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.