Expert: Philip Stahl - 3/22/2010

Question
i do mathematics in high school. and i have an understanding of conic sections and the formula for the ellipses and circles and so on. and i was wondering if the equation of the earths elliptical orbit around the sun can be calculated and if so, what is this equation?

thankyou for your time.

Answer
Hello,

Yes, the Earth's orbit can be calculated- by obtaining what we call the orbital parameters. Let's just focus here on Earth's orbit as an ellipse.

From careful measurements we know the eccentricity e = 0.0167 (Which can also be found in a Table of Planetary data).

We also know the mean radius (or semi-major axis) a = 1.000000 AU (astronomical units)

The semi-minor axis b =  a[(1- e^2)]^1/2

The standard equation of the ellipse for a planet is expressed(cf. 'Textbook on Spherical Astronomy' - 6th ed., W.M. Smart, p.99):

r =  p/ {1 + e cos (theta - f)}

where r is the radius vector, p = b^2/a =  a(1 - e^2), e is the eccentricity, f is the true anomaly (= 102.9 deg) and theta is the isntantaneous angle between the planet's actual position and the direction of the periastron or perihelion. (Thus, for a given day you need to obtain theta, say from a table).

Obtaining theta, you then procure the instantaneous radius vector, r.

Hope this helps!