This all-in-one online Kepler’s Third Law Calculator performs calculations using Kepler’s third law of planetary motion, which relates the mass of a star and the planet’s distance from the star (the elliptical semi-major axis) to its orbital period. You can enter the values of any three parameters in the fields of this calculator and find the missing parameter.
Kepler’s Third Law of Planetary Motion
Kepler’s laws of planetary motion, formulated by the German mathematician and astronomer Johannes Kepler in the early 17th century, revolutionized our understanding of the motion of celestial bodies.
These laws served as the foundation for Isaac Newton’s subsequent work on universal gravitation and laid the foundation for modern celestial mechanics. Kepler’s discoveries were a major leap forward in the field of astronomy and significantly influenced our understanding of the structure and behavior of the solar system.
Kepler’s third law, often called the harmonic law, a cornerstone in the field of celestial mechanics, establishes a mathematical relationship between a planet’s orbital period and its average distance from the Sun. It states that the square of a planet’s orbital period is directly proportional to the cube of its average distance from the Sun.
Derivation of Kepler’s Third Law
The easiest way to derive the relationship between the orbital period of a planet and its distance from the Sun is in the case of a circular orbit of motion.
Using Newton’s law of gravitation and assuming that the mass of the planet is much less than the mass of the Sun, this relation can be found by setting the centripetal force equal to the gravitational force:
$$mr {\omega^2} = G \cdot \frac{m \ M}{r^2},$$
where
• \(M\) is the mass of the central body (the Sun),
• \(m\) is the mass of the orbiting body (the planet),
• \(r\) is the radius of the circular orbit,
• \(\omega\) is the angular velocity of the orbital motion,
• \(G\) is the gravitational constant, its value is equal to 6.6743 × 10-11 N·m²/kg².
Then, expressing the angular velocity \(\omega\) in terms of the orbital period \(T\) and after rearranging, we arrive at Kepler’s third law:
$$mr \left ( \frac{2\pi }{T} \right )^2 = G \cdot \frac{m \ M}{r^2},$$
$$T^2 = \left ( \frac{4\pi^2 }{GM} \right ) r^3$$
In the case of general elliptical orbits, instead of circles, and also taking into account rotation around the general center of mass of the system, one can obtain a similar formula in which instead of the circular radius \(r\), the semimajor axis \(a\) of the elliptical trajectory of one mass relative to the other is used, and also instead of mass \(M\), \(M+m\) is used. Mathematically this Kepler’s third law formula can be expressed as follows:
$$\frac{a^3}{T^2} = \frac{G(M+m)}{4 \pi^2}.$$
However, since the masses of the planets are much smaller than the mass of the Sun, they can be neglected.
$$\frac{a^3}{T^2} \approx \frac{GM}{4 \pi^2}.$$
This is the formula used in our Kepler’s Third Law Calculator.
Note that for ease of use of our calculator, the masses of objects can be determined not only in the usual units of mass measurement, but also in the masses of the Sun or the Earth. And distances can be specified also in astronomical units au = 1.495978707×1011 m.
Example of use of Kepler’s 3rd law:
The planet Venus has a period of about 224.7 days, how far is it from the Sun?
Let’s enter the data into our online calculator. As the mass of the star, we specify 1 Sun, and the orbital period of 224.7 days. By clicking ‘Calculate’ button we will immediately get the result: semimajor axis = 0.7233 au.
Application and Limitations of Kepler’s Third Law
Kepler’s third law formula serves as a fundamental tool in astronomical calculations and observations. Knowing the orbital period of a celestial body and its average distance from the central mass, astronomers can infer key properties such as the mass of the central body.
However, the limitations inherent in Kepler’s third law must be recognized. One notable limitation is that it applies mainly to two-body systems where the mass of the central body is much greater than the mass of the orbiting body.
In addition, the law assumes that orbiting bodies move along elliptical trajectories with a central mass at one of the centers, which is not always true in complex multi-body systems or under the influence of external perturbations.
Despite these limitations, Kepler’s third law remains an important tool of celestial mechanics, providing invaluable insights into the dynamics of planetary motion and serving as a springboard for further exploration of the cosmos.
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