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Kepler’s Laws: Examples and Diffeq

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Contents:

  1. Overview of Kepler’s Laws
  2. Differential Equation for Kepler’s Laws

Kepler’s Laws: Overview

Kepler’s Laws of planetary motion are:

  1. Planets revolve around the sun in an elliptical orbit; the sun is at one of the two foci.
  2. The line that joins the sun to a planet sweeps out equal areas in equal times.
  3. A planet’s squared orbital period is directly proportional to the cube of the semi-major axis of its orbit.< /li>

1. Kepler’s First Law

Planets revolve around the sun in an elliptical orbit; the sun is at one of the two foci.

kepler's first law

The foci are labeled F1 and F2 (blue planet) and F3 and F3 (red planet).



2. Kepler’s Second Law

A radius vector that joins the sun to a planet sweeps out equal areas in equal times.

kepler's second law

Two months of the year are shaded in (labeled a1 and a2). Although they appear to be different in size, they have the same time period (one month, in this example), so they do have the exact same area.


3. Kepler’s Third Law

A planet’s squared orbital period is directly proportional to the cube of the semi-major axis of its orbit.
The third of Kepler’s laws allows us to compare the speed of any planet to another using a planet’s period (P)—the time it takes to go around the sun relative to the stars—and it’s average distance (d) from the sun. The period is is related to any planet’s average distance (d) from the Sun by the following equation:

P2 = k*d3

In word, that’s the period squared (P*P), divided by the average distance cubed (d*d*d), is equal to a constant.
This fact is true for every planet, no matter how far it is from the sun or how long/short its period.

Using algebra, the above equation can also be rewritten as:
P2 / d3 = k
This is important, because it allows us to easily find the value for k, which will be the same constant for every planet in our solar system:
The Earth’s period P = 1 year; d = 1 AU*, so:

P2 / d3 = k
= 12 / 13 = 1


The constant, k, is 1.

*Using astronomical units keeps the equations simple. An explanation of astronomical units is beyond the scope of this article, but you can read a simple explanation here.

Example

Now you know “k”, you can find out the distance of any planet from the sun, if you know it’s orbital period. For example, the orbital period of Mars is 1.88 years, so:
1.882 / AU3 = 1
d3 = 3.53 AU3
= 1.52 AU
Mars is 1.52 AU From the Sun.

Kepler’s Laws and Differential Equations

All three of Kepler’s laws result from the following differential equation:
kepler's laws diffeq

Where:

  • r = r(t) is the vector from the sun to the planet
  • m = planet’s mass
  • || = magnitude or length of the vector

k has a different value in this differential equation, due to Newton discovering that k related to the mass of the planet and the mass of the sun.

Kepler’s Laws: References

CalTech. How far away is the sun? Retrieved June 6, 2019 from:http://coolcosmos.ipac.caltech.edu/ask/8-How-far-away-is-the-Sun-
Koupelis, T. & Kuhn, K. In Quest of the Universe. Retrieved June 6, 2018 from: https://books.google.com/books?id=6rTttN4ZdyoC
Rogawski, J. (2007). Multivariable Calculus. Macmillan.

CITE THIS AS:
Stephanie Glen. "Kepler’s Laws: Examples and Diffeq" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/keplers-laws-examples-and-diffeq/
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