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epl draft New insights into black bodies F. J. Ballesteros(a) Astronomical Observatory of the University of Valencia (OAUV) - Ed. Instituts d’Investigacio´, Parc Cient´ıfic, C/ Catedra´tico José Beltrán 2. E-46980 Paterna, Valencia, Spain 2 1 PACS 44.40.+a–Blackbody radiation 0 PACS 95.75.Mn–Image processing in astronomy 2 Abstract –Planck’s law describes theradiation of black bodies. The studyof its properties is of n special interest, as black bodies are a good description for the behavior of many phenomena. In a J this work a new mathematical study of Planck’s law is performed and new properties of this old acquaintanceareobtained. Asaresult,theexactform forthelocusinacolor-color diagrams has 0 1 beendeduced,andananalyticalformulatodeterminewithprecision theblackbodytemperature of an object from any pair of measurements has been developed. Thus, using two images of ] the same field obtained with different filters, one can compute a fast estimation of black body M temperatures for every pixel in the image, that is, a new image of the black body temperatures I for all the objects in the field. Once these temperatures are obtained, the method allows, as a . consequence, a quick estimation of their emission in other frequencies, assuming a black body h p behavior. These results providenew tools for data analysis. - o r t s a [ Introduction. – Many interesting phenomena emit- findanalyticallythevalueofthetemperaturefromthisra- 2ting thermal radiation follow approximately the theoret- tio. One has to use numerical methods to solve the equa- vical curve of a black body: hot stars, volcanic lava, the tion for T [2,3], orfit measurements into a Planck’scurve 9 cosmic microwave background or the infrared emission by means of a minimization procedure [4] (or if the max- 0 from the Earth, for instance. The determination of their imum of emission can be determined taking more points, 8 1temperature and other properties is a frequent topic, and Wien’s law can also be used). .Planck’slawbecomesasourceofinformationaboutthem. In this paper we show a new mathematical study on 1 0 Planck’s law gives the spectral distribution of electro- Planck’s law for black body radiation [5], finding some 2magnetic emission for a black body at a given tempera- new properties. As a result we have deduced the general 1ture. Whentheblackbodyismuchbiggerthanthedetec- form of the stellar locus straight region in a color-color : vtion element field of view, obtaining its temperature from diagram (where stars with a behavior closer to a black ia measurement is trivial: although the intensity arriving body are). We have also developed a new, fast and accu- X to the detector from anarea unit decreases inverselywith rate analytical formula for obtaining the temperature of r athe square of the distance, the amount of area observed a black body from this intensity ratio measured at two increases proportionally with the square of the distance, wavelengths. This method allows one to produce thermal and both effects compensate each other (as in infrared images of objects taken through two filters, and as a con- land-surface temperature measurement from space, [1]). sequence to generate synthetic images of these objects at When this is not the case, the received flux changes inany other wavelength, following a black body behavior. verselywiththesquareofthedistance,hencethedistance to the object (and its size) matters. If this distance is Color-colordiagram as a power law. – Inacolor- unknown, it should be enough to obtain the ratio of the color diagram, many stars fall into a rather linear stellar intensities measured at two different wavelengths, as it is locus. This is because stars approximate blackbody radi- independentofthedistanceandsizeoftheemitterandde- ators. But although it is well known that black bodies in terminesunambiguouslythetemperature. Butthen,what a color-color diagram align into a straight line, the exact one get is a transcendental equation: it is not possible to form of the formula they