Determine the maximum of the Planck distribution (for the three dimensional case) as a function of the frequency and the wavelength. Show that this is possible if we maximize the function x"/(e – 1) for a = 3 and a 5 respectively. This means solving the equation x = a(1 - e-*), which can be done in an iterative way Xp = a(1 -en-1), starting from x1 = 1 (stop after 5 iterations). Verify Wien law, Amar T =const., and comment on the fact that we find two different constants in the two approaches. We know that the sun produces the largest amount of radiation around the wave- length 5 x 10-5 cm. Using the results previously obtained, determine: • the temperature of the sun; • the amount of energy produced, knowing that the main mechanism of produc- tion of such energy is the transformation of hydrogen into helium, and that this reaction stops when 10% of the hydrogen has been converted. A good approxi- mation is to take the whole mass of the hydrogen equal to the mass of the sun (use Einstein relation E = AMc²); • the lifetime of the sun. You can use the following numerical constants: h= 6.625 x 10-27 erg s; c=3x 1010 cm s-1; og = 5.67 x 10-5 erg cm-2 s- K-4; Run = 7 x 1010 cm; Mun = 2 x 1033 g; k = 1.38 x 10-16 erg K-!. %3D

Determine the maximum of the Planck distribution (for the three dimensional case) as a function of the frequency and the wavelength. Show that this is possible if we maximize the function x"/(e – 1) for a = 3 and a 5 respectively. This means solving the equation x = a(1 - e-*), which can be done in an iterative way Xp = a(1 -en-1), starting from x1 = 1 (stop after 5 iterations). Verify Wien law, Amar T =const., and comment on the fact that we find two different constants in the two approaches. We know that the sun produces the largest amount of radiation around the wave- length 5 x 10-5 cm. Using the results previously obtained, determine: • the temperature of the sun; • the amount of energy produced, knowing that the main mechanism of produc- tion of such energy is the transformation of hydrogen into helium, and that this reaction stops when 10% of the hydrogen has been converted. A good approxi- mation is to take the whole mass of the hydrogen equal to the mass of the sun (use Einstein relation E = AMc²); • the lifetime of the sun. You can use the following numerical constants: h= 6.625 x 10-27 erg s; c=3x 1010 cm s-1; og = 5.67 x 10-5 erg cm-2 s- K-4; Run = 7 x 1010 cm; Mun = 2 x 1033 g; k = 1.38 x 10-16 erg K-!. %3D