## Kepler’s Laws

### Table of Content

- What is Planetary Motion?
- Kepler’s Three Laws
- The First Law Statement
- Kepler’s 1st Law vs Copernicus Model
- Second Law Statement
- Third Law Statement
- Derivation of Universality of Kepler’s third Law of Motion

## What is Planetary Motion?

**Johannes Kepler** was the astronomer who published the three laws of planetary motion. He used the data on the planets collected by a fellow astronomer named **Tycho Brahe** to develop these three planetary laws. Tycho Brahe was a flamboyant naked eye astronomer who had a gold nose to replace his when he had lost in a rapier tool. The first two was in 1609 and the third one was 1619. These three laws came up by Johannes Kepler to define the mechanics of **planetary motion.**

## Kepler’s Three Laws of Motion

**1. Law of Orbits:** All planets are moving in an elliptical orbit, with the sun at one focus.

**2. Law of Areas: **A line that connects a planet to the sun cleans out equal areas in an equal time.

**3. Law of Periods: **The square of the planet period or the time it takes for the planet's orbit the sun is proportional to the cube of the semi-major axis. This means that the further planet is always away from the sun the longer it takes to orbit and the closer a planet is to the sun the quicker it orbits.

## The First Law Statement

**Law of orbit **states that all planets are moving in an elliptical orbit with the sun at one focus. Now, this allows there to be a lot of different sizes of orbits but all planets orbit in ellipses. Now look at some key pieces of an ellipse there is also the major axis which passes through the center of the ellipse and is long diameter of the ellipse it’s also called the transverse diameter. A minor axis passes through the center of the ellipse as well and is the shortest diameter. There is also semi-major axis which goes from the center of the ellipse to the longest edge and the semi-minor axis which goes from the center of the ellipse to the shortest side. There also two focus and remember that the sun is always at one focus. Now let’s talk about orbits and ellipse there are two other terms that can be helpful they are the perihelion and the aphelion. The perihelion is when the planet is closest it ever gets to the sun and the aphelion is the furthest the planet gets from the sun.

## Kepler’s 1st Law vs Copernicus Model

Let’s see how can we compare Kepler’s first law with the **Copernicus model**. Copernicus told that the sun and the planets revolve in circular orbit but Kepler said that it is not circulated instead it is an ellipse. How do these two theories contradict each other? The answer is no because the circle is nothing but a special case of an ellipse. So, we can say that what Copernicus suggested was not incorrect but he didn’t give a general explanation for all the orbits. He only spoke about one special case because the circle is just a special case of an ellipse. So in general Kepler’s law stated that all the planets revolve in elliptical orders. Now it is clear to all of you why it is called the **Law of Orbits**.

In this law, it talks about the orbital motion of the planet and that kind of orbit does the planets follow to travel around the sun and hence, that is why it is known as **Law of Orbits**.

What basically happens when a circle is a special case of an ellipse. A circle comes out of an ellipse and it has two focus. Those two focus of the ellipse merges and which is known as the center of the circle and the semi-major axis becomes the radius of the circle. The semi-major axis, half part of a circle becomes the radius of the circle and the two focus combine and becomes the center of the circle.

## Second Law Statement

It’s popularly known as the **Law of Areas**. The reason behind this is this law stated the line of joining a planet to the sun wipe out equal areas in equal intervals of time. It means whenever the first block clarified the fact that the planet will revolve in an elliptical orbit. The second law stated that the area covered by the planet while revolving around the sun will be equal in equal intervals of time that means the rate of change of area with time will be constant.

A straight line is joining the sun and a planet pull out equal areas in equal intervals of time. It means if it took one month for the earth to go from one focus point to another focus point in its orbit, it takes one month for the earth to go from another focus point to focus point end in his orbit and notice that this area was exactly the same as this area.

That means since this line is so much longer this didn’t have to be quite as long as for the area’s to be equal. So, as a result, the planet must be going slowly the first focus before you get faster the second focus.

So, the law is about the area’s being equal for an equal period of time but the main thing is that the planet is speeding up and slowing down in its orbit in order for this to happen. The main thing, it showed that there is some kind of relationship of the planet’s orbit around the sun.

## Third Law Statement

The third law is popularly known as the **Law of Periods**. This law state that the square of the time period of a planet is directly proportional to the cube of the semi-major axis of the orbit. Let’s assume any planet is revolving and earth is revolving around the sun. Now the time period of the planet that is the time period of earth that is the time it takes to complete one revolution around the sun the square of the time period. Let's assume T^{2 }is instantly proportional to the cube of the semi-major axis of the orbit. We already mention it whenever we talked about a semi-major axis which distance do we mean. We mean half of the major axis. So, time period of the planet square of the time period of the planet is proportional to the cube of the semi-major axis.

The equation:

**T ^{2} ∝ α^{3}**

Here is the semi-major axis. Now, we think it’s clear to all of you, why is it called the **Law of Periods**? Because it talks about the relationship of the time period of planet that is why it is known as the **Law of Periods**.

## Derivation of Universality of Kepler’s third Law of Motion

Let’s assume the planet mass is m

Mass of sun is M

Now according to newton’s law of gravitation, there is a sun and a planet which is revolving throughout the sun. So, there has to be a force of gravitation between the planet and the sun.

So

Here r is the distance between the planet and the sun. At the same time since it is moving in an elliptical orbit, there has to be the centripetal force which helps the planet to move around the sun in a circular path.

So that centripetal force (would be nothing but mv^{2}/r

Now since the planet is moving around the sun in a stable orbit that means these two forces are balancing each other.

**F = Fc**

So, the force F due to gravitation should be equal to the centripetal force.

Here v is the velocity of the planet that is the distance moved by the planet part by the time taken. The distance traveled means the circumference of the circle. In this case, in order to derive Kepler’s third law, we have made this assumption that the planet moves in a circular path and circular path is nothing but a special case of an ellipse. So, we take the assumption as a consideration to prove the law. Therefore, velocity of the planet v would be the margin of the path divided by the time taken.

Here the mass of the sun is constant, G is the universal gravitational constant and 4π^{2} is definitely constant. So, the entire thing is a constant. So, this proves that,

**K = **Kepler’s constant(m^{3}/s^{2})

**r = **is the mean radius of the orbit(m)

**T = **Period of orbit (s)

In general, for ellipse T^{2} ∝ α^{3}, where a is the semi-major axis.

So any planet follows this relationship, it means is there is a relationship between how much the planet is from the sun and how long it’s going to take to go throughout the sun. This formula is good for any system if we use the earth at the center of the objects, in the center control the system just like the sun, you could say control our solar system. So, we have one constant for it if we put the earth here at the center and we have the moon way around the earth or any satellite it must also follow this law. But the constant will be different because the earth is at the center, it’ll be a constant based on that system and it would be a different number. If we put a satellite or the moon around the Jupiter then it would follow this law.

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