# Apparent (clearly false) contradiction - Kepler's Third Law

In summary: An object in geosynchronous orbit has a speed of ca. 3 km/s. Therefore, an object in geosynchronous orbit is not "traveling faster" than an object in low Earth orbit. In summary, when considering a satellite in geosynchronous orbit, its speed is zero across (relative to) Earth's surface. Using Kepler's third Law, we can derive that the velocity of a satellite in geosynchronous orbit decreases as the radius from Earth's center increases. This decrease in velocity is necessary to maintain a constant orbital period and match the Earth's rotational period. There is no anomaly, as geosynchronous orbit occurs at the specific height where this balance is achieved.

## Homework Statement

When considering a satellite in geosynchronous orbit, its speed is zero across (relative to) Earth's surface.
From Kepler's third Law: T2=(4π2r3)/(GM), we can derive that v2=GM/r

This would tell us that as the radius of a satellite to Earth's centre increases, its velocity decreases by a squared amount.

My Physics Class realized that, for the period of Earth and consequently the satellite to be constant, an increased radius from Earth's centre would require the satellite to travel at a faster velocity.

We could not explain this apparent anomaly and were clearly not accounting for some crucial factor.

Any help at explaining where we are wrong would be appreciated.
Thanks :)

## The Attempt at a Solution

This would tell us that as the radius of a satellite to Earth's centre increases, its velocity decreases by a squared amount.

That's right.

My Physics Class realized that, for the period of Earth and consequently the satellite to be constant, an increased radius from Earth's centre would require the satellite to travel at a faster velocity.

That's also right.

We could not explain this apparent anomaly and were clearly not accounting for some crucial factor.

There is no anomaly. Geosync orbit occurs at the height where the orbital period of the satellite is exactly enough to match the Earth's rotational period. Below this height a satellite travels too fast. Above this height and a satellite travels too slow.

My Physics Class realized that, for the period of Earth and consequently the satellite to be constant, an increased radius from Earth's centre would require the satellite to travel at a faster velocity.
To travel at a faster velocity than the rotational velocity of the Earth's surface. Not to travel at a faster velocity than an object in low Earth orbit (which is what your equations tell you). The surface of the Earth rotates at ca. 1600 km/h (444 m/s) at the equator. An object in low Earth orbit has an orbital speed of ca. 8 km/s.

## What is Kepler's Third Law?

Kepler's Third Law, also known as the "Harmonic Law," states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. In simpler terms, this means that the farther a planet is from the sun, the longer it takes to complete one orbit around it.

## Why is Kepler's Third Law considered a contradiction?

Kepler's Third Law is considered a contradiction because it appears to contradict Isaac Newton's Law of Universal Gravitation, which states that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. According to this law, planets that are farther from the sun should have longer orbital periods, but Kepler's Third Law states that they should have shorter orbital periods.

## How did Kepler discover his Third Law?

Johannes Kepler discovered his Third Law through his observations of the planets' orbits and his mathematical calculations. He analyzed the data collected by his predecessor, Tycho Brahe, and noticed a pattern between the orbital periods and distances of the planets from the sun. After years of trial and error, he formulated his Third Law to accurately describe this pattern.

## What evidence supports Kepler's Third Law?

There is a vast amount of evidence that supports Kepler's Third Law, including observations of the planets' orbits, mathematical calculations, and experiments. For example, when scientists observed the orbits of the outer planets such as Uranus and Neptune, they found that their orbital periods followed the same pattern as predicted by Kepler's Third Law.

## How does Kepler's Third Law contribute to our understanding of the solar system?

Kepler's Third Law is a fundamental principle in our understanding of the solar system. It explains the relationship between the distance of a planet from the sun and its orbital period, and this information has been used to discover and study new planets in our own solar system and beyond. It also played a crucial role in the development of Newton's Law of Universal Gravitation, which helped us understand the forces that govern the motion of celestial bodies in the universe.

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