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As I understand it, Kepler used the orbital period of Mars, along with observational data of Mars' and the sun's position in the sky to derive the orbits of Earth and Mars. (As described, here: https://faculty.uca.edu/njaustin/PHYS1401/Laboratory/kepler.pdf)

But how did he determine the orbital period of Mars without already knowing the orbit of the Earth?

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2 Answers 2

He knew the orbital period of Earth. You can do this by observing the sun's motion in the sky over a year. (And then you know that the orbital period of the earth is a year.)

Now from earth, you can observe Mars from one Opposition (https://en.wikipedia.org/wiki/Opposition_(planets)) to another. You can find out the time ($T_\text{relative}$ it takes for Mars to do this, using which, you find the relative angular velocity between the orbital motions of earth and Mars ($\omega_\text{relative}=2\pi/T_\text{relative}$). This angular velocity is the difference between the angular velocities of earth and mars. Using this and knowing the orbital period of earth, you can find the orbital period of mars.

$2\pi/T_\text{relative}=2\pi/T_\text{earth}-2\pi/T_\text{Mars}$

Hope this is clear.

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Kepler did know the orbital period of the Earth, Mars, Venus, Mercury, Jupiter, and Saturn. He had access to observations going back centuries, including Brahe's most excellent data.

What couldn't know was the absolute size of the Earth's orbit, so the best he could do was give the size of the plantets' orbits in terms of the Earth-Sun distance, 1 A.U. Nobody else could do better for centuries.

Using the transit of Venus across the face of the sun, timing the instant the edge of Venus "touches" the edge of the Sun from two different locations on Earth a known distance apart makes a parallax measurable, and the absolute distance between the Earth and Venus a matter of trigonometry. Kepler's Law lets us convert all the other relative distances into absolute distances. I believe it was 1769 when this was first accomplished.

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This is all true, but this doesn't answer the question that was asked. –  Takku Jul 7 at 10:15

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