K^2

The only times I outright forgot to add parachutes, I was testing landers, and I always go for >1 TWR, so I was able to do a safe-ish landing. I have had all of the other mishaps, however. Making staging mistakes, putting them on the wrong action group, opening them too early and having them ripped off. I once tried for a multi-module ship with parachutes connected to the command module via a docking port. Well, turns out these ports aren't strong enough to support a command module. Splat!

The question is how long is the "long run". For martingale specifically, it's not long enough. If you happen to have 2n minimum bets worth of money, you can absorb n-1 losses. On n-th, game is over. In other words, your expectation for a single round of martingale is winning 1 minimum with probability of 1-21-n, or losing it all with probability 21-n. Situation changes if you manage to double your starting capital, requiring 2n consecutive wins. In other words, the odds of you successfully doubling your capital are 1 - (1-21-n)n. For large enough n, that expression goes to zero. So a way of making money it is not. On the other hand, say you have a million dollars in your savings. And say all you want to do is win enough money to get yourself some drinks. You can pretty much do this whenever you want. Yes, there is a tiny chance that you'll lose your savings, but it's more likely that you die on the way to casino, so it's a reasonable risk. Yeah, collecting interest is more efficient with this sum of money, but we are talking about a principle. The above basically works for the same reason that the game in OP's post doesn't. Outcome of martingale is a loss in the long term, but it keeps winning in short, despite the game being weighted against you. But what's interesting is that while martingale brakes down pretty fast, given limited resources, the game in OP's post takes an absurdly long time to pay off if you have to pay each time you play. So the question is whether this behavior is something that can depend on the betting tactic. Can there be a betting strategy that allows you to keep winning for a longer time on a significantly smaller budget? The answer is probably no, but I might be missing something. The idea is that game's average winnings cannot depend on the betting strategy. So in order for you to win small amounts in many games, the strategy has to allow for you to lose all that money in one go. And that means the game will be budget-limited, and you can't really deviate from martingale in any significant way.

That's the sort of thing that would make me wish I lost consciousness.

Transmitters, receivers, and testing equipment will bring it up to $100k easily. The rest depends on exactly what it is meant to do up there. That's per unit. 3U is 3 units, hence the name. It will cost around $100k for launch, perhaps rounded up a bit. Keep in mind that suitable orbit is also important, so it's going to narrow down the market. And you were planning to do telemetry how? You need to hit an object over 300,000km away with a satellite that needs more than a year to reach the necessary velocity, during which its orbit is going to become more and more elongated. There is simply no way in hell you are placing the apogee where you need it without correction in transit, and you aren't going to make corrections in transit without telemetry. You could, normally, make final adjustments using a camera, recognizing stars, and watching when they become blocked by the Moon, assuming you can actually write the software that does all that, but unfortunately, with thrusters you can actually get for this sort of dV, you won't have enough thrust to make a correction that late in flight. I suggest you take a look at history of unmanned lunar fly-by missions and see how complex they were, and how much development they took. Your estimates are naive. And that's not even mentioning the fact that you took a launch estimate for the wrong size sat.

It's much, much harder with real atmosphere. If, like in KSP, atmosphere was absolutely static, then even with all of the other complexities, one could do much better than now. But we have weather and all sorts of associated movements in atmosphere. Down here, the pressure can vary by a few percent. Up here, fluctuations can be even higher, and that can throw off a shallow re-entry trajectory by a number of revolutions, making a crash location almost impossible to predict.

I don't see why. It's one of the strategies that clearly fails for reasons stated in the article and ones I've mentioned in my post. There are much more clever variations on this that allow you to play at lower risk given table limits, etc, and they still fail in the real world.

This is dramatically reduced with modern warp methods. Think about it this way, all that energy being released has to be put in there first. Where in the world would you get that much energy from? So fortunately or unfortunately, depending on your point of view, a realistic warp ship would emit a fairly modest amount of radiation. Gah. This guy hurt my brain. Momentum conservation means that the same amount of momentum goes into recoil. 0.04c to a 600T slug against a 100kT ship gives 72km/s of recoil.