follow has never been deduced until now. A color index is the difference in magnitude in (a)E-mail: [email protected] two bands, which corresponds to the ratio of the respec- p-1 F. J. Ballesteros tiveintensities,asthemagnitudeisalogarithmicfunction of the intensity. Hence, a linear locus is equivalent to a power law in the space of intensity ratios. Planck’s law has the general form: xn I =C (1) ex/T −1 where C is a constant, x = hν/k = hc/kλ (being ν the frequency and λ the wavelength), T is the temperature of the black body, and n is an integer whose value is linked to the flux units used, according to the following rules: intensity units (proportional to) n photons/frequency unit 2 Fig.1: Dots: datafromstarsinNGC6910,takenwithUniver- energy/frequency unit 3 sity of Valencia’s telescope TROBAR with Stromgren filters; gray and black dots stand respectively for data before and af- photons/wavelengthunit 4 ter correction of atmospheric extinction, exposure times and energy/wavelengthunit 5 filterbandwidth. Solidline: theoreticalcurvefrom eq. (7)for When we represent I /I vs. I /I for black bodies a b c d blackbodies,wheretwopointshavebeenstressed(whitedots) with different temperatures (where Ii is the intensity at at T =3000 K and T =∞. wavelength/frequency i at a given temperature T), these intensity ratios follow a power law such as: Substituting (6) and (4) in eq. (2) we obtain the power I /I =A(I /I )α (2) law we were looking for, which is a good fit of data from c d a b black bodies. Note that we can interrelate black body whereAandαarecoefficientstobe determined. First,to intensitiesinonlythreespectralpositions,justbychoosing obtainαwe willuse a low temperatureapproximation,as d=b,andsoI =I ,x =x (asmallerreductioncannot d b d b α is mainly determined by this regime. Substituting eq. be afforded as in that case we would only obtain a trivial (1) into (2) and applying logarithms, α gives: identity). Hence, I can be written as: c α= Ln(cid:16)Lxxndncneexxdxxc//nanTTee−−xx1a1b//(cid:17)TT−−−11Ln(A) (3) Ic =(cid:18)xxcb(cid:19)n−1(cid:18)xxab(cid:19)(n−1)xxac−−xxbbIb(cid:18)IIab(cid:19)xxac−−xxbb (7) b (cid:16) (cid:17)   Inthelowtemperatureslimit,T ↓↓,then(x/T)↑↑,thus Datachecking. – Aseq. (7)hasbeenobtainedusing ex/T −1 ≈ ex/T. Substituting, and taking into account severalapproximations,its accuracy is relative: it is good that the terms of the form x /T dominate, we obtain the forinterpolations(whenspectralpositioncliesinbetween i exponent of the power law, i.e. the slope for a color-color aandb)butnotsogoodforextrapolations. Nevertheless, diagramc−d vs a−b for black bodies. Note that it only itisquicktocompute,rathercompactandkeepsthemain depends on the frequencies/wavelengths considered: featuresofthepowerlaw. Thusitcanbe usefulasis,taking into account its limitations, to check if astronomical α≈ Ln xxndnc +xTd−xTc−Ln(A) ≈ xTd−xTc = denaotaughhavsteabrseeinntphreopimeralygersedwuitcheda(gporoodvibdleadckthbaotdtyhebreehaavre- (cid:16)Ln (cid:17)xxnna +xTb−xTa xTb−xTa (4) ior). Intheexampleinfig. 1,graydotsaredatafromthe b 1 − 1 starsin the opencluster NGC 6910,once the imageswere (cid:16) (cid:17) = νc−νd = λc λd νa−νb λ1a−λ1b onlycorrectedofadditivecomponentsandflat-fielded,and blackpointsarethesamedataaftercorrectingothermulti- For A, let us consider the high temperatures limit, as plicativefactorsasatmosphericextinction,exposuretimes values in this regime tend to crowd together, reaching a and filter band width. Solid line is the theoretical curve fixedpointwhenT →∞. Now(x/T)↓↓,andthusex/T ≈ from eq. (7), which fits perfectly the black dots, showing 1+x/T. Therefore the intensity ratios become: that the data reduction has been correctly done. Althoughsimilarinphilosophytomethodsthatfitstars I xn xd xn−1 I xn−1 c = c T = c a = a (5) to a stellar locus, as the Stellar Locus Regressionmethod Id xnd xTc xdn−1 Ib xbn−1 [6], it is very different. SLR is a method for calibra- tion, giving magnitudes as output, but only applicable to Resolving eq. (2) for A it gives: a given set of filters for which a standard