Gravity Equations Formulas Calculator

Science Physics Kepler's Third Law

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Problem:

Solve for satellite orbit period.

G is the universal gravitational constant
G = 6.6726 x 10^-11N-m²/kg²

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	gravitational force exerted between two objects
	mass of object 1
	mass of object 2
	distance between the objects

	satellite orbit period
	satellite mean orbital radius
	planet mass

	gravitational acceleration
	planet mass
	radius from the planet center

	escape or critical speed
	planet mass
	planet radius

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Kepler's Third Law

Kepler's third law, the Law of Harmonies, is the final piece in the trio of laws governing celestial bodies' motion within our solar system. Conceived by the German mathematician and astronomer Johannes Kepler in the early 17th century, these laws revolutionized our understanding of planetary motion. In addition, they laid the groundwork for Isaac Newton's theory of gravitation. This article will provide an in-depth look into Kepler's third law, derivation, and significance in modern astronomy.

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Background

Kepler's third law was published in 1619 in his book "Harmonices Mundi" or "The Harmony of the World." It built upon two earlier laws that Kepler had formulated based on his analysis of Tycho Brahe's meticulous astronomical observations:

Kepler's third law expanded upon these principles and provided a mathematical relationship between the orbital periods and average distances of planets from the Sun.

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Statement of Kepler's Third Law

Kepler's third law declares that a planet's orbital period (T) squared is directly proportional to the semi-major axis (a) cubed of its elliptical orbit. Mathematically, it can be expressed as:

T² ∝ a³

To convert this proportionality into an equation, a constant of proportionality (k) is introduced:

T² = k * a³

This constant is the same for planets in our solar system, meaning the ratio of T² to a³ is consistent across all planetary orbits.

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Derivation of Kepler's Third Law

Kepler derived the third law using empirical data and his profound understanding of geometry. However, Newton's laws of motion and the universal law of gravitation can obtain a more rigorous derivation.

Newton's Second Law of Motion declares that the total force (F) acting on an object is equal to its mass (m) times acceleration (a):

F = m*a

Newton's Universal Law of Gravitation defines the gravitational force between two bodies as:

F = (G*m1*m2) / r²

Where:

By equating the centripetal force required to maintain the planet's circular motion with the gravitational force acting on the planet, we can derive the relationship between the orbital period and the semi-major axis:

T² = (4 * π² * a³) / (G * M)

Where M is the mass of the central body (in this case, the Sun).

For planets within the same solar system, the constants (4 * π²) / (G * M) remain the same, reducing the equation to the original form:

T² ∝ a³

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Applications of Kepler's Third Law

Kepler's third law has played an essential role in understanding the solar system and beyond. Some of its applications include: