Question

Earth’s orbit is an ellipse with eccentricity 0.0167. Thus, earth’s distance from the sun and speed as it moves around the sun varies from day to day. This means that the length of the solar day is not constant through the year. Assume that 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?

Answer 1

Angular momentem and areal velocity are constant as earth orbits the sun

At perigee r_p²ω_p =  r_a²ω_aat apogee.

If ‘a’ is the semi-major axis of earth’s orbit, then r_p = a (1 - e) and 

r_a = a(l + e)

Let ω be angular speed which is geometric mean of ω_p and ω_a and corresponds to mean solar day,

If ω corresponds to 1° per day (mean angular speed), then ω_p = 1.034^o per day and ω_a = 0.967 per day. Since 361° = 14hrs: mean solar day, we get 361.034^o which corresponds to 24 hrs 8.14'' (8.1'' longer) and 360.967° corresponds to 23 hrs 59 min 52'' (7.9'' smaller).

This does not explain the actual variation of the length of the day during the year.