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Radiative Flux

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            Radiative Flux



Useful Equations


Solution (A) The Solar Constant


Solution (B) Calculating the Flux Density of an Object Observed Through a Filter

To find the Johnson B-band flux of a star, in this case, the Sun, use the following formula. The value of the fν zero-point for the B filter is given in Table 2 of Section 2.3 in the text. Convert this to SI units: 4063*10^(-26) W m-2 Hz-1. The absolute magnitude of the Sun in the B filter is -26.10. Insert the relevant values and solve for fB of the Sun. You will need to use the text found on to complete this problem.




solve(-26.10 = -2.5*log[10](f[B]/(4063*10^(-26))), f[B])




The Sun's B-filter flux is approximately 1.12 * 10-12 W m-2 Hz-1 (= 1.12 * 10-12 N)


To calculate the total energy received by the Earth from the Sun in this band, multiply this figure times the central frequency of the B-band (λ = 0.438 * 10-6 m, from Table 2) and then by half the area of the Earth (the hemisphere facing the Sun) (radius = 6.371 * 106 m.)


nu = c/lambda


nu = 2.9979*10^8*Unit('m')/(Unit('s')*(.438*10^(-6)*Unit('m')))

nu = 0.6844520548e15/Units:-Unit('s')


or about 6.8 * 1014 Hz. So the luminosity at Earth is





The total luminosity of the Sun in the B band is taken at 1 AU (= 1.496 * 1011 m).


L[B, sun] = evalf(2*(6.8*10^14/Unit('s')*1.12)*10^(-12)*Unit('W')*Unit('s')*Pi*(1.496*10^11*Unit('m'))^2/Unit('m')^2)

L[B, sun] = 0.1070951962e27*Units:-Unit('W')


or approximately 1.07 * 1026 W.


The total luminosity of the Sun at all wavelengths (bolometric luminosity) is taken to be approximately 3.862 * 1026 W., as given above.



Solution (C) Luminosity, Flux, and Temperature

Luminous flux is related to temperature by the formula:


F := sigma*T^4


where σ is the Stefan-Boltzmann constant (5.67 * 10-8 W m-2 K-4). Therefore, luminosity may be expressed as:


L := 4*Pi*r^2*sigma*T^4


Knowing the luminosity and radius of the Sun, we can calculate its effective surface temperature, as follows:




evalf(Constant(sigma, units))



L := 3.862*10^26*Unit('W')



r := 6.95508*10^8*Unit('m')




solve(L = 4*Pi*r^2*evalf(Constant(sigma, units))*T^4, T)

5785.569728*(Units:-Unit('W')*Units:-Unit('m')^2*Units:-Unit(('kg')/(('s')^3*('K')^4))^3)^(1/4)/(Units:-Unit('m')*Units:-Unit(('kg')/(('s')^3*('K')^4))), (5785.569728*I)*(Units:-Unit('W')*Units:-Unit('m')^2*Units:-Unit(('kg')/(('s')^3*('K')^4))^3)^(1/4)/(Units:-Unit('m')*Units:-Unit(('kg')/(('s')^3*('K')^4))), -5785.569728*(Units:-Unit('W')*Units:-Unit('m')^2*Units:-Unit(('kg')/(('s')^3*('K')^4))^3)^(1/4)/(Units:-Unit('m')*Units:-Unit(('kg')/(('s')^3*('K')^4))), -(5785.569728*I)*(Units:-Unit('W')*Units:-Unit('m')^2*Units:-Unit(('kg')/(('s')^3*('K')^4))^3)^(1/4)/(Units:-Unit('m')*Units:-Unit(('kg')/(('s')^3*('K')^4)))



We take the first solution, rejecting the negative and imaginary solutions:














Cox, A. (Ed.) (2000). Allen's Astrophysical Quantities, 4th ed. New York: AIP/Springer.