Intensity of solar radiation at earth orbit

[tomelwood]

Homework Statement

I am try to understand how the value of 1370 watts per metre squared for the intensity of sunlight hitting the Earth has been derived. Is there a way to see this mathematically?

Homework Equations

I understand about how the intensity is related to Poyntings vector, by taking it's average over the wavelength (ie multiplying it by 0.5) but I don't see how this has anything to do with the radius of the orbit.

The Attempt at a Solution

The equation for I is 0.5*c*ε_0 * E^2_0 where E_0 is the amplitude of the Electric part of the EM wave, and ε_0 is the permitivity of free space 8.85*10^-12.
How do I calculate the E_0 value? And does that vary with radius? I can't imagine that it does, so I'm back to square 1! I'm trying to calculate the radiation pressure on a surface at differing distances from the sun, so knowing how and why the intensity changes would be greatly helpful
Any help would be greatly appreciated.

 

[D H]

I am try to understand how the value of 1370 watts per metre squared for the intensity of sunlight hitting the Earth has been derived. Is there a way to see this mathematically?

The solar constant is not derived. It is an observed quantity.

 

[tomelwood]

So rather than calculating the solar constant from the sun's power, rather the power of the sun is calculated from the observed solar constant. OK. I have since discovered that I can relate the power of the sun to the intensity of the sunlight passing through a sphere of radius r, simply by dividing it by the area of said sphere. I presume this is how the sun's power was calculated in the first place. If this is correct, then it should serve the purpose that I require. Hopefully!

In addition to this, I have read in a journal that the forces acting on a solar sail inclined at an angle θ to the normal are the force of gravity from the sun (Fq) (we're in free space, away from planets etc) and the force exerted by the sail perpendicular to it's surface (Fs)
This journal then constructs the equations f motion thus:
(-Fg + Fs cos θ)/m = du/dt - v^2/r ; where u is radial velocity and v is tangential velocity (making v^2/r angular velocity?) and r is distance from Sun, and m is total mass of spacecraft
and
(-Fs sin θ)/m = dv/dt + uv/r

I can't fathom how these have been arrived at. They appear to be resolving horizontally and vertically, but why is the radial component involved in the vertical resolution and vice versa?

Any hints/tips would be great, or links to other resources that explain the same thing.

Many thanks.