Name: kepler’s planet distance law
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Earth's Place in the Universe

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Kepler's Third Law of Planetary Motion

Stellen Sie sich einen Planeten vor, der die Sonne in einer elliptischen Umlaufbahn umkreist. Wenn der Planet Zeit T benötigt, um einen Umkreis um die Sonne zu vollenden, wird seine Winkelgeschwindigkeit ausgedrückt als zwei Pi geteilt durch T.
Die Zentripetalkraft, die auf den umkreisenden Planeten wirkt, wird als Produkt seiner Masse, als Quadrat seiner Winkelgeschwindigkeit und seiner durchschnittlichen Entfernung von der Sonne ausgedrückt.
Wenn man Omega ersetzt, kann die Zentripetalkraftgleichung in Bezug auf die Umlaufperiode des Planeten und die große Halbachse der Umlaufbahn erhalten werden.
Die Gravitationskraft der Sonne liefert die Zentripetalkraft, mit der der Planet die Sonne umkreisen kann. Wenn man die Terme ersetzt und neu anordnet, ist das Quadrat der Umlaufperiode gleich dem Produkt aus einem konstanten Term und dem Würfel der großen Halbachse der Umlaufbahn.
Daher ist das Quadrat der Umlaufperiode eines Planeten proportional zum Würfel der großen Halbachse der Umlaufbahn. Dies ist im Volksmund als Keplers drittes Gesetz der Planetenbewegung bekannt. Die Proportionalitätskonstante, die die Gravitationskonstante und die Sonnenmasse umfasst, wurde später abgeleitet, als Newton sein Gravitationsgesetz aufstellte.

keplers drittes gesetz der planetarischen bewegung

Im frühen 17. Jahrhundert postulierte der deutsche Astronom und Mathematiker Johannes Kepler drei Gesetze für die Bewegung der Planeten im Sonnensystem. Im Jahr 1909 formulierte er seine ersten beiden Gesetze basierend auf den Beobachtungen seiner Vorgänger Nikolaus Kopernikus und Tycho Brahe. Allerdings veröffentlichte er im Jahr 1918 sein drittes Gesetz der planetarischen Bewegung, das eine präzise mathematische Beziehung zwischen der durchschnittlichen Entfernung eines Planeten von der Sonne und der Zeit, die er benötigt, um die Sonne zu umkreisen, angibt. Es besagt, dass

<img alt="Equation1" src="/files/ftp_upload/12756/12756_Equations_1.png" style="text-align: center; height: 15px;" />

Die Proportionalitätskonstante für dieses Gesetz wurde viel später abgeleitet, nachdem Newton das universelle Gravitationsgesetz aufgestellt hatte. Allerdings formatierte Kepler dieses dritte Gesetz der planetarischen Bewegung basierend auf der Beobachtung von galileischen Monden, die den Jupiter umkreisen.

Dieses Gesetz legte die Verhältnisse der durchschnittlichen Entfernungen jedes Planeten von der Sonne zu der durchschnittlichen Entfernung der Erde von der Sonne fest, wie in Tabelle 1 dargestellt.

Tabelle 1: Verhältnis der planetarischen Entfernungen von der Sonne im Vergleich zur Entfernung Erde-Sonne.

<div class="overflow">
<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;">Verhältnis der Planetenabstände</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>
</div>

Der tatsächliche Wert für die Entfernung der Erde von der Sonne wurde zum ersten Mal während eines Venustransits mit Hilfe der Parallaxenmethode berechnet. Mit diesem Wert der astronomischen Einheit wurden die tatsächlichen Entfernungen zu allen anderen Planeten ermittelt. Dieses Gesetz ist sehr nützlich, um die Entfernungen aller Satelliten um die jovianischen Planeten nur durch Beobachtung ihrer Umlaufzeiten abzuschätzen.

Dieser Text wurde adaptiert von <a href="https://openstax.org/books/university-physics-volume-1/pages/13-5-keplers-laws-of-planetary-motion">Openstax, University Physics Volume 1, Abschnitt 13.5 Keplers Gesetze der planetarischen Bewegung.</a>

Keplers Drittes Gesetz der planetarischen Bewegung

Im frühen 17. Jahrhundert postulierte der deutsche Astronom und Mathematiker Johannes Kepler drei Gesetze für die Bewegung der Planeten im Sonnensystem. 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

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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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