SolarPhysics DOI:10.1007/•••••-•••-•••-••••-• Eclipses observed by LYRA - a sensitive tool to test the models for the solar irradiance A.I. Shapiro1 · W. Schmutz1 · M. Dominique2 · A.V. Shapiro1,3 2 (cid:13)c Springer•••• 1 0 Abstract We analyze the light curves of the recent solar eclipses measured 2 by the Herzberg channel (200–220nm) of the Large Yield RAdiometer (LYRA) n onboardPROBA-2.Themeasurementsallowustoaccuratelyretrievethecenter- a J to-limb variations (CLV) of the solar brightness. The formation height of the radiationdependsontheobservinganglesotheexaminationoftheCLVprovide 1 3 informationabout a broadrange ofheights in the solaratmosphere.We employ the 1DNLTE radiativetransfer COde for SolarIrradiance(COSI) to modelthe ] measured light curves and corresponding CLV dependencies. The modeling is R used to test and constrain the existing 1D models of the solar atmosphere, e.g. S the temperature structure of the photosphere and the treatment of the pseudo- . h continuum opacities in the Herzberg continuum range.We show that COSI can p accuratelyreproduce notonly the irradiancefromthe entire solardisk, but also - o the measured CLV. It hence can be used as a reliable tool for modeling the r variability of the spectral solar irradiance. t s a [ 1. Introduction. 1 v 6 The variability of the solar irradiance may have a direct impact on climate 4 (see e.g.the recentreviewsby Haigh,2007andGray et al.,2010).Althoughthe 5 measurementsandmodelingofthesolarirradiancewereunderthecloseattention 6 during the last decade, the complete picture of the solar variability is still far . 1 from being clear (see e.g. Harder et al., 2009; Haigh et al., 2010). Therefore the 0 launch of every new space mission devoted to the measurements of the solar 2 irradiance is able to provide a crucial piece of complementary information as 1 : well as to nourish the theoretical models. v i X 1 Physikalisch-MeteorologishesObservatoriumDavos,World r RadiationCenter,7260DavosDorf,Switzerland a email:[email protected] email:[email protected] email:[email protected] 2 RoyalObservatoryofBelgium,Ringlaan3B-1180Brussel, Belgium email:[email protected] SOLA: shapiroetal_eclipses.tex; 1 February 2012; 4:03; p. 1 Shapiroetal. 1.2 (a) January 15, 2010 e 1.0 c n dia 0.8 a r Ir 0.6 e v ati 0.4 el R 0.2 0.0 5 6 7 8 9 Time [h] 1.2 (b) July 11, 2010 e 1.0 c n dia 0.8 a r Ir 0.6 e v ati 0.4 el R 0.2 0.0 17 18 19 20 21 22 Time [h] Figure1. RelativevariationsoftheirradianceasmeasuredbytheHerzbergchannelofLYRA. TheintervalsofzerointensityoccurduringtheoccultationswhenPROBA-2passestheEarth shadowinthewinterseason.Thepanelsshowtwoeclipsesseparatedbythethreeoccultations on January 15, 2010 (a) and four eclipses on July 11, 2010 (b). The periodicabrupt changes oftheirradiancelevelareduetothespacecraftmaneuvers. In this paper we analyze the first measurements of the Large Yield RA- diometer (LYRA) (Hochedez et al., 2006; Benmoussa et al., 2009) onboard the PROBA-2 satellite launched on November 2, 2009. Up to now LYRA has ob- served several solar eclipses (see Fig. 1). During the eclipse the Moon consecutively covers different parts of the solar disk. The light curve of the eclipse depends on the CLV of the solar brightness andonthegeometryoftheeclipse(theangularradiioftheSunandtheMoonas wellas the minimum distance betweentheir centerswhichis reachedduringthe maximum phase of the eclipse). If the geometry of the eclipse is known and the distribution of solar brightness has radial symmetry then the light curve of the eclipsecanbeusedtoretrievetheCLVofthesolarbrightness.Letusnoticethat the assumption ofthe radialsymmetry is well-justified for the January15,2010 eclipse as the solar activity level was very low (according to the USAF/NOAA SOLA: shapiroetal_eclipses.tex; 1 February 2012; 4:03; p. 2 EclipsesobservedbyLYRA data the total sunspot area was about 0.025% of the full solar disk). The CLV ofthesolarbrightnessprovideavaluableinformationaboutthesolaratmosphere (Allende Prieto, Asplund, and Fabiani Bendicho,2004;Koesterke, Allende Prieto, and Lambert, 2008)anddeterminetheirradiancevariationsonthetime-scaleofthesolarrota- tion(Fligge, Solanki, and Unruh, 