TheBlackbodyFraction,InfiniteSeriesandSpreadsheets*DUNCANLAWSONSchoolofMathematicalandInformationSciences,CoventryUniversity,Coventry,CV15FB,UK.E-mail:d.lawson@coventry.ac.ukThispapershowshowPlanck'sblackbodyequationandtheresultingformulafortheblackbodyfraction can be used as a `real world' example to motivate undergraduate engineers in their study ofmathematics. A spreadsheet can be used to give students familiarity with some of the properties ofPlanck'sspectral distribution. Theproblemof calculatingtheblackbodyfractioncanbesolvedrelativelyeasilyusingaspreadsheet providedsomesimplemathematical analysisiscarriedoutfirst. The mathematical techniques which are required include change of variable, infinite binomialseries,integrationbypartsandsumsofgeometricprogressions.INTRODUCTIONALL ENGINEERS should be fluentin a range ofmathematical techniques. However, there is often amotivation problem when teaching mathematics toengineering undergraduates as they fail to appreci-ate the value of the material they are being taught.One way to address this is through the use of `realworld' applications of themathematics. Planck'sequationforthespectraldistributionoftheemis-sive power of a blackbody [1] is an excellentexample of a piece of `real world' mathematicsthat can illustratetheusefulnessofmanydifferentmathematicaltechniques[2].Inthispaperwewillconcentrateontwokeyaspects:1. How a simple mathematical transformation canreducethecomplexityofaproblem.2. The usefulness of infinite series in allowingevaluationofanotherwiseintractableintegral.Planck'sequationisnotenormouslycomplicated,but it is complexenoughthat students will gainlittle knowledge of the nature of the functionsimply by looking at its formula. They needtospend some time gaining familiarity with thenatureofthefunctionbeforeproceedingtosomeanalysis.Thisfamiliaritycouldbegainedbyusinga computer algebra systemsuch as Maple orMathematica but there are considerable advan-tagesinusingaspreadsheet togivestudentsthisbasicfamiliarity. Inparticular, theoverwhelmingmajority of engineering undergraduates are at easewithusingaspreadsheet toperformavarietyofcalculationsinothercontexts. Thereismuchlesswidespread fluency with specific mathematicalsoftware. Furthermorestudentsaremoreaccept-ingofresultstheygainfromspreadsheetsastheyusuallyhaveareasonablegraspof theprocessesthe spreadsheet is implementing. When using acomputer algebrapackagetheyoftenfeel unsureabout the routines the software has used to achievethe results it presents and therefore feel moreinsecureaboutthestatusoftheseresults.Inwhatfollowswewilldiscussthebasictheoryof Planck's equation, the tasks students can under-taketogainfamiliaritywithit and`discover' forthemselves some of its key properties. Then we willprogresstoamoredetailedanalysisoftheblack-body fraction function and demonstrate howachange of variable (suggested by findings fromtheearlier stages of investigation) cantransformthe apparentlyintractable integral intoone thatcanbeevaluatedbyusinginfiniteseries. At thispoint it is sensible to use a spreadsheet to evaluatetruncatedversions of this infiniteseries. Furthermathematicsrelatedtothesumsof infiniteseriescanalsobe usedtodetermine error bounds fortruncating the series after a finite number of terms.BASICFAMILIARITYWITHPLANCK'SEQUATIONPlanck's equation for the spectral distribution oftheemissivepowerofablackbodyisgivenby:eb``. T 2C1`5expC2,`T 11In this equation ` is the wavelength, the constantsC1andC2aregivenby:C1 hc2 0.59552197 108Wjm4m2sr1andC2 hc,k 14387.69 jmKwhere h is Planck's constant, 6.6260755 1034J s;kisBoltzmann'sconstant,1.380658 1023J