Question

According to Keplers 3rd Law, how would a planets orbital period change if the planets...

a. eccentricity doubled?

b. mass halved?

c. distance from the sun quadrupled?

Answer 1

where is the orbital period of the planet and is the semi-major axis of the orbit.

if eccentricity is doubled

then e=c/a (a must be halved) 2e=c/a/2 let (a/2=a')

so

p^2/p'^2=a^3/a'^3 (p' and a' are new time periods and lengths)

solving we get

p'=p/2sqrt(2) or p/2.82

b)

For circular orbits, Kepler's 3rd Law is also commonly represented as

or simply

t^2=cr^3/m

where c=proportinality constant

let t' and r' and m' be new constants

t^2/t'^2=(cr^3/m)/(cr^3/m') but (r=r')

t^2/t'^2=m'/m and m'=m/2

t'=sqrt(2)*t or 1.41(t)

c)

t^2=cr^3

let t' and r' be new constants

r'=4r

t^2/t'^2=r^3/r'^3

t^2/t'^2=r^3/64r^3

t'=8t