the measurement of solar diameter and limb darkening function with the eclipse observations

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Solar PhysicsDOI: 10.1007/•••••-•••-•••-••••-•

The Measurement of Solar Diameter and Limb

Darkening Function with the Eclipse Observations

A. Raponi1 · C. Sigismondi2 · K. Guhl3 ·

R. Nugent4 · A. Tegtmeier5 ·

c© Springer ••••

Abstract The Total Solar Irradiance varies over a solar cycle of 11 years andmaybe over cycles with longer period. Is the solar diameter variable over timetoo? We introduce a new method to perform high resolution astrometry of thesolar diameter from the ground, through the observations of eclipses by recon-sidering the definition of the solar edge. A discussion of the solar diameter andits variations must be linked to the Limb Darkening Function (LDF) using theluminosity evolution of a Baily’s Bead and the profile of the lunar limb availablefrom satellite data. This approach unifies the definition of solar edge with LDFinflection point for eclipses and drift-scan or heliometric methods. The methodproposed is applied for the videos of the eclipse in 15 January 2010 recordedin Uganda and in India. The result shows light at least 0.85 arcsec beyond theinflection point, and this suggests to reconsider the evaluations of the historicaleclipses made with naked eye.

Keywords: Solar Diameter, Limb Darkening Function

1 Sapienza University of Rome, P.le Aldo Moro 5 00185,Roma (Italy) email: [email protected] Sapienza University of Rome; ICRA, International Centerfor Relativistic Astrophysics, P.le Aldo Moro 5 00185, Roma(Italy); University of Nice-Sophia Antipolis (France); IstitutoRicerche Solari di Locarno (Switzerland); GPA ObservatorioNacional, Rio de Janeiro (Brasil). email: [email protected] IOTA, International Occultation Timing Association,European Section and Archenhold Sternwarte, Alt-Treptow 1D 12435 Berlin (Germany) email: [email protected] IOTA, International Occultation Timing Association, USSection email: [email protected] IOTA, International Occultation Timing Association,European Section email: [email protected]

SOLA: LDFSoPhys.tex; 19 September 2011; 0:31; p. 1

http://arxiv.org/abs/1109.3559v1

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Solar PhysicsDOI: 10.1007/•••••-•••-•••-••••-•

c© Springer ••••

Keywords: Solar Diameter, Limb Darkening Function

1. The variability of the solar parameter

Despite the observation of the changing nature of the solar surface dates backto the 17th century, the idea of the immutability of the solar luminosity wasleft only recently. ”Solar constant” was in fact the name that Pouillet (1838)gave to the total electromagnetic energy received per unit area per unit oftime, at the mean Sun-Earth distance (1AU). Today we refer to this parametermore correctly as Total Solar Irradiance (TSI). The difficulties in measuring theTSI through the ever-changing Earth atmosphere precluded a resolution of thequestion about its immutability prior to the space age. Today there is irrefutableevidence for variations of the TSI. Measurements made during more than twosolar cycles show a variability on different time-scales, ranging from minutes upto decades, or more. Despite of the fact that collected data come from differentinstruments aboard different spacecrafts, it has been possible to construct ahomogeneous composite TSI time series, filling the different gaps and adjustedto an initial reference scale. The most prominent discovery of these space-basedTSI measurements is a 0.1% variability over the solar cycle, values being higherduring phases of maximum activity (Pap, 2003).

The link between solar activity and TSI has certainly paved the way for adeeper understanding of the solar physics, and for a debate on the role of theSun in Earth’s climate. For example, the overlap between the Little Ice Age ofthe 17th century and the Maunder Minimum is a remarkable coincidence.

Presently, the most successful models assume that surface magnetism is re-sponsible for TSI changes on time-scales of days to years (Solanki and Krivova, 2005).Modeled TSI versus TSI measurements made by the VIRGO experiment aboardthe Solar and Heliospheric Observatory (SOHO), between January 1996 andSeptember 2001, show a correlation coefficient of 0.96 (Krivova et al., 2003).However, significant variations in TSI remain unexplained after removing the ef-fect of sunspots, faculae and the magnetic network (Pap, 2003; Kuhn et al., 2004).Other possible causes of the variation of the TSI may involve the global param-eters of the Sun: temperature and radius1.

