# Conservation Laws and Kepler’s Planetary Laws

Conservation of Momentum: We noted last time that Newton’s third law naturally holds the conservation of motion, so let’s find the connection together. When no external forces are applied to a body, then there is nothing to change the motion of that body, right? (Newton’s first law!). This is the same as a body with zero acceleration. Acceleration is the measure in the change of velocity, so if that change is zero, then this means the velocity of this body is constant, or unchanging. When the net force is zero, that means there is no change in motion – which means MOTION IS BEING CONSERVED! This is referred to as the conservation of momentum.  Momentum is always conserved, no matter what!

Angular momentum: We have actually already talked a little bit about the conservation of angular momentum! It is precisely what binds the planets to orbit our Sun! The rotating motion generates a twisting force we call “torque”. As long as the net force (or net torque) is zero, then angular momentum is conserved.  The conservation of angular momentum is the reasoning behind Kepler’s second law!

Energy: In essence, energy is what makes matter move! We like to categorize the types of energy to clarify how matter is moving. Kinetic energy is the energy of motion. Falling rocks, orbiting planets, dancing ballerinas, are all examples of matter that has kinetic energy. Then there is radiative energy which is synonymous to “radiation”. Radiation is the energy that light carries and light can definitely use this energy to change motion of other objects. The energy of light is what makes it possible for us to see, it warms our planet, and plants use it to live! And then there is energy that is stored, or potential energy. This energy can be converted later into energy that moves the object. Yes, that means potential energy can turn into kinetic energy.  The most common type of potential energy in astronomy is the potential energy from gravity! The gravitational potential energy of a body depends on its mass and how far it can move as a result of gravity.  Even energy is conserved! In fact, energy cannot be created nor destroyed. This means any energy a body gains is really just energy that was stolen from somewhere or something else.

Before Isaac Newton published his three laws of motion and discovered the law of gravity, we were already getting a glimpse into this new world of physics thanks to Johannes Kepler.

Kepler established three planetary laws of motion purely through observing the planets in the night sky. Kepler’s first law is that the planets travelled in elliptical orbits. Then, there was something else he noticed: planets in their elliptical orbits didn’t move at a constant speed. The planets actually sped up as they approached the Sun, and slowed down as they moved away from it. Technically Kepler’s second law states “planets cover equal areas in equal time intervals”, which turns out to also describe the conservation of angular momentum! Kepler’s third law is the empirical relationship between the orbital period of a planet and its distance from the Sun. It wasn’t until Newton came along 100 years later that we understood why Kepler’s laws were true. Particularly Kepler’s third law, which was incomplete. Kepler’s third law, which points out a relationship between the orbital period and the distance from the Sun, is now known as the Law of gravity!

Law of gravity: The universal law of gravitation describes how gravity behaves and can be summarized in the following three ways:

1. Every body is attracted to another body through the force of gravity.
2. The strength of the gravitational force is directly proportional to the product of the masses.
3. The strength of the gravitational force is inversely proportional to the square of the distances between the masses.

This means the farther two bodies are from each other, the weaker their gravitational force is on one another. The closer they are, the stronger the gravitational force is. This is precisely why the planets revolve around the Sun at different speeds.

Want to know more?

Check out Chapter 4: Making Sense of the Universe in the textbook The Cosmic Perspective by Bennet, Donahue, Voit,  and Schneider, Seventh Edition, 2014. Pgs 125-144