Name: kepler's equal-area planet motion
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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

Considere um planeta em uma órbita elíptica ao redor do sol movendo-se de A para A1 com um deslocamento angular de dθ no tempo dt. Então, a área do setor dA coberta no tempo dt seria metade do produto da distância radial do planeta e do arco rdθ. A velocidade do setor dA/dt seria então igual à metade de r2dθ/dt.
Aqui, dθ/dt é a velocidade angular do planeta. Assim, multiplicar a equação pela massa do planeta dá a velocidade do setor em termos da velocidade angular do planeta.
No entanto, mr2ω é o momento angular do planeta. Portanto, a velocidade do setor é igual à razão entre o momento angular do planeta e duas vezes sua massa.
A segunda lei de Kepler afirma que se um planeta leva o mesmo tempo dt para se mover de B para B1, então a área do setor sob o arco BB1 seria igual à área sob o arco AA1.
Portanto, a velocidade do setor é constante, o que implica que o momento angular permanece conservado.

a segunda lei do movimento planetário de kepler

No início do século XVII, o astrônomo e matemático alemão Johannes Kepler postulou três leis para o movimento dos planetas no sistema solar. Sua primeira lei afirma que todos os planetas orbitam o Sol em uma órbita elíptica, com o Sol em um dos focos da elipse. Portanto, a distância de um planeta em relação ao Sol varia ao longo de sua revolução ao redor do Sol.

Enquanto em órbita elíptica, a energia total do planeta é conservada. Portanto, o planeta desacelera quando está no apogeu e acelera quando está no perigeu. Essas conclusões levaram Kepler a enunciar sua segunda lei, que diz que o vetor raio do planeta percorre áreas iguais em tempos iguais. Isso significa que se um planeta leva o mesmo tempo para viajar de A para A1 e depois de B para B1, então as áreas AF1A1 e BF1B1 são iguais, como mostrado na Figura 1.

<img alt="Figure1" src="/files/ftp_upload/12755/12755_Figure_1.png" style="text-align: center; height: 350px;" />

Como a área é constante em um determinado intervalo de tempo, a velocidade setorial do planeta permanece constante. Como a velocidade setorial é proporcional ao momento angular do planeta, a segunda lei de Kepler implica que o momento angular de um planeta em órbita elíptica é conservado.

Este texto foi adaptado 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>

A Segunda Lei do Movimento Planetário de Kepler

No início do século 17, o astrônomo e matemático alemão Johannes Kepler postulou três leis para o movimento dos planetas no sistema solar. Dele...

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KEY TERMS AND CONCEPTS

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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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acceleration due to gravity on earth

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acceleration due to gravity on other planets

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Measuring Gravity on Distant Planets

Gravity near Earth’s surface is measured as acceleration due to gravity. On Earth, students can test it with simple experiments. On other planets, that kind of direct experiment is not possible. Instead, astronomers use observations to find a planet’s gravity. They study how the planet affects nearby objects, especially its satellite. The key idea is that the acceleration depends on the planet’s gravity, not on the object’s mass. To make the calculation, scientists track a satellite’s path...

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kepler's second law of planetary 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...

kepler's third law of planetary motion

Kepler’s Planet Distance Law

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space-time curvature and the general theory of relativity

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Einstein’s Gravity and Mercury’s Orbit

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key terms and concepts

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 - Fisica apresenta a teoria e a aplicação de tópicos fundamentais da física, incluindo cinemática, dinâmica, mecânica dos fluidos, estática, ondas e acústica, com vídeos animados concisos e de fácil compreensão. Além disso, os vídeos de cientistas em ação demonstram a aplicação de conceitos relacionados em experimentos de pesquisa clássica e original realizados nos laboratórios de todo o mundo hoje em dia.

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Motion and Stability- Forces and Interactions

PS.Phy.2.4

Pennsylvania State Academic Standards

Earth and Space Science

Space Systems

3.3.9-12.A

3.3.9-12.D

Weather and Climate

3.3.9-12.E

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Earth’s Place In The Universe

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SC.912.E.5

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SC.912.E.5.10

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PHYS.5.H

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South Carolina College and Career Ready Science Standards

Earth and Space Science (ESS)

E-ESS1-4.

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New York State Science Learning Standards

Earth and Space Sciences

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Georgia Standards of Excellence

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SAST1.1

Kepler's First Law of Planetary Motion

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. He ...

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. His ...

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Light as Energy

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Nuclear Fusion

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First Law of Thermodynamics

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Energy Dynamics

The Food Chain
Energy is one of the most important abiotic factors in an ecosystem and organisms in an ecosystem are connected by the flow of energy and ...

Use models to illustrate how the sun produces energy.

ESS.EES.1.3

Explain how Earth's position relative to the sun influences conditions on Earth.

Kepler's Equal-Area Planet Motion

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