Name: kepler’s planet distance law
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Kepler's Third Law of Planetary Motion

Wyobraźmy sobie planetę krążącą wokół Słońca po orbicie eliptycznej. Jeśli planeta potrzebuje czasu T na wykonanie jednego okrążenia wokół Słońca, jej prędkość kątowa jest wyrażona jako dwa pi podzielone przez T.
Siła dośrodkowa działająca na krążącą planetę jest wyrażona jako iloczyn jej masy, kwadratu jej prędkości kątowej i średniej odległości od Słońca.
Podstawiając omegę, równanie siły dośrodkowej można uzyskać w kategoriach okresu orbitalnego planety i półosi wielkiej orbity.
Siła grawitacyjna Słońca zapewnia siłę dośrodkową, dzięki której planeta krąży wokół Słońca. Podstawiając i przestawiając warunki, kwadrat okresu orbitalnego jest równy iloczynowi stałego składnika i sześcianu półosi wielkiej orbity.
Dlatego kwadrat okresu orbitalnego planety jest proporcjonalny do sześcianu półosi wielkiej orbity. Jest to popularnie znane jako trzecia zasada ruchu planet Keplera. Stała proporcjonalności, która obejmuje stałą grawitacyjną i masę Słońca, została wyprowadzona później, gdy Newton sformułował swoje prawo grawitacji.

trzecia zasada ruchu planet keplera

Na początku XVII wieku niemiecki astronom i matematyk Johannes Kepler postulował trzy prawa ruchu planet w Układzie Słonecznym. W 1909 roku na podstawie obserwacji swoich przodków, Mikołaja Kopernika i Tycho Brahe, sformułował swoje pierwsze dwa prawa. Jednak w 1918 roku opublikował swoje trzecie prawo ruchu planet, które podaje dokładny matematyczny związek pomiędzy średnią odległością planety od Słońca a czasem potrzebnym na obrót wokół Słońca. Stwierdza, że

<img alt="Równanie1" src="/files/ftp_upload/12756/12756_Equations_1.png" style="text-align: center; height: 15px;" />

Stałą proporcjonalności dla tego prawa wyprowadzono znacznie później, po tym jak Newton ustalił uniwersalne prawo ciążenia. Jednak Kepler sformatował tę trzecią zasadę ruchu planet w oparciu o obserwacje księżyców galileuszowych krążących wokół Jowisza.

Prawo to ustaliło stosunek średnich odległości każdej planety od Słońca do średniej odległości Ziemi od Słońca, jak pokazano w Tabeli 1.

Tabela 1: Stosunek odległości planet od Słońca w porównaniu do odległości Ziemia-Słońce.

<table style="margin-left:auto; margin-right:auto; border-collapse: collapse;">
	<tbody>
		<tr>
			<td style="color:black; border:1px solid black; text-align:center;">Materiał</td>
			<td style="color:black; border:1px solid black; text-align:center;">Współczynnik odległości planetarnej</td>
		</tr>
		<tr>
			<td style="color:black; border:1px solid black; text-align:center;">Merkury</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;">Wenus</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;">Ziemia</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;">Mars</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;">Jowisz</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;">Saturn</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;">Uran</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;">Neptun</td>
			<td style="color:black; border:1px solid black; text-align:center;">30,180</td>
		</tr>
	</tbody>
</table>

Rzeczywista wartość odległości Ziemi od Słońca została po raz pierwszy obliczona podczas tranzytu Wenus metodą paralaksy. Korzystając z tej wartości jednostki astronomicznej, uzyskano rzeczywiste odległości do wszystkich pozostałych planet. Prawo to jest bardzo przydatne do oszacowania odległości wszystkich satelitów wokół planet Jowisza, po prostu obserwując ich okresy czasu.

Ten tekst jest adaptacją <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>

Trzecia zasada ruchu planet Keplera

Na początku XVII wieku niemiecki astronom i matematyk Johannes Kepler postulował trzy prawa ruchu planet w Układzie Słonecznym. W...

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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

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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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Kepler’s Planet Distance Law

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