Name: kepler's equal-area planet motion
Uploaded: 2025-05-22T09:05:41.000Z
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Kepler's Second Law of Planetary Motion

Si consideri un pianeta in un'orbita ellittica attorno al sole che si muove da A ad A1 con uno spostamento angolare di dθ nel tempo dt. Quindi, l'area del settore dA coperta nel tempo dt sarebbe la metà del prodotto della distanza radiale del pianeta e dell'arco rdθ. La velocità del settore dA/dt sarebbe quindi uguale alla metà di r2dθ/dt.
Qui, dθ/dt è la velocità angolare del pianeta. Quindi, moltiplicando l'equazione per la massa del pianeta si ottiene la velocità del settore in termini di velocità angolare del pianeta.
Tuttavia, mr2ω è il momento angolare del pianeta. Pertanto, la velocità del settore è uguale al rapporto tra il momento angolare del pianeta e due volte la sua massa.
La seconda legge di Keplero afferma che se un pianeta impiega lo stesso tempo dt per spostarsi da B a B1, allora l'area del settore sotto l'arco BB1 sarebbe uguale all'area sotto l'arco AA1.
Pertanto, la velocità del settore è costante, il che implica che il momento angolare rimane conservato.

la seconda legge di keplero del moto planetario

All'inizio del XVII secolo, l'astronomo e matematico tedesco Giovanni Keplero postulò tre leggi del movimento planetario nel sistema solare. La sua prima legge afferma che tutti i pianeti orbitano intorno al Sole in orbite ellittiche, con il sole ad uno dei fuochi dell'ellisse. Di conseguenza, la distanza di un pianeta dal Sole varia durante la sua rotazione intorno al Sole.

Mentre in un'orbita ellittica, l'energia totale del pianeta è conservata. Pertanto, il pianeta rallenta quando è all'apogeo e accelera quando è al perigeo. Queste conclusioni portarono Keplero ad affermare la sua seconda legge, che il vettore del raggio del pianeta esce fuori in aree uguali allo stesso tempo. Questo significa che se un pianeta impiega lo stesso tempo per viaggiare da A ad A1, e poi da B a B1, allora le aree AF1Aa e BF1B1 sono uguali, come mostrato nella Figura 1.

1B1 are equal, as shown in Figure 1.



<img alt="Figure1" src="/files/ftp_upload/12755/12755_Figure_1.png" style="text-align: center; height: 350px;" />

Siccome l'area è costante in un dato intervallo di tempo, la velocità del settore del pianeta rimane costante. Visto che la velocità del settore è proporzionale al momento angolare del pianeta, la seconda legge di Keplero implica che il momento angolare del pianeta in un'orbita ellittica è conservato.

Questo testo è adattato da <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 seconda legge di Keplero del moto planetario

All'inizio del XVII secolo, l'astronomo e matematico tedesco Johannes Kepler postulò tre leggi per il moto dei pianeti nel sistema solare. Suo...

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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

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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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