Reflected solar radiance seen by satellite
We derive the integral expression for the reflected solar flux F (W/m 2 ) seen by a sensor deployed on a satellite circling the earth at elevation h t . The sensor’s responsivity is that of a Si photodiode, (0.3, 1.1) µm. We first ignore atmospheric absorption, but account for light emanating from all angles, zenith up to parallel to earth surface. The expression, which holds for all “Solar–Earth–Sensor” (elevation) angles φ e , corrects a long-used equation in the USNDS program, that models “background irradiance,” Ramsey [3], Eq.(2.2). Flux striking the sensor is proportional to the sea-level solar radiance I 0 and the average earth albedo A. Results are displayed for h t = 2.02 · 10 4 km. The integral varies with angle φ e , i.e., F = F(φ e ). When sensor and sun are aligned with earth center, F = F(0) = 0.620 I 0 AR 2 , where R = r e /(r e + h t ) and r e is the average earth radius. For the specified h t , |φ e | < 180 - θ 0 deg, where θ 0 = 13.90. For larger angles, F = 0 since the sensor cannot see any part of earth illuminated by the sun. The flux F(φ e ) decays as ϵ 3.5 when φ e = 180-θ 0 - ϵ and ϵ→ 0, i.e., as the sensor enters darkness. Our ϵ 3.5 result differs from Ramsey’s [3], Eq.(2.2), which has ϵ 4.3 . We conclude Eq.(2.2) is incorrect for two reasons: the erroneous decay and, more importantly, that it vanishes only at φ e = 180. Our result, Eq.(14), presents a simple approximation to F(φ e ), accurate to better than 0.5%. The approximation blends Ramsey’s expression for small φ e with an ϵ 3.5 decay for large angles. Assuming a solar radiance I 0 = 1120, W/m 2 Wikipedia [7], and albedo A = 0.3, NASA, yields F(0) = 3.8 W/m 2 , which is 80% less than the leading term in Ramsey [3], Eq.(2.2). If atmospheric absorption is not ignored our integrand is modified by the factor exp(-τ N atm ), where τ is the atmosphere’s optical depth and N atm accounts for slant path absorption. The result is spectrally dependent. Absorption significantly decreases the flux as φ e → 180 - θ 0 .