Kepler's Laws

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. The Earth goes round the Sun once a year. Now picture a comet. It swings in to half the Earth's distance from the Sun, then flies out to thirty five times the Earth's distance. How many years before it comes back? Hard to guess, right? And where on its path is it fastest? Today you will answer both. Kepler's three laws come up often in JEE gravitation. So, let's begin.

0:31Laws 1 and 2: an ellipse, and equal areas

So, law one, the law of orbits. Each planet moves in an ellipse, a stretched circle, with the Sun at one focus, not at the centre. In the figure, the orange dot marked Sun is pushed to one side. The closest point, on the right, is perihelion; the farthest, on the left, is aphelion. The long axis is marked two a, so its half, small a, is the semi-major axis. Keep it in mind, okay? Now, law two, the law of areas. Join the Sun to the planet with a line. As the planet moves, this line sweeps out area. Kepler found that it sweeps equal areas in equal times. In the figure, the two shaded slices are marked A one and A two. If they are swept in the same time, their areas are equal. The sketch is not drawn to scale, but the rule is exact. Right? Now look at the shapes. Slice A one, near the Sun, is short and wide. Slice A two, far away, is long and thin. Same time, so same area. So tell me, why must the planet move faster near the Sun? Yes, exactly! Near the Sun the slice is short, so to have the same area it must be wider: the planet covers a longer arc in the same time. So it moves fastest at perihelion and slowest at aphelion, just as the figure says. With me?

1:54Why equal areas? Angular momentum

But why are the areas equal? The rate at which area is swept, d A over d t, is called the areal velocity. It equals capital L over two m, where capital L is the planet's angular momentum about the Sun, and small m is its mass. Okay? Now, gravity is a central force: it always pulls straight towards the Sun. So capital L stays constant; the double arrow means implies. Constant L, so constant d A over d t: equal areas in equal times. Clear? Now use it at the two ends of the long axis. There, the angular momentum is just m times v times r. Same capital L at both ends, so v p r p equals v a r a; the small p and a written below mean perihelion and aphelion. So, v p over v a equals r a over r p. Got it? So v times r is the same all round the orbit, right? No, no, wait! That is a common trap. It works only at perihelion and aphelion, where the velocity is at right angles to the line from the Sun; the upside down T sign means at right angles. Elsewhere the velocity is slanted, and m v r is not the angular momentum. Okay?

3:09Law 3: the law of periods

Now, law three. But first, small a. Small a is the semi-major axis: half the long axis of the ellipse. Closest plus farthest distance makes the whole long axis, two a. So small a is r p plus r a, over two. Right? Now, a farther planet takes longer to go round, of course. So, twice as far means twice the period? No, more than that! Kepler found that T squared is proportional to a cubed. Capital T is the period, the time for one full orbit, and the little fish-shaped sign means proportional to. The full form: T squared equals four pi squared over capital G M, times a cubed. Capital G is the gravitational constant, and capital M is the mass of the Sun, the body at the focus. With me? Quick check. For a circle, small a is just the radius r. Put a equal to r and take the square root, the tick-shaped sign: capital T equals two pi, root of r cubed over capital G M, the circular-orbit period from earlier. It matches! Okay? Now, one condition. Four pi squared over capital G M is the same number only for orbits around the same central mass. So, for the planets of the Sun, capital T is proportional to a to the power three over two. Clear?

4:31Energy and speed on an ellipse

Two more results. On an ellipse, the total energy E is minus capital G M m over two a: the circular-orbit energy, with r replaced by small a. Okay? And the speed at perihelion: v p equals the square root of, two capital G M r a, over, r p times, r p plus r a. It comes from energy conservation and v p r p equals v a r a. Right? Now let's test it. What if r p equals r a? Then the ellipse is a circle of radius r. Put both equal to r: the top is two capital G M r, the bottom is r times two r, and we get root of capital G M over r: the orbital speed, v o, from earlier in this chapter. Perfect! Got it?

5:17Example 1: period of a planet 4 times as far from the Sun

Now, the first example. Here, a planet's mean distance from the Sun is four times the Earth's. Find its period. Pause the video, and try it first. Done? Okay, let's check. Both go round the same Sun, so capital T is proportional to a to the power three over two. The mean distance plays the role of small a, and the Earth's period is one year. Okay? Now, small a is four times the Earth's, and four cubed is sixty four. So the planet takes sixty four years, right? Oh no, careful! Sixty four is T squared, not the period itself. Students often forget this root. Clear? So, capital T is four to the power three over two. Root four is two, and two cubed is eight. Eight years. That's it!

6:08Example 2: comet period from perihelion and aphelion

Now, back to our comet. Its closest and farthest distances from the Sun are zero point five and thirty five A U, the astronomical unit, which is the Earth's mean distance from the Sun. Find its period. Pause and try it. Hint: which distance goes in as small a? Done? Okay, let's check. The comet goes out to thirty five A U, so small a is thirty five, right? No, no, wait! Thirty five to the power three over two is about two hundred and seven years; the wavy sign means approximately. Wrong! Thirty five is the farthest distance, not the semi-major axis. Okay? Small a is half the long axis: closest plus farthest, over two. Zero point five plus thirty five is thirty five point five, and half of that, seventeen point seven five A U, is small a. With me? So, capital T is seventeen point seven five to the power three over two: about seventy five years. Small a is in A U, so the period comes out in years, like the Earth's one year. So our comet returns about once in seventy five years. Puzzle solved! And it moves fastest when closest to the Sun. That's it!

7:25Quick revision: Kepler's three laws and orbit energy

So, quick revision. Law one: an ellipse, with the Sun at one focus. Law two: capital L is conserved, so d A over d t is constant: equal areas in equal times. At perihelion and aphelion only, v p r p equals v a r a. Law three: T squared equals four pi squared over capital G M, times a cubed, with small a the average of the closest and farthest distances; and the energy is minus capital G M m over two a. Now go and try the practice problems on this page. All the best!