Recently, monitoring the spectrum of the quiet atmosphere at the center ofthe solar disk during thirty years at Kitt Peak, Livingston and Wallace (2003)

1dealing with the radius is equivalent to dealing with the diameter, but in this work we referto the former or to the latter depending on the context

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Figure 1. The various total irradiance time series are presented on the upper panel, thecomposite total solar irradiance is shown on the lower panel (Pap, 2003).

have shown a nearly immutable basal photosphere temperature during the solarcycle within the observational accuracy, i.e. dT = 0± 0.3K.

The solar radius is the global property with the most uncertain determinationfor the changes over a solar cycle. The most accurate measures (with accuracybetter than 0.1 arcsec) are still far from an agreement (Djafer, Thuillier, and Sofia, 2008).The lack of agreement between near simultaneous groundbased measurementsat different locations suggests that atmospheric contamination is severe. More-over the lack of coherence of the set of the values obtained from differentobservers can also be explained by the lack of a common strategy: differentinstrumental characteristics, different choice of wavelength and different pro-cessing methods. The measurements from space are very few and they alsohave some sources of doubt. The Michelson Doppler Imager (MDI) on boardSOHO satellite indicates a change of no more than 23 ± 9 mas (milliarcsec-onds) in phase with solar activity (Bush, Emilio, and Kuhn, 2010), while theSolar Disk Sextant (SDS) (Sofia, Heaps, and Twigg, 1994), during five flightsbetween 1992 and 1996, measured a variation of 0.2 arcseconds in anti phasewith respect to the solar activity (Egidi et al., 2006). Although milliarcsecondsensitivity was achieved for the SDS, its results seem to be too large accord-ing to the studies on helioseismology, in particular to the f-mode frequencies(Antia et al., 2000; Dziembowski, Goode, and Schou, 2001).

The variability of the solar diameter over periods longer than a solar cy-cle is even more enigmatic. Ribes and Nesme-Ribes (1993) reported the mea-sures performed at the Observatoire de Paris from 1660-1719 taken by Frenchastronomers, including Jean Felix Picard, and the Italian director GiovanniDomenico Cassini. They measured a solar diameter 7± 1 arcsec larger than thestandard value of 1919.26 arcsec (Auwers, 1891) that is the modern reference


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Figure 2. (a) Point Spread Function for several values of Fried’s parameter r0 accordingto the Kolmogorov turbulence model, through a telescope having D = 10 cm; (b) effect ofatmospheric turbulence on the solar limb (Djafer, Thuillier, and Sofia, 2008).

for the absolute measures of the solar diameter. As we will see in section 5, thishigher value is somehow confirmed by observation of historical eclipses. Althoughthis strengthens the hypotesis of a larger diameter in the past, the discussion onthese absolute measures must be made with care, as we are going to show.

2. Observing the limb darkening function

Solar images in the visible wavelength range show that the disk centre is brighterthan the limb region. This phenomenon is known as the Limb Darkening Func-tion (LDF). The inflection point position of the LDF is the conventionallyaccepted definition of the solar edge. In this article we adopt this definition.

2.1. Seeing effect

In spite of the improvements in the measurements, there is always the uncer-tainty resulting from the effect of Earth’s atmosphere (seeing) and this effectis considered to be the main source of the discrepancies among the diameterdeterminations. The effect, as showed in Fig. 2, is the smoothing of the steepnessof the profile on the edge, moving inwards the inflection point.

