Astro 201 - Lab Manual Drafts

Alex Parker

October 26, 2009

1 Greenhouse effect

1.1 Indroduction

The temperature of the surfaces of planets plays an important part on whether or not they could be habitable. Forexample, most life as we know it requires liquid water to survive. So, if the surface of a planet is hotter than theboiling point of water, or colder than the freezing point, it would be uninhabitable to this kind of life. Astronomersrefer to the range of distances that a planet around a given star could have liquid water on its surface as the“Habitable Zone.” Planets’ atmospheres can have significant effects on surface temperatures by altering the wayincoming radiation (sunlight) is absorbed and reflected. For example, if an atmosphere freely allows visible light topass through it, but traps infrared light (thermal radiation), the atmosphere will have a warming effect. This effectis commonly referred to as the “greenhouse effect”, because it operates on the same principle that makes greenhouseshot.

This experiment compares the greenhouse effect by Earths atmosphere with a nearly pure CO2 atmosphere likeVenus’ (CO2 exhibits a strong greenhouse effect).

1.2 Blackbody Temperature

What governs the temperature of the surface of a planet?

• A planet orbiting a star will intercept and absorb some light and become warm.

• The warm planet must radiate this heat away or it will become hotter and hotter.

• Thus, the planet will need to radiate all the energy it receives to keep a stable temperature. Therefore:

ENERGY ABSORBED = ENERGY RADIATED or

Ein = Eout (1)

An object that maintains its temperature through this sort of energy balance is called a “Blackbody.” Therefore,the following derivation find the so-called “Blackbody Temperature” of the surface of a planet - the temperature itwould have if it behaved like a Blackbody (without any atmosphere).

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1.2.1 Derivation of Blackbody Temperature of a Planet

The ENERGY ABSORBED by the planet will depend on the distance between the planet and star D, the luminosityof the star L, and the radius of the planet r. It will also depend on how much of the sunlight is absorbed by the planet,which is determined by the planets reflectivity or albedo. We will express the fraction of light absorbed (which is definedby 1 - albedo) with the letter F, which will range from 0 (perfectly reflective, no sunlight absorbed) to 1 (perfect absorber,all sunlight absorbed).

Ein =FLπr2

4πD2(2)

The ENERGY RADIATED by the planet will depend on the temperature of the planet T and the radius of the planetr. There is an additional constant factor, called the Stefan-Boltzmann constant that is required to convert temperaturesto energy radiated. We will represent it with the letter B, where B = 5.6697× 10−8 Watts/meter2/Kelvin4

Eout = 4πBr2T 4 (3)

Combining Eqns. 1, 2, and 3, we find that FLπr2

4πD2 = 4πBr2T 4

Since r2 appears on both sides, we can cancel it. If we solve for T we get:

T =(

FL16πBD2

) 14

(4)

We can combine 16πB into one numerical term A to simplify the equation:A = 16πB = 2.85× 10−6 Watts/meter2/Kelvin4

Substituting A into Eqn. 4 we get the following simple equation for the Blackbody temperature of a planet:

TBB =(

FLAD2

) 14

(5)

Constants for Eqn. 5Term Meaning ValueA 16π×Stefan-Boltzmann constant 2.85× 10−6 Watts/meter2/Kelvin4

L Solar Luminosity 3.84× 1026 Watts

Planet-specific valuesTerm Meaning Venus Earth MarsF Fraction of light absorbed by planet 0.4 0.6 0.8D Planet-Sun distance 0.73 AU 1.00 AU 1.52 AU

1 AU = 1.50× 1011 meters

Table 1: Factors for Eqn. 5 for the planets Venus, Earth, and Mars. In order to use Eqn. 5, D must be expressedin meters, so the conversion from AU to meters is included. For using Figure 1, D is given in AU .

