Geometric Universe

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A: See figure.

Slide 36

Q: What does Ptolemy’s geocentric model look like?

The epicycle is used to explain prograde/retrograde motion

The epicycle center rotated uniformly about equant E, instead of the center of deferent M.

The Earth is located off-center at the eccentric.

Distances EM = MO.

This is the geocentric model that agrees quantitatively with observations of the time. From the time of Apollonius to Ptolemy, planetary theories changed gradually from qualitative to quantitative science.

A:

Slide 37

Q: Ptolemy model looks quite different from Kepler’s, why did it work so well?

The reasons are:

the elliptical orbits of the planet are close to a circle

the eccentric takes the role of a focus, approximating Kepler’s first law

Ptolemy’s equant has the effect of approximating Kepler’s second law

Using Tycho’s data, Kepler refitted Ptolemy’s model, which gave a maximum error of only 8’ for Mars.

Since uncertainty for Tycho’s data is only 1’, Kepler was forced to give up the circles.

Eccentric/Sun

Equant

Comparing the elliptical orbit of Mars (red) to a circle (blue).

A: Ptolemy was approximating Kepler’s law, without knowing it.

More:

Slide 38

Q: How to transform between geocentric and heliocentric models?

The two models are equivalent if constructed as shown. The vectors pointing from the Earth to the planet are always the same between the two models.

Copernicus used his own observation as well as Ptolemy’s data to obtain parameters to his model.

The precisions of the two models are the roughly same.

Earth

Sun

Slide 39

Q: One arcmin is about the size of a HK$1 coin in 88 m away, how did Tycho Brahe achieve this accuracy without telescopes?

A: Great care for accuracy, a whole lifetime of pursuit, and a lot of support.

More:

He was the first one to notice the problem relating observation accuracy and have the ability to improve on them. He improved the sight with a slit design, and also added gradual scale to improve reading. Very large instruments help measuring smaller angles, but they requires stronger materials and mechanical parts. To support Tycho’s work, the King of Denmark granted him the estate of the island Hven, on which he built world’s best observatory called Uraniborg.

Left: The sight’s aligned horizontally if the star can be seen just on the CBGF edge and ADHE edge at the same time. The vertical alignment can be found similarly. For solar alignment, sunlight is allowing to pass thru the hole in the front and fall on a circle drawn on the ABCD plate.