experimental stellar locus has been defined. On the contrary, our ap- A= Ic Ib α = xcn−1 xbn−1 xxac−−xxdb (6) plication is completely general, valid in principle for any Id (cid:18)Ia(cid:19) xdn−1 (cid:18)xan−1(cid:19) set of filters. It only needs that some of the stars behave p-2 New insights into black bodies oneofablackbody,eq. (1). Thusitistemptingtoidentify τ with a temperature (it has temperature units too). Indeed, the behavior of τ is surprising. In the low temperatures regime (when the black body’s maximum x max < x ,x ), τ estimates correctly the temperature, τ ≈ T, a b but in the high temperatures regime (x > x ,x ), it max a b tendstoτ ≈2T,ascanbeseeninfig. 2,solidline,where we have used data from black bodies at many temperatures, estimating τ from their intensities at wavelengths 440and540nmasanexample(althoughthesamebehav- Fig. 2: Solid line: ratio between the magnitude τ of eq. (9), ior can be find with any other pair of spectral positions). calculatedfrom theintensitiesofseveralblackbodiesat wave- Taking this property into account, we are going to de- lengths 440 and 540, and the real temperature of these black bodies. Dashed line: the same but for τ′ following eq. (11). fine a new estimator for the temperature of a black body. Dotted line: theaverage of both magnitudes, eq. (12). The trickwill be to averageτ with a modificationofitself ′ (namely τ )thatwillcoincide withτ for the lowtempera- tureregimeandthatwillbenegligiblerespecttoτ forthe moreorlessasblackbodies(whichisfrequent). Itisgood high temperature regime. This can be afforded by chang- forcheckingdata reduction, but aseq. (7) dealswith flux ing the exponent n−1 by n−γ (where γ is a numerical ratios, magnitude information cannot be recovered. Any value that has to be fitted to assure such behavior, and position of the black dots along the line in fig. 1 could that depends on the spectral positions a and b used): work (it will be equivalent to changing the temperature x −x of the stars). There is in principle no way to discriminate ′ b a τ = (11) among all the possibilities. Nevertheless, some limits can Ln Ia xb n−γ be imposed, because there is an absolute upper limit, eq. Ib xa (cid:20) (cid:16) (cid:17) (cid:21) (5), corresponding to the asymptotic limit for high tem- For 440and 540nm, a goodchoiceof γ is −1.711(see fig. peratures (not black body beyond this point is allowed), 2, dashed line). Then, our temperature estimator T˜ will andonecanalsosetbyhandalowerlimit(asforexample, be the average of both, T˜ =(τ +τ′)/2: T =3000 K, as M stars differ rather from a black body). Although it cannot be directly used to calibrate data, T˜ = xb−xa 1 + 1 (12) if we manage to have data well calibrated for two given 2 Ln Ia(xb)n−1 Ln Ia(xb)n−γ filters by other means, the method could indeed be used (cid:20) Ib xa Ib xa (cid:21) for calibrate data in other filters, as α depends only on In the low temp(cid:2)erature re(cid:3)gime,(cid:2)τ′ ≈ τ, t(cid:3)hus, T˜ ≈ ′ theeffectivewavelength. Thiscanbeusefulformultiband 2τ/2= τ ≈T. In the high temperature regime, τ <<τ, surveys. The method can also be used to measure the ef- thus, T˜ ≈τ/2≈T. As can be seen in fig. 2, dotted line, fective wavelength midpoint of a unknown filter by using the discrepancy with the real T is minimal. twowellknownfilters. Thiscanbeusefultorecoverinfor- Notethatfortwogivenspectralpositionsaandb,γ has mation on the filters used for the case of old data, when to be carefully chosen in order to ensure the best results, logsarenotconservedandfilterinformationhasbeenlost. as γ is a function of x and x (a function that has to be a b invariant under the interchange a ↔ b). Now the art lies The temperature of a black body. – For the pur- inchoosingasuitablefunctionγ whichgivesgoodresults. poseofobtainingdirectlythetemperatureofablackbody Among the studied candidates it has been found that the from an intensity ratio, we take eq. (7) as starting point. function which works best in the whole spectrum is: There,giventwofixedspectralpositionsaandb,I varies c only with xc. Therefore, eq. (7) can be