Answer to Question #169018 in Astronomy | Astrophysics for n

Question

Sunspots are cooler spots on the Sun’s surface that have a temperature of about 3000-4500 K compared with the average surface temperature of 5800 K. Use this information to explain why calculations involving the inverse square law for the Sun may be inaccurate.

The star Rigel is found 773 light years away in the constellation of Orion. It has a surface temperature of 11000 K and an intensity of 3.7 x 10-8 Wm-2.

i) Calculate the wavelength of the peak in the black body radiation curve of Rigel (2)

ii) Sketch a black body radiation curve for Rigel, ensuring that you label peak wavelength.

iii) Sketch a black body radiation curve for Rigel, ensuring that you label peak wavelength.

Answer 1

In the first approximation, we will need to calculate separately the luminosity of the spots and the luminosity of the surface without spots. Since the luminosity of the surface element depends on the fourth degree of temperature, the luminosity of different parts of the solar surface will differ significantly (by $(4500/5800)^4 \approx 0.36$ times and less). It is also necessary to take into account that the spots are not visible from all points, so the illumination at the same distance in different directions will be slightly different.

i) According to the Wien's law, the peak corresponds to $\lambda_{\max} \approx \dfrac{0.0029}{T} = 2.64\cdot10^{-7}\,\mathrm{m} = 264\,\mathrm{nm}.$

ii), iii) Let us draw the Planck curve for the star as a blackbody with temperature T=11000 K.

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

Here we use units of $\mathrm{W}\cdot \mathrm{m}^{-2}\mathrm{\cdot m^{-1}\cdot sr^{-1}}$ , abscissa is the wavelength in meters.