That's kind of my point. You won't. What you can find is a high inclination launch to LEO. Say, something en-route to ISS. The ion drive will have to push the rest of the way. You'll have to do 12° of inclination change, requiring about 500m/s of dV, followed by a 2.3km/s apogee push. Because you are already in stable LEO, you can do it over many revolutions, so low thrust isn't a problem. And 3km/s of dV on a small ion drive is entirely doable. You can get an electrospray thruster with ISP of up to 1,300s. So you'd only need 26% of your sat's mass, or just over a kg in propellant for this. Another kg of mass is taken up by the engine, leaving you with 2g of payload. Entirely reasonable. Thrust is a bit more problematic. At 9W, you'll get only 1mN of thrust. Hauling the 4kg, it will take over four months of thrusting to reach the target orbit. Realistically, way longer because you have to time the puffs. But who cares, right? This isn't enough for orbit to decay significantly, and there is enough overhead in above to make corrections. And it will be the computer doing the pushing, so it's just launch and forget until it's time to collect the data. There are certainly faster ways to get this done, but not cheaper ones.

My wife was given this as a problem in a probability class. The way it was stated was, "a) Find expectation of the winnings, Should you play this game if it costs you $100 to play?" I was a bit surprised, to be honest, it kind of rocked everything I thought I understood about statistics. Namely, that an "average" value might be absolutely and totally useless. Basically, yeah, you win on average, given enough time, but unless the opportunity cost is very small, say $10, you just can't physically play enough games to get that infinite tail end on your side. Ever since I learned of this possibility, I've been wondering if it's possible to play this the other way. In every game of luck in the Casino, house always wins. No matter what strategy you use to bet in, say, roulette, the expectation value is always going to be in favor of the house. In fact, regardless of betting strategy, in a game of luck the expectation value cannot change! But this problem shows that it doesn't necessarily matter. Could there be a betting strategy which, while letting Casino win on average, still lets you get the money over any reasonably finite number of games? It seems that some betting strategies do work on this principle, but they still have rather short fluctuations. So while you can use them to get a bit of extra money in short term, you'll never make any significant amount. I hope I didn't spoil too much for people still trying to solve it.

There might be a reason for in-space fighting if we develop a warp drive. If it's anything like Alcubierre Drive, we'll have to pre-program a target location. A ship in transit would be virtually impossible to intercept, but equally, a ship in transit will be incapable of maneuvering except in a pre-determined way. This leads to some very interesting tactics. If a ship is heading straight at you under warp bubble, it will not be detectable until it comes out. This means that rather than engaging enemy in BVR combat, it will be far more beneficial to warp right in and attack the enemy at short range before they realize what's going on. What's the point of attacking something in space? Well, the primary target would be any stations in orbit. Even if we stay mostly planet-bound, interplanetary ships will remain topside. This means lots of transfer stations, both for military and civilians. Secondary are any ships protecting "fixed" targets or docked for resupply, and these will also be plentiful. That means lots targets. Any rock hurled at a station from BVR can be shot down. A ship warping in at beam weapons range is far more effective. Lasers and particle beams can hit the hull of the target literally at the same time as the signal announcing arrival of the enemy is received disabling a lot of defenses. This means that even automated defense systems would have no time to react. Once defenses are down, railguns and missiles can pound the target to scrap at a more leisurely pace. This also leads to fighters/drones to being a more viable defense. Any exposed weapons are likely to be fried in the first seconds of battle. Beam weapons aren't going to penetrate hulls worth crap, however, so launching something well armed and maneuverable from hangars might be the only chance a station or a large ship have in case of an ambush like this. The rest is going to be about early detection and tactics. If a ship under warp is detected by a remote station, a courier ship can beat it to the destination and allow for early warning. One of the features of Alcubierre Drive is that it's not terribly flexible in terms of trajectory. If you observe a warp bubble in transit, you can make a pretty good estimate on destination and arrival time, but still within a certain error. Intelligence and tactics might let you engage an attacking ship the moment it exits. Just start firing at it at the ETA. You'd only score lucky hits, but it can be more than enough to tip the balance. At any rate, engaging targets from BVR robs either side of any advantage. You want ships close so that the speed-of-light delays don't mess with your targeting, and you want to engage enemy any time they are not ready. So you are going to get a lot of exciting space combat. Of course, all of this assumes that we get warp figured out, and right now, there are some very significant puzzles with it. Unfortunately, the math and theory are far ahead of engineering on this. We know space-time configurations that can do it. We even have some guesses on how to get the curvature right. We do have some experiments running to confirm warp field works the way we think it does. But getting it strong enough requires more energy than we can produce right now. On the plus side, what we've thought was the minimum required energy for interplanetary drive has been reduced from equivalent of supermassive black hole to a few hundred kg. This is a step from absolutely impossible to potentially remotely feasible. And the warp drive physics has only been around less than two decades. I'm not going to promise warp travel in near future, there are many challenges and even philosophical puzzles, but I would not be surprised if we have some basic prototypes within century. And then, who knows?

