Many times, in making some calculations about star flight, someone will write that they are assuming a constant acceleration, sometimes equal to one g, sometimes not, and then start constant deceleration halfway through the flight. Thus they arrive at the destination star with zero velocity. Roaring by your target star at a fraction of light speed is considerably worse than missing your exit on the freeway. You can't make a U-turn and get back.
The real problem is that no starship captain would ever use his ship this way. A starship can be thought of as having two masses, the propellant mass and the payload, which is everything else, including the propulsors, the energy source, the habitat or the probe equipment, the communications gear, and everything else. Back in the Apollo days, chemical rockets were pretty much the only thing around, and so they were used. In this situation, you didn't have any separated propulsion power supply and propellant, they were the same. The propellant was a single solid or a pair of liquids, and it came with all the chemical energy you could pack into it. You burned it at some rate, and voila, you took off into space.
Now there are many options for stellar propulsion, some more hypothetical than others. But chemical rockets aren't the likely option. That doesn't matter for the arguments here, except for figuring out some practical cases, which we won't do.
The point of this post is that you are wasting your engine if you choose this trajectory. Suppose you want to get to the destination star at the earliest possible time. You start out, using your engine flat out. After you have gone some distance, and accumulated some speed, your propellant mass will be lower. If you continue to use your engine to the max, your acceleration will increase, as now the same thrust is pushing a smaller total mass, meaning the payload and the remaining propellant mass. So if you want to have the constant acceleration trajectory, you need to throttle back your engine, proportionally to the total mass of the ship as it exists. What this means is that you have an engine, but you are not using it to the maximum effect. You are not going to get to the target star in the minimum time.
The best you can do for a given mass of propellant is to run the engine at the maximum until you reach a particular point in the trajectory where you need to reverse thrust, and decelerate back down to approximately zero speed, when you hopefully will be at the target solar system. The time when you reverse thrust is not half way through the voyage, nor is it at the midpoint of the travel line stretching from the origin to the destination. It will occur much later than halfway through the trip. One way of looking at the situation is that at the reverse point, you don't have nearly as much mass to decelerate as you did to accelerate, as you have been throwing it all away out the engine. You have consumed more than half of the propellant by the reverse point.
By the time just before you arrive at the target star, you will have consumed all the propellant you designated for the trip. You might have saved some for some orbital maneuvers in the destination solar system, but that should be thought of as part of the payload for the interstellar travel portion. That means, your final deceleration will be with the same engine running at maximum output, but only having to slow down the payload plus a jot of residual fuel. The g load, i.e., acceleration, will be much greater at the end of the trip than at the beginning.
Consider a simple numerical example. There is only one critical parameter that needs to be considered, and that is the ratio of the payload weight, meaning everything but propellant, to the propellant weight. For a payload to propellant of 1 to 10, the turn-around time is about 76% of the total trip length. During the first three quarters of the trip, you are burning a lot of propellant accelerating the remaining propellant. After it is gone, you can decelerate much more readily.
You might ask, is the constant thrust trajectory the fastest way to get to another stellar system? It's obviously not. It's the economical way to get there. You could get a bigger engine, and blast yourself up to a high cruise speed, sit at that speed until you were close to the target solar system, and then decelerate in a hurry, using the bigger engine pushing against a smaller total mass. This trajectory, like all others that use the engine to its full power when thrusting, have a more abrupt slowdown than startup.
Having a larger engine comes with a cost. The cost is in the payload weight of the engine, as a larger thrust engine will have more mass, and therefore more payload, so the fuel you will have to carry will be larger. But that is not all. Since we are not using chemical rockets, there has to be a source of energy to ramp the propellant up to exhaust velocity. This could be a reactor, fission or fusion, or something more exotic. But whatever it is, it takes mass and having more thrust means more mass for the power source, not just the engine itself.
Recall that starships are going to be expensive, and while it might be great to want your starship to cruise at a high velocity, it might not be affordable. We are, in this blog, thinking about alien civilizations, and a starship would likely be a definitely noticeable item on their budget. Would they want to save some time in getting it to its destination faster? Why? It is going to be generations to get there anyway, and what would it matter if it was two or three or five or ten? They would still have to put the project into mothballs once the launch happened, and then rebuild an organization to use the data once the destination was achieved by the starship and was transmitted back.
So, if time is not of the essence in star travel, the constant thrust trajectory is likely to be the one chosen. This has some very, very interesting implications for us. A ship traveling silently and invisibly without any power, because it is cruising at high speed rather than rationing its propellant, would be hard to detect. A starship traveling under continuous power would possibly give off some signs. A starship that was accelerating would not look like anything else, as proper motion of anything doesn't do that. So, once noticed, it would be recognizable. That's good for us, if we ever spend a lot of time developing our observational powers for starships.
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