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 Xn = a(1 -e n-1), starting from x1 =1 (stop after 5 iterations). Verify Wien law, Amax 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= 3 × 1010 cm s-l; og = 5.67 x 10-5 erg cm-2 s- K-4; Run = 7 x 1010 cm; Mun = 2x 1033 g; k = 1.38 x 10-16 erg K-.
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 Xn = a(1 -e n-1), starting from x1 =1 (stop after 5 iterations). Verify Wien law, Amax 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= 3 × 1010 cm s-l; og = 5.67 x 10-5 erg cm-2 s- K-4; Run = 7 x 1010 cm; Mun = 2x 1033 g; k = 1.38 x 10-16 erg K-.
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Transcribed Image Text: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-!.
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