K^2

I don't think there is a very good way to do this by the numbers, but a way to adjust position is this. 1) Place satellite in geo-stat. 2) If it's too far ahead, raise the orbit somewhat. 3) If it's too far behind lower the orbit. 4) Once satellite is over desired position, return satellite to geo-stat. This might take a few iterations if you overshoot. You might also want to adjust the altitude in several steps to reduce how much you overshoot by. This is not an exact science without some sort of ground tracking. But you should at least be able to make corrections this way, and that should allow you to place the satellite where you want with some effort.

Can you cite me a single proposed resolution of Olber's paradox that breaks down under c -> inf?

Things would be boring. For starters, it'd take absolutely forever to get anywhere. Without relativistic effects you could hope to cover what, a few hundred light years in human life time? With relativity, you can travel between galaxies. And I'm not even talkinga bout warp drive which could improve on that dramatically. That's a total non-sequitur. There is no correlation between speed of light and inverse R-squared law. It's the later and the fractal nature of the universe that prevent you from going blind.

Again, what do you mean by rotation? Coriolis force doesn't seem to be necessary. It contributes, but we can have weather without it. Uneven heating is crucial however. And no, we don't have any empirical data for planets with one without the other. How could we?

I thought I remembered something about there being multiple cells of high/low. I wasn't quite certain if that was the cause for Westerlies, though. Good picture to have in mind. Thanks, westair.

What do you mean "didn't rotate"? If it faced the Sun with the same side all the time, there wouldn't really be much of weather at all. Just tremendous winds blowing along the surface across the day/night terminator. The sun-ward side would be close to 150°C generating major thermal updrafts. Most of the water would be accumulated as mountains of ice on the night side. Liquid water would only exist underneath all that ice due to tremendous pressure. Atmosphere would be a mix of carbon dioxide and nitrogen and quite a bit thinner than it is now. If the Earth really had zero angular momentum, then it would have a 1 year long day-night cycle. That wouldn't be quite as bad, but pretty close. The most interesting consequence is that you'd have liquid water on the surface. Water released from melting ice on the "morning side" would be blown to the day side by the aforementioned winds. Now, there is not enough energy in sun light released over the year to boil away all of that water. So the temperatures will only reach boiling in a very small region if at all. This means the temperatures well under 100° through most of the day side, and that means atmospheric pressure would probably be much closer to what we have now, with composition dominated by nitrogen gas. Water vapor carried by upper atmosphere to he night side would also form quite fantastic clouds and rather powerful blizzards once all that moisture is brought down. This I might actually call weather. Finally, we can look at one more possibility. Suppose, Earth orbited a much colder Sun at such a close proximity that it had a "year" that is comparable to our day in duration. Then it could have day-night cycle comparable to ours without rotating. Of course, this situation can't last very long, as an object orbiting that close to a star would quickly become tidally locked, but until that happens, such a planet would have weather patterns very similar to our own. The only effect that's missing is the Coriolis effect, and it's not that important for the weather. Mostly, you'd lose prevailing winds. So no Westerlies or Trade winds. Coriolis effect also helps form cyclones. Note that we'd still have these. We'd just end up having about an equal number of clockwise and counter-clockwise cyclones in both hemispheres. That can lead to some interesting phenomena whenever two cyclones rotating in opposite directions would happen to collide. But otherwise, the weather would be much the same. P.S. Note, I'm not saying the climate would be the same. GROOV3ST3R's note is valid for that last scenario. Changes in wind patters would certainly shift climate zones about.

Yup. As of .20, the first module that can hold crew - will. But you can still have extra pods that will start out empty. Or if you already have a rocket you like, just strap an external chair on it. Though, re-entry with these is a bit tricky. I've lost one kerbal to a parachute opening.

The time travel paradoxes only appear in classical mechanics. Any field theory completely resolves grandfather paradox. I can go into details, but there is a lot of math. In practice, the easiest way to think of it is the alternate time lines explanation. E.g., if somebody travels back in time to prevent a catastrophe that led to him being sent into the past in the first place, you can think of it as there being two worlds, one where things are bad and one where things are good. Person from the "bad" world comes back and ensures that the "good" world can exist. It's kind of up to interpretations whether that's what is actually going on, but if you use this sort of logic to sort through what would actually happen, you should get the right outcome. In terms of where we are with understanding of time travel, the particular field theory we use to resolve paradoxes is Quantum Field Theory. And because we understand how time travel works in confines of General Relativity, we can actually solve a certain class of time travel problems on paper. Specifically, ones where the time machine exists regardless of any choices made, but whether somebody uses it to go back in time can depend on the circumstance. Unfortunately, if the very creation of time machine is subject to changes you make by time-traveling, the exact solutions are not known. Using the aforementioned tools, we would need theory of Quantum Gravity to do that. But in general, it should behave the same way. Again, I'm oversimplifying a lot of this. Just explanation of how time travel in GR works is rather involved, because there are all sorts of rules and limitations. But this is the gist of it.

Keep in mind that anything that's further than certain distance from your main ship isn't affected by physics. So even if you have parachutes deployed on these stages, unless they are within a few km, they'll smash into ground as if neither parachutes nor even atmosphere are even there.

There is no accepted theory that requires extra dimensions nor is there any experimental evidence of such.

There is no such thing as escaping the universe. You cannot reach any part that is outside. Mostly, because there is no outside. Thing of the universe less as empty space in which galaxies are flying apart and more along the lines of the surface of a balloon that's being inflated.

There are ionic liquid lubricants that can form an almost perfect air seal, hardly evaporate even in vacuum, and allow for very low friction. Of course, the joints have to be built to within a few microns for this to be effective. So while the solution is conceptually simple, it is extremely expensive.

