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• Thanks to the meticulous astronomical observations of his colleague and employer Tycho Brahe,

• Johannes Kepler was able to test several rival hypotheses for how the Sun and the planets

• are arranged in the Solar System, eventually leading to his three laws of planetary motion.

• In 1609, he published the first two laws in a book called Astronomia Nova, which focused

• on the movements of the planet Mars. Mars was something of a conundrum - its observed

• motions didn't match any of the proposed models of the solar system,

• which involved circular orbits.

• Kepler's First Law states simply that Mars travels in an elliptical orbit, with the Sun

• at one focus of the ellipse. Although he chose to list it first, Kepler only came to this

• conclusion after figuring out hissecondlaw, which says that if you draw a line from

• the Sun to Mars, and wait a fixed amount of time, that line will sweep out a certain area

• as Mars moves along its orbit. What Kepler noticed was that this area is exactly the

• same no matter where in the orbit you are.

• This is often phrased as Kepler's “equal area in equal timelaw, and this law works

• because Mars doesn't move at a constant velocity - it speeds up the closer it gets

• to the Sun. So if Mars is approaching perihelion, the point in the orbit nearest to the Sun,

• it's traveling faster than if it's at aphelion, the point that's farthest away.

• In the first case, the line connecting Mars to the Sun is very short, but because the

• planet is moving faster, it covers a lot of distance. In the second case, the line segment

• is much longer, but Mars also moves more slowly. Either way, the area swept out in a fixed

• amount of time is the same.

• Kepler and his contemporaries could see that Mars doesn't move at a constant rate, but

• they didn't know why. The inverse relationship that Kepler proposed between distance from

• the Sun and orbital velocity could explain the puzzling observations of Mars' movements,

• but only if the orbit is an ellipse. A circular orbit would mean no change in distance from

• the Sun with time, and thus the velocity would be constant as well.

• These two statements--that

• Mars travels in an elliptical orbit and that its speed varies so that the Mars-Sun line

• sweeps out equal areas in equal time--were generalized to include all planets in 1621,

• and they constitute Kepler's first and second laws of planetary motion.

• The 2nd Law, it turns out, is also a consequence of the conservation of angular momentum (which

• was not a concept known to Kepler in the seventeenth century). Angular momentum is a measure of

• the amount of rotational motion in a body or system of bodies, like Mars and the Sun,

• and in the absence of outside forces, it's a fixed quantity. This implies a tradeoff

• between the distance at which Mars orbits and its velocity -- like Kepler noticed. Just

• as an ice skater spins faster after pulling her arms close to her body, Mars has to move

• faster when it gets closer to the Sun. Kepler's statement that the area swept out by the Mars-Sun

• line is constant is equivalent to the statement that angular momentum is a constant as well

• -- that is to say, that it's conserved.

Thanks to the meticulous astronomical observations of his colleague and employer Tycho Brahe,

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# Kepler’s Second Law of Motion (Astronomy)

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joey joey posted on 2021/04/11
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