2.2. Instrumental effect

Instrumental effect depends on the characteristics of each telescope.The FWHM of the PSF is proportional to λ/D, where λ is the wavelength of

the observation and D is the pupil diameter. This is similar to the FWHM of the


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Figure 3. COSI model predictions of the inflection point position as a function of wavelengthfor the continuum (lower solid line) and for the case when Fraunhofer lines are taken intoaccount (upper ”zig-zag” lines corresponding to outer inflection point when observed withvery narrow band-pass) (Thuillier et al., 2011).

atmospheric PSF represented in Fig. 2, that is proportional to λ/r0. Then for agiven value of the wavelength, if D decreases, the FWHM of the PSF increases,the inflection point moves inwards, and consequently the calculated diameterdecreases (Fig. 2b) ”mutatis mutandis”. Thus, two instruments with different Dwill measure different solar diameters if this instrumental effect is not taken intoaccount.

2.3. Dependence on wavelength

The position of the solar limb depends on wavelength not only through the PSF(instrumental or atmospheric) but also on layers at different quotes.

The dependence of the solar limb on the wavelength has a sign opposite tothat seen for the PSF: longer wavelengths have the inflection point outwards.The effect of the presence of Fraunhofer lines is illustrated by Thuillier et al.(2011). The authors reconstructed the limb profile for different wavelengths,including the continuum and the centre of spectral lines, with the Code forSolar Irradiance (COSI) (Haberreiter, Schmutz, and Hubeny, 2008). The resultson the prediction of inflection point position is shown in Fig. 3.

2.4. Asphericity

The asphericity is the observed variability of the radius with the solar lati-tude. The asphericity of the Sun, and in particular the oblateness: f = (Req −Rpol)/Req, has great implications on the motion of the bodies around the Sunthrough the quadrupole moment (Sigismondi and Oliva, 2005; Sigismondi, 2011).It can also provide important informations on solar physics as the other globalparameters.

Measuring the radius of the Sun for several solar latitudes may provide ameasure on asphericity. Conversely, for monitoring the variation of the solarradius, one must take into account that different solar latitudes may give different


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radii because of the asphericity, anyway not more than 20 mas according toRozelot et al. (2004).

2.5. Solar surface magnetic structure

The inflection point position depends also on solar active region involved. This isdiscussed in detail by Thuillier et al. (2011). Because of the lack of observationson this topic, the authors make use of some models of solar atmosphere. Despitethe differences between the results of the models, its clear a common trend:the presence of faculae displaces the inflection point outward with respect toits location predicted by the quiet Sun models, while the sunspots displace itinward.

The observation of the LDF in conjunction of these phenomena can helpto discern between the models, providing useful information on the solar at-mosphere. But if the goal of the measurement is the monitoring of the solardiameter, one has to avoid to measure in solar active regions.

3. The method of eclipses

3.1. Comparison between observation and ephemeris

With the eclipse observation we are able to bypass some problems that affectthe measurement of solar diameter. The atmospheric and instrumental effectsthat distort the shape of the limb (see section 2.1 and 2.2) are overcome by thefact that the scattering of the Sun’s light is greatly reduced by the occultationof the Moon, therefore there are much less photons from the photosphere to bepoured, by the PSF effect, in the outer region.

A decisive breakthrough in the determination of the solar diameter by theeclipse observation was made thanks to David Dunham that proposed to observethe Baily’s beads in connection with lunar profile data.

The Baily’s beads, from Baily (1836) who first explained the phenomenon,are beads of light that appear or disappear from the bottom of a lunar valleywhen the solar limb is almost tangent to the lunar limb.

It is not their positions to be directly measured, but the timing of appearingor disappearing. In fact the times when the photosphere disappears or emergesbehind the valleys of the lunar limb, are determined solely by the positions ofthe solar and lunar limb and the lunar profile involved at the instant, bypassingin this way the atmospheric seeing.

The International Occultation Timing Association (IOTA) is currently en-gaged to observe the eclipses with the aim of measuring the solar diameter. Thisis facilitated by the development of the software Occult 4 by David Herald2.

