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Homework answers / question archive / In the absence of any internal heat sources, the temperature of a solid planet with no atmosphere adopts a value such that it is in radiative balance

In the absence of any internal heat sources, the temperature of a solid planet with no atmosphere adopts a value such that it is in radiative balance


In the absence of any internal heat sources, the temperature of a

solid planet with no atmosphere adopts a value such that it is in radiative balance. That is, energy comes in via visible light and this energy is absorbed by the surface atoms that are made to vibrate faster and hence re-radiate this energy back into space as infrared radiation. Hence, we can treat the Earth as a perfect black body, which means that all the absorbed energy (from visible electromagnetic radiation) is eventually re- emitted, but at a different wavelength. We also assume (although with some reservation) that the energy absorbed by the Earth is quickly distributed around the surface of the Earth because of the Earth's rapid rotation.
6. Now we take the energy absorbed from the Sun (EA), and use it as the energy radiated by the whole Earth, and turn the Stefan-Boltzmann equation (from I, part 2) around to calculate the black body temperature for the Earth (TE) ignoring the effects of the earth's atmosphere. That is, let EA = sAETE4 where AE is the surface area of the Earth, and TE is the black body temperature of the surface of an atmosphere-less Earth and solve for TE. Subtract 273 from your answer to change it to degrees Celsius (°C).  Now convert this answer to degrees Fahrenheit (°F). (Recall that °F = 1.8 (°C) + 32).

7.  Water freezes at 0 °C or 32 ° F and boils at 100 °C or 212 °F.  Could liquid water exist on the Earth if your calculated TE is correct?  Since you know that liquid water exists on the surface of the Earth, do you think that the average temperature of the Earth is colder or warmer than that which you just calculated?

IV.  The Role of the Atmosphere: the Greenhouse Effect
The primary reason your derived temperature above is off is because the atmosphere's effect on planetary surface temperatures has been ignored! Water vapor (H2O), carbon dioxide (CO2) and methane (CH4) in the atmosphere trapsthe infrared radiation emitted from the surface and thus prevents some the heat from being radiated back into cold space. Water vapor in the atmosphere keeps the infrared radiation from being lost into space.  This is the greenhouse effect. Like the glass of a greenhouse, the atmosphere lets visible light in but keeps the longer wavelength infrared radiation from escaping back into space, thus warming the Earth.  There is, however, a small Catch-22:  clouds in the atmosphere can make the days cooler by reflecting light back into space, but CO2 is transparent to visible light and opaque to infrared radiation, hence it plays the role of the greenhouse 'glass.'  Methane (CH4) is also a very efficient greenhouse gas.

8.  The fraction of the infrared energy emitted by a real body compared to that emitted by a black body of the same temperature is called the emittance (e, epsilon).  We can find the Earth's average emittance by using the Earth's real temperature and modifying the Stefan- Boltzmann equation to find the Earth's emittance:  e = (EA)/(4πRE2s Treal4). Calculate e for Earth from this formula, given that the actual average temperature at the surface of Earth is Treal = 12 °C (Recall to use the Stefan Boltzmann equation, you must employ the absolute Kelvin temperature scale).

9.  The value of e for the Earth, as discussed above, is largely a consequence of the amount of CO2 in the atmosphere.  If an increase in the CO2 of the atmosphere decreased e by 1.5%, it should make a difference of about 1 °C in the Earth's average temperature. Consider what this does to the solid fraction of H2O (i.e., ICE) on the surface of the Earth. How does the amount of ice that covers the surface of the Earth affect the albedo (reflectivity) of the Earth? Discuss some ways to prevent the CO2 concentration in the atmosphere from increasing. 

V.  Comparing Planetary Atmospheres:  Venus and Mars, Greenhouse Extremes

For a planet that is slowly rotating or one for which the greenhouse effect is not important, and that has an emittance equal to its absorptance, we can calculate a theoretical subsolar (high noon) temperature (Tss).  Equating the luminosity of the solar black body to that of a black body sphere with a radius as big as the planet's orbit and performing a little algebra:  T2ss = (RSun/dplanet) x TSun2.  Because Venus rotates slowly this is the temperature that the surface directly underneath the Sun should reach, in the absence of an atmosphere.
10.  Taking dVenus = 1.08 x 1011 m, RSun = 6.96 x 108 m, and TSun = 5800 K, calculate Tss, Venus.

11.  How does this compare with the 700 K temperatures measured by Venus probes on the surface of Venus? Based on the discrepancy between Tss,venus and the actual value, what might you conclude about the presence of greenhouse gasses on Venus?

12.  Mars has a thin atmosphere that is mostly CO2. Use the formula above to calculate Tss,Mars with dMars = 2.28 x 1011 m.  

13.  How does your calculated temperature compare with the observed subsolar temperature of Martian soil of 300 K?  Comment on the apparent importance of the greenhouse effect on Mars today.

14. Presently liquid water is NOT stable on the Martian surface. The pressure is so low (about 1/100th of the Earth's surface pressure) that liquid water is not stable. Instead H2O (ice) goes directly to the gaseous state, H2O (gas). This process is called sublimation. In fact, on Earth frozen (icy) CO2 does the same thing: solid CO2 ice sublimes directly to CO2 gas, something you may be already familiar with (if you have ever been to a rock concert!). Now the question: If liquid water is not stable TODAY on the surface of Mars how can one explain the Rover data returned in mid 2004 consistent with the existence of giant lakes on Mars in the past? The evidence for these lakes is that certain rocks called evaporites are present as mineral precipitates in cracks and vugs of Martian volcanic rocks. The idea is that liquid water at some point flowed through these volcanic rocks and when the water evaporated, mineral deposits were left behind. A typical mineral of this type is Epsom salt, MgSO4. It is found on earth in regions where saline lakes develop. What do the findings of evaporites in Martian rocks indicate about the Martian atmosphere in an earlier epoch? What can you say about the ancient Martian atmosphere in light of these recent findings? How does the 'greenhouse effect' tie into these ideas?
[Hint: consult your textbook]

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