K^2

I'm pretty sure resupply missions are far more frequent. Besides getting food, water, oxygen, and CO2 scrubbers up there, ISS needs quite a bit of fuel to maintain its orbit. A lot of these missions are unmanned Progress launches. Looking at the list of unmanned flights to ISS, I'm counting 8 missions in 2013. Four of these are Progress missions. Seeing how these are launched with Soyuz rockets, and rides on Soyuz manned vehicles cost NASA $60M/seat, and taking into account the fact that Soyuz still carries cargo, we are talking about hundreds of millions per Progress launch. Even if NASA only fronts half for Russian launches, and with other vehicles being cheaper, this can still easily add up to something in the $1b neighborhood just for the unmanned resupply launches. And then there are manned launches, equipment and experiments that need construction/preparation on the ground, all of the personnel that has to do with tracking and planning of missions. $3b/year is actually starting to look quite cheap. As for it getting more expensive, Dragon is just entering the stage. NASA has been relying almost exclusively on Russians to deliver cargo in bulk. Given general inflation, economic situation in Russia, and Russian political climate, these costs are bound to keep climbing until NASA can switch most of its cargo to Space X or other contractors. That will happen, and cost of maintaining ISS operations will probably go down. That's big part of the reason why these programs are being financed. But we're not there yet.

Depends on your choice of frame. But you are looking at some added complexity either way. For the "conventional rocket", the easiest thing to do is work out fuel consumption for constant acceleration in rocket's frame, then integrate over constant proper acceleration in map frame to get actual delta-v or delta-proper-v, depending on what you are looking for. You also have to keep in mind that the way delta-v is added is using relativistic addition formula. If you want, I can track down or re-derive the correct formulation of rocket formula for relativistic speeds. If you scoop up propellant from interstellar medium, however, things might be a bit more complicated. I haven't really worked that out, but the way I'd attack this problem is still from the rocket's frame. So I'd treat the medium as incoming at relativistic speeds and work out energy consumption required to accelerate it.

You can get decent density by using a Bussard scoop. But it isn't a new idea.

Can't do that, either. You can use relativistic mass with mc2 to get the total energy. And if you want kinetic energy, you can subtract mc2 of the rest mass, but that will just give you the relativistic formula that you linked. Plugging relativistic formula into classical kinetic energy will give you wrong result, because classical formula is just a leading term of power series expansion of the relativistic one.

As soon as you put down your strawman and make an actual argument. You've claimed that scientists need to, and I quote, "go way out into the crazy lands" to come up with a new theory. I challenged that assertion, to which you replied with a bunch of quotes saying that intuition and creativity matter. Since I never claimed otherwise, that's just a strawman argument. Now, in order for you to prove your assertion, you need to provide an example of just one scientific theory which would be constructed as you claim. In order for us to "continue the discussion like adults," you need to provide such an example. Otherwise we are stuck with proving a negative. Oh, and you've been on a thin line with arguing from ignorance at a few points. So as long as you are going to keep to your logical fallacies, I'll keep my argument from authority and ad-hominems. And if you really do want to try arguing like adults, your thesis only leaves you one avenue of attack. So again, we're waiting for an example.

This is a bit out of my comfort zone, but it's my understanding that despite having a dense, massive core, the O V stars have a very low density outer shells, due primarily to high temperatures and immense amount of energy they radiate. The O V Wikipedia article gives the following useful bits of info. The O2 would be on the hot, massive, luminous side of this spectrum. So it looks like I was right on the money with it being a million times more luminous. The article even links to BI 253 as an example of an O2 V. So these things do exist, it turns out. And the article on stellar evolution notes the following. And everything points to the O2 V being in the 60-90 solar masses range, so it'd definitely fit in this category. Seeing how it's going to be losing atmosphere already, I can't imagine how putting a brown dwarf right next to it is going to do anything but make this way worse. On the other hand, having something with 60+ solar masses right next to a brown dwarf can't be good for the dwarf, either. So I don't really know which is going to happen. If the dwarf is going to be stripping off the outer shells of the star. If so, will that make the dwarf go nova, or just slow it down enough to be consumed by the star. Or is the brown dwarf just going to be ripped apart by tidal forces and that'd be the end of it. One thing I'm not picturing is the brown dwarf peacefully orbiting that star.