rewritten as: γ =5−9[|Ln(xa/xb)|+3]−0.252 (13) I =C′ xmc (8) Of course, other possible choices of γ deserve consider- c exc/τ ation. But with this one, the estimation of black body temperature from I /I is very precise in the whole spec- with a b xb−xa tral range, with a precision under 1% for most cases (the τ = (9) n−1 bigger error happens when x lies in between x and Ln Ia xb max a Ib xa xb; here in some cases, the error can rise up to ∼ 5%). (cid:20) (cid:16) (cid:17) (cid:21) Although in many inverse problems one finds often unex- and pectedlystrongnumericalinstabilities,thisisnotthecase. ′ 1 n−1 xb (n−1)xa−−xbxb Ia xa−−xbxb Thesolutionisawell-behavedformula,veryrobust. Asan C =Ib (10) example, although γ of eq. (13) was in principle designed x x I (cid:18) b(cid:19) (cid:18) a(cid:19) (cid:18) b(cid:19) to assure a good behavior in the visible spectrum, let us wheremisaninteger. Exceptforthe lackofa”−1”term use data coming for the Cosmic Microwave Background in the denominator, the formula for I is analogue to the spectrum measured by FIRAS on board COBE [7]: c p-3 F. J. Ballesteros Fig. 3: Images of the Sombrero Galaxy taken by the ACS WFC instrument on board NASA/HST, with the filters F435W, F555W and F625W respectively (theimages are not reproduced here in linear scale in order to show all the richness of details they have), and the image of equivalent black body temperatures for the galaxy (bottom right, in linear scale), obtained assuming a black body behavior for each pixel (black means a ”temperature” smaller than or equal to 4000 K; white, greater thanorequalto10000K).Notethatthisdoesnotaffordforarealtemperatureofthedust;itseemsthatthiscolorisproduced byunresolved stars in theforeground, which givea bigger bluecomponent to thelight from thedust. at 10 cm−1, I =200 MJy/sr. Thermal imaging. – Equations (12) and (13) pro- at 20 cm−1, I =10 MJy/sr. videatoolto obtainquickly,fromtwoimagesofthe same From this data, using eqs. (12) and (13) we get for the part of the sky taken through two filters, a new image of CMB a temperature estimation of T˜ =2.72 K. equivalentblackbodytemperatures,oncetheimageshave Statistical fluctuation can affect the estimation, being been properly flat-fielded, calibrated, etc. (such that the more sensible to it when x and x are closer or when samegraytoneonbothimagesisproportionaltothesame a b the black body temperature is higher. For example, a fluxintensity)andaligned. Aseqs. (12)and(13)areana- SNR of 10 in I and I , becomes a SNR in the estimated lyticalformulae,theircalculationisfast,soitcanbedone a b black body temperature of: 10 for a 6000 K black body for every pixel. This way we get a new image, where gray measured at 440 and 640 nm; 6 for a 9000 K black body tones are proportionalto the temperature that eachpixel at the same filters; 6 for a 6000 K black body measured should have if fluxes were coming from a black body. at 440 and 540 nm. But note that this is not a problem Asanexample,thismethodhasbeenappliedtotheim- of the estimator but of the black body curve itself. ages of the Sombrero galaxy obtained by NASA Hubble Alltheformerrelationshipswerededucedformonochro- Space Telescope, with the ACS WFC instrument. Con- matic emission. When observingblack bodies throughfil- cretely the filters F435W, F555W and F625W have been ters, these relationships are right for narrow filters (once used, see fig. 3. dividing the intensities by the bandwidth), but for very To obtain an estimation of temperatures, we just need broadfilters as Johnson-Morganones, the linear behavior twofilters, thus three possible combinationsare available: in the color-color space becomes slightly curved (as the a thermal image using F435W and F555W, another with effectivewavelengthofthefilterchangeswiththeshapeof F435W and F625W, and still another with F555W and the black body). Nevertheless,they still workratherwell. F625W. Of course, the sky areas corresponding to these As a direct application of eq. (12), it can be use to pixels are not real black bodies, and they have statistical