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Ptolemy

Kepler ’ s Laws of Planetary Orbits: All deduced from observations 1. Orbits are ellipses 2. Plane Velocity Varies as it orbits 3. P 2 = A 3. Ptolemy. Copernicus – doesn ’ t quite get it right – still needs epicycles. Tycho ’ s Insane Model. Kepler ’ s Ultimate Resolution: 3 laws.

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Ptolemy

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  1. Kepler’s Laws of Planetary Orbits:All deduced from observations1. Orbits are ellipses2. Plane Velocity Varies as it orbits3. P2 = A3

  2. Ptolemy

  3. Copernicus – doesn’t quite get it right – still needs epicycles

  4. Tycho’s Insane Model

  5. Kepler’s Ultimate Resolution: 3 laws

  6. An Ellipse:

  7. 1st Law: All orbits are ellipses. Circle vs Ellipse

  8. Kepler’s Dilemma • Kepler’s working notes show that he thought he must have made an error when he first made his measurements.  Not until late in his process of discovery did he abandon the idea of a circular orbit for Mars.  

  9. Kepler Writes • "My first error," he relates, "was to suppose that the path of the planet is a perfect circle, a supposition that was all the more noxious a thief of time the more it was endowed with the authority of all philosophers, and the more convenient it was for metaphysics in particular.

  10. Kepler Writes • “Clearly, then, [what is to be said] is this: the orbit of the planet is not a circle, but comes in gradually on both sides and returns again to the circle’s distance at perigee.  They are accustomed to call the shape of this sort of path an oval.”

  11. Used Tycho’s Positional Data for Mars

  12. Important Notes • This is Good Data – 10 orbital periods • The Y-axis (basically the position of Mars) has a large amplitude – easy to measure • In real life, the “eccentricity” (deviation from a circle) of Mars orbit is 9% - after 10 orbital periods (20 years of data) the difference between a perfect circle and an ellipse is pretty clear

  13. Eccentricity of Mars orbit means variations in Earth-Mars distance at opposition

  14. Eccentricity Varies • Due to gravitational interactions with Jupiter so that Orbits change (but slowly) • Kepler got “lucky” to be alive at time of maximum eccentricity

  15. Kepler’s Second Law – Equal areas in Equal Times This requires the planet to move fastest when its closest to the sun

  16. Copernicus realized that orbital period of Mars was larger than Earths

  17. Kepler Thought Experiment • Is there a relation between orbital period and distance from the sun.? • If Planet A is twice as far away as Planet B from the sun and both have the same orbital velocity around the sun, then Planet B takes twice as long to orbit the sun (since the circumference is twice as big) as Planet A • This disagrees with both Tycho’s data and Kepler’s own second law which shows the orbital velocity is not constant

  18. Resolution: Kepler’s 3rd Law • From extant data, Kepler establishes the proportionality: P = orbital period measured in units of years A = orbital distance, measured in units of AU (1 AU = 1 Astronomical Unit = distance from the Earth to the Sun

  19. Kepler has no clue • Why his third law (the harmonic law) actually works – he just knows it agrees with the data • Example Calculation

  20. Kepler’s Laws are a purely empirical and precise description of the orbits of the planets But, he has no dynamical basis for understanding why these laws exist.

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