What is the period of Neptune based on this information?

Through research on the internet the period of Neptune is approximately 165 years (the time it takes for it to travel around the sun once), however I don't know how to arrive at such a number through the information I was given in the following problem:





If the Earth's average distance from the Sun is 1.496x10^11 m, and Neptune's average distance from the Sun is 4.497x10^12 m, find the period of Neptune.





Please show your work.|||Use this version of Kepler鈥檚 3rd law:





T虏 = a鲁





Where T is the period in years and a is the semi-major axis (mean distance will do) of the planet鈥檚 orbit in astronomical units (AU). Dividing Neptune鈥檚 distance by earth鈥檚, you鈥檒l get Neptune鈥檚 distance in AU:





a(Neptune) = 4.497x10^12 m / 1.496x10^11 m


= 30.1AU





Therefore:





T = 鈭歔(30.1)鲁]


= 165





Hope this helps.|||There's a formula, but you learn more physics from practicing at deriving them than by rote memorization. So, start with the basics and assume circular orbits.





The acceleration on a planet at distance r from the Sun is GM/r虏, were M is the mass of the Sun, and G is the gravitational force constant ("big G").





The acceleration required for circular motion is v虏/r.





So in orbit GM/r虏 = v虏/r, or v = sqrt(GM/r).





The circumference of the orbit is 2蟺r, so the orbital speed is also 2蟺r/T, where T is the orbital period.





2蟺r/T = sqrt(GM/r)


T = [2蟺/sqrt(GM)] sqrt(r^3)


T = [2蟺/sqrt(GM)] r^(3/2)





So T is a constant [2蟺/sqrt(GM)] times the square root of r. If you remembered that the period is proportional to the 3/2 power of the distance, then you could skip the above. The exact constants don't matter in this problem, because...





If T1 is the orbital period of the Earth at radius r1, and T2 is the period of Neptune at radius r2, then:


T1 = [2蟺/sqrt(GM)] r1^(3/2)


T2 = [2蟺/sqrt(GM)] r2^(3/2)


T2/T1 = (r2/r1)^(3/2)


T2 = T1 * (r2/r1)^(3/2)





The constants have disappeared and you have a simple calculator exercise.


T2 = 1 year * ( 4.497e+12 / 1.496e+11)^(3/2)





Edit: Oops, hit Submit instead of Preview too soon...wasn't finished fixing error. Should be better now.