Name: kepler's equal-area planet motion
Uploaded: 2025-05-22T09:05:41.000Z
Duration: 1 min 29 s
Channel: JoVE

K12 Standards

north-carolina-k-12-science-standard-course-of-study

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

太陽の周りを楕円軌道で移動し、A から A 1 に移動する惑星が、時間 dt で dθ の角変位を持つとします。その場合、時間 dt でカバーされるセクター面積 dA は、惑星の半径距離とアーク rdθ の積の半分になります。その場合、セクタ速度 dA/dt は r2dθ/dt の半分に等しくなります。
ここで、dθ/dtは惑星の角速度です。したがって、方程式に惑星の質量を掛けると、惑星の角速度に関するセクター速度が得られます。
ただし、mr2ω は惑星の角運動量です。したがって、セクター速度は、惑星の角運動量とその質量の2倍の比率に等しくなります。
ケプラーの第二法則は、惑星がBからB1に移動するのに同じ時間dtかかる場合、アークBB1の下のセクター面積はアークAA1の下の面積に等しくなると述べています。
したがって、セクター速度は一定であり、角運動量が保存されたままであることを意味します。

ケプラーの惑星運動の第 2 法則

17 世紀初頭、ドイツの天文学者で数学者のヨハネス ケプラーは、太陽系の惑星の運動に関する 3 つの法則を仮定しました。 彼の第一法則は、すべての惑星は太陽の周りを楕円軌道で公転し、太陽はその楕円の焦点の 1 つにあると述べています。 したがって、惑星と太陽の距離は、太陽の周りを公転するたびに変化します。

楕円軌道上では、惑星の総エネルギーは保存されます。 したがって、惑星は遠地点にあるときは減速し、近地点にあるときは加速します。 これらの結論により、ケプラーは第 2 法則、つまり惑星の動径ベクトルは同じ時間に同じ面積を掃引するという結論に至りました。 これは、惑星が A から A1 へ、そしてB へ移動するのに同じ時間がかかる場合を意味します。 B1 に、次にエリア AF1A1< 図 1 に示すように、/sub> と BF1B1 は等しいです。

<img alt="Figure1" src="/files/ftp_upload/12755/12755_Figure_1.png" style="text-align: center; height: 350px;" />

一定の時間間隔では面積が一定であるため、惑星の扇形速度は一定のままです。 扇形の速度は惑星の角運動量に比例するため、ケプラーの第 2 法則は、楕円軌道上の惑星の角運動量が保存されることを意味します。

このテキストは<a href="https://openstax.org/books/university-physics-volume-1/pages/13-5-keplers-laws-of- Planetary-motion">から引用されたものです。 Openstax, University Physics Volume 1, Section 13.5 Kepler’s Laws of Motion.</a>

17世紀初頭、ドイツの天文学者で数学者のヨハネス・ケプラーは、太陽系内の惑星の運動について3つの法則を仮定しました。彼の。。。

Education

JoVE Core

Physics

物理学

Unit 1

ユニット1

Gravitation

万有引力

KEY TERMS AND CONCEPTS

主要用語と概念

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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newton's law of gravitation

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Newton's gravity links the fall of objects on Earth with the motion of the Moon and planets. People once thought the downward pull seen near Earth was unique to our planet. Later, Nicolaus Copernicus proposed the heliocentric theory, which placed Earth and the other planets around the Sun, with the Moon orbiting Earth. Isaac Newton connected these motions in the 17th century. He reasoned that the same attraction that keeps the Moon on its path around Earth also makes an apple fall from a tree.

gravitation between spherically symmetric masses

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gravity between spherical bodies

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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...

apparent weight and the earth's rotation

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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

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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

physics

JoVE Core 物理学は、運動学、力学、流体力学、静力学、波動、音響などの基礎的な物理学の理論と応用を、簡潔でわかりやすいアニメーション ビデオで紹介します。さらに、科学者の実際のビデオでは、世界中の今日の研究所で行われている古典的および独自の研究実験における関連概念の適用を示しています。

ESS.EES.1.2

motion-and-stability-forces-and-interactions

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

Louisiana Student Standards for Science

High School

Earth Science

Earth’s Place In The Universe

HS-ESS1-1

HS-ESS1-2

HS-ESS1-3

HS-ESS1-4

Virginia  Standards of Learning

HIgh School

ES.2

ES.2.b

ES.2.c

PH.3

PH.3.b

Minnesota Academic Standards in Science

2.2.1 9E.2.2.1.1

Arizona Science Standards

HS.E2

HS.E2U1.16

HS+E.E2

HS+E.E2U1.14

new-jersey-student-learning-standards-for-science

New Jersey Student Learning Standards for Science

Alabama Course of Study Science

9-12.ESS.1.2.a

9-12.ESS.1.2.d

9-12.ESS.1.3

Texas Essential Knowledge and Skills

Astronomy

112.48.c.9.

112.48.c.9.B.

Next Generation Science Standards

Earth’s Place in the Universe

Florida Department of Education

SC.912.E.5

SC.912.E.5.6

SC.912.E.5.10

SC.912.E.5.11

PHYS.5

PHYS.5.H

south-carolina-college-and-career-ready-science-standards

South Carolina College and Career Ready Science Standards

Earth and Space Science (ESS)

E-ESS1-4.

new-york-state-science-learning-standards

New York State Science Learning Standards

Earth and Space Sciences

HS-ESS1-4.

Georgia Standards of Excellence

SAST1

SAST1.1

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

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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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