Problem set #4 Biology 463

Working individually, correctly answer the following questions. To get credit for your answer, you must show all work in a legible and organized way.

Scenario - A new planet has been discovered in our solar system - Zebulon 1. From the comfort of your spaceship you analyze the radiant emissions from this planet and discover that the wavelength of maximum emission is 1000 nm. Assuming the planet behaves like a blackbody, answer questions 1-4.

1) What is the frequency of maximum emission?

2) What is the energy content of a mole of photons at the wavelength of maximum emission?

3) What total rate at which EM radiation is being emitted from each square meter of theplanet/atmosphere system?

4) From previous research, you know that 40% of the radiation emitted by Zebulon 1 is prevented from reachingspace because it is either absorbed or scattered by the planet’s atmosphere. You’ve also determined that the materials in the atmosphere attenuate planetary radiation with an efficiency of 0.05 km-1 (i.e., the extinction coefficient = 0.05 km-1). How thick is the planet’s atmosphere?

5) Read pages 65-66 & 69-72 in Consider a Spherical Cow (see attached information), and complete Exercise 3 onpage 72 (Note: you may need to use an equation presented in class).

6) Go the following web site: http://climatemodels.uchicago.edu/isam/ then select the “About this model” tab. Readthe “Overview” & “How To” material, then watch the Video Introduction. After you have completed this, then use the model to do the following:

a. Determine what would happen between 2015 and 2100 to the concentration of CO2 in the atmosphere, and theglobal temperature, if CO2 emissions by burning fossil fuels and land use stayed at 2015 levels. Include the graphical output from your experiment. What causes this result?

b. It is thought that a total increase of 1.5o C might avoid the worst consequences of global warming. Use the modelto find an emissions scenario that meets this goal. Include the graphical output from your experiment. What does your scenario tell us about what society needs to be doing?

c. Use the model to explore a compelling question of your own. Explain why the question is interesting, and howyou used the model to examine your question. Include the graphical output from your experiment, and provide an accurate summary of the important implications of the results you found.

NOTE: Include a screenshot of the graphical output for each part of question 6.

To receive credit your answers must be emailed to me as a PDF file by the date indicated on the syllabus. Answers received later than this deadline will not be accepted.

B.Thermodynamics and EnergyTransfer

Thermodynamics is pn'maril'y the study of energy and its m'teractionwith matter. Energy can take many forms: the energy an object hasby v1r'tue of its motion is called kinetic energy; the kinetic energy ofmolecules movm'g randomly is called heat energy; the energy re-quired to Mt or shove an object that is acted on by such forces asgravity and friction is called work; the energy avall'able for release bydroppm‘g an elevated object is called potential energy; the energyavailable for release by burning substances such as coal is calledchemical energy; and chemical energy, when released, often takes theform of heat energy.

The ways energy can interact with matter are 11k'ewise numerous.Energy can warm matter, melt it, freeze it, b01l' it, expand or contractit, scramble or unscramble it, or mutate it. The results depend onwhat k1n'ds of energy and matter are m'volved. Despite the apparentcomplexity, however, a few snn'ple and um'versal physical laws havebeen discovered that allow us to make sense of the whole story.Among these are the laws of thermodynanu'cs. The first law statesthat the total amount of energy m’ a closed system (i.e., one whichenergy cannot enter or leave) remam's constant, regardless of thetransformations among energy types withm’ the system. This lawplays a central role m' modehn'g both meteorological phenomena andphysiological aspects of animal m'teractions with their env1r'onment.17The second law, which describes constram'ts on the conversion of onetype of energy to another, is' particularly useful for evaluating the

17. Schmidt-Nielsen (1972) has wn'tben a deligh‘tful and mf'ormative book on animalphysiology and physics. An excellent m‘Wducho’n to the subject of thermodynamics isthe terse text by W' jw‘; by Morowitz (1970) is also highlyrecommended. ' '

66 TOOLS OF THE TRADE

performance of technological devices designed to convert One for

1x‘ problems that follow Illustrate Varim ofenergy to another. The first 5 ' I

last problem 1n this seChbn ous1applications of thermodynamics. 'The .

the multiple scattering of solar radratron 1n the atmosphere, mum, on

concepts about energy transfer in the atmosphere that are Widatesused in climate research.

ell

THERMODYNAMICS AND ENERGY TRANSFER 69

13. How Hot Is Planet Earth?

What is the temperature of planet Earth? This isa complex question because Earth’s temperaturevaries with time and especially with location. 80,to be more specific, imagine a space travelerviewing our planet from afar. What temperaturewould she say Earth is?

