Name: kepler's equal-area planet motion
Uploaded: 2025-05-22T09:05:41.000Z
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North Carolina K-12 Science Standard Course of Study

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Earth and Environmental Science

Earth's Place in the Universe

ESS.EES.1.1

Kepler's Second Law of Planetary Motion

Stellen Sie sich einen Planeten vor, der sich auf einer elliptischen Umlaufbahn um die Sonne von A nach A1 bewegt und eine Winkelverschiebung von dθ in der Zeit dt hat. Dann wäre die Sektorfläche dA, die in der Zeit dt abgedeckt wird, die Hälfte des Produkts aus der radialen Entfernung des Planeten und dem Bogen rdθ. Die Sektorgeschwindigkeit dA/dt wäre dann gleich der Hälfte von r2dθ/dt.
Dabei ist dθ/dt die Winkelgeschwindigkeit des Planeten. Wenn man also die Gleichung mit der Masse des Planeten multipliziert, erhält man die Sektorgeschwindigkeit in Bezug auf die Winkelgeschwindigkeit des Planeten.
mr2ω ist jedoch der Drehimpuls des Planeten. Daher entspricht die Sektorgeschwindigkeit dem Verhältnis des Drehimpulses des Planeten zum Zweifachen seiner Masse.
Das zweite Keplersche Gesetz besagt, dass, wenn ein Planet die gleiche Zeit dt benötigt, um sich von B nach B1 zu bewegen, die Sektorfläche unter dem Bogen BB1 gleich der Fläche unter dem Bogen AA1 wäre.
Daher ist die Sektorgeschwindigkeit konstant, was bedeutet, dass der Drehimpuls erhalten bleibt.

keplers zweites gesetz der planetenbewegung

Im frühen 17. Jahrhundert postulierte der deutsche Astronom und Mathematiker Johannes Kepler drei Gesetze für die Bewegung der Planeten im Sonnensystem. Sein erstes Gesetz besagt, dass alle Planeten die Sonne in einer elliptischen Umlaufbahn umkreisen, wobei sich die Sonne in einem der Brennpunkte der Ellipse befindet. Daher variiert die Entfernung zwischen einem Planeten und der Sonne während seiner Umrundung um die Sonne.

Während einer elliptischen Umlaufbahn bleibt die Gesamtenergie des Planeten erhalten. Daher verlangsamt sich der Planet, wenn er sich am Apogäum befindet, und beschleunigt sich, wenn er sich am Perigäum befindet. Diese Erkenntnisse veranlassten Kepler, sein zweites Gesetz aufzustellen, wonach der Radiusvektor des Planeten in gleichen Zeiträumen gleiche Flächen aufsucht. Das bedeutet, dass, wenn ein Planet die gleiche Zeit benötigt, um von A nach A1 zu reisen, und dann von B nach B1, die Flächen AF1A1 und BF1B1 gleich sind, wie in Abbildung 1 dargestellt.

<img alt="Figure1" src="/files/ftp_upload/12755/12755_Figure_1.png" style="text-align: center; height: 350px;" />

Da die Fläche in einem bestimmten Zeitintervall konstant ist, bleibt die Sektor-Geschwindigkeit des Planeten konstant. Da die Sektor-Geschwindigkeit proportional zum Drehimpuls des Planeten ist, folgt aus Keplers zweitem Gesetz, dass der Drehimpuls eines Planeten in einer elliptischen Umlaufbahn erhalten bleibt.

Dieser Text basiert auf <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 Bewegung.</a>

Keplers Zweites Gesetz der Planetenbewegung

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

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kepler's equal-area planet motion

Kepler's second law explains how a planet moves faster and slower in its orbit around the Sun. Johannes Kepler, a German astronomer and mathematician in the early 17th century, developed three laws for planetary motion. His first law says that planets travel in elliptical orbits, with the Sun at one focus of the ellipse. Because the planet-Sun distance changes during the orbit, the planet does not move at the same speed all the time.

In an elliptical orbit, the total energy of the planet is conserved. As a result, the planet slows down at apogee, the point farthest from the Sun, and speeds up at perigee, the point closest to the Sun. Kepler used this idea to state his second law. The law says that the radius vector, the line from the Sun to the planet, sweeps out equal areas in equal times.

This means that if a planet takes the same time to travel from A to A1 and from B to B1, then the areas AF1A1 and BF1B1 are equal. Because the area swept out in a given time interval stays constant, the planet's sector velocity also remains constant. Sector velocity is linked to angular momentum. For that reason, Kepler's second law shows that the angular momentum of a planet in an elliptical orbit is conserved.

Kepler's Equal-Area Planet Motion

gravitation

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Learn how Kepler's second law explains changing planet speed, equal areas in equal time, and why angular momentum stays conserved in an elliptical orbit.

Kepler's Equal-Area Planet Motion | JoVE K12

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Kepler's Equal-Area Planet Motion

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