Question

What is the mathematical equation used to calculate the distance between earth and the sun at any given day of the year?

Answer 1

A good approximation to calculating the distance from the sun is to use Kepler's first law.

Explanation:

The Earth's orbit is elliptical and the distance `r` of the Earth from the Sun can be calculated as:

`r = (a(1-e^2))/(1-e cos theta)`

Where `a=149,600,000km` is the semi-major axis distance, `e=0.0167` is the eccentricity of the Earth's orbit and `theta` is the angle from perihelion.

`theta=(2 pi n)/365.256`

Where `n` is the number of days from perihelion which is 3rd January.

Kepler's law gives a fairly good approximation to the Earth's orbit. In actual fact the Earth's orbit is not a true ellipse as it constantly being changed by the gravitational pull of the other planets.

If you want a really accurate value you need to use numerical integration data such as NASA's DE430 data. This data consists of a large number of coefficients for a series of polynomial equations which have been derived from observations and satellite data.