Kepler’s Second Law explained with Calculus
Did you know Kepler’s second law or as it’s more famously known, the equal area in equal time law can be simply demonstrated using calculus?
Firstly, what is the equal area in equal time law, and how does this apply to planetary motion? This is one of Kepler’s 3 laws governing planetary motion and it states that the area swept by a planet in a set duration of time will always be a constant. This can seem tough to visualise so let’s get the help of a diagram:
Let’s take the yellow circle in the above graph as the sun with the green circles representing a planet at different points in time. As you’ll notice most obviously, sector A is much shorter and wider than sector B. Despite this, when calculated you will find that these sectors are of the same area. This sounds like an interesting coincidence but there is in fact physics behind this phenomenon.
To figure out how this is happening, let us consider why the planet is even moving in an orbit at all. This is due to the force of gravity exerted by the sun on the planet. Rather than being attracted in a straight line to the sun, the planet travels in a roughly elliptical shape since it is traveling at a certain velocity. To break down its motion in this orbit we must bring in principles of circular motion.
To find the change in area we use,
As you can see equation 1 on the bottom right corner of figure 1 has far too many unknowns to serve us any good. This is where angular momentum, or L, comes into the picture as it is the cross product of r, v, & m. Since there’s no m in figure 1, our working would look a little something like this:
Great, now we’ve reduced the number of variables but what is L and how do we find it?
To answer that question we need to zoom out and ask the bigger question of what force is moving our planet anyway? Gravity, that was an easy one. The second thing to consider here is the radius of our orbit or the distance between the sun and the planet.
In an equation combining both gravity and the radius, we have,
Torque is calculated by T = rFsin(ø) where r is the distance between the sun and the planet, F refers to the force of gravity experienced by the planet and theta refers to the angle between d and F. If you were to draw a line from the planet to the sun representing r, you would most likely realise it is parallel to the direction of the force of gravity. This makes the angle between them 0. Since sin(0) = 0, we can conclude that there is no torque in this case.
That’s okay but how do we find L you may ask? here’s another formula for torque to address that.
Now we know for sure that L, or the angular momentum of the planet is a constant. We inherently knew that the mass of the planet is a constant. Let’s plug our new-found information into figure 2,
From figure 4, we can see that the change in area divided by the change in time has to be some constant since a planet’s mass and angular momentum are constant forever. This means that the area swept by a planet per unit time will be constant, thereby explaining Kepler’s 2nd Law.
The idea for this article was from one of you, the readers of this blog for whom I am very grateful. Comment a topic if you’d like for me to try and explain it. As always, keep reading my articles to learn more about physics and stay curious.
