What equation do astronomers use to calculate the sun-earth distance?
The Easiest is S = V . t
The Easiest way to get the distance between the Sun and the Earth is using the Equation of Motion. S = V.t. For this we need the Time that a Photon takes to reach Earth from the Sun's Surface and the Speed of light in Vacuum. Once we have these we can put these in the distance equation. Below is how it works.
The Time that a photon takes from the Surface of the Sun to Reach Earth = t = 8 minutes and 19 seconds = 499 seconds.
Speed of Light in Vacuum = V = 300,000 km/ sec.
Distance = V . t
Distance = 300000 x 499
Distance = 149,700,000 km
Distance = 149 Million Km.
Please note that this is the average distance between the Sun and the Earth since the Orbit is an ellipse so the time for a Photon to reach Earth also changes with distance and Vice Versa.
Kepler's 3rd law relates a planets orbital period
By observing the positions of the planets we can easily determine the orbital periods for them in AU.
Now we need one more piece of information to determine the actual length of an AU. The easiest way of doing this is to find the distance between Earth and Venus. This was originally done using parallax. Now we can measure the distance to a high degree of accuracy using radar. Radio waves are bounced off Venus and the time taken for the return journey gives the distance.
Using Kepler's law we know that Venus is 0.73 AU from the Sun. So, the distance between Earth and Venus is 0.27 AU. Using measurements we can determine that the distance between Earth and Venus is about 42,000,000km. From that we can determine that 1AU, which is the distance of the Earth from the Sun, is about 150,000,000km.