The technique consists to look at the time of appearance of the beads and tocompare it with the calculated positions by the ephemeris using the software

2www.lunar-occultations.com/iota/occult4.htm


www.lunar-occultations.com/iota/occult4.htm

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Figure 4. Total eclipse in Uganda (2◦ 41’ 19.8” N, 32◦ 19’ 2.9” E, 15 January 2010) filmedby Richard Nugent (upper panel). Lunar profile by Kaguya (low resolution) plotted by Occult4 software (lower panel). In abscissa the Axis Angle (AA) is the angle around the limb of theMoon, measured eastward from the Moon’s North pole. The distance in arcsec from the meanlunar limb is in ordinate. The curved line is the standard solar edge, that is the edge of theSun with the standard radius and with position by ephemeris calculation.

Occult 4 (Sigismondi, 2009). The simulated Sun by Occult 4 has the stan-dard radius: 959.63 arcsec at 1AU (Auwers, 1891). The difference between thesimulations and the observations is a measure of the radius correction withrespect to the standard radius (∆R), within the accuracy on the ephemeris(few milliarcseconds).

In the lunar polar regions due to the geometry of the eclipse the beads lastfor more time (Sigismondi, 2009). Observing in the lunar polar regions we arealso able to avoid the measurement in the solar active regions that could affectthe measures (see section 2.5). In fact the solar active regions do not appear insolar latitude higher than ∼ 40◦, and the maximum offset between the lunar andthe solar pole is ∼ 9◦ in Axis Angle (by Occult 4 calculation).

In grazing eclipses the number N of the beads can be high, providing Ndeterminations of photosphere’s circle.

However this approach conceives the bead as an on-off signal. In this way oneassumes the Limb Darkening Function as a Heaviside profile, but it is actuallynot. Different compositions of optical instruments (telescope + filter + detector)could have different sensitivities and different signal/noise ratios, recording thefirst signal of the bead in correspondence of different points along the luminosityprofile. This leads to different values of the ∆R (see Fig. 5). An improvement ofthis approach has to take into account the whole shape of the Limb DarkeningFunction and thus the actual position of the inflection point.

3.2. Numerical calculation

The shape of the light curve of the bead is determined by the shape of the LDF(not affected by seeing) and the shape of the lunar valley that generates thebead. Calling w(x) the width of the lunar valley (i.e. the length of the solar edge


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Figure 5. Beads magnitude evolution: the height of the solar limb above the valley is on theabscissa. The various square dots represent different type of telescopes. [25, p] means 25 cmopening with projection of the image. [25, ND4] the 25 cm telescope with a filter of NeutralDensity of transmittance 1/10000; ND5 stands for 1/100000, and so on.

visible from the valley in function of the height x from the bottom of the valley),and B(x) the surface brightness profile (i.e. the LDF), one could see the lightcurve L(y) as a convolution of B(x) and w(x), being | y | the distance betweenthe botton of the lunar valley and the standard solar edge (see Fig. 4), settingto 0 the position of the standard edge (the 0 is not the inflection point Positionbecause its position is our goal).L(y) =

∫B(x)w(y − x) dx

The discrete convolution is:L(m) =

∑B(n)w(m − n)h

where n, m are the index of the discrete layers corresponding to x, y coordinateand h the layers thickness.

Thus the profile of the LDF is discretized in order to obtain the solar layersB(n) of equal thickness and concentric to the center of the Sun. In the shortspace of a lunar valley these layers are roughly parallel and straight.

The lunar valley is also divided in layers of equal angular thickness. Duringa bead event every lunar layer is filled by one solar layer every given interval oftime. Step by step during an emerging bead event, a deeper layer of the solaratmosphere enters into the profile drawn by the lunar valleys (see Fig. 6), andeach layer casts light through the same geometrical area of the previous one.

Being: A1, A2..An the area of lunar layers, from the bottom of the valley goingoutward; B1, B2..Bn the surface brightness of the solar layers (our goal) from theouter (dimmer) going inward; L1, L2.. Ln the value of the observed light curvefrom the first signal to the saturation of the detector or to the replenishment ofthe lunar valley. One has:Lm = B1Am +B2Am−1 + ..+BmA1 =

∑m

n=1Bn Am−n+1

To derive the LDF profile:B1 = L1/A1

B2 = [L2 − (B1 · A2)] /A1


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Figure 6. Every step in the geometry of the solar-lunar layers (up) corresponds to a giveninstant in the light curve (down). The value of the light curve is the contribute of all the layers.

