Kepler’s Second Law of Motion (Astronomy)

By | December 7, 2019


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 his “second”
law, 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 time” law, 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.

20 thoughts on “Kepler’s Second Law of Motion (Astronomy)

  1. Kushal Saitia Post author

    The background music, the narration and animations are enough for people not at all interested in Astronomy( flat earth society) would even binge watch this series.
    Coming to my second point: They are so binge worthy!!

    Reply
  2. Alex Anthony Post author

    And Kepler figured this out in 1609? Imagine if he had been born in the 20th century!

    Reply
  3. Veglia Borletti Post author

    your beauty is far beyond anything i ever encountered before

    Reply
  4. Amit upadhyay sultanpur amethi Post author

    Very nice .i am Indian

    Reply
  5. Jb Jb Post author

    Maybe just a coincidence that speed also seems to increase going into sharper turn of ellipse revolution just like driving your car into a sharper turn,.

    Reply
  6. Brian Meyrick Post author

    Brilliant ..so 1st law is geometric … 2nd is 'dynamical'. I never realised that.

    Reply
  7. Jim Keller Post author

    there is no place on the web that explains how Kepler actually derived his second law…the actual technique. If there is a site I have yet to find it…and I have looked for a good long time. I think that very few people actually know how. If you ask they will derive the laws using Newton….after the fact of Kepler's life

    Reply

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