This is fine for maneuvering, but you aren't going to do any significant orbit changes with that. You need about 500m/s just to transfer to Molniya assuming you had correct inclination to begin with. If you are shooting for GEO, the budget is closer to 2km/s from LEO. Keeping in mind that 1U is capped at 1.33kg, and the maximum you can have is a 3U at 4kg, your options are very limited. The only viable way to do significant orbit changes with a cube sat is by using ion drives. These are energy-limited, and with a solar cell you can get as much as you need over time. So yes, if you need to collect data near GEO and transmit it to a base on Earth, by far your best bet is a 3U with an ion drive in Molniya orbit. If I was contracted to build this, I'd be quoting something in the $300k - $400k range, depending on some specifics. This is $100k for launch, $100k-$200k parts and anything I have to special-order, and $100k commission for the work. Later includes nav and comm software, testing, etc. If anyone's interested, send me a PM with details and I'll get you a more precise quote. There is no way OP is building it for less.

Yeah. They demo "zero gravity drill" in this video, which is meant for asteroid missions. They show it as man-operated, but it can be part of a "lander" as well. Something that attaches itself to the surface and drills in to study the rock.

Same as it does during normal reentry. Into the atmosphere. The trouble is that during normal reentry, your ship has to be in close contact to the air flows it superheats, so some of the heat rubs off. If you brake against plasma using magnetic fields, density of plasma anywhere near your ship can be almost zero. Make the ship shiny to reduce absorption of the radiated heat, and you don't have to worry about it. And like I said, magnets would have to be superconductive. No heating there.

High power rockets gimball the entire engine. You already have to cool the nozzle with the fuel. Any bends would simply burn through. There is no good way to gimball just the nozzle. Now, with pumps, you have options. Most of the engines I've seen have the pumps attached to the combustion chamber/nozzle assembly rigidly, but you can, at least in theory, have them on rocket proper.

You might not even need to. The ship passes through ionosphere, where it does a great deal of braking. For reference, ionosphere starts at about 85km, while Space Shuttle lost most of its orbital speed between 30 and 120km. I can't say exactly where most of the heating is going to take place, but it seems to be close to the ionosphere boundary.* So if you have a strong enough magnetic field, you can brake against ionosphere without exposing your ship to the flow. This can dramatically cut on the heatting of the ship during re-entry. In part, because you can brake at higher altitude where density is lower. Unfortunately, I was able to do only very rough estimates, but it still seems like the weight of the magnets is going to be too big of a problem. HTC superconductors just can't take these kinds of fields, and resistive magnets will require the kind of power you can't reasonably produce on the ship we can launch to space. So you'd be stuck with standard superconducting magnets, which are very heavy and have to be kept at 2K, which requires cooling systems that make them way heavier still. It's a really neat idea, and if we learn to generate huge magnetic fields with something much lighter, it'd be the way to go. One potential I can see here is trying to use charged plasma flows to generate even stronger magnetic fields. Unfortunately, my brain can't handle this sort of E&M/Plasmodynamics problem. Edit: * Ok, I looked into the wrong column. Peak heating is actually between 40km and 70km. Velocity is down to half at about 50km. So this is quite a bit bellow ionosphere, but like I said bellow, with a magnetic field, you can start braking at a much, much lower density without worrying about burning up. So I still think this is workable, pending super-magnets. Edit2: Damn, this chart is weird. You know what, can somebody double check my numbers? Here is the chart.

