That just won't fly. You need an entire host of support structures and a hell of an anchor. It makes other proposed megastructures like a school science project in comparison.

Launch loop is way more practical than space elevator could ever be. There is zero reason for us to ever build space elevators.

I'd like to hope that shortage of acetylene in Titan's oceans is due to metabolism rather than geological process. So fingers crossed for us changing that statement in the next few of decades, and adding methane-carbon-??? to this list. But for now, this is definitely true. Hence the liquid water being pretty much the definition for habitable zone. The reason that I'm suggesting that Earth would be frozen on the surface on the outer rim of habitable zone is because we allow for a bit of an overrun to allow for something with a very dense atmosphere and very strong greenhouse effect to maintain water just above freezing, when it would have been an icicle otherwise. Which means Earth would actually have 100% ice coverage on the very edge of habitable zone.

Lets rephrase it a little bit. Lets say that in the first group, probability of having a trait is p. We need to ask two questions. What is the probability distribution of p? What is the probability of the second group has the same probability distribution? Both questions are answered with Bayesian Statistics. To start with P(p) = 12 p(1-p)²/4. (General formula: (a+b+1)!/(a! b!) pa(1-p)b, for a that have trait and b that do not.) Probability that second group has the same p is given by ∫P(p) (1-p)³ dp = 2/7. (General formula: B(a+1, b+n+1)/B(a+1,b+1), where n is number of elements in second set, and B(x,y) = (x-1)!(y-1)!/(x+y-1)! is the Beta function.) We can also run this thing backwards. Probability distribution for odds of second group having a trait is 4 (1-p)³, and odds of first group matching it are ∫P(p) 3 p(1-p)² dp = 2/7. Which is unsurprisingly the same. So the odds that these two groups are the same are 2/7, giving you an estimate of 5/7 that there is a difference between them. Which is, indeed, significantly better than a coin toss. Keep in mind, though, that this is just the best estimate on given information. It's going to have very large uncertainty.

OP specifically mentioned going to the outer edge of the hab zone, and that means permafrost on any world without significant greenhouse effect. Combine that with going to inner edge on the other side of the swing, and I cannot imagine any life that would survive on land. I should, perhaps, specify terrestrial life. But if we imagine life that's drastically different to our own, our definition of habitable zone might not be applicable either.

Educated guesses, since experiments in this regard are rather limited. (You can "fake" a time loop in some optical setups with lasers, but it's not exactly the same.) Nonetheless, all of the theory that doesn't blow up seems to suggest that observer would encounter history that's only consistent so far as his own timeline is concerned. It's easiest to interpret via MWI, because you can think of contradictions as observer simply encountering different parallel timelines, but the gist is the same in Copenhagen. In other words, no fading photographs, Marty McFly style. Instead, you'll just have a piece of documentation or memory from your other visit to the same period that doesn't match what's currently happening. If you subscribe to Copenhagen interpretation, or any other interpretation involving collapse, from perspective of other observers, a traveler who steps out of a CTC wormhole might as well be a random quantum fluctuation. Astonishingly unlikely one, but that's the sort of stuff you get in your QM when you construct time loops. Honestly, time travel will ever only make complete sense to the traveler.

4 bar - depends on composition, but if it's similar to Earth's, it should be fine. Cloud cover might end up being denser, which means even warmer temperatures on the surface, but so long as life adapts to that form the start, I don't see a problem. Eccentricity is a bigger concern. Instead of conventional seasons due to axis inclination, which are balanced between the two hemispheres, you are going to end up with sizable global temperature swings. All of the surface is going to drop bellow freezing near the aphelion and climate will turn very hot and very humid at perihelion. I'm pretty sure you can forget about land life under these conditions. But oceans should still remain habitable. Surface will probably bloom with algae every "summer", and that will provide sufficient nutrients for more complex life to winter in the depths. Could be an interesting world, but with no land life and reduced biodiversity, odds of it harboring something intelligent are greatly reduced.

In terms of intuitive understanding, this is about relationship between coordinate systems, though. People keep talking about "vertical" and "horizontal", which are local concepts. Hence confusion. And yes, we should try and limit this particular discussion to inertial frames for that very reason, but I still caution about telling someone they are flat out wrong about centrifugal effect being a factor. Choice of frames is an important caveat and should be pointed out. There is also the fact that you cannot always choose a frame that is inertial everywhere, but that is a separate story.

Again, depends on the coordinate system. If you write out equations of motion in polar coordinates, there is a centrifugal term. I'm trying to avoid this discussion for Fez' benefit, but you should be aware of it.

Not in any inertial frame of reference.

Other way around. Van Allen belts are formed due to solar wind getting trapped in Earth's magnetic field. And yes, as kerbiloid pointed out, it's particle radiation, not EM radiation, since EM radiation isn't affected by mag fields.

What kind of a ceiling are you using for space? I am getting guaranteed full coverage of the sky above 60km with 332 stations. 12 on verts and 16 on each face of an icosahedron mapped to sphere. 1.1 * (4pi)/(pi arccos(R/(R+h))²) This gives me the low end estimate based on solid angles. The factor of 1.1 accounts for the hex packing. This is about 250. Next possible packing is with 4 stations between any pair of the 12 verts. This puts total at 332. Of course, the underlying assumption is that the surface is sufficiently smooth not to have any other kind of obstruction.

Don't let them read Machiavelli, though, or they'll realize what you're really up to.

