Name: kepler's equal-area planet motion
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K12 Standards

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North Carolina K-12 Science Standard Course of Study

9-12

Earth and Environmental Science

Earth's Place in the Universe

ESS.EES.1.1

Kepler's Second Law of Planetary Motion

Considérons une planète sur une orbite elliptique autour du soleil se déplaçant de A à A1 ayant un déplacement angulaire de dθ en temps dt. Ensuite, l'aire du secteur dA couverte en temps dt serait la moitié du produit de la distance radiale de la planète et de l'arc rdθ. La vitesse du secteur dA/dt serait alors égale à la moitié de r2dθ/dt.
Ici, dθ/dt est la vitesse angulaire de la planète. Ainsi, en multipliant l'équation par la masse de la planète, on obtient le secteur vitesse en fonction de la vitesse angulaire de la planète.
Cependant, mr2ω est le moment angulaire de la planète. Par conséquent, la vitesse du secteur est égale au rapport entre le moment angulaire de la planète et deux fois sa masse.
La deuxième loi de Kepler stipule que si une planète met le même temps dt pour se déplacer de B à B1, alors l'aire du secteur sous l'arc BB1 serait égale à l'aire sous l'arc AA1.
Par conséquent, la vitesse du secteur est constante, ce qui implique que le moment angulaire reste conservé.

deuxième loi de kepler sur le mouvement des planètes

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. Sa première loi affirme que toutes les planètes orbitent autour du Soleil dans une trajectoire elliptique, le Soleil étant l'un des foyers de l'ellipse. Par conséquent, la distance d'une planète par rapport au Soleil varie au cours de sa révolution autour du Soleil.

Lorsqu'elle est en orbite elliptique, l'énergie totale de la planète est conservée. Par conséquent, la planète ralentit lorsqu'elle est à l'apogée et accélère lorsqu'elle est au périgée. Ces conclusions ont amené Kepler à formuler sa deuxième loi, selon laquelle le vecteur de rayon de la planète parcourt des aires égales en temps égal. Cela signifie que si une planète met le même temps pour parcourir la distance de A à A1, puis de B à B1, alors les aires AF1A1 et BF1B1 sont égales, comme le montre la figure 1.

<img alt="Figure1" src="/files/ftp_upload/12755/12755_Figure_1.png" style="text-align: center; height: 350px;" />

Étant donné que l'aire est constante dans un intervalle de temps donné, la vélocité sectorielle de la planète reste constante. Étant donné que la vélocité sectorielle est proportionnelle au moment angulaire de la planète, la deuxième loi de Kepler implique que le moment angulaire d'une planète en orbite elliptique est conservé.

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>

Deuxième loi de Kepler sur le mouvement des planètes

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. Son...

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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

Newton’s Law Behind Planet Motion

Newton’s law of gravitation links falling objects, the Moon’s orbit, and planetary motion. Before Newton, many people thought the sky and Earth followed different laws. Kepler later described the three laws of planetary motion, but their connection to a deeper force was still unclear. Isaac Newton made that connection in 1665 to 1666. He showed that the same phenomenon explains why objects fall to Earth and why the Moon moves around Earth. He also found that the Moon’s centripetal acceleration...

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kepler's second law of planetary motion

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kepler's third law of planetary 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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JoVE Core Physique présente plusieurs branches de la physique fondamentale grâce à des vidéos animées concises et faciles à comprendre ; cela inclut: la cinématique, la dynamique, la mécanique des fluides, la mécanique statique, les ondes et l'acoustique. En parallèle, les vidéos de scientifiques à l'action démontrent l'application de concepts associés à ces domaines dans des expériences de recherche classiques et originales réalisées dans des laboratoires modernes du monde entier.

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Motion and Stability- Forces and Interactions

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Weather and Climate

3.3.9-12.E

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Georgia Standards of Excellence

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Kepler's Equal-Area Planet Motion

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