how to measure the distance of the sun from the earth?

Answer 1

Aristarchus of Samos estimated the distance to the Sun to be about 20 times the distance to the moon, whereas the true ratio is about 390. His estimate was based on the angle between the half moon and the sun, which he estimated as 87°.

According to Eusebius of Caesarea in the Praeparatio Evangelica, Eratosthenes found the distance to the sun to be "σταδιων μυριαδας τετρακοσιας και οκτωκισμυριας" (literally "of stadia myriads 400 and 80000"). This has been translated either as 4,080,000 stadia (1903 translation by Edwin Hamilton Gifford), or as 804,000,000 stadia (edition of Édouard des Places, dated 1974-1991). Using the Greek stadium of 185 to 190 metres, the former translation comes to a far-too-low 755,000 km, whereas the second translation comes to 148.7 to 152.8 million km (accurate within 2%).

At the time the AU was introduced, its actual value was very poorly known, but planetary distances in terms of AU could be determined from heliocentric geometry and Kepler's laws of planetary motion. The value of the AU was first estimated by Jean Richer and Giovanni Domenico Cassini in 1672. By measuring the parallax of Mars from two locations on the Earth, they arrived at a figure of about 140 million kilometres.

A somewhat more accurate estimate can be obtained by observing the transit of Venus. This method was devised by James Gregory and published in his Optica Promata. It was strongly advocated by Edmond Halley and was applied to the transits of Venus observed in 1761 and 1769, and then again in 1874 and 1882.

Another method involved determining the constant of aberration, and Simon Newcomb gave great weight to this method when deriving his widely accepted value of 8.80" for the solar parallax (close to the modern value of 8.794148").

The discovery of the near-Earth asteroid 433 Eros and its passage near the Earth in 1900–1901 allowed a considerable improvement in parallax measurement. More recently very precise measurements have been carried out by radar and by telemetry from space probes.

While the value of the astronomical unit is now known to great precision, the value of the mass of the Sun is not, because of uncertainty in the value of the gravitational constant. Because the gravitational constant is known to only five or six significant digits while the positions of the planets are known to 11 or 12 digits, calculations in celestial mechanics are typically performed in solar masses and astronomical units rather than in kilograms and kilometres. This approach makes all results dependent on the gravitational constant. A conversion to SI units would separate the results from the gravitational constant, at the cost of introducing additional uncertainty by assigning a specific value to that unknown constant.

Answer 2

Triangulation.

By coordinating the efforts of multiple observers over the face of the planet, one can simultaneously measure the apparent angle to the center of the solar disk (careful -- NEVER look directly at the sun without special equipment). Once one knows those angles from different locations on the surface of the earth, it becomes a relatively simple geometry problem (assuming one knows accurately the distances between the observation points).

Answer 3

Answer 2 suggests one type of triangulation; baseline between measurement points known, angles measured. Here's another, with target size known, target angle measured. If you know the diameter d of the sun, measure its angular width theta from Earth. The accurate formula is then:
theta = 2 * sin(0.5 * d / Distance)
Distance = 0.5 * d / arcsin(0.5 * theta)
The small-angle approximation formula is:
theta = d / Distance
Distance = d / theta
Since the actual angle of the sun from Earth is about 0.5 degree, the small-angle equation is very accurate.

Answer 4

Tulshidas Ji, in hanuman chalisha Yug Sahatra yojan par bhanu lilyo tahi madhur phal janu

Answer 5

Use a six inch scale ruler.