Global Journal of Science Frontier Research, A: Physics and Space Science, Volume 23 Issue 1

where a is the Stefan-Boltzmann constant, i.e., 33 2 4 15 c k a  π = . (32) ( ) ∫ ∫ ∞ ∞ =         −         − = 0 2 0 1 2 1 3 2 2 1 1 2 2 2 2 1 1 θ ω θ ω ω ω ω π    e d e d c I ( ) '' 2 ' 2 3 2 2 2 II c π  , (33) ( ) ( ) ( ) ∫ ∫ ∞ ∞ = Γ = − =         − = 0 3 3 1 3 3 1 1 2 1 3 3 1 0 1 2 1 ' 2 1 1 1 1 1     θ ζ θ θ ω ω θ ω z z e dy y e d I y ∙2∙1.5498≈3.1 3 3 1  θ , (34) where ( ) z Γ is the gamma function [17] , ( ) ... 16 1 9 1 4 1 1 1 1 + +++= = ∑ ∞ = k z K z ζ ≈1,5498; ( ) ( ) ( ) ( ) .1 2 1 ; 1 = Γ= Γ Γ=+Γ z z ∫ ∞ =         − = 0 2 '' 2 1 2 2 θ ω ω  e d I ( ) ( ) [ ] ∞→ = ∞ +− +− = − ∫ y y y y e y e dy 0 2 0 2 2 1 ln 1 2   θ θ . (35) As seen, the integral (35) is integrated by quadratures but it divergences within the interval ∞− 0 , namely, ( ) [ ] ∞→ +− +− = ∞→ = y y y e y Tk I 0 '' 2 1 ln  , Tk y 2 ω  = , (36) however, we can overcome the difficulties that have arisen. Indeed, we obtain the infinity for zero in the bottom limit of the integral (35). But we can take a number ϑ in the bottom limit of (35) instead zero. The number has to be very close to zero at a given temperature taking into account of the energy, by which we neglect. It must be much less than the whole energy radiated by the black body in this direction, i.e., << ϑ ( ) [ ] ∞→ = +− +− y y y e y ϑ 1 ln . (37) Here we should note that a conscious inaccuracy was made in the classical method of obtaining Stefan- Boltzmann’s law. As known, according to this method, the integration takes place in the space of positive (quantum) integers. But they form a continuum for large values. If the integers are small, there is a discrete series, and we cannot formally integrate. If nevertheless we are doing that, we have: ∫ ∞ =         − = ϑ ω ω 1 '' 2 Tk e d I  ( ) ( ) [ ] ∞→ = ∞ +− +− = − ∫ y y y y e y Tk e dy Tk ϑ ϑ 1 ln 1   =  Tk Λ . (38) As a result, we have for 2 I : ( ) ( ) 2 3 1 33 2 2 3 1 '' 2 ' 2 3 2 2 238 .0 2 2 1.3 2 2 TaT c II c I Λ = = ≈   π θ θ π , (39) and 1 Year 2023 35 © 2023 Global Journals Global Journal of Science Frontier Research Volume XXIII Issue ersion I VI ( A ) The Heat Transfer by Radiation under Relativistic Conditions