Name: kepler’s planet distance law
Uploaded: 2023-04-30T11:07:15.000Z
Duration: 1 min 18 s
Channel: JoVE

K12 Standards

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

NC 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 Third Law of Planetary Motion

楕円軌道で太陽を周回する惑星を考えてみましょう。惑星が太陽の周りを1周するのに時間Tが必要な場合、その角速度は2πをTで割ったもので表されます。
軌道を周回する惑星に作用する求心力は、その質量、角速度の2乗、および太陽からの平均距離の積として表されます。
オメガを代入すると、求心力方程式は、惑星の軌道周期と軌道の半長軸に関して求めることができます。
太陽の重力は、惑星が太陽の周りを公転するための求心力を提供します。項を代入および再配置すると、軌道周期の二乗は、定数項と軌道の長半径の3乗の積に等しくなります。
したがって、惑星の公転周期の二乗は、軌道の長半径の三乗に比例します。これは、ケプラーの惑星運動の第3法則として広く知られています。重力定数と太陽の質量を含む比例定数は、ニュートンが重力の法則を述べたときに導き出されました。

ケプラーの惑星運動の第 3 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。 1909 年、彼は先祖であるニコラウス コペルニクスとティコ ブラーエの観察に基づいて最初の 2 つの法則を策定しました。 しかし、1918 年に彼は、惑星の太陽からの平均距離と太陽の周りを公転するのにかかる時間との間の正確な数学的関係を与える惑星運動の第 3 法則を発表しました。

と記載されています。

<img alt="Equation1" src="/files/ftp_upload/12756/12756_Equations_1.png" style="text-align: center; height: 15px;" />

この法則の比例定数は、ニュートンが万有引力の法則を確立した後、ずっと後に導かれました。 ただし、ケプラーは、木星を周回するガリレオ衛星の観察に基づいて、この惑星運動の第 3 法則を定式化しました。

このこの法則は、表 1 に示すように、太陽からの各惑星の平均距離と地球の太陽からの平均距離の比率を確立しました。

表 1: 地球と太陽の距離と比較した、太陽からの惑星の距離の比率。

<table style="margin-left: auto; margin-right: auto;">
	<tbody style="border: 1px solid black;">
		<tr>
			<td style="color: black; border: 1px Solid black; text-align: center;">素材</td>
			<td style="color: black; border: 1px Solid black; text-align: center;">惑星間の距離の比率</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px Solid black; text-align: center;">水星</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.387</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px Solid black; text-align: center;">金星</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.723</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px Solid black; text-align: center;">地球</td>
			<td style="color: black; border: 1px solid black; text-align: center;">1.000</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px Solid black; text-align: center;"> 火星</td>
			<td style="color: black; border: 1px solid black; text-align: center;">1.520</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px Solid black; text-align: center;">木星</td>
			<td style="color: black; border: 1px solid black; text-align: center;">5.200</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px Solid black; text-align: center;">土星</td>
			<td style="color: black; border: 1px solid black; text-align: center;">9.570</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px Solid black; text-align: center;">天王星</td>
			<td style="color: black; border: 1px solid black; text-align: center;">19.170</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px Solid black; text-align: center;">海王星</td>
			<td style="color: black; border: 1px solid black; text-align: center;">30.180</td>
		</tr>
	</tbody>
</table>

太陽から地球までの距離の実際の値は、視差法を使用して金星の太陽面通過中に初めて計算されました。 この天文単位の値を使用して、他のすべての惑星までの実際の距離が取得されました。 この法則は、その期間を観察するだけで、木星の周りのすべての衛星の距離を推定するのに非常に役立ちます。

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">から編集されたものです。 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。で。。。

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 planet distance law

Kepler’s third law links a planet’s average distance from the Sun to the time it takes to orbit the Sun. This rule gives a clear mathematical relationship between orbital distance and orbital period, which is the time for one full revolution. Johannes Kepler developed this law in the early 17th century after building on the observations of Nicolaus Copernicus and Tycho Brahe.

Kepler first formed his first two laws from those earlier observations. He later published the third law of planetary motion in 1918. The proportionality constant for the law was found much later, after Newton established the universal law of gravitation. Kepler also shaped this third law from observations of the Galilean moons orbiting Jupiter.