So the part where I described historical context and how it was logically built up from ground up you just missed? You are conjecturing. You do not understand how the theory is built, so you assume that neither did people who built it until they did, in fact, build it. That's not how science works. I'm telling you that as a person who is actually involved in modern research, and I'm entirely prepared to discuss this on a case-by case basis. But even when I do that, all you get to dig into is your false conjectures without absolutely a shred of support for it. You are making stuff up and arguing from ignorance rather than try and listen to somebody who knows what he's talking about. Again, I'm entirely happy to spend time clarifying any particular point, but you just keep repeating your conjecture over and over without any room for discussion. This does not make your point sound any more valid.

ZetaX mentioned it briefly already, but it follows neatly from electrodynamics. Ever wondered why it's called "Lorentz Factor" and "Lorentz Transformation" if Einstein was the one who came up with Special Relativity? Most of the ideas behind Special Relativity were already floating around in scientific community. Einstein's contribution was primarily in organizing it all, and in building a self-consistent set of axioms from which SR follows. No easy task, and quite a creative one in places, but it was not a leap of logic by any means. When Special Relativity got published, it was instantly accepted specifically because it was not a leap of logic. Note, by the way, that it was work of Poincare and Minkowski that lead to formulation of SR in terms of unusual transformations of space-time, which is how we know it today. Einstein's original work read more along the lines of, "If we postulate that speed of light is the same in every frame of reference, as Maxwell's equations and Michelson-Morley experiment suggests, we get transformation laws for coordinates that are consistent with what we know from electrodynamics." There was nothing revolutionary about the suggestion. Revolution came from what followed logically from that. The fact that Lorentz Transform is just a hyper-rotation in space-time. This gives you Poincare symmetry of space-time. This leads to stress-energy tensor being the conserved charge by Noether's Theorem. And this launches us straight into General Relativity. Of course, the final pieces only fell into places by the late 50's, early 60's. Yang and Mills came up with the way to build a field theory from the Lie group symmetries in Lagrangian. That lead to formulation of General Relativity as a Yang-Mills theory on Poincare group, which finally closed the gap between General Relativity and Quantum Field Theory and opened the road towards the Grand Unified. But that's a whole another story.

Hypothesis is never crazy. Hypothesis is always based on prior observation and prior knowledge of the world. Crazy hypothesis is statistically guaranteed to be useless. There are not enough man-hours in the age of human kind to think up enough of them to yield any sort of progress. Again, give me an example. Any example where you think the hypothesis has had to be far-reaching, and I will show you how it was not so.

No. Expansion is accelerating. If you somehow managed to stop it, and then let go, it'd still expand. Why gravity suddenly turns repulsive on grand scale is the million dollar question in cosmology right now. GR predicts that to this repulsion corresponds a so-called dark energy, and there is bloody lot of it in the universe, except, for some reason, nowhere where we can see it. So yeah, a riddle wrapped in a mystery inside an enigma.

All of science has been built in incremental steps. Again, there are no exceptions to that. There have never been any leaps, or any shifts into "crazy lands". It has always been rational increments. If you'd like to provide a single counterexample, I'll be happy to explain what you've overlooked.

Relative to what? But if you are far enough from gravity sources for gravity to be negligible, then your acceleration will also be negligible, and you'll stay very close to whatever velocity you've been at.

I'm not sure you understand what "theory" means in context of science. Again, absolutely not a single new theory has been constructed the way you are saying. If you actually spend a bit of time studying theory and history behind it, you will find that no matter how crazy the theory sounds by itself, when you take it in context of how it came to be, you'll see that it has been arrived at very methodically. There are no exceptions to it, because it is simply impossible to build a good theory, consistent internally and with observation, without following through systematically. Impression that any particular theory came about otherwise can only come from ignorance of the theory itself and of its history. So if you have such illusions about any branch of science, I urge you to spend a bit of time learning more about it.

Where does he even once mention faster than light?

And by light speed, he clearly means near-light speed. No, actually, it does not. All has to do with the way proper acceleration works. As it turns out, once you are going really, really fast, speeding up gets way easier.

