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and interplanetary space travel in general
Don't fence me in ... So here it is ...
Vr M r+fuel
______ = loge _________
Vex M r (no fuel)
I prefer stating it in this form because I believe it is easier to memorize as two ratios. Here, Vr is the final velocity of the rocket after acceleration (taking the initial velocity as "0") ... divided by ...
= loge ... means that left side of the equation is equal to the natural logarithm (ln) of the right hand ratio, which is ...
M r+fuel ... the mass of the rocket plus the mass of its fuel divided by
This equation (and variations) has stood the test of time and experience and should be applicable to all rocket type situations wherein the astronauts take their fuel along with them and use it to "push off of". If linear momentum conservation is correct (and I believe it must be for sound logical reasons), then, no one can go to another star system without carting along an enormous mass of fuel to push off of in order to accelerate ... and ... very ... frustratingly important ... another huge mass of fuel to use to decelerate at the destination. So the above rocket equation would generally have to be modified to include more than a double load of fuel.
I have seen estimates of the amount of fuel needed to go to another star. It isn't pretty. In fact, it's downright depressing even if one factors in an anti-matter drive. That would be the most efficient engine possible given that we are to take the fuel with us. Of course, they figure to go near light speed which is (to my mind) rather stupid. Last week I was thinking about this and decided that it might be easier for engineering purposes to accelerate very heavy particles to near light velocity so as to "m-load" them (give them additional mass by way of relativistic increase). This would not be as efficient as a pure anti-matter drive but might be easier to handle. Visualize stripping one electron from a heavy uranium atom and using its subsequent charge to whip it up to 10 times its normal mass. Then sling it out as exhaust and send the extra electron along with it by means of a little lightning rod on the ass end of the ship to maintain a neutral charge on the hull. If this were feasible, we might detect such exhaust as very, very heavy cosmic ray particles. Hmmmmm ... Let's make some calculations Now understand that if you can send any size rocket, you can send two of 'em. And you can hook them together and make one big one, etc. Similarly, you could make a hundred airplanes and tie them altogether to make one big one. So don't be intimidated by size. Everything here scales up nicely in linear fashion. It's the velocity and what it takes to get there that is the bone of contention ... not the size of the craft. So, we only need to consider the ratio of final velocity to exhaust velocity and then afterwards take a look at the tonnage ratio requirements. Let's imagine a plasma engine with an exhaust velocity of 80% of light velocity (to keep the problem within reasonable limits). And we'll make the final rocket velocity say 10% of light velocity. Actually, that's about all I'd hope for. We need minimally a velocity of about 1/1000 c in order to contemplate going to another solar system at all because it doesn't do any good to go to the train station to catch a train if that train doesn't stop there and goes whizzing by at 100 mph. Stars generally have speeds on that order but it's too damn slow for my liking as a final velocity. So the left side ratio is 1/8. So, what number will we get when we raise "e" (2.718281828...) to the 1/8 power? And the answer is about 1.133. So, our vehicle + fuel to empty vehicle ratio is about 1.133 / 1 This appears to me to be a technical problem rather than a theoretical one. Slightly over ten percent of the vehicle's mass must be devoted to fuel ... bearing in mind that this IS NOT A CHEMICAL ROCKET. Such a reaction will never give an exhaust anywhere near 80% of light velocity. We also must take enough fuel to stop when we get there ... and ... we have to accelerate that as well ... so ... let's triple the fuel load to be on the safe side and take it to 30% of original vehicle mass. Note: We won't take fuel to get back because we have "confidence", i.e. if we are brazen enough to go at all ... then ... we've got the 'chutzpah' to assume that we can get more fuel at our destination (no problema). Tsiokovsky's equation is just a restatement of linear momentum conservation for a body ejecting a stream of matter instead of one big chunk. In the above example, from where the rocket starts off, an observer would see a rocket ... at the end of the acceleration ... going at 10% of light velocity. Considering the mass of the rocket as "1", its momentum (mv) is equal to ".1". So, the ejecta going in the opposite direction must sum to an opposite momentum of just ".1".How about 50% of light velocity for the rocket and 80% for the exhaust? Answer: 5/8 = ln 1.86/1 That's a bit too much for me. 46% of the initial mass must be fuel and that doesn't include stopping ... and ... that's just to get there five times faster, i.e. instead of taking a typical 100 years to get to a star 10 light years distant, we only take 20 years. The engineering requirements would appear to be too great a price to pay for that small result.
