PURDUE UNIVERSITY

EAS 105-THE PLANETS

Prof. Robert L. Nowack

 

Lecture 4

 

 

Prior to Copernicus: PTOLEMY MODEL of the SOLAR SYSTEM (Geocentric)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The Second Century A. D. scholar Ptolemy (above) formalized the long-held view that the Sun, Moon and planets traveled in perfect circles around the Earth and inside a sphere of fixed stars.  To explain the planets’ noticeable deviations from the theoretic orbits, Ptolemy suggested that the planets also moved in small circles, called epicycles, at the same time they revolved around the Earth.

 

After Copernicus: HELIOCENTRIC MODEL of the SOLAR SYSTEM

 

 

 

 

 

 

 

 

 

 

 

 

 

Copernicus challenged the traditional view of the Solar System in the 16th century.  He moved the Sun to the center and placed the Earth, with its companion Moon, among the revolving planets.  Although he was correct in his essential reference frame, Copernicus did not realize the planets had elliptical orbits.  He retained Ptolemy’s fanciful epicycles and perfect-circle orbits.

 

 

 

Planetary Motions and Orbits

 

Tycho Brahe (1546-1601)

-     Established astronomical observatory (for visual observations).

-     For 20 years he made some of the most accurate measurements of astronomical objects up to that time.

-     constructed a star chart.

-     observed the continuous motions of the Moon and the planets.

-     Unfortunately he was not an advocate of the Heliocentric hypothesis.

 

Johannes Kepler (1571-1630)

He was an assistant to Tycho Brahe.  However, only after Tycho's death could Kepler have access to enough data to make progress on his investigations of planetary motions.  (Kepler was a Copernican.)  Kepler's most detailed study was that of Mars.

 

 

 

 

After trying a number of planetary motions, he came up with his First Law:

 

1)   Each planet moves in a path shaped like an ellipse with the Sun as one of the foci.

 

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Aside: One can draw an ellipse by using a string and 2 tacks.

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Ellipse

 

(r,q) coordinates of a path around ellipse a is the semi-major axis

 

e is the eccentricity for:  0 < e < 1 and e = 0 for a circle

 

 

 

 

Kepler found Mars’ eccentricity to be ~ 0.1 (the modern estimate is 0.093).  For Earth, e = 0.017.

 

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Aside:  An ellipse is one of several curves determined by a plane intersecting a cone.

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Kepler's Second Law

 

2)   Planets move faster when closer to the Sun and slower when farther away. 

 

Planets orbit the Sun in such a way that the area of the ellipse that is swept out as it travels is a constant for constant time intervals.  Equal areas swept out in equal time.

 

 

 

 

In order for equal areas to be swept out in equal times, a planet must move slightly faster when closer to the Sun and slightly slower when it is farther away from the Sun.

 

Kepler's Third Law

 

(3)  The time it takes for a planet to complete one orbit (1 period ) squared is proportional to the average distance a, between the Sun and planet cubed.

 

Period x Period ~ a x a x a     Where a = semi-major axis

 

This equation requires the Period in years and a in A. U.  Thus, if the period is known, the distance from the Sun (or visa versa) can be estimated.

 

 

Example:  For Mars,    a = 1.524 a.u.

                                                a3 = (1.524) x (1.524) x (1.524) = 3.54

                                                Period = Sqrt (3.54) = 1.88 years

 

Kepler believed in the underlying harmony of the worlds.

 

 

Planet

a (a.u.)

P (yr)

Mean Orbit Speed km/s

Eccentricity e

Inclination to ecliptic plane

Titius-Bode prediction

Mercury

.387

.241

    47.9

    .206

  7.004

    .4 Au

Venus

.723

.615

    35.0

    .007

  3.394

    .7 Au

Earth

1.

1.

    29.8

    .017

  0.0

    1 Au

Mars

1.524

1.881

    24.1

    .093

  1.85

    1.6 Au

    2.6*

Jupiter

5.203

11.862

    13.1

    .048

  1.308

    5.2

Saturn

9.534

29.456

    9.6

    .056

  2.488

    10.

Uranus

19.191

84.07

    6.8

    .046

  .0774

    20.

Neptune

30.061

164.82

    5.4

    .01

  1.774

    39.

Pluto

39.529

248.6

    4.7

    .248

  17.15

    39.

 

*This distance is in the asteroid belt.

 

In 1766, Johann Titius attempted to explain the distances of the planets (the Titius-Bode relation) by a mathematical relation.  Although it doesn't quite fit, it was historically important in the search for new planets.

 

>  Hershel in 1781 discovered Uranus by adding a planet where it should be.

>  In 1800's, the asteroid belt centered on 2.6 A.U. was found.

 

Galileo (1564-1642)

He accepted the Copernican System.  He used a small telescope to discover that:

1)   Venus goes through phases just like the Moon.

2)   Jupiter has satellites of its own.

 

An odd twist of fate is that the naked eye can't see the moons of Jupiter, but a good pair of binoculars can.

 

Galileo also investigated the Laws of Motion.  Law of Inertia - bodies resist changes in motion.

 

In 1632, Galileo wrote a book "Dialogue on the Two Great World Systems" - an argument for the Copernican System.  As a result of this book, he was called forward to the Roman Inquisition.  To save his life, he pleaded guilty to charges of heresy and recanted his Copernican view.  He was placed under house arrest for the last 10 years of his life.  His book was on the index of prohibited books until 1835.  In 1980, a re-examination of evidence against Galileo was done, and finally led to his exoneration by the Catholic Church.