2000).Additionallythechangesofthespectral solarirradianceduringtheeclipsesareimportantforstudyingtheEarth’satmo- sphereresponse(Kishore Kumar, Subrahmanyam, and John,2011;Sumod et al., 2011). Shapiro et al., 2010 showed that the 1D NLTE radiative transfer COde for SolarIrradiance(COSI)(seeHaberreiter, Schmutz, and Hubeny,2008)allowsto calculatethesolarspectrum(125nm–1µm)fromtheentiresolardiscwithahigh accuracy.InthispaperweuseCOSItocalculatetheCLVofthesolarbrightness and compare them with ones deduced from the eclipse light curves as observed by LYRA. We show that the measured CLV provide an important constrains onthe UV opacities and the temperature structure of the solaratmosphere.We come up with a model which allows to accurately reproduce the measurements. We restrict ourselves to the modeling of the eclipse profiles and CLV in the Herzberg channel of LYRA. The solar irradiance in the Herzberg contin- uum range (200–220 nm) is of especial importance for the climate modeling as it directly affects the ozone concentration and stratospheric temperature (Brasseur et al., 1987; Rozanov et al., 2006; Shapiro et al., 2011c). The proper modeling of the formation of this radiation in the solar atmosphere is a base for the variability modeling and for the irradiance reconstruction to the past (see e.g. Krivova,Solanki, and Unruh, 2011; Vieira et al., 2011; Shapiro et al., 2011a). Weareawarethatthe1Dmodelsdonotnecessarilyreflecttheaveragephysi- calpropertiesoftheinhomogeneoussolaratmosphere(Uitenbroek and Criscuoli, 2011). Thusthemaingoalofthispaperisnottolearnthenewfactsaboutthedynamic 3D solar atmosphere but rather to develop a reliable semi-empirical tool for modeling the solar irradiance variability. In Sect. 2 we deduce the empirical CLV from the LYRA observations of the January 15, 2010 eclipse. In Sect. 3 we compare these CLV with ones calcu- lated by COSI and discuss the constrains on the temperature structure of the solar atmosphere (Sect. 3.2) and UV opacities (Sect. 3.3). The main results are summarized in Sect. 4. 2. Empirical center-to-limb variations as deduced from the LYRA observations PROBA-2 evolves on a dawn-dusk heliosynchronous orbit, with an altitude of 720kmonaverage,whichallowsaquasi-permanentobservationoftheSun.The spacecraftaccomplishesafullorbitinabout100minutes.Incaseofsolareclipse, it is therefore not unusual that it crosses the eclipse zone more than once. Lyra data in Herzberg channel are acquired, for all three units, by experimental PIN detectorsmadeofdiamond.Suchdetectorshaveproventobeverystablewithre- spect to temperature variations (Benmoussa et al., 2004). The nominalcadence SOLA: shapiroetal_eclipses.tex; 1 February 2012; 4:03; p. 3 Shapiroetal. 1.2 1.2 (a) Transit 1 (b) Transit 2 1.0 1.0 0.8 0.8 e fil 0.6 Unit 2 0.6 o r P Unit 1 0.4 0.4 0.2 0.2 0.0 0.0 4.9 5.0 5.1 5.2 5.3 5.4 9.0 9.1 9.2 9.3 1.5 1.5 (c) Transit 1 (d) Transit 2 1.0 1.0 % 0.5 0.5 n, o ati 0.0 0.0 vi e -0.5 -0.5 D -1.0 -1.0 -1.5 -1.5 4.9 5.0 5.1 5.2 5.3 5.4 9.0 9.1 9.2 9.3 Time [h] Time [h] Figure 2. Theprofilesofthe05:00(a)and09:00(b)transitsretrievedfromthelevel1data aswellasthedeviationsbetween theseprofilesandtheprofileretrievedfromthelevel2data oftheunit2(c,d). Theshadedareaonpanelcindicatestheestimatederrorrange.Thedotted lines correspond to the widest part of the error range in the time interval between 5 h and 5.3h(i.e.excludingtheproblematicfeatureafter5.3h). of acquisition is 20 Hz. A more detailed discussion of the in-flight performance of LYRA is given in Dominique et al., 2011. The first light curve of the eclipse event was obtained by LYRA on January 15, 2010. The eclipse was shortly preceded by LYRA first light on January 6, 2010andwasthelongestannularsolareclipseofthemillennium.Itwasobserved onthegroundfromAfricaandAsiaandwasseenasapartialfromthePROBA- 2. The eclipse lasted more than 6 hours, so the PROBA-2 passed through the Moon’s shade three times. However the intermediate transit could not be observed due to the simultaneous occultation (i.e. it was shaded by the Earth). The raw (level 1) data collected by the Herzberg channel of LYRA during this eclipse are presented in panels a and b of Fig. 2. The plotted