By far the SIMPLEST way to make liquid fuel on Mars would be to build pressurized greenhouse pools on the surface and grow algae. Just pump in CO2, pump out oxygen, and use organic material algae generate to make ethanol. No complicated chemical processing required. No energy other than solar required. The downside is that this would have to be built over significant area, and keeping it pressurized might be tricky. (Pressure needs high enough for water not to boil at room temperature.)

You can't really prove 1 + 1 = 2 in the framework of standard algebra, because in the frame work of standard algebra 1 + 1 is the definition of 2. More specifically, a sum of any two integers must be an integer. (You may replace integer with any other ring, including rationals, reals, etc.) So there exists an integer that is equal to 1 + 1. We assign label "2" to that integer. You could build algebra differently, starting with a different set of definitions, but then you really have to specify what your axioms are before you can ask for anything to be proven. Standard way to define natural numbers, however, is via addition of a unit. 2 ≡ 1 + 1 3 ≡ 2 + 1 4 ≡ 3 + 1... So long as we agree that no element repeats (for example, I could have set 0 = 4 + 1, in which case we have numbers 0-4 with a Z5 algebra) you get the set of natural numbers. Once you throw in the negative elements, you have your ring of integers. Therefore anything + 1 is just matter of definition. But then if you want to prove 2 + 2 = 4 in that framework, you actually have to construct a proof, albeit, a simple one. P.S. Yes, there is a set-theory method for constructing a set of natural numbers, in which case 1 + 1 = 2 is a derived result. But that's not a terribly natural way of approaching counting, unless you happen to be a set theorist.

Stochasty, yes, I've derived a general form for optimal ascent by minimizing a functional. Assuming exponentially decreasing density (which is true in KSP) the optimal ascent requires you to always travel at the free-fall terminal velocity. (This is the same as trivial result for constant density.) So until acceleration becomes significant in upper atmosphere, your TWR should be exactly 2 and you should be climbing vertically. I have not been able to work out the optimal gravity turn yet. I need to write a very specialized program to solve for that, and I've been lazy, but for Eve it does not matter. Your ascent must start with vertical climb out of the atmosphere.

The framework of mathematics is universal. Particular algebras are not. Some people are convinced that 2+2=4 is a universal truth. It is not. It's a consequence of the way operations and relations are defined, and these are not universal. In fact, people study many different algebras. 2+2=1 is just as valid in Z3 for example. What people normally understand as addition and multiplication is covered by algebraic rings and fields. Though, even with the relationship between the two operations locked, you still have quite a bit of freedom, such as with the Z3 example. Individual operations are studied by group theory and there you have even more possibilities because you no longer have to worry about distributive properties.

The fact that gravity doesn't behave like a force anymore at this point. Objects move through space-time at a constant speed. The speed of light. What changes is how much of that is along spatial directions. Because gravity is properly described as space-time curvature, the maximum local velocity will always be the speed of light. By the way, you can exceed speed of light when passing a rotating black hole. Your local speed will still be limited to c, but you can be moving FTL relative to remote objects due to frame dragging.

This is basically along the same lines as question that someone asked about grabbing onto the space station while passing it by on a re-entry trajectory. I'm guessing, that thread got lost with many others in the crash. Basically, yes, relative velocities are too high. But if, say, you used a really long and really strong bungee chord to attach yourself (or your ship) to the station, if nothing breaks, you'll end up with a weighted average of the two velocities (weighted by mass due to conservation of momentum). Whether that puts you and the station on re-entry path or if you stay on orbit will depend on exact trajectories and masses. But station's orbit will most certainly change.

after the fact. Naturally, that would let them filter what does and does not become public, but to merely remark "lets roll" during the liftoff of the first human launch must have taken some restraint. And there are no do-overs there. So I'm guessing, the cosmonauts were also instructed to keep language clean in event that party does decide to make the recordings public for PR purposes.

That's the black sphere result, isn't it? Albedo of most objects results in lower values because they tend to be "ligher" in visible light and "darker" in infrared. Of course, the atmosphere tends to drive the equilibrium the other way.

That doesn't really make a difference. Logic is similar to pendulum fallacy. Now, if control systems varied thrust of individual engines to maintain balance, then spreading engines out would help.

There are a number of issues with VTOL. Biggest one is that the vectoring on engines becomes less and less efficient as the center of thrust is moved higher up towards center of gravity. If center of thrusts ends up above, you actually get engines actively trying to flip the craft over. So first thing is that if there is room at all, move the lifter jets down. Second issue is that gimballing initiates when you fire a stage. If you just activate the engines via action groups, you get no gimball control. So fire all of your engines in one stage, then shut down these you don't need with action groups. When you re-activate these with action groups, they'll work as supposed to. Finally, as fuel is used up, center of mass shifts. Your VTOL might start out perfectly balanced and hover steady, and then a few seconds later start turning over. Before you can properly react, the thing is flipping over. You have to make sure that fuel is being used up symmetrically around center of mass. Not an easy task, but that's the only way you'll get a good VTOL. P.S. Which mod is that cockpit from?

That is the correct relationship for a black body, by the way. Distance should be measured from center, by the way. So if you have distance from surface, add the radius in.

Point mass approximation only works for approximately spherical objects. For a perfectly spherical object, it is exact. Edit: Sorry, didn't notice the post right above mine.

Nozzle diameter also depends on the fuel. You'll probably only get it by trial and error. And avoid metal parts. Once you start getting close to reasonable nozzle sizes, you WILL have explosions.