K1and c is the speed of light in a vacuum2.97792458 108ms1. * Accepted14July2004.984Int.J.EngngEd.Vol.20,No.6,pp.984990,2004 0949-149X/91$3.00+0.00PrintedinGreatBritain. # 2004TEMPUSPublications.Afirst step towards gaining familiarity withPlanck's function is to plot its graph for anumber of different temperatures, for example1000K, 2000Kand4000K. ThisiseasilydoneinaspreadsheetasshowninFig.1.It is a simple matter to produce a figure showingthegraphoftheblackbody function fora numberoftemperatures.Figure2showssuchafigureforthe temperatures 4000K (the curve with the highestpeak), 3500K, 3000Kand2500K(thecurvewiththelowestpeak).The curves drawn all have the same basic shape:they start from zero, rise to a peak and then decaybacktozeroasthewavelengthincreases. Asthetemperature increases the position of the peakmoves to the left. It is interesting to note thelocation of the peak. By looking at the datapointsusedtodrawthegraphs, agoodestimateof the peak wavelength can be obtained. Thefindings from a selection of temperatures aregiveninTable1.The data points used to plot the graphs arespacedat intervals of 0.05 jm. For the cases of1000Kand2000KthepointsindicatedinTable1areclearlyhigherthantheirneighboursoneitherside whereas at 4000Kthe values at 0.7 and0.75 jmarealmostidentical.The data in Table 1 indicates that as thetemperatureisdoubledsothepeakwavelengthishalved. In other words `pT is a constant (where `pis the wavelengthat whichthe peakonthe eb`curve occurs). This is known as Wein's law [3]. Thefactthatthelocationofthepeakontheeb`curvedoes not dependon`andTindependentlybutonlythecombination `Tsuggeststhatitmightbeinterestingtoplot thesecurvesagainst `Tratherthan against `. A spreadsheet which allows two ofFig.1. SpreadsheettoplotPlanck'sblackbodyfunctionforaselectedtemperature.Fig.2. Theblackbodyfunctionfor4000K(highestcurve),3500K,3000Kand2500K(lowestcurve).Table1. Wavelengthatwhichthepeakvalueofeb`occursTemperature(K) Peakwavelength(jm)1000 2.92000 1.454000 0.70.75TheBlackbodyFraction,InfiniteSeriesandSpreadsheets 985these curves (for different values of T) to be plottedinshowninFig.3.In the spreadsheet shown in Fig. 3, values for thetwotemperaturesareenteredincellsC6andE6.The values in column A are fixed values of `T. Thevalues of ` in column B are calculated fromcolumnAbydividingbythe value of T1. Thenthevaluesofeb`incolumnCarecalculatedfromPlanck's Equation (1) using the values of ` incolumnBandthevalueofT1.ColumnsDandEare similar except they use T2. The graphs areplotted by selecting the data in columns A, CandEandthenproducinganXYscatter.The curves confirm what has already beenobserved, namely that the peak occurs at thesame value of `T. Lookingat the datausedtoplot thesegraphsit isclearthat thepeakoccurssomewherebetween`Tvaluesof 2800 jmKand3000 jmK.Theconstantlocationofthepeakandthefactthat the curves have the same shape suggest that itmight be possible to find a scaling which willcollapseall thecurvesontoeachother. Thisisinfactwhatweobserveifweploteb`/T5against `T.AspreadsheettodothisisshowninFig.4.This spreadsheet is the same as the one shown inFig. 3exceptthatthevaluesincolumnsCandEhavebeendividedbyT51andT52respectively.Thegraphappears toshowonlyonecurve, but thatis because the values in columns Cand Eareidenticalandsothesecondcurveexactlyoverlaysthefirst.The above shows that the function eb`(`,T)/T5isnot a functionof `andTindependently but aFig.3. Spreadsheettoploteb`curvesagainst `T.Fig.4. Spreadsheettoploteb`/T5curvesagainst`T.D.Lawson 986function of the product `T. This is seen quiteeasilymathematicallybyworkingfromPlanck'sequation(1):eb``. TT5