data were corrected for the dark current which is still present in the original data. The 05:00 UTC transit of the January 15, 2010 eclipse (hereafter first transit) was simultaneously observedby the LYRA units 1 and2 (the back-upand standard acquisition units, accordingly), while the 09:00 UTC transit (hereafter second transit)only bythe unit2.Thedropsofthe irradianceafterthe firstandbefore the second transits correspond to the occultations (see panel a of Fig. 1). SOLA: shapiroetal_eclipses.tex; 1 February 2012; 4:03; p. 4 EclipsesobservedbyLYRA The level 1 data are uncalibrated. The calibrated (level 2) data are also available for the community. For the analysis presented below, we will therefore use the profiles retrievedfrom the level 2 data. These data are correctedfor the temperature effects, degradation and the dark current. Let us note that level 2 data always refer to the unit 2 measurements, while the measurements from the back-up units 1 and 3 are normally used for the calibration and currently only available in their uncalibrated version (Dominique et al., 2011). Originally the level 2 data were correctedfor the degradationby adding a time-dependent offset (to allow a better analysis of the solar flares). The offset shifts the zero- level of the irradiance and leads to the erroneous profiles of the eclipses and solarvariability(Shapiro et al., 2011d).Thereforeforouranalysistheoffsetwas removed from the level 2 data. The blue curves on the panels c and d of Fig. 2 represent the deviations betweenthelevel1andlevel2data.Thezero-levelofbothdatasetswascorrected as discussedabove.One can see that the level2 data are almostidenticalto the originalmeasurementsofthestandardacquisitionunit2. Thisconfirmsthatthe temperature correction for the diamond detectors is very small. From now on wewillnotdistinguishbetweenthe level1andlevel2dataoftheunit2.Onthe contrary there is a significant deviation between the data collected by the unit 1 and unit 2 during the first transit. Both units were carefully calibrated in the groundfacilities so their different responses are connected with the degradation which affects the sensitivities of the units. The standard acquisition unit 2 was opened almost constantly, while the back-up units 1 and 3 were opened only occasionally.As a result on January15, 2010unit 2 degradedapproximatelyby 15% while unit 1 did not yet show any noticeable degradation. The differential behavior of the units allows us to estimate the error range of the measured profiles. We defined the uncertainty that it is twice the difference between the profiles as observed by the unit 2 and the unit 1 (see panels c of Fig. 2). This approachdoesnotincludeanysystematicdeviations,commontobothunitsbut it measures the degradationdue to the space exposure. The arbitrary character of this estimate does not affect the results presented below. The second transit was only observed by the unit 2 so for its analysis we will use the error range defined from the first transit. The profile of the irradiance variation during the eclipse depends on the CLV of the solar brightness. For example a strong decrease of the brightness towards the solar limb would lead to a smaller residual irradiance during the maximal phase of the eclipse and accordingly to a deeper eclipse profile. Thus, the observedprofilesallowtodetermine the CLV.Forsimplicity weadoptedthe widely used polynomial parametrization of the CLV (see e.g. Neckel and Labs, 1994; Neckel, 2005): I(µ) = A µi, (1) i I center i X where I is the disk-centerintensity, µ is the cosine ofthe heliocentric angle center andA arefreeparameters.The solardiskwasdividedintothe thirteensuppos- i edlyuniformconcentricringsandthebrightnessofeachringwascalculatedwith the help of the Eq. (1) using the µ value of the ring’s mean circle. We checked SOLA: shapiroetal_eclipses.tex; 1 February 2012; 4:03; p. 5 Shapiroetal. 