Comets are known to have very eccentric far from circular or

Comets are known to have very eccentric (far from circular) orbits around the Stars. Imagine a comet with a mass of 4,481 kg that orbits the Sun in an eccentric elliptical orbit. We know that this comet’s perihelion (the closest distance to the Sun) is 0.099 AU, and its aphelion (the farthest distance from the Sun) is 18.8 AU. 1 AU is the mean distance between the Earth and the Sun and it is equal to 150 million km. If this comet has been to its perihelion in 2017, how many years later should we expect it go farthest from the Sun, that is, how many years later will it visit its aphelion? Enter with a precision down to 0.1 year or better. (The mass of the Sun is 1.99x10^30 kg. )

Solution

Time period of the body in elliptic orbit is given by
T = 2*pisqroot(a^3/mu)
where a = semimajor axis
in this case, perhilion, x = 0.099 AU
Aphellion, y = 18.8 AU
then
a = (x + y)/2 = 9.4495 AU
now 1 AU = 150*10^9 m
then a = 1.417425*10^12 m

and for sun mu = standard gravitational parameter = 1.327*10^20
hence T = 2*pi*sqroot(a^3/mu) = 920437998.8649 seconds = 29.166 years

so time taken to go from perihilion to aphelion = T/2 = 14.58346 years