1T52C1`5expC2,`T 1

2C1`T5expC2,`T 12whichshowsthatwehaveafunctionof`T.THEBLACKBODYFRACTIONThe total emissive power is found by integratingthe spectral emissive power over the entire spectrum:ebT

10eb``. Td` oT43where o 25k4/15c2h35.67051108Wm2K4istheStefan-Boltzmann constant.Theemissivepowerinabandofthespectrum,say from `1 to `2, is found by integrating Equation(1)from``1to` `2:eb.band`1. `2. T

`2`1eb``. Td` 4It is clear fromthe above definition that theemissive power of the bandfrom0to`2is thesum of the emissive powers of the bands from 0 to`1andfrom`1to`2.Thereforewehave:eb.band`1. `2. T eb.band0. `2. Teb.band0. `1. T 5FromEquation (5) we see that the emissivepowerof a band anywhere in the spectrumcan becalculatedifweknowtheemissivepowerofeachbandstartingfromzerowavelength.We define the blackbody fraction, F0. `. T, ofthe wavelength band from 0 to `, to be the fractionofblackbodypowerintheband.FromEquations(1),(3)and(4)thisgives:F0. `. T eb.band0. `. TebT

1oT4

`02C1`5expC2,`T 1d` 6There is no obvious closed form for the integral ontheright-handsideof Equation(6) andsosomecomputational method is needed to evaluate it.Jain [4] used the numerical integration routineswithinMathematicatocalculatesomeblackbodyfractions,theseareshowninTable2.Although the use of a sophisticated piece ofmathematical software is appealing it has somedisadvantages. Primarily the computed answersare not particularly accurate (as will be shownlater). Nodoubt withsuitable choice of optionswithinthepackagemoreaccuratevaluescouldbecalculated but as there is no indication of the errorinthesevaluesit wouldbedifficult tobecertainwhenanaccuratevaluehadbeenachieved.Thecomplexityof theintegral canbereducedby making a simple mathematical transforma-tion. This illustrates the value of carrying outsome mathematical analysis before rushing intoevaluation.ANALTERNATIVEAPPROACHTOCALCULATINGTHEBLACKBODYFRACTIONWe have already seen that in the theory ofspectral radiative emission the combination ofthe variables `andTintoasingle variable `Tsometimes hassignificance. This gives anindica-tion of a way to proceed with the integral inEquation (6). Before we do that we need to restateEquation (6) in a more mathematical precisemanner. The way it is giveninEquation(6) ishowit is presented in many expositions of thetheory, howeverit ismathematicallysloppy. Thesymbol `isbeingusedintwodistinctlydifferentways. Firstlyit specifiesoneof theargumentsofthe functionFandit is inthis capacity that itappears as the upper limit on the integral. Howeverit alsoappears as theintegrationvariableintheintegrand.So,forexample,ifwewishedtocalcu-latetheblackbodyfractionfortheintervalfrom0to 0.5 jm F0. 0.5. T we would integrate from 0to 0.5 but leave all the `'s in the integralunchanged. Toavoidthis confusing double useof the symbol `, we will restate Equation (6) usingiastheintegrationvariable:F0. `. T eb.band0. `. TebT

1oT4

`02C1i5expC2,iT 1di 7At this point it appears that the blackbody fractiondepends on ` (in the upper limit of the integral) andT(inthedenominatorofthecoefficient oftheintegraland the denominator of the exponent in theintegrand) completely independently. However,following the approach of Chang and Rhee [5], wetaketheTinsidetheintegralsigntoobtain:F0. `. T 1o

`02C1iT5expC2,iT 1Tdi8Table2. BlackbodyfractionsinselectedbandsfromJain[4]Temperature(K) Band00.38 jm Band00.76jm2500 0.00017 0.052095000 0.05129 0.4429510 000 0.44342 0.83909TheBlackbodyFraction,InfiniteSeriesandSpreadsheets 987Changing to an integration variable of iT, we canwriteequation(8)as:F0. `. T 1o

`T02C1iT5expC2,iT 1diT

1o

`T02C1x5expC2,x 1dx 9Equation (9) shows clearly that F0. `. T does notdepend on ` and T independently, but only on theproduct `T (which only appears in the upper limitoftheintegral).Inrecognitionofthis, henceforthweshallrefertoF0. `. TasF`T.Further progress can be made by making afurtherchangetotheintegrationvariable, settingz C2/x,thisleadsto:F`T 2C1oC52

C2,`T1z5expz 1C2z2 dz

154

1C2,`Tz3ez1 ezdz 10wherethedefinitionsofC1,C2and ogivenearlierhavebeenusedtosimplifythecoefficient of theintegral andthe numerator anddenominator ofthequotientintheintegrandhavebeenmultipliedby ezto enable expansion of the denominator usingthebinomialtheorem.Thefactor 1 ez

1intheintegrandcanbewritten as an infinite series using the binomialexpansion 1 x1 1 x x2x3. . . whichconverges for |x|