1.2 3 (a) (b) 1.0 2 ] 2 V -0 L 0.8 1 1 alized C 0.6 LV, [unit 2 0 m C r 0.4 - -1 o V N L C 0.2 -2 0.0 -3 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 R / R R / R Sun Sun Figure 3. (a) Empirical CLV deduced from the profile of the first transit as observed by the unit 2 (orange curve); (b) the deviations between the CLV plotted on panel (a) and the CLV deduced from the profile of the first transit as observed by the unit 1 (red curve) and CLVdeducedfromtheprofileofthesecondtransitasobservedbytheunit2(bluecurve).The empiricalCLVdeducedfromthetwoextremeprofilescorrespondingtotheedgesoftheerror rangeinFig.2delimittheCLVerrorrange(theshadedarea). thatsuchdivisionofthesolardiscallowstocalculatetheirradianceprofileswith the accuracy better than 0.02% and thus is sufficient for our purposes. Simple geometrical calculations allow to obtain the profile of the eclipse for each set of the coefficients, assuming that the time dependencies of the angular distance between the Sun and the Moon as well as of their angular sizes are known. For the both transits of the January 15, 2010 eclipse we searched for the set ofthecoefficientsA whichleadtothebestagreementwiththeobservedprofiles i and minimized the error E defined as: E = Pi −Pi 2, obs emp i X(cid:0) (cid:1) where P is the observed profile and P is the empirical profile calculated obs emp fromthe Eq.(1).For the firstandsecondtransitsthe summationwasdone over thetimeintervalsbetween 5hand5.3handbetween9hand9.25hrespectively. The minimization was performed applying the additional condition of the monotonous CLV. The CLV profiles were calculated using the second degree polynomial. The employment of the higher degree polynomial (up to sixth de- gree) had no visible effect. The CLV depend on the wavelength (Neckel, 1996; Hestroffer and Magnan, 1998).Thus,thecoefficientsA determinedbythe min- i imizationprocedurecorrespondtotheCLVofthesolarintensityconvolvedwith the profile of the LYRA Herzberg channel. The latter is a combined profile of the detector and filter. The resulting empirical CLV dependencies are presented in Fig. 3. The CLV dependency for the first transit is normalized to unity in the disk center, while all other CLV dependencies are normalized to give the same integral flux from SOLA: shapiroetal_eclipses.tex; 1 February 2012; 4:03; p. 6 EclipsesobservedbyLYRA 1.5 1.5 (a) Transit 1 (b) Transit 2 1.0 1.0 % 0.5 0.5 n, o ati 0.0 0.0 vi e -0.5 -0.5 D -1.0 -1.0 -1.5 -1.5 4.9 5.0 5.1 5.2 5.3 5.4 9.0 9.1 9.2 9.3 Time [h] Time [h] Figure4. Thedeviationsbetweentheprofilesasmeasuredbyunit2andascalculatedwith theempiricalCLV.Theerrorbars(dottedlines)arethesameasinFig.2. the entire disk. The errorrange of the CLV was estimated performing the mini- mization procedure to the two maximal error profiles of the first transit. So the shaded area in the Fig. 3 is constrained by the two CLV dependencies which correspond to the edges of the shaded area in the panel c of Fig. 2. The deviations between the profiles as measured by LYRA and calculated with the empirical CLV are shown in Fig. 4. The deviations can be attributed to the limited accuracy of the measurements (note that they are within the estimated error bars) and to the violation of the radial symmetry of the solar brightness. One of the sources of the asymmetry of the solar brightness is the inhomegenous structure of the quiet Sun, which consists from several bright- ness components (Fontenla et al., 1999). The small amplitude of the deviations supports the analysis presented below. 3. Comparison with modeling To calculate the theoretical CLV we employed the 1D NLTE radiative transfer code COSI developed by Hubeny, 1981; Hamann and Schmutz, 1987; Schmutz, Hamann, and Wessolowski, 1989; Haberreiter, Schmutz, and Hubeny, 2008;Shapiro et al.,2010.COSIsimultaneouslysolvesthestatisticalequilibrium and radiative transfer equations in the spherically symmetrical geometry. The temperature and density structures of the different components of the solar atmosphere were taken from Fontenla et al., 1999, while the electron density and all level populations were self-consistently calculated in the NLTE. 3.1. Solar irradiance in the Herzberg continuum range ThepropercalculationoftheradiativetransferintheHerzbergcontinuumregion is sophisticated by two factors. Firstly, the continuum opacity in this region is stronglyaffectedbytheNLTEoverionization(seee.g.Shchukina and Trujillo Bueno, 2001; Short and Hauschildt, 