Question

Is it possible to plot a graph of time for one revolution around the sun against distance from the sun for all the planets? Why?

Answer 1

Yes, it is possible to plot distance against time for all of the planets.

Explanation:

A good approximation for the obits of planets is to use Kepler's laws of planetary motion.

To calculate the distances using Kepler's laws you first of all need to know the semi-major axis distance of the planet `a` in AU and the eccentricity of the orbit `ε`. These can be found from this NASA JPL Web site http://ssd.jpl.nasa.gov/txt/p_elem_t1.txt

We can now calculate the orbital period `T` using Kepler's third law `T^2=a^3`.

To calculate the distance of the planet at time `t`, we first calculate the mean anomaly `M=(2πt)/T`. We now need to calculate the eccentric anomaly `E` by solving Kepler's equation `M=E-εsinE`. Note that Kepler's equation has no analytic solution and has to be solved numerically using Newton's method or equivalent.

Now we can calculate the distance form the Sun `r` using the equation `r=a(1-εcosE)`.

Kepler's laws give quite a good approximation to the planets orbits, but planets don't actually follow true elliptical orbits due to the gravitational pull of the Sun and other planets. Get get a more precise value for planetary distances requires computing distances using power series. The VSOP87 data gives quite accurate positions for the planets. Really accurate positions can be calculated using the DE430 data from NASA JPL.