In the 1960s, the space race was well under way. Both the United States and Russia were trying to figure out how to put people on the moon. Another obvious place to compete in space is to send the first space craft to another planet. It sounds a little complicated, to send a space craft to another planet, but I'm actually here to tell you a, a dirty little secret. The dynamics, the orbital dynamics of sending a space craft to another planet are actually pretty simple. And the reason they're pretty simple is because all you have to deal with is gravity. And when you're in, flying through interplanetary space, all you have to do with, deal with, is the Sun's gravity. Eventually you get close to Mars, you have to deal with Mars' gravity. You have to leave the Earth, you had to leave, deal with Earth's gravity. But once you've done those two things, that flight through space is pretty easy. It's efficiently easy that we can at least calculate some simple things about it just right now. First question you might ask yourself is how would you get from Earth to Mars? Well let's remember that, here let's put the sun here in the middle again. Earth is here. Mars is out here. And let's actually write down the distances because these are going to be important. The distance from the Sun to the Earth, average distance, is defined to be one AU. One astronomical unit. And in these same units, in these same units, Mars is 1.524 astronomical units. So, Mars is about 50% further away than the Earth is. Is this when you launch your spacecraft? Well, okay. Let's figure out how you do it. You launch your spacecraft here and you head straight towards Mars. But of course Mars has moved along this way. And so maybe you should launch in this direction and eventually intersect with Mars. If you start to iterate this long enough you realize that the easiest thing to do, the least energetic solution is actually not to launch towards Mars at all, but to launch perpendicular to the Earth's orbit, with just a slightly higher velocity than the Earth's orbit. Let's look at how that looks. If this is the Earth's orbit, I'm going to draw the whole thing now. Here is the orbit of Mars. The way you get from Earth to Mars using the least amount of energy and using the least amount of energy is important because you have this large spacecraft that you're trying to get to Mars, usually your limiting factor is how much energy you can generate from his rocket. So the least energetic way to get form the Earth, Earth is right here, to Mars is a flight that goes, just barely hits the orbit of Mars. And if it continued on, didn't stop at Mars, it would come back to the Earth. Now, you could do a lot more things. You could say, I'm going to go faster, and I'm going to try to go this way. But imagine what you've done now. You have intersected Mars, and you still have a high velocity as you're going through here. So you would take yourself way out through here to the outer part of the solar system. And back in, if you continued on after Mars. To get on an orbit like that requires a lot more energy than this one that's just barely leaving the Earth's orbit, and just barely getting to Mars. As you might remember, from orbital mechanics, or just some basic physics. The important things that matter in an orbit are the semi major axis. And the excentricity of the semi major axis, this is an elliptical orbit now. The semi major axis is the average of the closest and the furthest points from the orbit. The closest points of the orbit is called the parhelion, furthest point from the orbit is called the aphelion and they're simply related mathematically by q parhelion. And this is the typical thing we call use for parahealian is a the semi major axis times 1 minus the eccentricity, and Q, it's terrible that astronomers use this notation, but they do, is, which is the appelline is a times 1 plus e. So, if you know the semi-major axis you can figure out the eccentricity. If you know the locations where these parahelia and aphelia are, and, in fact, you do, because you want the parahelian to be Earth's orbit, which is one AU, so that is going to be 1AU. You want the aphelion to be the Martian orbit, which is 1.524AU. You can solve these two for the semi major axis. Of course we know the semi major axis is just the average of these two. So the semi major axis of your new orbit that you're putting this thing on is AU, is 1.262AU. So you have added energy to the orbit by moving it to a higher semi-major axis. As you might remember from basic physics, orbital mechanics, adding energy to an orbit causes the semi-major axis to increase. Decreasing energy causes it to crease, decrease. And so, you see that this orbit is the one where you add the least amount of energy and still make it to Mars. Again, you could do this orbit sure, but now you're semi-major axises is much larger and your energy is much larger, which means you need a huge rocket to get there. The benefit is you get there faster. That has never been an important enough benefit that anybody's ever done it that way. Instead, every single mission to Mars has taken an approach that moves from here over to here, lands at that point. How long does that flight take? Well, it's, it's half of the orbital period of this new orbit of the rocket. The orbital period of an object is equal to the semi-major axis to the three-halves power. This is simply one of Kepler's laws, one of Kepler's three laws. The semi-major axis of the Earth, and this is, of course, in, semi-major axis in AU, orbital period in years. Here's the semi-major axis of the Earth is one AU, and so the period is one year. That's nice. Semi-major axis of Mars as you remember is 1.524. So it takes Mars 1.88 years to go around the sun. And the semi-major axis of this is 1.262 so it takes this object that, your spacecraft, about 1.4 years or 17 months to go all the way around the Sun. But you don't go all the way around the Sun. You go to Mars. So you're only doing half of your orbit. So it takes you about eight and a half months to get from the Earth to Mars. And if you're ever paying attention to Martian flights. The next one is not going to be for a while so you had to've been paying attention recently. But they always launch something like nine months before they get there. They take about that long of a time to get there. So when do you launch? You don't launch when the Earth is right here and Mars is right here. Because, then in nine months, what's going to happen? Well, in nine months your spacecraft is going to be here, but where is Mars going to be? It will only have gone about 40% of the way around it. It'll be around right here. And your space craft will fly by and say Mars wasn't here. So instead, you launch when Mars is about right here. The Earth has not quite cut up, caught up with Mars yet, and then Mars is right here when the spacecraft arrives there. Or if you think about it, the spacecraft is actually moving slower then Mars at this point which is why its going to start to fall back towards the sun. And so its not really that Mars is there when the spacecraft arrives, the space craft is there when Mars arrives. And how do you actually do it, really all you do is you have the the rocket is in orbit. Getting into orbit, okay, that's a little bit complicated. I told you that the orbital dynamics is easy because all you have to worry about is the sun. Rockets, going through the Earth's atmosphere? Much more complicated. We're not going to worry about that. We're going to assume that somebody else takes care of that and we are now in orbit around the Earth. Once you're in orbit around the Earth, all you do is calculate the velocity that you need to have, and you know the direction you need to go, perpendicular to the sun, and then that's all you do. You, you blast your rocket, it gives you that exact velocity, and you coast. The spacecraft moving to Mars, we sometimes think of them as, burning their rocket the entire time, but they don't. It's easier to think of it as they do one blast of the rocket, and they coast for nine months until they get there. In practice, somewhere around in here, they take a look where they are, and they say, oh you know, I'm off by a little bit this way, a little bit this way, and they do a mid-course correction or two to perfectly align themselves. But other than those small corrections, they're really just coasting. They're really just resting and waiting until Mars grabs them. And then they have to do something else because of course, if they got here, and nothing else happens, well, if we, if we, aimed perfectly we would smash right into Mars which would be entertaining but not very scientifically useful. We've actually done that before a few times. If we slightly missed and we kept on going around the sun we'd come back to the Earth or at least the position where the Earth was when we launched it. And so instead what we do is get too close to Mars and then slow down use rockets to slow ourselves down in getting to orbit around Mars at this location. In principle relatively simple once you have figured out rockets, and the Space Age, and all these things. Let's look at how it worked in practice.