2005), so the proper NLTE calculations are neces- sary.Secondly,theimmensenumberofweak,mostlyspectrallyunresolvedlines, SOLA: shapiroetal_eclipses.tex; 1 February 2012; 4:03; p. 7 Shapiroetal. formtheso-calledUVlinehazeinthisregion.Thecreationofthecorrectlinelist whichwouldincludeallpossiblespectrallinesisataskofatremendousdifficulty. Although considerable progress was reached during the last few decades, there is still only 1% of the UV lines which are measured in the laboratory, while all remaining lines are predicted only theoretically (Kurucz, 2005). As the conse- quence the existing line list are not complete and underestimate the opacity in theUVspectralregion(Bus´a et al., 2001;Haberreiter, Schmutz, and Hubeny, 2008; Short and Hauschildt, 2009). Toaccountforthe missingopacityShapiro et al.,2010multiplied the contin- uum opacity coefficient k (λ) by the wavelength dependent coefficient f (λ): c c k′(λ)=k (λ)·f (λ). (2) c c c Theemployedmultiplicativecoefficientisastepfunctionofthewavelengthwith thestepequalsto1nm.ItwasempiricallydeterminedsothattheUVirradiance calculatedbyCOSIequalstotheirradianceasmeasuredbySOLSTICE(SOLar- STellar IrradianceComparisonExperiment,see McClintock, Snow, and Woods, 2005)onboardtheSORCEsatellite(Rottman, 2005)duringthe2008solarmini- mum.TheadditionalopacitywhichisnecessarytoreproducetheSOLSTICE/SORCE irradiancedoes notexceedafew percents ofthe totalopacityincluded inCOSI. Itis only necessaryin the 160–320nm spectralregion.Shortwardof160nmthe total opacity is mainly dominated by the photo-ionization continuum opacity, whilelongwardof320nmtheexistinglinelistsareaccurateenoughtoreproduce the irradiancewithbetterthan5%accuracy(andwithbetterthan2%accuracy longward of 400 nm). Asduringthesolarminimumthe solarirradiance(exceptthe extremeUV)is dominatedbythequietSun,thesolaratmospheremodelC(averagesupergranule cell interiormodel) from Fontenla et al., 1999was used for the calculations.Let us however note that it would be possible to be in agreement with the SOL- STICE/SORCE measurements while using another temperature profile by ad- justing additional opacities. Thus the SOLSTICE/SORCE measurements alone does not allow to constrain the temperature structure of the solar atmosphere. InSect. 3.2we showthatalthoughsimultaneousadjustment ofthe temperature structure and additional opacities does not affect the irradiance from the entire solar disc, it significantly changes the CLV. The comparison with the measured CLV allows to choose the most suitable for the CLV calculations model. Shapiro et al.,2010assumedthatthecoefficientoftheadditionalopacitydoes notdependonheightinthesolaratmosphere.Thisimpliesthatitdoesnothave any dominant source as the concentration of every particular ion or molecule is height dependent. In Sect. 3.3 we show that the CLV provide important complementary informationwhich allow to reevaluate this assumption and help to understand the nature of the additional opacity. 3.2. Test of the temperature structure To test the sensitivity of the CLV to the change of the temperature structure weperformedthecalculationsemployingthemodelsforseveraldifferentcompo- nents of the solar atmosphere: model A (faint supergranule cell interior), model SOLA: shapiroetal_eclipses.tex; 1 February 2012; 4:03; p. 8 EclipsesobservedbyLYRA 1.2 10 (a) (b) 1.0 5 ] 2 V -0 L 0.8 1 alized C 0.6 LV, [unit 2 -05 m C r 0.4 - No A C E P LV -10 C 0.2 -15 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 R / R R / R Sun Sun Figure 5. ThesameasFig.3butwiththeCLVdependencies calculatedforthefaintsuper- granule cell interior(A), average supergranule cell interior (C), quiet network (E) and plage (P). C (average supergranule cell interior), model E (quiet network) and model P (plage).ThetemperatureanddensitystructuresweretakenfromFontenla et al., 1999. The models A and E correspond to the cold and warm components of the quiet Sun, while the Model C represents the spatially averaged quiet Sun. Shapiro et al.,2010showedthatthecalculationswiththelattermodelcanrepro- ducespectralirradiancemeasuredbySOLSTICE(upto320nm)andSIM(Solar Irradiance Monitor; Harder et al., 2005) (from 320 nm onward) onboard the SORCE satellite during the 2008 solar minimum, as well as SOLSPEC (SOLar SPECtral Irradiance Measurements; Thuillier et al., 2004) during the ATLAS 3 mission in 1994 with high accuracy. They used the model C to calculate the f (λ) factor(seeEq.