ATSC 5002 Atmospheric Radiation Homework
Solid angle, Spectroscopy, Stefan-Boltzmann law, Planetary equilibrium temperature
1. Show that the solid angle under which the earth is seen by a satellite is given by 2π[1- √(2Reh
+ h2) / (Re + h)], where Re=radius of earth, h=height of satellite above the earth’s surface.
2. Using Bohr's theory and the fundamental constants calculate the binding energy of the
hydrogen atom, and the electron radius, for the ground state. What wavelength and frequency
of photon is required to ionize hydrogen?
3. Calculate the wavelength of the following emission lines from hydrogen: Lyman-, Lyman, Balmer-, Balmer-, Paschen-, Paschen-.
4. a) From Planck's theory derive the Stefan-Boltzmann law.
b) Calculate the value of the Stefan-Boltzmann constant, , based on your derivation.
5. Use Stefan-Boltzmann law to derive the mean radiative equilibrium temperature for a planet.
Use this formula (set up in IDL) to calculate (using Solar_temp=5800 K, Solar radius=7 x 105
km) the mean radiative equilibrium temperature for:
a) Earth for albedo = 0 and albedo =0.31.
b) Venus, albedo=0.78
c) Mars. albedo=0.17
d) Compare these temperatures with the actual mean temperatures of the Earth, Venus, and
Mars. Discuss any differences.
6. The height of geostationary orbiting satellites is about 35,000 km. Calculate the temperature
of such a satellite in the earth-satellite system. Use 288 K for the earth’s temperature and
assume the satellite is a black body.
7. a) Show that δTe/Te, the variance of the Earth’s equilibrium temperature (Te) varies as δr/(2r),
the variance of the Earth Sun distance (r).
b) In today’s elliptical Earth orbit, r varies by 3.3% from perehelion (January 3) to aprehelion
(July 5). What would be the change in Te for this orbital change?
c) What is the change in the amount or radiation received by the earth?
8. Given the solar constant of 1366 W m-2 and an earth sun distance of 1.5 x 108 km, calculate the
equilibrium temperature of the sun.
9. Of the total radiation emitted by the Sun (T=5800 K) and the Earth (T=288 K) calculate the
fraction of solar radiation which is emitted at λ > 4 µm and the fraction of Earth radiation
emitted at λ < 4 µm.