At the level of analysis in this problem, a precise answer to the ques-tion “How hot is Earth?” is derived. However, the answer will not beof direct relevance to our experience on Earth’s surface. In later prob-lems (III.6—9), more reah'stic models will be used to calculate morepertinent information, such as Earth’s average surface temperature.These models w11'l build upon the simple “core model” presentedhere.

Earth, continually receiving energy from the sun, would get hotterand hotter 1f' it did not radiate energy back to space. This insight sug-gests that an energy conservation equation, balancing the rate of en-ergy flow arriv1n'g at Earth against the rate of energy flow departing,might allow a calculation of the earth’s temperature. To carry out thisbalancm'g act, the m'put and output rates are needed. Because onlym'puts and outputs are involved at this stage, you can ignore the com-plexities of atmosphen'c convection and the hydrocycle, which serveto redistribute energy on the wet, airy', planetary skin but do not af-fect the overall energy balance of the planet. Only radiant energy canbe transmitted to any appreciable degree through the nearly emptyspace beyond Earth’s atmosphere.

Consider, fir'st, the solar m'put. On a flat disc perpendicular to theunim'peded sun’s rays and 150 X 106 km from the sun, where Earthresides, the solar flux is 1,372 W/mz. For future reference, this impor-tant number w111' be denoted by the symbol, (2. Across an averagesquare meter above the atmosphere the flow of solar energy will notbe this large. On the m'ght side of Earth the flow w11'l obviously bezero, and only on the pom't of Earth directly under the sun (for ex-ample, the equator at high noon on the equin'ox) w111' the energy fluxfrom the sun equal 0. The flux above the atmosphere, averaged overday and m'ght and over all latitudes, 15' equal to 0/4. To see why thisis' so, note, first, that the ratio of the area of a sphere to the area of adisc of the same radius iskmlhr 12/1112 = 4. Thus, 1f' you imagine thesun’s rays makm'g dots on the ima'gmary' disc as they cross it, andthen stretch the,_cl1&cfl_m' am.'md , ,sph.,.ere, the dots wfll‘ be one fourth

Emil“. of outgom'g energy, per unit area‘ . at the earth from space, you

/'U l‘UULb' U!" [H]: l'KAUb

7rr2A

A -= 41rr2

1372 W/m2 ofsurface perpendicularto sun’s rays- 9

4——1372W/m2 of

(2Earth’s surface =- 4—

Figure 11-13 The solar flux on Earth. The factor of four arises because the surface areaof Earth is four times the area of a disc with a diameter equal to that of Earth. The solar

flux on Earth, averaged over all latitudes and longitudes, is (2/4. The sun’s rays do notof course, strike the Earth from behind. Rather, Earth turns and the radiation is distrib-uted over all longitudes.

would see a moon-hk'e glow on its sunlit side. This is reflected solarradiation. Earth has a reflection coefficient, called the albedo, of about0.30, which means that about 30% of the m'cident solar flux is re-flected directly back to outer space. (Most of this radiation is reflectedby Earth’s clouds rather than the planet’s surface.) The remaining70% of the solar flux is absorbed, either in the atmosphere or by ma-ten'als at Earth’s surface. This absorbed solar energy is converted tomfr'ared energy by the absorbing material. It is then emitted fromEarth and radiated away to space 111' a form called “infraredradiation.”

Thus, the outgom'g energy takes two forms: reflected solar energy,and infrared radiation by objects that have absorbed such energy. Therate at which an object radiates energy away from itself depends onits temperature; the hotter the object, the greater the rate of radiation.Objects that are max1m'ally effective radiators of energy are calledblackbodies. (A glowm'g lump of coal is a good example.) Such objectsare also perfect absorbers, which is why they are called black. TheStefan-Boltzmann Law describes the relation between the tempera-ture of such objects and their' rate of radiation. It states:

P = GT4. (1)

Here, P is the radiant energy output, with units of energy per unittime per unit area, and T is the absolute temperature of the object inkelvm's. The fundamental constant, a, is given by

(r = 5.67 X 10’8 J/m2 sec K4 (2)

