Earth's orbit is an ellipse with eccentricity 0.0167. Thus the earth's distance from the sun and speed as it moves around the sun varles from day-to-day. This means that the length of the solar day is not constant through the year. Assume that the earth's spin axis is normal to its orbital plane and find out the length of the shortest and the longest day. A day should be taken from noon to noon. Does this explain variation of length of the day during the year?

Explanation

`implies` Let , m = mass pf earh
`v_p` = velocity of earth at perigee
`v_a` = velocity of earth at perigee
`omega_p` = angular velocity of earth at perigee
`omega_a` angular velocity of earth at apogee

Angular momentum and areal velocity are constant.
`:. r_p^2omega_p = r_a^2omega_a " "....(i)`
If a is the semi - major axis of the earth.s orbit, then
`r_p =a (I-e) and r_a = a(1+e) " "....(ii)`
`:. (omega_p)/(omega_a) =((1+e)/(1-e))^2` [From equ. (i) and (ii) ]
`:. omega_p/omega_A= 1.0691 ( :. e = 0.0167)`
Let `omega` be angular speed which is geometric average of `omega_p and omega_a` and corresponds to average solar day ,
`:. ((omega_p)/(omega_a))(omega)/(omega_a))=1.0691`
`:. (omega_p)/(omega) = omega/omega_a=1.034`
`:. omega_p = 1.034^@`
`:. omega_a = 0.967^@`
`361.034^@` which corresponds to 24 h , `8.14^@ (8.1 ".. longer..") and 360.967^@` corresponds to 23 h to 59 min 52.. (7.9 .. smaller)
This does not explain in actual variation of the length of the day during the year.