Question

Earth's orbit is an ellipse with eccentricity 0.0167. Thus, earth's distance from 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 & 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

Let mass of the earth be = M

Velocity of the earth  at perihelion = vp

Velocity of earth at aphelion = va

Angular velocity of earth at perihelion = wp

Angular velocity of earth at aphelion = wa

Distance between aphelion and perihelion = rp = a(1-e) and ra = a(1+e) -- 1

Since, the angular momentum and real velocity are constant as earth is orbiting -  

r²pw = r²aw -- 2

From equation 1 and 2

wp/wa = ( 1+e / 1-e) = 0.0167

= 1.0691

Now, w² = wpwa

wp/w = w/wa = 1.034  -- 3

Also, Ta + Tp/2 = 24

Ta + Tp = 48 -- 4

From equation 3 and 4

Tp = 24.8h