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1.2.2 Blackbody Temperature Exercises

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10

125

150

175

200

225

250

275

300

325

350

375

400

425

450

475

500

Distance from Sun (AU)

Tem

p (

Kelv

in)

0.2

0.4

0.6

0.8

1.0

F =

Figure 1: Blackbody temperatures vs. distance from the Sun, for different reflectivities. Based on Eqn. 5. Onlyvalid for Sun-like stars.

1. Use Figure 1 and the D and F values in Table 1 to determine the Blackbody temperature of Venus, Earth,and Mars.

2. Compare the results of Exercise 1 with the freezing point of water ( 0 degrees C or 273.15 Kelvin). Based ontheir Blackbody temperatures, could any of these planets could have liquid water on their surface?

3. Using the F = 0.6 line as the temperature of Earth if we imagine moving Earth to different distances from theSun, at what range of distances could the Earth have liquid water on its surface? In other words, at what innerD is the Earth’s Blackbody temperature equal to water’s boiling point (373.15 Kelvin), and at what outer Dis the Earth’s Blackbody temperature equal to the freezing point of water (273.15 Kelvin)?

4. Define the “Habitable Zone” around our Sun, using Figure 1. What is the minimum distance any planet (withany of the listed F values) can be without the surface being hotter than the boiling point of water (373.15Kelvin)? What is the maximum distance any planet can be without the surface being colder than the freezingpoint of water (273.15 Kelvin)?

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1.3 Greenhouse Effect

sunlight (visible radiation)

radiated heat from ground (thermal / infrared

radiation)

Absorbed energy from sunlight heats ground

Ground re-radiates absorbed energy

as infrared radiation

NO ATMOSPHERE(all thermal radiation escapes)

sunlight (visible radiation)

radiated heat from ground (thermal / infrared

radiation)

(atmosphere lets visible light through)

Absorbed energy from sunlight heats ground

Ground re-radiates absorbed energy

as infrared radiation

(atmosphere absorbs infrared, gets heated by re-radiated energy

from ground)

(atmosphere re-radiates absorbed energy, 1/2 up and 1/2 down.

upward-radiated heat from atmosphere

(thermal / infrared radiation)

downward-radiated heat from atmosphere(thermal / infrared

radiation)

Ground re-absorbs downward-radiated

heat from atmosphere

WITH ATMOSPHERE(some thermal radiation trapped)

Figure 2: Illustration of the principal of the greenhouse effect. On the left, incoming solar radiation is absorbed by theground and re-radiated to space. On the right, incoming solar radiation is absorbed by the ground and re-radiated,but then captured by the atmosphere. The atmosphere then re-radiates this energy, but both upward and back downtoward the ground, increasing the incoming radiation the ground sees over the case with no atmosphere.

From the results of the previous exercises, it should be clear that something is causing the Earth’s surface to bewarmer than it should be. This something is the Earth’s atmosphere!

Consider the addition of an atmosphere that blocks infrared (thermal) radiation but allows visible light through(Figure 2, right). Since incoming radiation from the Sun is mostly in the visible part of the spectrum, the originalEin is not decreased. However, the radiation re-radiated by the surface of the Earth is mostly in the infrared portionof the spectrum. So this radiation is absorbed by the atmosphere.

Now, the atmosphere has to obey Ein = Eout as well - but it can both radiate upward and downward. So half ofthe energy re-radiated by the atmosphere is radiated back at the surface of the planet.

Therefore, the surface not only sees the original Ein from the sun, it also sees 0.5× Eatm !Since the atmosphere sees the Eout from the surface of the planet, and it must re-radiate exactly this amount,

Eatm = Eout(surface), so:Ein(surface) = Eout(surface) = Esun + 0.5× Eatm

= Esun + 0.5× Eout(surface)

If you solve this for Ein(surface) you find:

Ein(surface) = 2× Esun (6)

So just by adding an atmosphere, the surface of the planet now sees 2 times the incoming radiation!Now, if we substitute this in our previous equations to find how this changes the surface temperature, we find:

T = (2)14 × TBB ≈ 1.19× TBB (7)

where TBB is the Blackbody temperature of the planet. This calculation only considers a single layer, perfectgreenhouse gas atmosphere - the simplest possible case. In reality, the effect depends on other factors, like theefficiency of the greenhouse gas and the number of atmosphere layers you consider.