B3 = [L3 − (B1 · A3 + B2 · A2)] /A1

B4 = [L4 − (B1 · A4 + B2 · A3 +B3 · A2)] /A1

and so on.The situation described above relates to an emerging bead. The same processcan run for a disappearing bead, simply plotting the light curve back in time.The LDF obtained in this way is a discrete LDF. The smaller is the thickness ofeach layer, the more is the resolution of the LDF. The angular thickness of thelayers, i.e. the sampling, depends on our knowledge about the error of the lunarprofile.

What we need for the deconvolution procedure is thus:

• the light curve of the bead event;• the area of each lunar layer involved;• the correspondence between L(y) and L(t), i.e. the motion of the solar limb

in the lunar valley.

4. An application of the method of eclipses

We studied the videos of the annular eclipse in 15 January 2010 realized byRichard Nugent in Uganda and Andreas Tegtmeier in India.

The equipment of Nugent was: CCD cameraWatec 902H Ultimate3; Matsukovtelescope (90/1300 mm); panchromatic filter Thousand Oaks, ND5.

The equipment of Tegtmeier was: CCD camera Watec 120N3; Matsukov tele-scope (100/1000 mm); Filter: I = IOTA/ES green glass based neutral ND4.

3http://www.aegis-elec.com/products/watec-902H spec eng.pdf3http://www.aegis-elec.com/products/watec-100N spec eng.pdf


http://www.aegis-elec.com/products/watec-902H_spec_eng.pdf

http://www.aegis-elec.com/products/watec-100N_spec_eng.pdf

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Figure 7. Limovie software. An instant of the light curve analysis is shown. The red circleselect the area where the light counts are recorded in function of the time. The pixels involvedin this area are πr2 = 380, where r is the settable radius of the red circle. The two blue sectionsof circle select the area where the background noise is recorded. On the right panel a 3D graphshows the intensity of the bead. This is useful in the choice of the radius of the red circle andin the evaluation of the saturation of the CCD.

Figure 8. Light curve of the disappearing bead at AA = 177◦. Increasing frames correspondto a decreasing time. The velocity of acquisition is 30 frames per second. The counts on the yaxis are the sum of the intensities of 380 pixels involved in the bead. Each pixel has 256 levelsof intensity (8 bits). A detail of the light curve is zoomed. The polynomial fits are shown. Thefit number in the right panel is the polynomial order.

4.1. Light curve analysis

Two beads located at AA = 171◦, 177◦ are analyzed for both the videos. Theyare disappearing for Nugent’s video and appearing for Tegtmeier’s video. For theanalysis of the light curve it is used a software specially realized for this purpose:the Limovie free software4.

The standard deviation σ of the background noise is calculated. Then 5σ aresubtracted to the light curve. The light curves of the disappearing beads areplotted back in time. The first positive value is considered the first signal. Thenegative values are set equal to 0. Then some polynomial fits are performed inorder to smooth the electronic noise (see Fig. 8).

4http://www005.upp.so-net.ne.jp/k miyash/occ02/limovie en.html


http://www005.upp.so-net.ne.jp/k_miyash/occ02/limovie_en.html

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Figure 9. The lunar valley at AA=177◦ is divided into 21 layers. The thickness of each layeris 30 mas.

4.2. The thickness of the layer

The lunar valley analysis is performed with the new lunar profile obtained bythe laser altimeter (LALT) onboard the Japanese lunar explorer Kaguya5. Theradial topographic error is estimated to be 4.1 m corresponding to 2.1-2.3 masat the distance of the Earth.

