There are several techniques I would like to discuss that allow you to put together more sophisticated interplanetary transfers. Not all of them can be reasonably described in terms of the previous post (basic interplanetary transfers). Nevertheless I would like to characterize them as well to some degree. First I will discuss aerobraking, then gravity assists, and finally continuous thrust trajectories that came with the advent of ion drives.
Aerobraking is the general process of reducing your velocity relative to a planet for free. Free here meaning you don't have to spend fuel in the process. The big downside is you need protection from the atmosphere of some kind unless you are performing a relatively gentle maneuver. The concept of aerobraking is diving into a planets atmosphere, and using the drag in order to slow yourself down. The atmosphere of a planet is effectively stationary relative to a planet, so it will keep trying to slow you down until you have effectively stopped relative to the planet. In actuality, it is stationary relative to the surface of the planet, so it will effectively have some velocity. This can be used to lower the apoapsis of your orbit, via principles discussed previously, or it can be for the purpose of leaving orbit entirely and landing on the surface. In general the amount of velocity you bleed off is related to how close you come to the surface of a planet in the process. Aerodynamics is extremely complex, so the amount of speed it is possible to bleed off can vary significantly. Different spacecraft will get different results.
I don't have quite as satisfactory an explanation for gravity assists. I will try to lean on your spatial intuition to give you some kind of idea. Imagine you are riding a bike into a valley, as pictured below:
You could expect yourself to come out the other side of the valley at more or less the same speed. Now imagine that the valley is moving forward at a substantial speed as you come into it. Gravity is strong enough to keep pulling you to the bottom of the valley, but you can feel the valley pulling you forward. By the time you reach the other side you have gained a bunch of speed from the movement of the valley. This is basically how a gravity assist works. You may have seen a diagram like this before (credit AllenMcC from wikipedia):
These are actually a pretty good metaphor for the effects of gravity. A cross section of that would resemble a valley. Travelling through that gravitational valley while Jupiter is moving around the sun allows you to pick up quite a bit of speed. Jupiter has been a common target of gravity assists, and by the law of conservation of energy is actually slightly slowed by every spacecraft that has utilized it. The amount of energy that Jupiter loses (and the amount that you gain) is equal to the gravitational effects of your spacecraft on the entire mass of Jupiter. A reverse gravity assist if you will. Jupiter is immense, so this amounts to quite a bit of energy that you are gaining.
The general utility of this sort of maneuver is that you can get some free forward velocity to throw you onto a higher orbit on your way to your destination. As to how exactly you would go about making use of that, that is a much more complicated question. It doesn't really map to the more simplistic stuff we discussed earlier. I'll close this section out with a diagram of the probe Rosettas trajectory, which made use of multiple consecutive gravity assists over the course of its ten year flight. (courtesy of the ESA)
The last thing that I wanted to discuss was a continuous thrust trajectory. Out of all of them it is probably the most alien to what we have discussed previously, in some ways. The trajectories overall resemble a basic interplanetary transfer, due to the excess of fuel that tends to be available (relatively few gravity assists and such). Ion drives have until recently been something of a fictional technology. Very high specific impulse, but very low thrust. Due to the low thrust, you have to fire your engines continuously for a long time in order to reach escape velocity and eventually ascend (or descend) to your destination. The reason this low thrust is considered tolerable is because of the specific impulse factor that I have mentioned. Imagine you are sitting on a skateboard and have three basketballs at your disposal. You want to get moving, and in this example can only propel yourself by throwing the basketballs. Will you choose to throw them really hard, or will you gently toss them in order to get yourself moving? The answer will probably be throw very hard. This is the general idea behind high specific impulse. Ion drives throw the fuel very very hard (at very high speed, at least compared to rockets), so you need less fuel in order to change your speed by a certain amount. The only downside is they can't throw lots of fuel at once, so there is relatively minimal thrust. You might be throwing basketballs really fast, but the multi-ton probe is hardly budging. The physical reason this works is because of the idea of impulse and momentum. When you throw something, you impart a certain momentum. The mass of the object multiplied by its speed. That momentum gets transferred back to you in the opposite direction per Newtons Third Law. The higher the speed, the more momentum you gain. This allows spacecraft to get a lot more momentum out of their fuel and do a lot more maneuvering over the course of their missions. The downside of the low thrust is that it greatly complicates entering and leaving gravity wells. Dwarf planets and asteroids are generally easier to deal with as far as that goes.
I implied that Ion drives entered use at some point. The recent Dawn mission utilized Ion drives, and was able to pack enough fuel to visit two dwarf planets over the course of its mission, entering into orbit of both (rather than a measly flyby). Ceres and Vesta, if you were wondering. You can look up lovely high resolution maps of both planets because of that mission. Here is a diagram of its continuous thrust trajectory, courtesy of NASA:
As you can see, the thrust wasn't entirely continuous, but it most definitely was not freefall.
To recap, we discussed the concept of aerobraking. Diving into an atmosphere to reduce your velocity relative to a planet, specifically relative to its surface. The idea of a gravity assist, as well as a metaphor to describe how it allows you to gain speed. Finally, continuous thrust trajectories were described, as well as the reason that they exist.
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