Speed of Earth on its orbit

Eccentricity = 0.0167

Winter solstice comes around time of perihelion (fastest speed). Summer solstice is around aphelion. Just a coincidence, precession causes this to change gradually, so this is just how it happens to be in current millennium.

v_peri/v_ap = v_max/v_min = (1+e)/(1-e) = 1.0167/0.9833 ≈ 1.034.

Probably you could google "perihelion speed earth orbit" and get an exact figure. I don't happen to know. Someone else here may have a better estimate. But on a back of envelope rough guess basis I would say that around winter solstice the speed is about 1.7% faster than average and around summer solstice the speed is about 1.7% slower than average.

You know the average is about 30 km/s, so you can work out the answer approximately.

 

Someone else may want to explain why the length of the day changes (that is not orbit speed, it is axis tilt and depends on your latitude, how far north you live)

I googled to get the speed:
Mean orbital velocity (km/s) 29.78
Max. orbital velocity (km/s) 30.29
Min. orbital velocity (km/s) 29.29
http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html

I probably do not understand your question. Are you wondering why there are fewer days between December 23 and March 23 than there are between March 23 and June 23? the "23" is just approximate here, maybe you would say 21 or 22 in some cases.
That is for a different reason. The sun is at a FOCUS of the ellipse, it is not in the CENTER of the ellipse. So the orbit path is actually longer from March to June than it is from December to March. There should be more days, and there are more days.
I am confused about what you are asking. Someone who understands better will hopefully help explain.

 

To a first approximation...

The length of the day correlates with the tilt of the north pole either toward the sun (summer in the northern hemisphere) or away from the sun (winter in the northern hemisphere).

The tilt of the north pole is constant but the direction to the sun changes at a reasonably constant rate. The sunward component of the tilt will therefore vary like a sine wave.

At the equinox (when the sine wave crosses the x axis) the sunward tilt will be changing rapidly.
At the solstice (when the sine wave is at a maximum) it will be changing slowly.

That's pretty much a tautology. The first derivitive (if it exists) of an arbitrary function will be zero at any local maximum. If the function is continuously differentiable, the first derivitive will be small near any local maximum.

 

Both the speed of the Earth in its orbit and its angular speed are roughly constant (to with a percentage point or so). Neither can account for the entire change in the rate of change in the time of sunrise or sunset from one day to the next.

Close your eyes and draw a wavy line on a piece of paper from left edge to right edge. Look at the slope of that line near one of its extreme points. Look at the slope of that line between successive extreme points. Which slope is shallower? Which slope is steeper.

 

When you drew that wavy line on a piece of paper, did you carefully make it match the length of the daylight hours in successive days as a function of time?

Did it nonetheless display the property that its slope was small near the top of each peak and near the bottom of each valley and that the magnitude of its slope was generally larger in between?

Could you, perhaps, conclude that this is then a general property of [suitably smooth] graphs and not a property specific to the graph of daylight length as a function of time?

http://en.wikipedia.org/wiki/Mean_value_theorem
http://en.wikipedia.org/wiki/Maxima_and_minima
http://www.proofwiki.org/wiki/Derivative_at_Maximum_or_Minimum