Question

1. An infrared radiometer on a satellite measures radiation emitted from Earth's surface in the "atmospheric window" part of the spectrum (8.5 - 12.5 km). (a) Assuming no clouds between the satellite and surface, what is the equivalent blackbody temperature of the surface if the observed radiance at 10 km is 8.0 W m-2 pm- sr-1? (b) If, instead, there were clouds between the satellite and surface, provide a plau- sible range for the altitude of the tops of these clouds, and justify your guess.

Answer 1

By Planck's law, spectral radiance per unit wavelength λ is given by

${\displaystyle B_{\lambda }(\lambda ,T)={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}},}$

where k_B(Boltzmann constant) = 1.38×10⁻²³ J/K ,

h (Planck constant)= 6.626×10⁻³⁴ J.s

and c(speed of light in the medium) = 3 x 10 ⁸ m/s

S.I. is of B_λ are W·sr⁻¹·m⁻³

B_λ = 8 W·sr⁻¹·m⁻²$\mu$m^-1 = 8 x 10⁶  W·sr⁻¹·m⁻³

and λ = 10 $\mu$m = 10 x 10^-6 m = 10^-5 m

So , By above formula

| ره) ( G2 ( x64 یا , روم، 5 . 38 اع x 2ر x 3x1 رو3-8x10 3x . 3x16 ا گر 5-10) 1 وا x 9 . 264 ا ا ب ا ۲ 4 - م اہا. ۱۹۹۵ به . x 1 = 9 . 3 ( « « ] ac 2 ا . 10 85 ه. و 14 = 1 - وه . 150 = . و - و 156 . 3 ) 23 : e ,و ۷۰۰۰۰۰ - T ا وا ، ۱۹۹۵ ا واہ.284 = T ا : = F gn ( 5 .

Average Earth's temperature is about 287.  Air temperature drops 9.84 K per 1000 m of altitude. Hence we can say that Cloud height should be around 1 Km.