Respuesta :
Hotter Sun: Suppose the surface temperature of the Sun were about 12,000K, rather than 6000K.
a. How much more thermal radiation would the Sun emit?
b. What would happen to the Sun’s wavelength of peak emission?
c. Do you think it would still be possible to have life on Earth? Explain.
Answer:
a) To calculate the emitted power per square meter we need to use Stefan-Boltzmann’s Law,
that is,
𝸠= 𝜍𝑇
4
đť‘Šđť‘•đť‘’đť‘źđť‘’,
𝸠= đť¸đť‘šđť‘–𝑡𝑡𝑒𝑑 đť‘𝑜𝑤𝑒𝑟 đť‘ťđť‘’đť‘ź 𝑠𝑞𝑎𝑢𝑟𝑒 𝑚𝑒𝑡𝑒𝑟 đť‘śđť‘“ 𝑠𝑢𝑟𝑓𝑎đť‘đť‘’
𝑇 = 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 đť‘–đť‘› đťľđť‘’𝑙𝑣𝑖𝑛 𝑎𝑛𝑑
𝜍 = 𝑆𝑡𝑒𝑓𝑎𝑛 â’ đťµđť‘śđť‘™đť‘ˇđť‘§đť‘šđť‘Žđť‘›đť‘› đť¶đť‘śđť‘›đť‘ 𝑡𝑎𝑛𝑡 = 5.7 Ă— 10â’8
𝑤𝑎𝑡𝑡
đť‘š2 Ă— đťľ4
At 6000K,
Emitted Power = 𝜍 × (6,000)
4
At 12000K,
Emitted Power = 𝜍 × (12,000)
4
Now calculate the ratio of thermal radiation at the two temperatures to find out how much more
thermal radiation would the Sun emit,
đť¸12000đťľ
𝸠6000đťľ
=
𝜍 × (12,000)
4
𝜍 × (6,000)
4 =
12,000
4
6,000
4 =
12,000
6,000
4
= 2
4 = 16
â´
đť¸12000đťľ
𝸠6000đťľ
= 16
Therefore, emitted power at 12000K is 16 times the emitted power at 60000K.
b) To calculate the wavelength of maximum intensity we need to use Wien’s Law, that is,
𝜆𝑚𝑎𝑥 =
đť‘Ź
𝑇
đť‘Šđť‘•đť‘’đť‘źđť‘’,
𝜆𝑚𝑎𝑥 = 𝑤𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡𝑕 𝑜𝑓 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦
đť‘Ź = đť‘Šđť‘–đť‘’đť‘›
′
đť‘ đť‘𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 2,900,000 𝑛𝑚.đťľ
𝑇 = 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 đť‘–đť‘› đťľđť‘’𝑙𝑣𝑖𝑛
𝜆𝑚𝑎𝑥 =
2,900,000
𝑇
=
2,900,000
6,000 𝑜𝑟 483.33 𝑛𝑚
𝜆𝑚𝑎𝑥 =
2,900,000
𝑇
=
2,900,000
12,000 𝑜𝑟 241.67 𝑛𝑚
𝑅𝑎𝑡𝑖𝑜 =
𝜆max 𝑎𝑡 12,000đťľ
𝜆max 𝑎𝑡 6,000đťľ
=
2,900,000
12,000
Ă—
6,000
2,900,000
â´
𝜆max 𝑎𝑡 12,000đťľ
𝜆max 𝑎𝑡 6,000đťľ
=
1
2
Therefore, wavelength of maximum intensity at 12,000K is half the wavelength of maximum
intensity at 6,000K.
c) We know that the intensity of light emitted by a given object depends on its wavelength or
frequency. Here, the wavelength at 12,000K is reduced by half which means its frequency
has increased by half. In the EM spectrum, UV rays have shorter wavelengths. Therefore, at
this temperature, the Sun would emit copious ultraviolet light which cannot be suitable for
supporting life.
Chapter 5: #52
Doppler Calculations II. In hydrogen, the transition from level 2 to level 1 has a rest wavelength of
121.6 nm. Suppose you see this line at wavelength ay 121.9 nm in Star C and at 122.9 nm in Star D.
Calculate each star’s speed, and be sure to state whether it is moving toward or away from us.