Yes. If the force was constant, higher orbits would have to be actually faster. The fundamental problem you are having with picturing this is due to the frames of reference. I'm sorry I can't explain this better without throwing a bunch of math at you. But I can give you another analogy that might help you a little. Picture a rock that you've thrown at an angle. We'll ignore air resistance. Then rock travels along an almost perfect parabola until it strikes the surface again. This is what you normally picture as falling. Now imagine that earth surface wasn't there, and all of Earth's mass was concentrated at the center. (Neutron star densities would suffice, actually.) What would happen to the rock? Well, it'd still travel along the same curve where you've thrown it, but then it would drop down bellow, take a turn around the center, and come back up, completing an ellipse. In fact, it would be traveling along a classical, albeit very elliptical orbit. When you see something thrown into the air, and as it drops down, it does in fact merely traverses a very small section of that orbital motion.

Sure, and a single video conference takes up the entire bandwidth. Ok, maybe you can have ten with crummy quality and compression. This is not how communicators work in sci-fi. @fairytalefox Absolutely. And it's kind of a shame, because we are nowhere near the point where technology is too complex for everyone to know at least the basic principles and limitations behind everyday things. You could fit pretty much everything electronic into a single university course and everything mechanical into another.

First, let me point out that there is such a thing as hard sci-fi. There aren't many authors who try to stick to laws of physics and imagine how technology might actually work in the future, but there are some. These few examples aside, yes, all of these things were magic as written in the original setting. I mean, I can use Skype on my phone to talk to someone in another country face-to-face. This was originally accomplished with magic mirrors in many folk tales. Are these no longer magic? Just because we've learned to replicate the effect with technology, doesn't make their original concept any less magic. How were the range and bandwidth issues resolved in the old sci-fi stories featuring wireless phones? I'll tell you how, they weren't. They magically went away. Same with power consumption issues. In real life, these didn't go away until we invented concept of cells and constructed immense infrastructure of cell towers. Sure, the result is the same, you have a device in your pocket that lets you talk to or even see someone on the other side of the world. But it works completely differently from magic point-to-point communication of every sci-fi story ever. The fact that they called their magic "radio" doesn't make it any less magical, because radio communication doesn't work like that.

That's bad to throw as a blanket statement. For a moment, picture a perfect world with perfect energy to thrust conversion. Suppose that you have a drive with energy source separate from propellant. You will consume quantity of fuel dM to produce quantity of energy dE, at some constant dE/dM = kc², which is used to accelerate propellant of mass dm. Since we expect really high energy, we need to consider relativistic limit. As such, dp = sqrt(dE²/c²+2 dm dE) = sqrt(k²c² dM² + 2 k c² dm dM) For convenience, lets also define fraction x = dm/dM, so that dp = c dM sqrt(k² + 2kx). Now, we wish to maximize impulse gain, dp, while minimizing fuel+propellant we've expelled. Which is equivalent to maximizing sqrt(k² + 2kx)/(1 + x). This function has very interesting behavior. For "low" values of k, such as we see with nuclear reactions (k ~ 0.01), this function has a maximum at just bellow x = 1. (Wolfram Alpha Plot) Which means that the amount of propellant you should bring should be equal to the amount of fuel on board. (This assumes you'll be ditching spent fuel as you go along, by the way, so that the ratio stays constant.) So an Orion drive should have a damper roughly equal to the mass of the bomb. This behavior starts to change as fraction of energy gets closer to 1. At k = 0.5, you should already bring significantly less propellant (Wolfram Alpha Plot), but things only really get interesting when you get to k = 1, which is the case for matter-antimatter drive or a black hole drive. Here, with total conversion of fuel to energy, you get optimum at x = 0. In other words, you shouldn't bring any propellant at all and use radiation to propel yourself. (Wolfram Alpha Plot) So it's not that radiation is useless for thrust. It can be absolutely the perfect propellant. But its efficiency depends on your conversion ratio, and photon drives only make sense when conversion ratio is damn near 1.

A little bit less, but yeah. The actual formula is g*R²/(R+h)², where g is surface gravity, R is radius of the planet (i.e. distance of surface from CoM) and h is the altitude. You can compute the exact value for h = 80km, or whatever orbit you have in mind.

At the rate equal to the strength of gravitational field, which is going to be slightly weaker than surface gravity in the low orbit. In other words, a little less than g, unless your orbit is really high.

No, if you look at height and vertical velocity, you are inherently looking at it from a polar coordinate system. If you insist on doing it that way, besides the force of gravity, you must include the centrifugal term. Math gets a little tricky really fast if you are using anything other than Cartesian coordinates. In Cartesian coordinates, just picture a velocity vector for each space craft. The arrow for the faster one is way longer. Now shift the tips of these arrows down by the same amount. Clearly, the shorter one is more sloped downwards. That's what causes the slower satellite to drop while the faster one remains in orbit.

Huh. Not bad. Storing fuel in long tubes would be pretty inefficient in terms of fuel/tank weight, but at well over 400ks impulse (I'll check this number when I have a moment), it'd be shame to complain even if that ratio gets as silly as 1:1. Yeah, it's definitely a bit on the far side, but I would invest some research into it when we get to the point where we could make practical use of something like that.

Forget the pressure chamber and the nozzle. How would you store that fuel in the first place? There's also significant loss of ISP due to the weight of all that uranium, meaning temperatures and pressures are going to have to be absolutely insane to get an actual ISP leverage over, say, H2 NTR.

Both craft are going to accelerate towards the planet at exactly the same rate. But it's hard to gauge by just looking at, because instinctively, you're looking at altitude above ground as reference, and that inherently puts you into a different frame of reference, where you have to start thinking about inertial forces. Anyways, if you actually draw velocity vectors for both craft, and look at how velocities change, you'll see that the rate of change is the same.

It should be noted that ISP of an LH2/LOX rocket is already very close to the 500s limit on the NTR. Both operate very close to thermodynamic limits of conventional materials. So applications for a water NTR are going to be somewhat limited.