Computing the Solar Constant

Problem

What is the mean solar electromagnetic radiation (solar irradiance) per unit area of the Earth?

Solution

Identify: We need to compute the irradiance of the Sun at the distance the Earth is from the Sun.

Setup and Execute: We need to use the Stefan–Boltzmann constant:

(defun stefan-boltzman ()
  (let ((kb 1.380649e-23)
        (hbar (* 0.5 (/ 6.62607015e-34 pi)))
        (c 299792458.0))
    (/ (* pi pi (expt kb 4.0))
       (* 60.0 (expt (* c hbar) 2.0) hbar))))
(stefan-boltzman)

5.6703744191844294e-08

The luminosity of the Sun would be, theoretically:

(defun luminosity (r temperature)
  (* 4 pi
     (expt r 2.0)
     (stefan-boltzman)
     (expt temperature 4.0)))

(let ((sun-temperature     5772.0) ;; K
      (sun-radius        695700.0) ;; km
      (earth-distance 149596000.0) ;; km
      (km                  1000.0)
      (earth-radius        6378.1366)) ;; km
  (luminosity (* sun-radius km) sun-temperature))

3.827990903277097e+26

The irradiance on a point on the Earth would be

(defun irradiance (lumos distance)
  (* (/ lumos pi) (expt (* 2 distance) -2)))

(let ((sun-temperature     5772.0) ;; K
      (sun-radius        695700.0) ;; km
      (earth-distance 149596000.0) ;; km
      (earth-radius        6378.1366) ;; km
      (km                  1000.0)) ;; m/km
  (irradiance (luminosity (* sun-radius km) sun-temperature)
              (* km earth-distance)))

1361.197273723066

Compare this to the empirical value of the Solar constant which is about 1.361 kilowatts per square meter.