# How is the distance between earth and the sun measured?

Nov 23, 2016

The distance from the Earth to the Sun is measured using the position of another planet and trigonometry.

#### Explanation:

The distance from the Earth to the Sun can be determined by trigonometry and another planet.

To give an example Venus will be used and it will be assumed that the orbits of Earth and Venus are concentric perfect circles. The orbits are not perfect circles but it simplifies the calculations and is not too inaccurate.

First we need to know the distance ${d}_{e v}$ from Earth to Venus at their closest point. We can measure this by bouncing radar signals off Venus. The distance to Venus is

${d}_{e v} = c \frac{t}{2}$

Whee c is the speed of light and t is the time it takes for the radar signal to get to Venus and back.

The average value is ${d}_{e v} = 40 , 000 , 000 k m$
If the distance from the Earth to the Sun is ${d}_{e}$ and the distance from Venus to the Sun is ${d}_{v}$, then

${d}_{e} = {d}_{v} + {d}_{e v}$

The other piece of information required is Venus' greatest elongation $\theta$, which is the greatest angular separation of the Sun and Venus. At this point the Earth, Venus and the Sun form a right angle triangle. The average value is 46 °.

With some trigonometry we can determine that:

${d}_{e} = {d}_{e v} / \left(1 - \sin \theta\right)$

Plugging the numbers in gives ${d}_{e} = 142 , 251 , 000 k m$. This is quite close to the average actual value of ${d}_{e} = 150 , 000 , 000 k m$ given the assumptions made.

The actual distance can now be calculated to great accuracy using the NASA DE430 data which is calculated using numerical integration techniques based on observational data from satellites.