What's wrong with argument against cos?

[ecklstn36]

I am trying to compute the amount of energy received by the Earth from the sun, by integrating over the Earth's surface. I keep reading that the formula should be:

E = e-max * cos @ * area

e-max is the solar insolation,
@ is the angle between the sun and the perpendicular to the land, @=0 at noon (at equator)

It's the cosine that I am having trouble with. What is wrong about the following argument against a factor of cosine for the received energy?

Cosine changes fastest at cos @ = 0, and slowest at cos @ = 1. That means that at sunrise(SR) and sunset(SS), the amount of sun that the land gets is changing most rapidly, and at noon(N) least rapidly. When I picture a mental image of the Earth rotating about its axis, it seems to me that the change in amount of sun at SR, SS, and N should be the same, should all be at a minimum, and at +/- 45 deg should be at a maximum.

That would correspond to cos^2, not cos.

It also seems that the factor at +/- 45 should be half of the maximum, by symmetry. A factor of cosine means half of e-max occurs at @=60, which seems anti-intuitive.

 

[Matterwave]

I'm not sure your argument; however, the cosine term there isn't necessarily for rotation. The cosine term is just there because that's how fluxes work. If you want to compute the flux of a vector field through an area, you need to take the dot product of the vector with the area vector (defined to be the vector with length equal to the area and perpendicular to the area).

The definition of dot products means that there is a cosine term (and not a cosine squared term).

 

[ecklstn36]

The rotation is irrelevant to the argument. I might as well imagine the Earth to be standing still in this problem.

If you didn't know that calculating fluxes required a cosine term, if you didn't know this was a flux in the first place, how would you argue for a cosine term when trying to calculate the incoming energy for a piece of the Earth's surface? ie if you said

E = e-max * some factor * area

e-max is the solar insolation

how would you argue that "some factor" should be cosine?

Thanks again.