Let's try staying at 10% of light velocity for the payload and an exhaust velocity of say, 40%. What's the story here?
Anyway, you get the picture. You want the highest exhaust velocity possible and must accept a reasonable payload velocity. Ask too much and the equation will ruin you. To go to other stars we must be patient travelers. Scientists who "poo-poo" space travel just want to steer your attention away from the fact that it is entirely feasible and thus away from the dreaded "U" word which scares the bejeesus out of them, i.e. if we can do it ... others can ... and there are probably thousands of other inhabited planets which have developed in the past two billion years ... and we've probably developed in the middle of that pack ... and ...
It can't be ... therefore ... it isn't. Travel through this solar system I have two new modes of travel which might work in our solar system which I haven't seen anywhere else. They are no good for travel to other stars but might be good locally.
"The Big Ass Sling Shot"
And then there's the ... "The Very Big Fucking Rail Gun" Here the object is to accelerate the ship electro-magnetically to high velocity using the mass of the rail gun to push off of. Thus, no propellant is needed for the outward bound journey. After releasing the ship, the rail would go to a slightly lower orbit if we were aiming for Jupiter's moons for instance. Such a "thing" would be manufactured from materials mined on the moon and would be constructed away from the Earth by a few hundred thousand miles. It could be any length imaginable ... maybe 20 ... 50 ... 100 miles long?? We are now building suspension bridges on the scale of 5-10 miles in length. Whatever. S = 1/2 at2 ... where s is the distance, a is the acceleration and t is the time. So, at 1G (9.8 m/s2), and 10 kilometers long, it would take 45 seconds to get released from the rail and your speed would be a lowly 442 meters per second. That's about 1600 kilometers per hour. Let's try 3Gs and 20 kilometers long. Then it would take 37 seconds to get to the end of the rail and your speed would be 1084 meters per second or just about a kilometer per second (3900 kph). A respectable speed but we want more than ten times that much. Well, it's a good start, eh? And it might save a lot on that initial fuel requirement. The key to getting around in the solar system and galaxy Is ... Don't take the fuel with you. If at all possible push off on something other than the fuel you are carrying along. That's what makes the enterprise so exasperating. That damn conservation of linear momentum. But there is no way to get around that is there? I certainly don't know of any.
What about anti-gravity? I have no idea here. Maybe anti-matter will exhibit some weird behavior (like falling up in our gravitational field). Maybe it will go up when we spin it around ... eh? ... I doubt it :o( If one could push off on the gravitational field of the star from which you were coming and the one you were going to, it would be a piece of cake. Any nuclear reactor would provide the necessary power to get you to respectable percentages of light velocity. But I just can't see any way to do it. It's not forbidden by any conservation law I know of. How is it done? How do they do it? Maybe they don't. Maybe no one is here after all. Sure.  
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This gizmo consists of tethering the rocket to a larger mass and firing rockets in both the rocket and mass so as to rotate them around a common center of gravity. Fuel is fed to the rockets through tubes running the length of the tether originating from a tanker stationary at the point of rotation of the entire apparatus. Thus, we don't have to load the ships with excess fuel mass to get them going. When the desired velocity is reached, release both rocket and dead mass. One goes off to Mars ... the other goes down maybe into the sun or just into a much lower orbit. Or, better yet, send two expeditions at once ... one to Jupiter and the other to Venus ... just whatever might be worked out mathematically.