No. Anything dense enough to give you lift is dense enough to torch your glider. It's about the worst configuration you can come up with. The idea itself is not stupid, though. I had to do some math before I realized it wasn't going to work.

Oh, it's not nearly that bad. It used to be a Sun-synch, so its orbit is going to run roughly North-to-South, placing impact areas in narrow strips of land spaced about 7.5°. The area of possible impact zone is actually not that great, and that zone can be estimated rather well. Trouble is that the zone itself stretches to cover pretty much everywhere on Earth. I can't find the exact parameters, but since it's a Sun-sync, it's probably a dawn/dusk runner, so wherever it falls, it will be dawn or dusk there. Just stay indoors at these times. I doubt shrapnel big enough to punch through the roof is going to survive reentry.

... or math.

I don't think of that as a problem. Once you're in LEO, you have all the time in the world to get to the orbit you need. What you really care about is ISP. So if there is a good commercial option for a high ISP ion drive, then it's doable. But if it has to be a 3U, that's going to roughly tripple the launch cost. Then again, since 1U would probably be too small for this mission anyhow...

That's a good idea. Molniya should do the trick. It has a period of exactly half of sidereal day, meaning it can be right overhead at lowest approach once in every two revolutions, and its apogee is at 40,000km. That's a touch short of the OP's 50,000km, but maybe that's good enough. By the way, I mentioned earlier that there are relatively affordable cubesat launches, but I should have pointed out that it's for LEO. GEO is not going to be as cheap. Though, there was a project for a cubesat with a tiny ion drive. That could get to GEO or Molniya from a LEO launch.

You really wouldn't need much. At four revolutions per hour, just 4x faster than the minute hand on the clock, and 20m you'd already get about 100mN of pull. That corresponds to about 10 grams worth of weight on Earth. Not a whole lot, but if there is nothing to keep you properly anchored, it's enough. And the rotation could have been considerably faster without still being apparent.

Ah, I see what you mean. Get scalar curvature to be equal to trace over the stress-energy tensor. But yeah, as you point out yourself, that's going to be a coordinate-dependent relationship. Which, I suppose, is what OP is asking about in the first place, but it's still an oversimplification. Yeah, curvature due to a moving object is higher, but the real fallout is in how that effects geodesics, and it's a very different effect from simply making the source heavier.

There are. And I've seen quotes as low as $30k for a cubesat launch. It's pretty reasonably priced. You can do bursts. Dish won't do you much good on a cubesat. Since you are limited to 10cm on the side, even if the dish folds out, you'll have hard time getting something over 30cm in diameter. That's a 1GHz wavelength. Even at 5GHz you'll be lighting up a 10° cone. You can work with that in LEO, but from GEO you are doing hardly better than omnidirectional broadcast. Point is, if you want to "talk" to something in GEO, you need a much bigger sat.

Addition to. SuperDracos are designed to produce axial thrust. That's all you need for landings, so long as you can throttle them individually, but it won't do for docking. Similarly, you aren't going to be able to throttle down 120,000lb engine to anything suitable for precise docking maneuver.

There are two regimes. First in upper layers of the atmosphere, where density is very low. The in-falling object will collide with individual air molecules there. There is no hydrodynamics. Every molecule on your path you are going to hit. Here, the shape is absolutely irrelevant. Heat generation can be quite severe, but falling straight down you also pass this region pretty fast. Then you hit thicker atmosphere where air starts to behave as a fluid. The in-falling object generates a shock wave, and that helps reduce heat transferred to the body. On the other hand, air is denser, so way more energy is dissipated here. If there is a problem, it will happen here. And yeah, shape and size will matter. Slow down too gradually, you burn up. Slow down too fast, G-forces will get you. The later shouldn't be a problem for a falling human. The heat, though, I don't know. This is Mach 6, so it's going to be bad. But it could be brief enough to be survivable.