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1) The sun’s radius is approximately 696,000 km, and its surface temperature is approximately
5770 K. Assuming that the sun is a blackbody, how much mass does it lose every second?
Hint: you will need a fundamental relationship derived by Einstein.
If the sun is a blackbody, then its emissivity
ε
= 1.0. The Stefan-Boltzmann law
gives the energy emitted by a body as J =
εσ
T4 W m-2, where the constant
σ
= 5.67
x 10-8 W m-2 K-4. At a temperature of 5770 K, 6.3 x 107 J are emitted every
second from 1 m2.
If the sun is a sphere with radius 696,000 km, then its surface area is 4 π
(696,000 km)2 = 6 x 1018 m2. Multiplying this by the energy per m2, the sun emits
3.8 x 1026 J every second.
Einstein gives us e = mc2, where c is the speed of light, approximately 3 x 108 m/s.
The mass lost by the sun must therefore be e/c2. So 3.8 x 1026 J / (3 x 108 m/s)2
= 4.26 x 109 kg lost every second – about 4.3 million tons.
Incidentally, if you have trouble remembering the base units for energy, you can easily get it
from Einstein’s equation. mc2 has units of M L2 T-2., so 1 Joule = 1 kg m2/s2.
2) The earth’s mean temperature is approximately 288 K. Suppose that climate change raises
that to 290 K:
a) What is the percentage increase in the energy emitted by the earth? (Assume a constant
emissivity
ε
= 0.95)
The emissivity doesn’t affect this, since it appears in both numerator and
denominator. The percentage increase is 100% * (2904 – 2884)/2884 = 2.8%
b) What are the wavelengths at which maximum energy is emitted by (i) the 288 K earth,
and (ii) the 290 K earth?
According to Wien’s law, the wavelength at which the most energy is emitted, λ
λλ
λm, is
given by λ
λλ
λm = 2900 µm K / T. So the 288 K earth has λ
λλ
λm = 10.07 µm, and the
warmer earth has λ
λλ
λm = 10.00 µm, a decrease of 100% * (10.07 – 10.00) / 10.00 =
0.7%.
c) What are the implications of this change in wavelength, given the absorption spectrum of
the earth’s atmosphere (see, e.g.,
http://en.wikipedia.org/wiki/File:Atmospheric_Transmission.png)?
The earth radiates heat into space, and the wavelength at which the most energy
is emitted, λ
λλ
λm, happens to be a wavelength at which the atmosphere transmits
readily. If the earth warmed so much that lm shifted to a wavelength at which the
atmosphere did not transmit well – instead it reflected, or absorbed and re-
emitted the energy – then the earth would not shed its energy readily, increasing
the warming. But the shift in lm that would result from a 2 K increase in
temperature should have little to no effect.
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