THERMODYNAMICS AND ENERGY TRANSFER 71

To a good approximation, Earth is a perfect emitter and absorberof infrared radiation, and so Eqs. 1 and 2 can be used to describe theoutgoing infrared radiation.19 Thus, as shown in Figure 11-14, the in-coming and outgoing energy fluxes are:

Fin _- (3)PM = arr/4 + 07"“,

where the albedo is denoted by a. Equating incoming flux to outgoingflux,

0/4 = arr/4 + 0T4. (4)

Eq. 4 implies

(1 — (1)040 I (5)T4 ——

The solution to this equation is often denoted by To, a quantityreferred to as the blackbody temperature of Earth. Using the numen'-cal values of Q, a, and o, the result is To = 255 K.

To is not equal to the measured average temperature of Earth’s sur-face, which is about 290 K. Earth’s surface does emit m'frared radia-tion upward, but that radiation is largely absorbed in the atmosphere

\vi/

/ l \ /9

Fl” 4—

Fouf ‘ 4—80 + 0T4

Figure 11-14 The mfl'ow and outflow of energy on Earth. Straight arrows are solar fluxand Wiggly ones are mfr'ared radJa'tion. Arrow lengths are roughly proportional to ac-tual flows. Note that the total outflow from the polar regions is greater than the m'flow,whereas the outflow from the tropics is" less than the mfl'ow. Poleward flow of heat m‘the oceans and atmosphere on Earth accounts for this.

19. More generally, any object' radiates at a rate ecT‘ where e 15' a function of thewavelength of the radiation" mulls always & 1. An object is‘ a blackbody m' that portionof the spectrum of wavelengflismp‘ forwhxch' e = 1.

72 TOOLS OF THE TRADE

and then reradiated within' the atmosphere (see Problem 111.6). Radia-tion erru'tted higher in the atmosphere has a greater chance of beingradiated directly to outer space. The zone of the atmosphere that ra-diates directly to outer space consists of air from the mid-troposphereupward, and the density-weighted mean temperature of this zone isabout 250 K, which is quite close to our calculated value of 255 K.Observed from far away, the infrared earthglow has a spectrumroughly like that of a 250-K blackbody, and thus our assumption thatthis planet is thermodynamically "black" is not a bad one. It must beemphaSiz'ed, however, that Earth is nearly black only in the infraredrange of the spectrum, not in the spectral range of sunlight. A planetthat was a blackbody in' the solar spectrum would have zero albedobecause it would be a perfect absorber of sunlight; in contrast, Earth’salbedo in sunlight is about 30%.

EXERCISE 1: Verify that Eq. 5 yields the result: To = 255 K.

EXERCISE 2: Usm'g the procedures presented here, we calcu-late an mfr'ared blackbody temperature for Jupiter of 98 K; the mea-sured value is 130 K. It is the only planet with such a large discrep—ancy. The reason is that the gravitational collapse of Jupiter’s interiorproduces a Siz'eable amount of m'ternal heat energy. Estimate the in-ternal energy flux on Jupiter from the data above and compare it toEarth’s geothermal flux, which is deterrm'ned from data given in theAppendix' (VII. 1).

k‘EXERCISE 3: Throughout Earth’s history there have been cy-cles and fluctuations m' the planet’s average temperature. Changes inEarth’s solar orbit have been proposed as one cause of these varia-tions.20 While the most im'portant orbital modification of Earth’s cli-mate is most likely due to changes in" the orientation of the tilt ofEarth’s axis with respect to orbital perih'eh'on and aphelion (an effecttoo comph'cated to study with the globally averaged model presentedhere), a sun'pler phenomenon—change in the mean distance betweenEarth and the sun—can be studied. Assuming a circular orbit, by howmuch would the distance from Earth to the sun have to change inorder to bnn'g about a 1 K in'crease m' To? Later, in the second sectionof Chapter III, other possible causes of ch'mate alteration Will be dis-cussed. 1r

EXERCISE 4: Den've the fact that the sun’s blackbody temper-ature is' 5,750 K, given the values of Q and 0' quoted above, and themf'ormation from the Appendix' (IV) on the distance from the sun toEarth and the radius of the sun.

20. See Ruddiman' and McIntyre (1981) and Kerr (1983a) for further discussion of tlu'shypothesm'.