KeplerKepler’’s Laws of Planetary Orbits:s Laws of Planetary Orbits:All deduced from observationsAll deduced from observations1. Orbits are ellipses1. Orbits are ellipses2. Plane Velocity Varies as it orbits2. Plane Velocity Varies as it orbits3. P3. P2 2 = A= A33

PtolemyPtolemy

Copernicus – doesnCopernicus – doesn’’t quite get it t quite get it right – still needs epicyclesright – still needs epicycles

TychoTycho’’s Insane Models Insane Model

KeplerKepler’’s Ultimate Resolution: 3 s Ultimate Resolution: 3 lawslaws

axis major of length

points focus between distancetyeccentrici

An Ellipse:

11stst Law: All orbits are ellipses. Law: All orbits are ellipses. Circle vs EllipseCircle vs Ellipse

KeplerKepler’’s Dilemmas Dilemma

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

Kepler WritesKepler Writes

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

Kepler WritesKepler Writes

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

Used TychoUsed Tycho’’s Positional Data for s Positional Data for MarsMars

Important NotesImportant Notes

This is Good Data – 10 orbital periodsThis is Good Data – 10 orbital periods

The Y-axis (basically the position of Mars) The Y-axis (basically the position of Mars) has a large amplitude – easy to measurehas a large amplitude – easy to measure

In real life, the In real life, the ““eccentricityeccentricity”” (deviation (deviation from a circle) of Mars orbit is 9% - after 10 from a circle) of Mars orbit is 9% - after 10 orbital periods (20 years of data) the orbital periods (20 years of data) the difference between a perfect circle and an difference between a perfect circle and an ellipse is pretty clearellipse is pretty clear

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

Eccentricity VariesEccentricity Varies

Due to gravitational interactions with Due to gravitational interactions with Jupiter so that Orbits change (but slowly)Jupiter so that Orbits change (but slowly)

Kepler got Kepler got ““luckylucky”” to be alive at time of to be alive at time of maximum eccentricitymaximum eccentricity

Kepler’s Second Law – Equal areas in Equal Times

This requires the planet to move fastest when its closest to the sun

Copernicus realized that orbital period of Copernicus realized that orbital period of Mars was larger than EarthsMars was larger than Earths

Kepler Thought ExperimentKepler Thought Experiment

Is there a relation between orbital period and Is there a relation between orbital period and distance from the sun.?distance from the sun.?

If Planet A is twice as far away as Planet B from If Planet A is twice as far away as Planet B from the sun and both have the same orbital velocity the sun and both have the same orbital velocity around the sun, then Planet B takes twice as around the sun, then Planet B takes twice as long to orbit the sun (since the circumference is long to orbit the sun (since the circumference is twice as big) as Planet Atwice as big) as Planet A

This disagrees with both TychoThis disagrees with both Tycho’’s data and s data and KeplerKepler’’s own second law which shows the orbital s own second law which shows the orbital velocity is not constantvelocity is not constant

Resolution: Kepler’s 3Resolution: Kepler’s 3rdrd Law Law

From extant data, Kepler establishes the From extant data, Kepler establishes the proportionality: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

Kepler has no clueKepler has no clue

Why his third law (the harmonic law) Why his third law (the harmonic law) actually works – he just knows it agrees actually works – he just knows it agrees with the datawith the data

Example CalculationExample Calculation

KeplerKepler’’s Laws are a purely s Laws are a purely empirical and precise empirical and precise description of the orbits of the description of the orbits of the planetsplanets

But, he has no dynamical But, he has no dynamical basis for understanding why basis for understanding why these laws exist.these laws exist.