I'm sorry, but that's a load of crap. Understanding limitations of known models, and searching for ways to improve on them is completely different from simply throwing all the known rules out of the window and making stuff up. Not a single significant discovery has been made without basing it on foundation of work done before. From Newton, to Einstein, to modern day scientists, everything has been done taking previous works into account. Even something as seemingly revolutionary as General Relativity is based on the works of such people as Euler, Lorentz, and Reimann, to name but a few, who have laid the foundation for understanding space as something more complex than the Eucledian space people used to picture. I'm not going to call Michio Kaku a kook, because he does seem to be capable of separating scientific principles from fantasy, but general public doesn't seem to be capable of it, and he certainly comes off as one if you just watch any of his shows.

It helps to read the opening post, not just the title. Author compares it to particles in an accelerator, so he's obviously talking about very, very close to light speed, yet sub-light.

I would imagine so. Which is still rather dense compared to the outer layers of the star. Especially a young one with surface temperature of over 50,000K.

Not sure what you mean. Scenario you described is perfectly reasonable. What wouldn't happen is stick getting longer when it stops. After looking through it, Ladder Paradox page doesn't cover acceleration too well. (It only briefly mentions it in relation to another paradox.) I'll prepare some graphs to show what would happen when the object decelerates "almost" instantly from light speed. I'll use the ladder instead of the stick, just like in paradox, because spacing between rungs gives a really good picture of how it deforms.

Number of stars and other dense objects is so low, that you might as well just take your chances. Navigate far from the stars you know to the best of your ability, and take the chance with rest. No, you don't. Matter in the path of Warp Drive is still a problem. And we are talking about sub-light travel, not FTL.

The whole scenario is kind of crazy to begin with. I don't know why there would be a brown dwarf right next to an O V. Main sequence stars are hot really early in their life. So I have no idea why there would even be a brown dwarf, but supposing it was a capture, they'd tear at each other. And to be honest, I don't know which one would be winning. A dwarf has advantage due to much higher density, so it might be the brown dwarf that would be stealing gas from the star. Either way, though, you aren't going to have a peaceful co-habitation. I mean, that's basically how novas happen.

This is the sort of thing that happens when two nuclei collide at near-light speed. Collisions with absolutely anything, down to space dust, is going to be really, really bad. Navigation you can work around. Combination of inertial navigation and some other cues, like signals from pulsars, would get you to the right general location, and you can slow down well ahead of your destination to make corrections. Compared to all other issues, this is category of engineering problems. They might be tricky, but we have means of solving them. Actually accelerating to light speed and protecting ship from impacts are the more fundamental problems of near-light speed travel.

I don't think it matters. Tidal forces, etc.

It's called Ladder Paradox. The linked wiki article covers it pretty well.

The only calculators I know to do symbolic math are the Ti-89 and Ti-92. They are pretty much the same, but Ti-92 has a full qwerty keyboard and a larger screen. It's also larger and heavier overall. Ti-89 has the same body as the rest of the 8x series, so it's a bit more convenient on the go. The software on the two is pretty much the same. These will, indeed, do symbolic calculus, including doing things like taking indefinite integrals. They are also pretty good at factoring, solving systems of equations, and so on. Nowhere near as advanced as PC software, like, say, Mathematica, but it's handy. So if this is what you are looking for, you probably won't be disappointed. I've used both, and liked everything but the price. they are pretty expensive. The other 8x models only do some numerical calculus. They can do things like area under a curve, find local minima/maxima, and do tangent lines. In other words, they can handle calculus problems numerically, but not analytically. So if your interest is purely in solving numerical problems, you'll probably be able to learn to do everything even with a Ti-82. Edit: There is one more cool option for portable computing. If you have an android phone or tablet, you can get Octave on it. I don't know if there is a port for iOS devices as well, but there might be. Octave basically runs engine equivalent to Matlab and can handle pretty much all the same things. In terms of getting the most computational power on the go, this is miles ahead of any graphing calculator. Unfortunately, Octave doesn't do symbolic math. It's a purely numerical tool. This might also be your cheapest option even if you have to buy an Android device for it. Texas Instruments calculators are ridiculously expensive.