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Find the properties of material, which you need to make a rope for a lift from geostationary orbit to the Earth surface. Is such material available on Earth?

Make a measurement of the height of Sun above horizon at noon time and time from sunrise (middle of Sun disk) till its sunset. Then you can calculated theoretical duration of day and compare with reality and comment on differences.

Suggest several methods for estimating the mass of the Sun. Explain in detail each of them and estimate mass of the Sun.

The magnitude of binary star is changing with 4-day period with this sequence:

side minimum: m = 3.5

maximum: m = 3.3

main minimum: m = 4.2

maximum: m = 3.3.

The larger part of binary star has higher temperature then its escort. Assuming, that the Earth is in the orbital plane of binary star, calculate magnitudes of each component and ratio of its linear dimensions. Relation between magnitude $m$ and irradiation $E$ is given by

$m=-2,5\log(E⁄E_{0})$,

where $E_{0}$ is some defined value.

We have calculated spectral class of two stars forming binary star from reduced star spectra (from present spectral lines which does not change its position in time). From spectral class we have estimated its masses as 2 and 3 times the mass of the Sun. From the observation with telescope of focal length 3 m we know that the stars really orbit in constant angular distance of 5 angular minutes. One orbit takes 50 years.

Are you able to calculate distance of this binary star from the Sun? If yes, state which information you need and the result round accordingly. Comment on the precision of the result and how the uncertainty of input information (mainly the masses) impacts on accuracy.

Let's return back to 18th century, when the value of Newton's gravitation constant was not known yet. And the distance between Sun and the Earth was unknown as well. During this times Edmond Halley (astronomer who discovered that the comet in the sky in 1682 is the same as the one in 1456, 1531 and 1607) suggested to find distance Earth-Sun by measuring time of transit of Venus over the Sun circle. Unfortunately the Venus cross the Sun irregularly, in pair each eight years and then there is century-long break. During Halley life there was no crossing.

However, the idea was not forgotten and with the next crossing was successfully resuscitated and in 1761 a next scientific experiment was prepared. Scientists went to many parts of world, including Siberia, China, South Africa and Indonesia. It was the first international scientific experiment.

After all people have returned, it was concluded that no conclusion can be made. Ironically it was due to too many contradicting observation. Successful was captain James Cook in 1769 on one summit in Tahiti. After his return the astronomers were able to calculate the average distance between the Earth and the Sun as approximately 150 millions kilometers.

It is up to you to calculate distance Earth-Sun using only data known at the time: Earth diameter and orbit time of Venus and Earth.