The lunar valley has to be divided into layers as explained in section 3.The layer at the bottom of the valley (A1), is special because it defines thethickness of all the layers: its area, being more in contact with the lunar profile,is the most uncertain; moreover according to the algorithm in section 3.2, A1 isaffecting all Bn determinations. The choice of the thickness of the layer has tobe optimal: large enough to reduce Bn uncertainties, but small enough to havea good resolution of the LDF. We chose h = 30 mas for the lunar valley at AA= 177◦ and h = 73 mas for the lunar valley at AA = 171◦.

4.3. The motion of the solar limb

The motion of the Sun in the lunar valley is simulated with the Occult 4 software.Being ωy the vertical velocity of the solar limb with respect to the lunar limband being T the duration of the light curve considered, every layer is filled int = h/ωy and the number of the layers is N = int(T/t).

As an example for the Nugent’s video, from the bead at AA = 177◦ we obtain:ωy = 27.4 mas/s; t = 1.11 s; T ≈ 23.5 s; N = 21.

4.4. Results

A program in Fortran90 is performed to calculate the LDF points (Bn). Theprogram takes into account the polynomial fits of the light curve and 3 values

5http://wms.selene.jaxa.jp/selene viewer/index e.html


http://wms.selene.jaxa.jp/selene_viewer/index_e.html

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Figure 10. The luminosity profiles obtained are plotted and put together. The inner andbrighter parts are obtained from Tegtmeier’s video; the outer and weaker parts are obtainedfrom Nugent’s video. Panels (a) and (b) show respectively the luminosity profile concerningthe bead at AA = 177◦ and AA = 171◦. The luminosity profiles are normalized to the centerof the solar disk according to Rogerson (1959) for the inner parts, and in arbitrary way for theouter parts. The zero of the abscissa is the position of the standard solar limb with a radius of959.63 arcsec at 1 AU. The error bars on y axis are the 90% confidence level. The error barson x axis are the thickness (h) of the lunar layers. The solid line is an interpolation betweenthe profiles and gives a possible scenario on the position of the inflection point.


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for each lunar layer area (A, A+∆A, A-∆A), giving a distribution of values foreach point Bn. Figure 10 shows the results.

The resulting points show that the inflection point is clearly between thetwo profile obtained for each of the two beads. The saturation of the CCDpixels avoided to measure the luminosity function more inward for Nugent’svideo, while the low sensitivity avoided to measure the luminosity function moreoutward for Tegtmeier’s video. Therefore from these points it is impossible toinfer an exact location of the inflection point, but it is possible to deduce an upperand lower limit corresponding to the points that better constrain it: -0.190 arcsec< ∆R < +0.050 arcsec.

Another result comes out: as the external profiles are completely outside ofthe inflection point, it means that the CCD has captured light 0.6 arcsec (Fig.10a) and 0.75 arcsec (Fig. 10b) outside the edge of the sun. These are indeed thelengths of the two outer profiles. With respect to the standard range we observelight up to 0.85 arcsec externally to it (Fig. 10b).

The same eclipse was recorded by Adassuriya, Gunasekera, and Samarasinha(Sun&Geosphere 2011, in press). The video analysis they performed (with theclassical approach) led to a value of ∆R = 0.26 ± 0.18. This result doesn’tseem compatible with the possible range for the position of the inflection pointwe found. This shows that the solar radius defined by the classical method isdifferent than that defined by the inflection point of the LDF.

5. Historical eclipses

Total eclipses have been observed even in times prior to the birth of the telescope.Even if made with the naked eyes, their observations are of great interest forstudying the variations of the solar diameter over periods longer than a solarcycle.

The results for the following historical eclipses show a solar radius significantlylarger than the standard one, but they must be reconsidered in the light ofthe results obtained in the previous section: determination with the ”classicalapproach” can be misleading.

5.1. Clavius, 1567, Rome

Stephenson, Jones, and Morrison (1997) studied the observation of an annulareclipse made by the Jesuit astronomer Christopher Clavius in April on 9, 1567in Rome, they derived limits to the Earths spin rate back to that time. Theyattributed the appearance as a ring of the annular eclipse to the effect of the”inner corona” of the Sun. If the ring of that annular eclipse was instead a solarlayer inner to the inflection point position, the average angular radius of the Sunwould have been some arcsec larger than its standard value of 959.63 arcsec.Figure 12 shows that for the solar limb being higher than the mountains of theMoon, it should be ∆R > +4.5 arcsec.