(2))fortheadditionalopacitiesinthe160–320nmspectral c range. The same procedure of the f (λ) factor fitting was performed for the Mod- c els A and E. The Model A is colder than Model C so it yields smaller UV irradiance.ThusasmalleradditionalopacityisnecessarytoreproducetheSOL- STICE/SORCE measurements in the 160-320 nm spectral region. Accordingly the use of Model E leads to a larger additional opacity. The Model P yields so high irradiance that it is not possible to reach SOLSTICE/SORCE level by increasing the continuum opacity. Thus we didn’t recalculate the f (λ) factor c for the model P and left it the same as for the Model C. In Fig. 5 we present the calculated CLV for each of these models. The radi- ation which comes from the regions close to the solar limb is formed in higher and colder regions of the solar atmosphere than the radiation coming from the disc center. Hence the solar brightness is decreasing towards the limb. One can see that the colder the model, the stronger CLV it yields. This can be partly explained by the fact that the sensitivity of the Planck function to the tem- perature change is the decreasing function of the temperature. Thus the same change of temperature results in larger alteration of the Planck function and SOLA: shapiroetal_eclipses.tex; 1 February 2012; 4:03; p. 9 Shapiroetal. Table 1. Theoretical profilesvs.observedand empir- ical (calculated with the empirical CLV) profiles. The numbers are discrepancies calculated with the help of Eq. (3). The minimum values of the discrepancy are boldfaced. Passage1 Passage2 observed empirical observed empirical A 49.9 3.3 65.0 9.9 C 105.8 92.5 130.1 113.0 E 168.7 160.4 197.3 186.5 P 216.3 209.2 245.2 236.3 accordinglylarger change of the emergent irradiancefor the colder models. The completepicturedependsonthetemperatureanddensitystructureaswellason the opacity behavior in each of the solar atmosphere components. It is further sophisticatedbytheNLTEeffectswhichcausesdeviationsofthesourcefunction from the Planck function. Although due to the readjustment of the additional opacity the calculations with the Models A and E yield the same UV irradiance as the calculation with the ModelC,the correspondingCLVdependencies areremarkablydifferent.All differences between the theoretical and empirical CLV’s have a sudden drop at R/R ≈ 0.99 (see panel (b) of Fig. 5). The rings which correspond to these Sun points (see Sect. 2) have very small relative area, so the eclipse profiles are basicallyinsensitive to the changeoftheir brightness.Thus the reliabilityofthe empirical CLV for these points is very low. Interestingly, the calculations with the model C underestimate the CLV and are outside of the estimated error region. One of the possible reasons for this could be the erroneous assumption of the depth independency of the additional opacity coefficient f (see Eq. (2)). This possibility will be discussed in the c Sect. 3.3. On the other hand it is known that the 1D models can underestimate the anisotropy of the radiation field (Shapiro et al., 2011b; Kleint et al., 2011) andtheCLVofthesolarbrightness(Koesterke, Allende Prieto, and Lambert, 2008; Uitenbroek and Criscuoli, 2011). Hence the reported disagreement could be a signature of the general problems which are inherent to 1D modeling. The warmer Models E and P yield even weaker CLV than the Model C. At thesametimethecoldermodelAyieldstheCLVwhichareinagoodagreement with the empirical ones. The corresponding deviations between the calculated and measured profiles are shown in Fig. 6. The structure common to the both transits represents propertiesofthemodels,whileindividualstructureisduetothelimitedaccuracy ofthemeasurementsandviolationoftheradialsymmetryofthesolarbrightness. We can define the discrepancy between the profiles Pi and Pi as: 1 2 N ∆ =10−4 Pi−Pi 2/N , (3) 1,2 v 1 2 u i=1 ! u X(cid:0) (cid:1) t SOLA: shapiroetal_eclipses.tex; 1 February 2012; 4:03; p. 10