To calculate the distance of a planet, parallax can be used.
Parallax consists of putting a finger before the eyes. The background should not be uniform. Without moving the head or finger and looking first with one eye and then with another, you can see that the position of the finger in relation to the background changes. If we bring the finger closer to the eyes and look again first with one eye and then with the other, the two positions of the finger against the background will cover a larger part.
This is because between the two eyes there is a separation of several centimeters, so the imaginary line that joins the finger with one of the eyes forms an angle with the imaginary line that joins the finger with the other eye. If we extend these two imaginary lines to the bottom, we will have two points that correspond to the two apparent positions of the finger.
The closer to the eyes we put the finger, the greater the angle and the greater also the apparent displacement.
If the eyes were more separated, the angle formed by the two lines would increase more, and thus the apparent displacement of the finger against the background would be greater.
This can also apply to a planet. It is true that the Moon is so far away that we can not see any difference when we look with both eyes. But if we look at the Moon against the starry background of the sky, from two observatories some hundreds of kilometers away, we will notice something. From the first observatory we will see that one of the edges of the Moon is at a certain distance from a specific star, while in the second observatory the distance between the same edge and the same star will be different.
Knowing the apparent displacement of the Moon against the starry background and the distance between both observatories, the distance aided by trigonometry can be calculated.
This experiment could be done perfectly, because the apparent displacement of the Moon with respect to the starry background when changing the position of the observer is very large. Astronomers have normalized this displacement for the case that one of the observers is seeing the Moon on the horizon and the other just above his head. The base of the triangle will then be equal to the radius of the Earth and the angle with vertex on the Moon is the “equatorial horizontal parallax”. Its value is 57.04 minutes of arc, or 0.95 degrees of arc. A really appreciable displacement, because it is equivalent to twice the apparent diameter of the full Moon. It is a magnitude that can be measured with enough precision, and allows to get a good value for the distance of the Moon. This distance, calculated with the help of parallax, matches very well with the figure obtained by that old method based on the shadow projected by the Earth during a lunar eclipse.
Unfortunately, the conditions in the year 1600 did not allow to place the observatories at a sufficient distance, which, together with the enormous distance to which the planets are located, made the apparent displacement against the starry background too small to be measured. accurately.
Later, the telescope arrived, invented or reinvented by the Italian scientist Galileo Galilei . The telescope allowed an angular distance not detectable to the naked eye to be easily measured.
The planets with major parallaxes are the closest, that is, Mars and Venus. Venus, in its closest approach to the Sun passes so close to him that it is impossible to observe it, except in the transits, where it can be seen against the bottom of the solar disk. So, the only case to measure parallax was the planet Mars.
The first telescopic measurement of a planetary parallax was made in 1671. The two observers were Jean Richer, French astronomer, leading a scientific expedition to Cayenne, in French Guiana and the Italian-French astronomer Giovanni Cassini, who stayed in Paris. They observed Mars with the maximum possible simultaneity and took note of their position with respect to the closest stars. Calculated the difference of positions observed and known the distance from Cayenne to Paris, the distance of Mars was calculated at the time of measurement.
Once this was done, we already had the scale of the Kepler model, which would allow us to calculate all the other distances of the solar system. Cassini estimated that the distance between the Sun and the Earth was 140,000,000 kilometers, nine million kilometers below the actual figure, but an excellent result for the first attempt.
Later, more accurate measurements of planetary parallaxes were made. Some on Venus, on the occasions that pass right between the Earth and the Sun, and that can be seen as a small dark circle crossing the Sun’s disk. These transits took place in 1761 and 1769. If the transit is observed from two different observatories , it can be verified that the moment in which Venus comes into contact with the solar disk and also the moment in which it separates from it, which is the duration of the transit, varies from one observatory to another. Known these variations and the distances between the two observatories, you can calculate the parallax of Venus. Having this data, one can calculate the distance to Venus, and then the distance to the Sun.
The German astronomer Johann Franz Encke , in 1835, used the existing data of the transits of Venus to calculate the distance from the Sun, and the figure was 153,450,000 kilometers. It exceeded a bit of the actual figure, but only about 3,000,000 kilometers.
In order to obtain more exact values, the main difficulty was that Venus and Mars were seen by the telescope as tiny spheres, which prevented to fix accurately the position of the planet. Especially disappointing was Venus, because the thick atmospheric layer that it had produced optical effects that prevented seeing during the transit the exact moment of contact with the solar disk.
Suddenly an unexpected event happened. The Italian astronomer Giuseppe Piazzi, in 1801, discovered a small celestial body whose orbit was between Mars and Jupiter, and called it Ceres. It had a diameter of less than 800 kilometers. As the century progressed, hundreds of even smaller planets were discovered, all between the orbits of Mars and Jupiter. They were the asteroids. Sometime later, in 1898, Karl Gustav Witt, a German astronomer, discovered Eros, an asteroid far from the asteroid zone. A part of its orbit passed through that of Mars, and very close to Earth as well.
It was estimated that in 1931 Eros would approach Earth. It was a good opportunity to calculate the parallax. Since Eros is very small, it is estimated that its maximum diameter is 24 kilometers, and it has no atmosphere that could blur its contour, it would be observed as a bright spot and its position could be well calculated.
A large project was organized on an international scale. Thousands of photographs were taken and studied, and it was concluded, from the parallax and the position of Eros, that the Sun is a little less than 150,000,000 kilometers from Earth. This is an average, because the Earth describes an ellipse around the Sun, not a circumference. The perihelion, or minimum distance between the Earth and the Sun, is 147,000,000 kilometers and the aphelion, maximum distance of 152,200,000 kilometers.