estimatethetemperaturefromthewidelyusedcolorindex fluctuations too, so we will not get the same temperature B−V. Equation (13) is a generic fit which works well in estimations in the three cases. But nevertheless the three the whole spectrum and makes for you the selection of a images obtained are coherent: temperatures are indeed γ. But for a particular case, one can find a value of γ quite similar and keep a good agreement, showing that that worksstill better than the one produced by eq. (13). the method is robust. In fig. 3, the bottom right image After this fine tuning, the final result is: corresponds to the averagedimage of these three. In general this image shows what one should expect: T =4600 0.92(B−1V)+1.7 + 0.92(B−1V)+0.62 (14) dust features have an equivalent black body temperature (cid:16) (cid:17) lower than the rest of the galaxy, which means that light comparable to other results in the literature (for a com- from these zones is reddish, due to the reddening caused parison,see [8] where B−V vs. T data for different stars bythedust. InEarthwarddirection,lightfromthe center are represented together with several fittings). of the galaxy crosses the dust ring directly towards us, p-4 New insights into black bodies where i can be a or b. The best results are obtained by choosing the one closer to the target spectral position x. This equation (together with (12) and (13) to obtain T˜), gives much better results than eq. (7) and allows to ex- trapolatethebehaviorofblackbodiesatwavelengthsvery far away from the input spectral positions a and b. Equation (15) provides a quick way to identify diver- gencesfromblackbodybehaviororsimilitudeswithit. As an example we will use infrared data from NASA’s IRAS SkySurveyAtlas(ISSA). Concretely,imagesinthe12,60 and 100 Micron bands from the Galactic Plane Mosaic, ◦ centered at 90 of galactic longitude. Figure 5 first three Fig. 4: Triangles: average relative proportions among the panels show these IRAS images as insets at right, being intensities in the three HST images at the ”hot” zones of the themainpanelsazoomontheircentralzone,centeredap- dust ring (according to fig. 3 bottom right) and its fitting to prox. at Ra 22h, Dec +60◦, with a size of about 30◦x12◦, a black body of 7000 K (solid line). Circles: the same for the respectively at wavelengths 12, 60 and 100 Microns. ”cold”zonesofthedustring,anditsfittingtoablackbodyof In the 12 Microns band, the galactic plane is far from 5500 K (dashed line). a black body behavior, as the emission from polycyclic aromatichydrocarbonmoleculesdominatesinthegalactic plane at this wavelength, meanwhile in the 60 and 100 suffering a reddening more intense than the one coming Micron bands dominates the thermal emission from cold from the rest of the dust ring, where disperse (and hence dust, following a Planck distribution [10]. Given that in bluer)lightcontributionishigher. Andinfact,thefrontal the 60 and 100 Micron bands we do expect blackbody zone of the dust ring in the ”thermal” image is darker. behavior,wewillusetheimagesatthesebandstogenerate But inside the dust ring, a strangeand unexpected fea- asyntheticimageat12µm,andthendividetherealoneat ture appears: a narrowzone surrounding the galaxy,with 12µmbythesyntheticone. Hence,the excessofemission high equivalent black body temperatures. At first sight, inthisratioimagewillberepresentativeofthepresenceof one could think it is an odd artifact of the temperature hydrocarbons. This is what is shown in the bottom right estimation method, but indeed it is not. It is in the data: panel of fig. 5 (in logarithmic scale, rescaled). light coming from this narrow zone of the dust ring is in- We are not interested in the PAH emission itself but deed bluer than the one from the rest of the ring. The in the cluster of dark objects which seem behave as black averagerelativeintensitiesinthethreeHSTimagesinthe bodies. Theseobjectsseemtobeintheforeground,shield- ”hot” dust zone (see fig. 4) are higher in the F435W ing the background hydrocarbon radiation emitted from filter, followed by the F555W filter, and are lower