The law helped establish the ratios of the average distances of the planets from the Sun compared with Earth’s average distance from the Sun. For example, the distance ratios are 0.387 for Mercury, 0.723 for Venus, 1.000 for Earth, 1.520 for Mars, 5.200 for Jupiter, 9.570 for Saturn, 19.170 for Uranus, and 30.180 for Neptune. These ratios show how planetary orbits can be compared using Earth as the reference point.

The actual Earth-Sun distance was first calculated during a transit of Venus using the parallax method, which measures distance from an observed shift in position. Once this astronomical unit was known, the actual distances to the other planets could be found. The same law is also useful for estimating the distances of satellites around the Jovian planets by observing their time periods.

Kepler’s Planet Distance Law

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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gravitation between spherically symmetric masses

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Gravitational Potential Inside a Shell

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Point-Mass Approximation in Spherical Gravity

Newton’s law of gravitation gives the force between two point masses. For large spherical objects, such as the Earth, the Moon, and other planets, the same law can be used by treating each body as if its mass were concentrated at its center. This idea can be proven by comparing gravitational potential energy. The transcript shows that a hollow sphere of mass M and a point mass m give the same energy result as two extended solid spherical objects. To set up the proof, a tiny ring on the shell...

reduced mass coordinates: isolated two-body problem

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Center of Mass in the Two-Body Problem

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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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variation in acceleration due to gravity near the earth's surface

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How Earth’s Shape Affects Apparent Weight

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circular orbits and critical velocity for satellites

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Orbital Speed and Period in Circular Paths

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Satellite Orbits and Energy in Space

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kepler's second law of planetary motion

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kepler's third law of planetary motion

Kepler’s third law links a planet’s average distance from the Sun to the time it takes to orbit the Sun. This rule gives a clear mathematical relationship between orbital distance and orbital period, which is the time for one full revolution. Johannes Kepler developed this law in the early 17th century after building on the observations of Nicolaus Copernicus and Tycho Brahe. Kepler first formed his first two laws from those earlier observations. He later published the third law of planetary...

tidal forces

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schwarzschild radius and event horizon

Escape Velocity Limits and Black Holes

Escape velocity helps explain how black holes and event horizons form. It is the minimum speed a projectile needs to leave the surface of an object and eventually escape its gravitational field. In this context, escape velocity does not depend on the mass of the object being launched. The key idea is that no object with finite mass can move faster than the speed of light in a vacuum. When escape velocity is compared with that speed limit, the Schwarzschild radius becomes important. If an...

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How Astronomers Find Black Holes

Black holes are found through indirect astronomical evidence. They were first proposed in the 1920s, but they stayed outside observational astronomy until the 1970s. Their closest known cousins are neutron stars, which are made mostly of packed neutrons and are extremely dense. A neutron star can have the mass of the Sun while its diameter is only a few kilometers. Because of that, the escape velocity from its surface is close to the speed of light. Interest in black holes rose again in the...

principle of equivalence

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Weightlessness and Einstein’s equivalence principle are closely connected in gravity. A person in free fall under gravity, such as someone diving toward Earth from an airplane, would feel completely weightless. A person descending in a lift may feel partially weightless. The principle of equivalence says that an object in a uniform gravitational field and an object accelerating at a constant rate with no gravity are under the same experimental conditions. In other words, no experiment can tell...

space-time curvature and the general theory of relativity

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Einstein’s Gravity and Mercury’s Orbit

Einstein’s theory of gravity explains how space-time affects motion. In 1905, Einstein published his special theory of relativity. It says that no object can travel faster than the speed of light in a vacuum, which is the speed limit of the universe. Special relativity also changed the way scientists think about space and time. Two observers moving relative to each other do not agree on object length or on how time passes. Newton’s laws still work well for many cases, even for objects moving...

key terms and concepts

Explore how Kepler’s third law connects a planet’s orbit time with its average distance from the Sun, using Earth as the reference for planet distances.

Kepler’s Planet Distance Law | 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

Kepler's First Law of Planetary Motion

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Energy Dynamics

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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 Planet Distance Law

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