Name: kepler’s planet distance law
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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

Considere um planeta orbitando o Sol em uma órbita elíptica. Se o planeta leva tempo T para completar uma órbita ao redor do Sol, sua velocidade angular é expressa como dois pi dividido por T.
A força centrípeta que atua no planeta em órbita é expressa como os produtos de sua massa, ao quadrado de sua velocidade angular e sua distância média do Sol.
Substituindo ômega, a equação da força centrípeta pode ser obtida em termos do período orbital do planeta e do semi-eixo maior da órbita.
A força gravitacional do Sol fornece a força centrípeta para o planeta orbitar em torno do Sol. Substituindo e reorganizando os termos, o quadrado do período orbital é igual ao produto de um termo constante e ao cubo do semi-eixo maior da órbita.
Portanto, o quadrado do período orbital de um planeta é proporcional ao cubo do semi-eixo maior da órbita. Isso é popularmente conhecido como a terceira lei do movimento planetário de Kepler. A constante de proporcionalidade, que envolve a constante gravitacional e a massa do Sol, foi derivada mais tarde, quando Newton declarou sua lei da gravitação.

terceira 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. Em 1609, ele formulou suas duas primeiras leis com base nas observações de seus antecessores, Nicolau Copérnico e Tycho Brahe. No entanto, em 1619, ele publicou sua terceira lei do movimento planetário, que estabelece uma relação matemática precisa entre a distância média de um planeta ao Sol e o tempo que leva para orbitar o Sol. Esta lei afirma que:

<img alt="Equation1" src="/files/ftp_upload/12756/12756_Equations_1.png" style="text-align: center; height: 15px;" />

A constante de proporcionalidade para esta lei foi derivada muito tempo depois, após Newton estabelecer a lei universal da gravitação. No entanto, Kepler formulou esta terceira lei do movimento planetário com base na observação das luas galileanas orbitando Júpiter.

Esta lei estabeleceu as razões das distâncias médias de cada planeta ao Sol, em comparação com a distância média da Terra ao Sol, conforme mostrado na Tabela 1.

Tabela 1: Razão das Distâncias Planetárias do Sol em Comparação com a Distância Terra-Sol.

<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;">Planeta</td>
			<td style="color: black; border: 1px solid black; text-align: center;">Razão das Distâncias Planetárias </td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Mercúrio</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;">Vênus</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;">Terra</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;">Marte</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;">Júpiter</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;">Saturno</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;">Urano</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;">Netuno</td>
			<td style="color: black; border: 1px solid black; text-align: center;">30,180</td>
		</tr>
	</tbody>
</table>

O valor real da distância da Terra ao Sol foi calculado pela primeira vez durante um trânsito de Vênus usando o método de paralaxe. Usando esse valor da unidade astronômica, as distâncias reais para todos os outros planetas foram obtidas. Esta lei é muito útil para estimar as distâncias de todos os satélites ao redor dos planetas jovianos, apenas observando seus períodos.

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>

Terceira 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. Em...

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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

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kepler's third law of planetary motion

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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

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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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Kepler’s Planet Distance Law

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