in the the Galaxy. Some of them coincide with nebulae, clearly F625W one, fitting rather well into a 7000 K black body. visible in the three infrared images. But there are many Meanwhileinthe”cold”dustzones,theintensitypeaksin others (a selection has been marked with boxes) that can the F555W filter, fitting better into a 5500 K black body. hardly be made out in the infrared images, some nearly Of course this does not indicate the true temperature imperceptible (but it is clear that they are real objects of the dust, which is in fact much colder (almost an order and not artifacts, as they show a clear elliptical/circular of magnitude). But why is bluer the light coming from shape). Comparing with visible images of this region, we this region of the dust? The fact is that infrared Spitzer seethatapartoftheseobjectsarerelatedtostars(stand- images of this galaxy [9] show this zone of the dust ring ing out the bright one close to the center, whose location glowingbrightly in infraredlight, unveiling a disk of stars coincides with the supergiant λ Cephei), but not all of withinthedustring. Thisseemstoconfirmthatthe”hot” them appear in opticalimages(and vice versa,not allthe dustzoneisindeedthefingerprintofthisstellarformation stars have a counterpart in the ”blackbodyness” image). zone;someofthesestarscouldbeintheforegroundofthe In all these cases, their equivalent black body tempera- dust ring, sticking out from under the dust, unresolved ture happens to be very low: they are compatible with a but contributing to the color to the dust. black body at ≈50 K. What are these objects? Are these featuresdustcloudsshieldingthe galacticplaneemission? Synthetic images according to black bodies. – Does the supergiantλ Cepheihaveanenvelope surround- Once one has an estimation of the temperature of a ing it, perhaps due to its stellar wind? Are protostars black body from I and I using eq. (12), it is possible to a b thoseglobulesapparentlynotassociatedtoastar? Inany estimate its emission at any other wavelength, just using case, they seem to have a strong ”black body” signature eq. (1) and solving C from the known data: that makes them very evident under this procedure. xn exi/T˜−1 Conclusions.– Wehavedoneanewanalyticalstudy I =I (15) ixni ex/T˜−1! of Planck’s law and we have obtained some new proper- p-5 F. J. Ballesteros Fig. 5: First three panels: infrared images from NASA’s IRAS Sky Survey Atlas Galactic Plane Mosaic, centered at 22h, ◦ +60 , respectively for the 12, 60 and 100 Micron bands. Bottom right: ratio between the real IRAS image at 12 Microns and a synthetic image for the same wavelength generated from the IRAS images at 60 and 100 Microns, assuming a black body behavior(logarithmic scale). Thebrightertheimage, thebiggertheamountof emission from polycyclicaromatic hydrocarbon molecules. Main panelsareazoom ofthecompleteimages, shownatright. Boxesmarktheposition ofsomeobjects appearing clearly in thebottom right image butdim or absent in thethree infrared images. tiesinaformulawhichwaspublishedmorethanacentury ∗∗∗ ago. The two main outcomes of this analysis are the de- duction of the exact form of the power law lying inside I would like to thank Vicent Peris for his invaluable Planck’s law, which explains most of the behavior of the help in the data processing and useful brainstormings. stellar locus in a color-color diagram, and a new analyti- I would like also to thank Eusebio Llacer for his help calformulatoestimateblackbodytemperaturesfromthe with the English, and Vicent Martinez, Alberto Fernan- ratio of intensities measured at two different spectral po- dez, Bartolo Luque and Juan Fabregat for their useful sitions, which also provides a method to estimate black suggestionsandcomments. Iwanttoacknowledgesupport body emissions in any other spectral position. fromMICINNthroughCONSOLIDERprojectsAYA2006- 14056 and CSD2007-00060, and project AYA2010-22111- As a result, we have introduced a new tool for data C03-02. Alltheimagesforthispaperhavebeenprocessed analysis. 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