Astronomy "before the Common Era"

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Problem 1: Aristotle's geometry

Aristotle knew there were stars, that one can see in Egypt but not in Greece. What conclusions could be made from this observation?

Problem 2: Earth's radius

How could Eratosthenes in 250 BCE determine the Earth's radius with the help of a well?

Problem 3: distance to the Moon

How could Aristarchus (310-230 BCE) compute the distance to the moon, knowing that it sets in about two minutes, and the maximum duration of a lunar eclipse is three hours?

Problem 4: distance to the Sun

How can one calculate the size and distance to the Sun, while observing the phases and eclipses of the Moon?

Problem 5: naked eye limitations

Can one estimate how farther is the Sun than the Moon using only naked eye observations?

Problem 6: measuring distance to a satellite

How to determine the distance to an Earth's satellite (Moon, for example) or to the Sun, using only a chronometer?

Problem 7: orbits of planets

Ancient Babylonians knew, that apparent motion of Mars relative to the Earth is periodic with the orbital period 780 days (the synodic period of Mars). Tycho Brahe, in XVI century, measured the apperent Mars' trajectory with great precision. How could Johannes Kepler, using Brahe's data, accurately calculate the orbits of both Earth and Mars relative to the Sun and discover the laws named after him, which in time enabled Newton to biuld the classical theory of gravity?

Problem 8: speed of light

The orbital period of Io, one of the satellites of Jupiter, is equal to $42.5$ hours. In the 1670-ties Ole Rømer was measuring the time between two successive eclipses of Io when Earth moved on its orbit towards Jupiter and when it moved in the opposite direction. The noticed difference, about 20 minutes, allowed him to estimate the speed of light. Try to reproduce the reasonong and estimates of Rømer.

Problem 9: Doppler shift

In 1982 Doppler predicted the effect of change in the percieved frequency of oscillations when there is relative motion of emitter and detector. Doppler assumed that this effect can cause the difference in the color of stars: a star moving towards the Earth seems "bluer", while the one moving away "reddens". Explain why the Doppler effect cannot substantially change the color of a star.

Problem 10: absorption lines

In the beginning of the XXth century J.F. Hartmann, a German astronomer, was studying the spectra of double stars. The wavelengths of their spectral lines shifted periodically due to their relative motion, with the period equal to the orbital one. In the spectra of some of the binaries he also noticed there were absorption lines with wavelengths that did not change with time. What discovery did Hartmann make due to this observation?

Problem 11: lines of calcium

Originally the interstellar gas was discovered by its absorption of spectral lines of calcium. Does that mean that calcium is the dominating component of the interstellar medium?

Problem 12: getting rid of $c$

One of the creators of the Theory of Relativity, Henry Poincare, when speaking in 1904 (!) of the fact that the speed of light $c$ enters all the equations of electromagnetism, compared the situation with the Ptolemy's geocentric theory of epicycles, in which the Earth's year enters all the relations for the relative motion of celestial bodies. Poincare expressed hope that a future Copernicus would rid electrodynamics of $c$. Recall other examples of blunders of geniuses.