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

Question

Owen Atkinson · Accepted Answer

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

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.

Nova Adams · Answer

undefined The mathematical equation used to calculate the distance between Earth and the Sun at any given day of the year is represented by the formula:

$$r = a(1 - e^2) / (1 + e \cdot \cos(v))$$

where:
- $r$ is the distance between Earth and the Sun,
- $a$ is the semi-major axis of Earth's orbit,
- $e$ is the eccentricity of Earth's orbit,
- $v$ is the true anomaly, representing the angular distance of Earth from the vernal equinox.