Download Astr 138 Assignment #8

Astr 138 Assignment #8 - due Fri Apr 15, 2005
Wien’s Law and the temperatures of planets
We discussed the blackbody radiation laws in class, using proportional relationships. Wien’s Law relates the
wavelength of the brightest emission, λpeak , to the object’s temperature, T ,
λpeak ∝
1
.
T
Typically, we measure temperature in kelvin and wavelength in meters. Zero degrees kelvin is -273◦ C, and
the size of one degree on the kelvin scale is equal to one degree centigrade. The yellow color in the spectrum
represents light with a wavelength of about 5 × 10−7 meters, or 5 ten-millionths of one meter.
To change the “proportional to” relation to an “equality” relation, we multiply by a constant of proportion,
λpeak =
1
× constant;
T
in general, the values of constants of proportion are determined by experiment.
If we compare two different objects, then we divide this equation by itself and thus remove the constant:
λpeak,1
T2
=
,
λpeak,2
T1
where λpeak,1 and T1 are the values for object #1, and where λpeak,2 and T2 are the values for object #2.
You can see that if object #2 has a temperature twice as great as object #1, then object #2 has a spectral
peak at half the wavelength of object #1.
We can easily measure the wavelength of the peak of emission for any object, by passing its light through
a spectroscope. This information has been placed in the table, below. Using the information in the table,
compare each planet with the Earth and thus derive its surface temperature. For the gas giant planets, you
will be deriving the temperature near the tops of their atmospheres. The values for the Earth will be λpeak,1
and T1 in the equation above.
Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Avg Distance
from Sun (AU)
0.39
0.72
1.0
1.5
5.2
9.5
19
30
40
Blackbody Peak
(meters)
6.41 × 10−6
3.97 × 10−6
1.02 × 10−5
1.35 × 10−5
2.42 × 10−5
3.29 × 10−5
4.91 × 10−5
6.04 × 10−5
7.83 × 10−5
T (avg at surface)
(kelvin)
285
Requirements
1. Answer all questions on a clean sheet of paper or graph paper where appropriate. Either type or neatly
print/write your answers. Complete your assignment in a professional manner.
2. Complete the table, above, by using Wien’s Law to find the temperatures of each of the planets listed.
You must show at least one example calculation, carrying units through all your equations.
3. Find, on the web or in a textbook, the wavelength ranges of the major sections of the spectrum of light
(γ-ray, X-ray, ultraviolet, visible, infrared, microwave, and radio) and STATE where in the spectrum
each of the planets has its peak emission. For example, Earth’s peak at 1.02 × 10−5 meters occurs in
the infrared region.
4. Make a graph of planetary temperature on the ordinate (the vertical or “y” axis), and the planet’s
distance from the Sun on the abscissa (the horizontal or “x” axis). Plot the data for all nine planets.
Plot the data as points, do not join them with lines. Describe how planetary temperatures change with
increasing distance from the Sun.
5. Venus has its temperature raised by greenhouse gases, whereas Mercury does not, and on Mars the
thin atmosphere makes the greenhouse effect negligible. Make a second graph of T versus distance
that includes only the inner four terrestrial planets. Connect the Mercury and Mars points with a line,
and use this line to determine what the temperature would be on Venus in vacuum (i.e., if it were to
lose its atmosphere). By how many degrees kelvin does the greenhouse effect raise the temperature on
Venus?
[NOTES: (1) We cannot connect Mercury and Mars with a straight line unless we assume that the Sun is
the dominant source of energy for heating a planetary surface. You should think about why this assumption
must be made, and how the data validates it. (2) From the graph (and ignoring Venus), it is obvious that
the trend of temperature with distance is not accurately represented by a straight line, but a curved one.
You should think about whether we are underestimating or overestimating the magnitude of the greenhouse
effect on Venus when using a straight line interpolation for its temperature in vacuum.]