How Hot the Sun?

The total radiant energy flux at the Earth from the Sun, integrated over all wavelengths, is observed to be approximately 0.14 × 10⁷ erg cm^-2 s^-1. The distance from the Earth to the Sun, d, is 1.5 × 10¹³ cm and the solar radius, R_S, is 7 × 10¹⁰ cm. Treating the Sun as a “blackbody,” make a crude estimate of the surface temperature of the Sun (see Figure 1.1). To make the numerical estimate, you are encouraged to ignore all factors of 2’s and π's to express any integrals that you might have in dimensionless form, and to take all dimensionless quantities to be unity.

Figure 1.1

SOLUTION

The distribution of photons over the quantum states k with energy ε_k = hω_k is given by Planck’s distribution n_k (the Bose–Einstein distribution with chemical potential μ = 0).

(1)

where τ is the temperature of the radiation which we consider equal to the temperature of the surface of the Sun. To find the total energy, we can replace the sum over modes by an integral over frequencies:

(2)

where the factor 2 accounts for the two transverse photon polarizations. The energy of radiation dε_ω in an interval dω and unit volume is therefore

(3)

The total radiation energy density u = ε/V over all frequencies is

(4)

The integral with factor 1/π² is just a number which we can take ~ 1 (in fact it is π²/15). So

(5)

The energy flux J_T per a unit solid angle is

(6)

The flux that illuminates the Earth is proportional to the solid angle ∆Ω subtended by the Sun’s surface at the Earth:

 (7)

The radiant energy flux at the Earth is therefore

(8)

where T_S is the temperature of the Sun’s surface in K. Now we may estimate T_S:

(9)

(The actual temperature is about 6000 K.)