Planetary Temperatures Radiation, Albedo and the Greenhouse Effect

In this experiment you will explore the effects on the temperature of a planet of radiation from the Sun, the albedo or reflectivity of a planet’s surface and the greenhouse effect. You will calculate the effects that each of these processes have on a planet’s temperature then compare your calculations to actual measured temperatures of the planets.

Radiation The main source of energy in the solar system is radiation from the Sun. This is the main factor that affects the temperature of a planet. The Stefan- Boltzmann Radiation Law, states that the rate at which an object emits radiation through its surface area is proportional its temperature to the fourth power, J~ T4. As the Sun emits radiation out into the solar system, a planet will absorb just a small fraction of the radiation that has traveled out to its distance that depends on the planet’s size. An expression that can be derived for the temperature of a planet a distance, d, from the Sun based on the Stefan-Boltzmann Law, the Sun’s temperature of 5800 K, the Sun’s Radius of 6.95 x 108 m is

Tr=279/(d)^0.5 (1)

Use the orbital radius (a planet’s average distance from the Sun) of each planet as, d, found in TABLE 1, to calculate a theoretical temperature for each planet based on the radiation it absorbs from the Sun and record them in the column labeled Theoretical Temperature in TABLE 1 below. Raising a number to the 0.5 (one-half) power is the same as taking the square root, so the calculation amounts to 279 divided by the square root of the planet’s orbital radius. You may round off any decimals in your calculated temperatures to the nearest whole number.

Percentage Error Your temperatures are theoretical and must now be compared to measured or observed experimental temperatures for each planet found in the column labeled Observed Temperature in TABLE 1. Now, for each planet, calculate the percentage error that your theoretical temperature, T, is off from the observed temperature, To, with the formula;

([T-To] / To ) x 100% (2)

This is simply the difference between your theoretical and observed temperatures divided by the actual observed temperature. Note-Whether the number in the parenthesis [T-To] comes out positive or negative, just use it as positive in your calculation. We are mostly concerned with how far off the theoretical temperatures are from the observed temperatures, not whether they are too high or two low. Enter your % errors in the last column of TABLE 1.

TABLE 1 Planetary Temperatures from Radiation

Theoretical (calculated) Temperature Observed Orbital (from (measured) Radius r=radiation) Temperature Object Tr=279/(d)^0.5 To % error d (AU) Tr (K) To (K) % Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

Now examine the % errors you calculated. Any theoretical temperature that is <5% away from the experimental temperature can be considered a satisfactory value. This means that the planet’s temperature can be considered mostly due to radiation from the Sun. If the error is more than 5%, then we must consider other factors that affect a planet’s temperature. Circle any errors that you find that are greater than 5%. Albedo The albedo, a, of a planet is the percentage of the radiation incident upon the planet that the planet reflects back out into space rather than absorbs. Earth’s albedo is a=0.31, meaning that it reflects away 31% of the radiation that comes to it from the Sun. The albedo for each planet is found in the column labeled albedo in TABLE 2 below. Now, ONLY FOR PLANETS THAT HAD > 5% ERROR WHEN THEIR TEMPERATURE WAS CALCULATED BY RADIATION, use the following expression to calculate a new theoretical temperature for a planet;

Ta=Tr*(1-a)^0.25 (3)

First subtract the albedo, a, from 1 then raise that number to the 0.25 (one- fourth) power. This is the same as taking the square root twice. Then multiply your result times the theoretical temperatures you already have for radiation alone, Tr from TABLE 1. This will be theoretical temperature of a planet based on the effects of radiation and albedo. Enter your new Theoretical Temperatures (from albedo and radiation) in their column in TABLE 2. Again, do this ONLY for planets that had greater than 5% error in their temperature calculated from radiation only.

Now, for each planet that you calculated the effect of albedo on its temperature, use equation (2) to calculate the percentage error that your new theoretical temperature, Ta, in TABLE 2 is off from the original observed temperature in TABLE 1 and enter your new % errors in their column next to (to the right of) the theoretical temperatures from radiation and albedo in TABLE 2. Once again, any error 5% or less is acceptable, but any error much more than that must still be corrected for other factors.

TABLE 2 Planetary Temperatures from Radiation and Albedo Theoretical (calculated) Temperature (from a=albedo and r=radiation) Object Albedo Ta=Tr*(1-a)^0.25 % error A Ta (K) % Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

The Greenhouse Effect After a planet absorbs radiation from the Sun, it reemits it. Certain gases in a planet’s atmosphere, such as water vapor and carbon dioxide absorb some of this reemitted energy before the rest of it escapes back out into space. This is called the greenhouse effect. The greenhouse effect can be considered in our calculations by using the following formula

Tg=Ta*(1+)^0.25 (4)

The thickness of greenhouse gases, in equation (4), is a comparison between the amounts of greenhouse gases in a planet’s atmosphere to those on Earth (where  ). In Mars’ very thin atmosphere,  Venus’ very thick atmosphere  greenhouse effect in your calculation of the temperature of any planet that still had >5 % error in its theoretical temperature after considering albedo in TABLE 2. Enter your results in the column labeled Theoretical Temperatures (from albedo, radiation and greenhouse effect) in TABLE 3 below.

This time add the thickness, , to 1 then raise that number to the 0.25 (one- fourth) power by taking the square root twice. Then finally, multiply your result times the theoretical temperature you already have for radiation and albedo from TABLE 2. Also use equation (2) to calculate a new % error, how far Tg is off from the original To in TABLE 1, and enter it in the last column of TABLE 3.

TABLE 3 Planetary Temperatures from Radiation, Albedo and the Greenhouse Effect  Theoretical  (calculated)  Temperature  (from a=albedo, r=radiation and  g=greenhouse

effect ) Object Tg=Ta*(1+)^0.25 % error  % Venus 92 Earth 1 Mars 0.2

Questions

1-Which planetary temperatures could be satisfactorily calculated by considering radiation alone?

2-Which planetary temperatures were satisfactorily calculated by including the effect of albedo in addition to radiation?

How does albedo affect a planets’ temperature? To answer this, look at the change in the theoretical temperatures of the planets for which you included the effect of albedo.

Why does albedo have this effect on a planet’s temperature?

Why do you think Venus has such a high albedo? Why do you think Pluto does?

3-Which planetary temperatures were only calculated satisfactory by including the greenhouse effect in addition to radiation and albedo?

What affect does the greenhouse effect have on a planet’s temperature?

Why does the greenhouse effect affect a planet’s temperature this way?

How does the greenhouse effect on Earth compares to that on Mars? Why is this the case?

How does the greenhouse effect on Earth compare to that on Venus? Why is this the case?