304 ATTENUATION OF RADIO WAVES 



7.13. Thermal Noise Emitted by the Atmosphere 



General laws of thermodynamics relate the absorption characteristics 

 of a medium to those of emission. Good absorbers of radiation are also 

 good emitters, and vice versa. Thus, in the microwave region, the atmos- 

 phere is also a good emitter, as well as a strong absorber, of radiation. 

 We may, therefore, describe quantitatively both emission and absorp- 

 tion by the same parameter; namely, the absorption coefficient. 



The emission characteristics of any real body at a fixed frequency may 

 be compared to those of a blackbody at the same temperature. In the 

 microwave region, the noise intensity emitted by a blackbody is given by 

 the Rayleigh -Jeans law: 



i(.) = 2kT[-J (7.8) 



where \l/{v) = emitted blackbody flux density per unit frequency 

 V = frequency 

 T = absolute temperature, °K 

 c = the velocity of light, and 

 k = Boltzmann constant (1.38054 X 10-i« erg/°K) 



The emission per unit length along an actual ray path may now be ex- 

 pressed as 



Biu) = yiu) 4^ip) (7.9) 



where 7(1') = attenuation per unit length. Remembering that the fraction 

 of energy absorbed in a path length, ds, is given by the optical depth, 

 dr = y{v)ds, we may obtain the differential equation for transmission of 

 radiation through the atmosphere: 



^ = -/(.) + V(«^) (7.10) 



where liu) is the flux density per unit frequency. 

 The solution to this radiative transfer equation is 



liv) = i:lm{v) exp (- / "W) 



+ / 4^{v) exp {- \ dr) dr (7.11) 



