Welcome back, everyone. Well today's lecture is just a tiny bit of maths. And what we're really looking for, or geometry, at least. And what we just want to describe is a object that is extremely important for astronomy. And it's what's called the Ellipse. Now, ellipses are, you know? They're like circles in the sense that they're geometrical objects, or mathematical objects. But they play an essential role in astronomy. So in the mid 1600s, Johannes Kepler figured out, that all planets orbit on ellipses, they're motion can be described by ellipses. So that's why they're so important to astronomy. So we just want to sort of think a little bit about what ellipse is. So an ellipse kind of looks like a squashed circle where a circle just has one radius, there are sort of, two radiuses to describe an ellipse. One is called the semi-major axis, and that is the distance from the center. Out to the long, axis, and then the semi-minor axis is the distance from the uh,center to the short axis. And, but of course that's actually not, doesn't completely describe the ellipses. But they are important for the ellipses are what are called the foci, and the foci are these two points over here, in the circle the foci are actually at the center. But why are the focii important, because the sun is at the center of one of the focii for all of the orbits in the solar system. In fact, anything which is orbiting anything else the, the central object always will be at the focus of the elliptical orbit. So you know all orbits or ellipses. And the thing your orbiting is always going to be at one of the foci. Now there's this thing called the eccentricity which describes sort of how squashed the circle is. So mathematically the, the eccentricity is defined as the distance from the center. To the focus, divided by the center to the semi major axis. so what that means is, is that for a circle, since the focus is actually at the center, the eccentricity is zero, and for a really really sco, squashed circle, it's really a squashed ellipse. Which would be sort of like the orbit of a comet then they had eccentricities close to one. Okay. So remember the star's always at the focus and here's the interesting thing, as the star, so if this is the sun and this is the planet moving around on ecce, eccentric orbit. That was a very eccentric orbit by the way, I just described, most of the planetary orbits in the solar system are pretty close to circles. But the important thing to remember is that, as you get close to the star or the focus of your orbit, you speed up. So a typical orbit looks something like this. Zoom. You zoom around, slow down then zoom back in. So that's why comets as they come in start moving very quickly and then as they go out tend to spend, you know, a very long time out at the edges of their orbit.