Name: kepler’s planet distance law
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9-12

Earth and Environmental Science

Earth's Place in the Universe

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Kepler's Third Law of Planetary Motion

Considérons une planète en orbite autour du Soleil sur une orbite elliptique. Si la planète met le temps T pour effectuer une orbite autour du Soleil, sa vitesse angulaire est exprimée par deux pi divisés par T.
La force centripète agissant sur la planète en orbite est exprimée comme le produit de sa masse, du carré de sa vitesse angulaire et de sa distance moyenne au Soleil.
En substituant oméga, l'équation de la force centripète peut être obtenue en fonction de la période orbitale de la planète et du demi-grand axe de l'orbite.
La force gravitationnelle du Soleil fournit la force centripète pour que la planète orbite autour du Soleil. En substituant et en réarrangeant les termes, le carré de la période orbitale est égal au produit d'un terme constant et du cube du demi-grand axe de l'orbite.
Par conséquent, le carré de la période orbitale d'une planète est proportionnel au cube du demi-grand axe de l'orbite. Ceci est populairement connu sous le nom de troisième loi de Kepler sur le mouvement planétaire. La constante de proportionnalité, qui implique la constante gravitationnelle et la masse du Soleil, a été dérivée plus tard lorsque Newton a énoncé sa loi de la gravitation.

la troisième loi du mouvement planétaire de kepler

Au début du XVIIe siècle, l'astronome et mathématicien allemand Johannes Kepler a postulé trois lois sur le mouvement des planètes dans le système solaire. En 1909, il a formulé ses deux premières lois basées sur les observations de ses prédécesseurs, Nikolaus Copernicus et Tycho Brahe. Cependant, en 1918, il a publié sa troisième loi du mouvement planétaire, qui établit une relation mathématique précise entre la distance moyenne d'une planète par rapport au Soleil et le temps qu'il lui faut pour effectuer une révolution autour du Soleil. Cette loi stipule que:

<img alt="Equation1" src="/files/ftp_upload/12756/12756_Equations_1.png" style="text-align: center; height: 15px;" />

La constante de proportionnalité de cette loi a été dérivée beaucoup plus tard, après que Newton a établi la loi universelle de la gravitation. Cependant, Kepler a formulé cette troisième loi du mouvement planétaire en se basant sur l'observation des lunes galiléennes orbitant autour de Jupiter.

Cette loi établit les rapports des distances moyennes de chaque planète par rapport au Soleil par rapport à la distance moyenne de la Terre par rapport au Soleil, comme indiqué dans le tableau 1.

Tableau 1: Rapport des distances planétaires par rapport au Soleil comparées à la distance Terre-Soleil.

<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;">Matériaux</td>
			<td style="color: black; border: 1px solid black; text-align: center;">Ratio de distance planétaire</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Mercure</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;">Vénus</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;">Terre</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;">Saturne</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>

La valeur réelle pour la distance de la Terre au Soleil a été calculée pour la première fois lors d'un transit de Vénus en utilisant la méthode de la parallaxe. En utilisant cette valeur de l'unité astronomique, les distances réelles de toutes les autres planètes ont été obtenues. Cette loi est très utile pour estimer les distances de tous les satellites autour des planètes joviennes, simplement en observant leurs périodes de rotation.

Ce texte est adapté de<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's Laws of Motion.</a>

La troisième loi du mouvement planétaire de Kepler

Au début du XVIIe siècle, l’astronome et mathématicien allemand Johannes Kepler a postulé trois lois pour le mouvement des planètes dans le système solaire. Dans...

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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

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kepler's third law of planetary motion

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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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