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

9-12

Earth and Environmental Science

Earth's Place in the Universe

ESS.EES.1.1

Kepler's Second Law of Planetary Motion

Consider a planet in an elliptical orbit around the sun moving from A to A1 having an angular displacement of d&#952; in time dt. Then, the sector area dA covered in time dt would be half of the product of the planet&#39;s radial distance and the arc rd&#952;. The sector velocity dA/dt would then be equal to half of r2d&#952;/dt.
Here, d&#952;/dt is the planet&#39;s angular velocity. Thus, multiplying the equation by the planet&#39;s mass gives the sector velocity in terms of the planet&#39;s angular velocity.
However, mr2&#969; is the planet&#39;s angular momentum. Therefore, the sector velocity equals the ratio of the planet&#39;s angular momentum to two times its mass.
Kepler&#39;s second law states that if a planet takes the same time dt to move from B to B1, then the sector area under the arc BB1 would be equal to the area under the arc AA1.
Therefore, the sector velocity is constant, implying that the angular momentum remains conserved.

kepler's second 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. His first law states that all planets orbit the Sun in an elliptical orbit, with the Sun at one of the ellipse's foci. Therefore, the distance of a planet from the Sun varies throughout its revolution around the Sun.
While in an elliptical orbit, the total energy of the planet is conserved. Therefore, the planet slows down when it is at apogee and speeds up when it is at perigee. These conclusions led Kepler to state his second law, that the radius vector of the planet sweeps out in equal areas in equal time. This means that if a planet takes the same time to travel from A to A1, and then from B to B1, then the areas AF1A1 and BF1B1 are equal, as shown in Figure 1.
<img alt="Kepler's second law diagram, planets orbiting sun, equal area in equal time, orbital mechanics" src="/files/ftp_upload/12755/12755_Figure_1.png" style="text-align: center; height: 350px;" data-altupdatedat="1754904545000" data-alt="Kepler's second law diagram, elliptical orbit areas, equal time intervals, orbital dynamics." />
Since the area is constant in a given time interval, the planet's sector velocity remains constant. Since the sector velocity is proportional to the planet's angular momentum, Kepler&#8217;s second law implies that the angular momentum of a planet in an elliptical orbit is conserved.
This text is adapted from<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&#8217;s Laws of Motion.</a>

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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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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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...

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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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Motion and Stability- Forces and Interactions

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Earth’s Place In The Universe

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Kepler's Equal-Area Planet Motion

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