Average annual angle (Earth lattitude)

[cahill8]

Homework Statement

This is not the problem in itself but it is related to something I'm doing.
I need to find a formula for the average angle of a latitude on Earth w.r.t. the sun. e.g. if the Earth was not inclined by 23.5 degrees, the average angle would just be the lattitude. Instead, 30 degrees north is effectively between 30+23.5 degrees and 30-23.5 degrees. In this case, the average angle is simply 30 degrees but this changes if the lattitude is less than the rotational inclination.

e.g. at the equator (0 degrees), is between 23.5 degrees and -23.5 degrees. Just taking the average you get 0 degrees, where the average is half 23.5 (as going down to -23.5 is the same as going down to 0 and then back up to 23.5). Am I missing something simple?

To sum up I'm looking for a function where I can input a lattitude and get back an average effective angle (which should equal the lattide for lattitudes larger than the rotational inclination (23.5), half 23.5 for the equator, something in between for stuff in between. Can I do this logically or do I have to be more clever? (integral etc.)

 

[D H]

Averaged with respect to what?

To illustrate what I am talking about, consider the average velocity of something that starts from rest and undergoes uniform acceleration for some time t. The average velocity is at/2 when averaged with respect to time but 2at/3 when averaged with respect to distance.

The reason this is important is because the Earth's orbit is elliptical. This results in an asymmetry of the seasons between the Northern and Southern hemispheres. I suspect you will get something akin to an elliptic integral if you want to do time averaging.

 

[cahill8]

Sorry, I should have mentioned that R is constant (circular orbits).

I have so far said that an Earth moving around the sun with it's rotational axis at an angle to the sun-earth plane (23.5 degrees) is equivalent to the Earth being stationary at some point and rotating towards and away from the sun (a nodding action). I want to hopefully find that this is equivalent to the Earth stationary at some angle with respect to the sun-earth plane (measured from the middle of the earth). The 'some angle' is what I'm calling the average effective angle.

I'm not sure how to setup an integral for it though