Name: kepler’s planet distance law
Uploaded: 2023-04-30T11:07:15.000Z
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Earth and Environmental Science

Earth's Place in the Universe

ESS.EES.1.1

Kepler's Third Law of Planetary Motion

Consider a planet orbiting the Sun in an elliptical orbit. If the planet takes time T to complete one orbit around the Sun, its angular velocity is expressed as two pi divided by T.
The centripetal force acting on the orbiting planet is expressed as the products of its mass, square of its angular velocity, and its average distance from the Sun.
Substituting for omega, the centripetal force equation can be obtained in terms of the planet&#39;s orbital period and the orbit&#39;s semi-major axis.
The Sun&#39;s gravitational force provides the centripetal force for the planet to orbit around the Sun. Substituting and rearranging the terms, square of orbital period equals the product of a constant term and the cube of the orbit&#39;s semi-major axis.
Therefore, the square of a planet&#39;s orbital period is proportional to the cube of the orbit&#39;s semi-major axis. This is popularly known as Kepler&#39;s third law of planetary motion. The proportionality constant, which involves the gravitational constant and the Sun&#39;s mass, was derived later when Newton stated his law of gravitation.

kepler's third law of planetary motion

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. In 1909, he formulated his first two laws based on the observations of his forebears, Nikolaus Copernicus and Tycho Brahe. However, in 1918, he published his third law of planetary motion, which gives a precise mathematical relationship between a planet's average distance from the Sun and the amount of time it takes to revolve around the Sun. It states that
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The proportionality constant for this law was derived much later, after Newton established the universal law of gravitation. However, Kepler formatted this third law of planetary motion based on the observation of Galilean moons orbiting Jupiter.
This law established the ratios of the average distances of each planet from the Sun, to the Earth's average distance from the Sun, as shown in Table 1.
Table 1: Ratio of Planetary Distances from the Sun Compared to the Earth-Sun Distance.
<table style="margin-left: auto; margin-right: auto;">
	<tbody style="border: 1px solid black;">
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Material</td>
			<td style="color: black; border: 1px solid black; text-align: center;">Planetary Distance Ratio</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Mercury</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.387</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Venus</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.723</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Earth</td>
			<td style="color: black; border: 1px solid black; text-align: center;">1.000</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;"> Mars</td>
			<td style="color: black; border: 1px solid black; text-align: center;">1.520</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Jupiter</td>
			<td style="color: black; border: 1px solid black; text-align: center;">5.200</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Saturn</td>
			<td style="color: black; border: 1px solid black; text-align: center;">9.570</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Uranus</td>
			<td style="color: black; border: 1px solid black; text-align: center;">19.170</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Neptune</td>
			<td style="color: black; border: 1px solid black; text-align: center;">30.180</td>
		</tr>
	</tbody>
</table>
The actual value for the distance of the Earth from the Sun was calculated for the first time during a transit of Venus using the parallax method. Using this value of the astronomical unit, the actual distances to all other planets were obtained. This law is very useful to estimate the distances of all the satellites around the Jovian planets, just by observing their time periods.
This text is adapted from<a href="https://openstax.org/books/university-physics-volume-1/pages/13-5-keplers-laws-of-planetary-motion"> Openstax, University Physics Volume 1, Section 13.5 Kepler&#8217;s Laws of Motion.</a>

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. In ...

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kepler’s planet distance law

Kepler’s third law links a planet’s average distance from the Sun to the time it takes to orbit the Sun. This rule gives a clear mathematical relationship between orbital distance and orbital period, which is the time for one full revolution. Johannes Kepler developed this law in the early 17th century after building on the observations of Nicolaus Copernicus and Tycho Brahe.

Kepler first formed his first two laws from those earlier observations. He later published the third law of planetary motion in 1918. The proportionality constant for the law was found much later, after Newton established the universal law of gravitation. Kepler also shaped this third law from observations of the Galilean moons orbiting Jupiter.

The law helped establish the ratios of the average distances of the planets from the Sun compared with Earth’s average distance from the Sun. For example, the distance ratios are 0.387 for Mercury, 0.723 for Venus, 1.000 for Earth, 1.520 for Mars, 5.200 for Jupiter, 9.570 for Saturn, 19.170 for Uranus, and 30.180 for Neptune. These ratios show how planetary orbits can be compared using Earth as the reference point.

The actual Earth-Sun distance was first calculated during a transit of Venus using the parallax method, which measures distance from an observed shift in position. Once this astronomical unit was known, the actual distances to the other planets could be found. The same law is also useful for estimating the distances of satellites around the Jovian planets by observing their time periods.

Kepler’s Planet Distance Law

gravitation

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Kepler’s third law links a planet’s average distance from the Sun to the time it takes to orbit the Sun. This rule gives a clear mathematical relationship between orbital distance and orbital period, which is the time for one full revolution. Johannes Kepler developed this law in the early 17th century after building on the observations of Nicolaus Copernicus and Tycho Brahe. Kepler first formed his first two laws from those earlier observations. He later published the third law of planetary...

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key terms and concepts

Explore how Kepler’s third law connects a planet’s orbit time with its average distance from the Sun, using Earth as the reference for planet distances.

Kepler’s Planet Distance Law | JoVE K12

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Kepler’s Planet Distance Law

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