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1.3.1 Greenhouse Effect Exercises

1. Use Eqn. 7 to find the temperature of the Earth with this simple single-layer estimate of the greenhouse effect,using the Blackbody temperature you calculated in Exercise 1.2.1.1 .

2. With this simple estimate, would the Earth be able to sustain liquid water on its surface?

3. How does this agree with the yearly-averaged value of the temperature of the Earth, which ranges from 288 to293 K?

1.4 Testing the Greenhouse Gas Hypothesis

We have shown that if an atmosphere can cause a greenhouse effect, it will cause the surface of the planet to bewarmer than its blackbody temperature. Now we must devise a test to see whether or not atmospheric gasses cancause a greenhouse effect.

A frequently discussed greenhouse gas is Carbon Dioxide (CO2). To test whether or not it can cause a greenhouseeffect, we will have two sealed, clear-topped containers (see the figure below) - one we will fill with pure CO2 and theother we will fill with normal air (0.035% of which is CO2). We will expose both to simulated sunlight, and monitortheir internal temperature over time.

Based off the above description of the experimental setup, the Hypothesis below, and the diagram of the experi-mental setup, make your prediction for the outcome of the experiment.

1.4.1 Hypothesis

CO2 is a greenhouse gas, and will absorb infrared radiation while allowing visible light to pass.

1.4.2 Prediction

If this hypothesis is valid, what do you predict will happen to the temperature in the CO2 filled chamber comparedto the temperature in the air-filled container illustrated in Figure 3?

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AIR(most thermal radiation escapes)

100% CO2(most thermal radiation captured by CO2)

Light absorbers(black paper)

Light source

ther

mom

eter

therm

om

eter

visible light visible light

thermal radiation thermal radiation

Figure 3: Experimental setup for testing the greenhouse effect hypothesis.

1.5 Experimental Setup

Experimental Materials:

• 2 2-liter plastic bottles with labels removed, half-wrapped in black paper.

• 1 filled with CO2

• 1 filled with normal air

• 2 bottle caps with thermometer probes attached.

• 1 high-power light bulb in stand.

Procedure:

1. Check to make sure both your bottles are clean and dry, and have black backing paper attached to one side ofthem.

2. Make sure that your temperature probes are sealed in tightly, such that your CO2 has not leaked out.

3. Place both bottles on opposite sides of the light source, exactly equidistant from the light source, with un-papered sides facing the bulb.

4. Note the starting temperatures in each bottle in the “0 min” row in Table 2, and turn on light bulb. Note thetime as in the Time column of the “0 min” row in the table below.

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Time Temp (AIR) Temp (CO2)0 min:

5 min:

10 min:

15 min:

20 min:

25 min:

30 min:

Table 2: Worksheet for Greenhouse Effect Experiment. Please write the difference between your final temperatureson the whiteboard with your group number (indicate whether the CO2 or AIR bottle was hotter)

5. Every 5 minutes note the time and the temperature in each bottle in Table 2.

6. Continue for 30 minutes.

1.5.1 Greenhouse Effect Experiment Exercises

1. Did you see a difference in the temperatures as a function of time? Was your prediction validated?

2. Does this suggest that CO2 is a greenhouse gas?

3. What other tests would you like to do to make certain that your results were accurate?

4. In a column of Earths atmosphere with the same diameter as your bottle, there is roughly 0.036 kg of CO2.If the density of pure CO2 is 0.0018 kg/liter (at room temperature and pressure), how does the amount ofCO2 in the bottle compare to the amount in the atmospheric column? Would you expect a stronger or weakergreenhouse effect from this atmospheric column of CO2?

5. Compare the temperature differences on the whiteboard. Did every group see a similar trend? What was theaverage temperature difference?

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