by olog
There's nothing special about directions in space as such (maybe if you get into general relativity but I don't suppose you meant that). Mars is right in the direction where it seems to be. What I think you're asking is how the orbits work. That is, if you are on the Moon and want to go to Mars, with a spaceship that's at least somewhat realistic, then you of course don't point your ship towards Mars and just go.
You might be thinking that things in space go in a straight line so to get from A to B, you just point to B and go. And that's true if there's nothing else in space but everything in the solar system is in a big gravity field. So things don't go at all in straight lines, they go in what are called conic sections. There are three kinds of conic sections, ellipse (the circle is a special case of this), hyperbola and parabola. An object that's escaping the system is either on a parabolic (if exactly at the escape velocity) or a hyperbolic trajectory (if greater than the escape velocity). Everything else is in an elliptic orbit and we'll focus mostly on them now. An important thing to keep in mind is that if you don't use your engines or get close to another planet or moon or get inside the atmosphere of something, then you will stay in the same orbit forever. It doesn't spiral down to the object you're orbiting. It doesn't circularise itself. After a full orbit you return to the same spot you left from and at the same speed.
Orbits have a couple important properties. First the eccentricity. This means how circular the orbit is. Eccentricity 0 means that the orbit is perfectly circular, values between 0 and 1 mean more and more elliptic orbits as the eccentricity gets closer to 1. Eccentricity 1 is a parabolic orbit and values higher than 1 are hyperbolic.
The other important thing is altitude, or radius. If the orbit is perfectly circular then altitude is clear. If it's an ellipse then the altitude will vary at different points of the orbit and it's better to use something else. One way is to specify the radius at the lowest and highest point, called the periapsis and apoapsis, respectively. Or names such as apogee and perigee are used if you're on Earth orbit, or aphelion and perihelion on Solar orbit or a whole lot of other stuff for other planets but they all mean the same thing. Another way to specify the same thing is to use the semi-major axis. It is the "radius" of the ellipse at the most elongated point. Note that it's measured to the middle of the ellipse, not to the central body you're orbiting. In essence this is a number that roughly describes the radius of an elliptic orbit. For a circular orbit it is the actual radius.
There are other things too that relate to how the orbit is aligned three-dimensionally. Like is it an orbit that goes over the poles or one that goes along the equator and such things. But we best keep things two dimensional for now so we don't need them.
And when thinking of moving around in the solar system, or just between the Earth and the Moon, or even just between different Earth orbits, don't think at all about going in a straight line or maintaining your velocity or in what direction the gravity is and how it'll affect your velocity. Just think what your orbit is, that is the eccentricity (how elliptic it is) and what the altitude is. And of course, where on the ellipse you are now. Your velocity, both direction and magnitude, are fully determined by your orbit and your location in it. An object on a specific orbit always has the same velocity at the same point regardless of the mass of the object or anything else (it does depend on the mass of the central body though). Or when you intend to change orbits think of it the other way round, how your changed velocity will determine your future orbit.
So let's start with a circular orbit around the Earth. You make some kind of a change in your velocity by using your engines. The orbit you get to will be one that goes through the point where you used your engines. That should be intuitive since your current location must of course be a point of the orbit. So if you want to get to an orbit that doesn't intersect your current orbit, then you will necessarily have to do at least two engine burns. First to get to a temporary orbit that intersects your current orbit and the target orbit, this is called a transfer orbit. And then a second engine burn at the intersection of the transfer orbit and the target orbit to get to the target orbit.
So let's say we want to get from a low Earth orbit to a geostationary orbit which is a much higher orbit. They are both circular and of course do not intersect. In general circular orbits of different radius can naturally never intersect. First we want to turn our low circular orbit into an elliptic orbit that will intersect the higher geostationary orbit. To do this we accelerate along our orbital motion. That means tangential to the circle. Or parallel to the surface of Earth. You'll get a long way by just accelerating along the motion (prograde) or opposite the motion (retrograde) and never accelerating in any other direction. Accelerating prograde will always increase the altitude at the opposite side of the planet. In other words it increases the semi-major axis and in this case it also increases the eccentricity, that is makes the orbit more elliptic. We should burn the engines until the altitude of the orbit at the opposite side is just at the height of geostationary orbit, anything that intersects the geostationary orbit would get us there but any more is a waste of fuel. After that we just wait until we move along the orbit to that point, that is the point where our transfer orbit and the geostationary orbit intersect. There we'll do another engine burn to turn our transfer orbit into the desired geostationary orbit. This will again be prograde (along the orbital motion) increasing our speed and again raising the altitude at the opposite side of the planet. Again the semi-major axis is increasing but now the eccentricity is decreasing and our orbit is becoming more circular. When the altitude at the opposite side is also at the geostationary orbit altitude, we will be done and are on a circular geostationary orbit.
What we just did is a Hohmann transfer and it's the basic building block of all orbital transfers. You can drop your orbit by doing the opposite, first burn retrograde (against the motion to deccelerate) and then again retrograde at the opposite side. To get to the Moon you do pretty much the same thing but raise the orbital altitude all the way to the orbit of the Moon and just time it so that the Moon is there when you get close to its orbit. Only thing that'll be a bit different is that the gravity of the Moon will become significant at some point and you need to start thinking in terms of orbiting the Moon and not the Earth. Or to get to the Mars it's again the same thing but now we're orbiting the Sun and not the Earth. So you have to accelerate along the orbit around the Sun to raise your solar orbital altitude. You should try to do this when your orbit around the Earth goes the same direction to take advantages of your orbital speed around the Earth.
If you want to get an intuition about orbits in a fun way, I highly recommend trying the Kerbal space program game.
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