Derivation of Kepler's Laws of Planetary Motion

When Johannes Kepler derived his laws of planetary motion, he only used the observation of the planets to come up with the three laws. As a reminder:

1.     Planets have elliptical orbits, i.e. circular or oval and the Sun is at one focus.

Hankwang

2.     A line connection the Earth and the Sun will sweep out equal areas in equal times.  This was discussed earlier here.

Shashi M. Kanbur, Professor of Physics and Earth Science, SUNY Oswego

3.     The cube of a planet's semi-major axis distance equals the square of the planet's orbital period, as long as they are measured relative to Earth.  The distance is measured in astronomical units, AU, which is the average distance from the Earth to the Sun. The period is measured in Earth year's.  This was actually proved by Isaac Newton with calculus and physics.

Here, I am going to use Newtonian Mechanics to show how these three laws can be derived from physics.

Isaac Newton first explained the elliptical orbits. He stated that all orbits are what are called conic sections. There are four conic sections: circle, ellipse, parabola, and hyperbola.

From Wikipedia

As you can see, they are conic sections because they are different cross sections of a cone. Both a parabola and hyperbola continue on to infinity. Those orbits are typically for long period comets.

The second and third laws can be derived from the force due to gravity and from centripetal force. Centripetal force is the force required to keep an object moving in a circular path. It is given by:

Where: F_C is the centripetal force, m is the mass of the object, v is the orbital velocity, and R is the radius of the path.

Gravitational force is the force exerted by one object on another. It is given by:

Where: F_G is the gravitational force, G is the gravitational constant, M is the mass of the larger object, m is the mass of the smaller body, and R is the distance between the two.

Now, let’s assume that M is the mass of the Sun (M₈=2x10³⁰ kg) and R is the orbital radius of the planet. We can equate these two (i.e. F_C = F_G) and we have

Orbital velocity, v is also given by how long (the period P) the object takes to make one complete orbit (the circumference, C = 2πR), or in equation form:

If we plug the equation for v in the third equation, we end up with:

Or simplifying:

Note that m cancels out on both sides, and knowing that M is the mass of the Sun (a constant in our Solar System) and G is also a constant, we can see the left side of the equation is a constant. In other words, as R increases, P also increases, or as R decreases, P decreases. So when a planet is closer to the Sun, its period is shorter, i.e. it is moving faster (v increases for smaller P and R). This explains Kepler’s second law.

To simplify this for Kepler’s third law, we have to do some math. We know that M₈=2x10³⁰ kg and G is 6.67x10^-11 m³/kg*s². Solving the right side of the last equation:

Looks gross, doesn’t it. Let’s make it simpler for us. Recall that Kepler’s Law states that R is in AU and P is in Earth years.

• 1 AU = 150x10⁹ m


• 1 year = 3.16x10⁷ seconds (3600 seconds in an hour, 24 hours a day, 365.25 days a year)
  
  
  

And setting R to a (semi-major axis), we see that:

Just as Kepler stated.