Because this observation was made with naked eyes, a more careful studyon this eclipse has to take in account the angular resolution of an eye pupil.


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Figure 11. Eclipse of April 9, 1567 simulated with Occult 4.0 software. View from CollegioRomano (lat = 41.90◦ N, long = 12.48◦ E) where probably he made his observation. Thelunar limb’s mountains are plotted in function of the Axis Angles (the angle around the limbof the Moon, measured Eastward from the Moon’s North pole). 1◦ in the Axis Angle (abscissa)corresponds to 16 arcsec at the mean lunar distance. The solid line under the mountains is thestandard solar limb. The figures are the Northern (left) and the Southern (right) semicircle.

According to the formula for the angular resolution ρ = 1.22λ/D, and takinginto account a pupil diameter D ∼ 2 mm (day vision), one gets ρ ∼ 50 arc-sec. Details smaller than 50 arcsec are not visible on the Moon profile. Thismeans that the ring of the annular eclipse could have not been complete butdivided by mountains no more than 50 arcsec, that is ∼ 3◦ in Axis Angle. Forexplain the observation of a complete ring with naked eye, for this eclipse, isthus sufficient ∆R ∼ +2.5 arcsec, that remains a surprising value. Accordingto this proof of a larger solar diameter Eddy et al. (1980) assumed a secularshrinking of the Sun from the 17th century to the present. The observation ofClavius was the subject of several studies and publications: even Kepler askedClavius to confirm it was a solar ring, rather than diffuse appearance, that hewould attribute to the lunar atmosphere, but Clavius always confirmed what healready wrote (Clavius, 1581). The interpretation for an annular eclipse is stilldebated in scientific publications.

5.2. Halley, 1715, England

Edmund Halley attempted to measure the size of the umbral shadow by observ-ing the total eclipse of 1715 in England. Halley as secretary of the Royal Astro-nomical Society collected the numerous reports of this observation (Halley, 1714).His idea was to associate the duration of the eclipse with the position for eachobserver, in order to assess the size of the shadow of the Moon on the Earth.But from these observations we can obtain also interesting informations aboutthe solar diameter.

In the present work, thanks to the Occult 4 software and the new data onlunar profile we are able to reanalyze the 1715 eclipse data. In particular weconsider the observations on the Southern and Northern path of totality, wheretimings are not required to deduce the diameter of the Sun. We simply requireto know where the observers were and if they saw a total or a grazing eclipse toinfer an upper or a lower limit of ∆R from each eyewitness.

Table 1 shows the observations we consider. The first is located in the North-ern limit of the shadow, the second and the third in the Southern one. Accordingto Occult 4 we have from the observers in the Southern limit: +0.85 arcsec


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Table 1. Eclipse in 1715, England. Observation in the Northern (Darrington) andSouthern (Bocton Kent and Cranbrook Kent) limit of the umbra shadow. Positioncoordinates by Dunham et al. (1980).

Location of Position How appeared the Sun Lunar - Solar

observation coordinates in the instant of limb position

maximum occultation by Occult 4

Darrington 53◦, 39’, 50.4” N ”Point like Mars” + 0.34 arcsec

358◦, 44’, 09.6” E

Bocton Kent 51◦, 17’, 16.8” N ”Point like Star” + 0.85 arcsec

0◦, 56’, 16.8” E

Cranbrook Kent 51◦, 06’, 03.6” N ”Duration instant” + 0.94 arcsec

0◦, 31’, 44.4” E

< ∆R < +0.94 arcsec. The error, due to the uncertainties on the observation’scoordinates is 0.2 arcsec.

Because of the uncertainties on ephemeris, the center of the solar disk couldhave been in a different position with respect to the simulated standard Sun.Thanks to the observation in the Northern limit, an eventual error on ephemeriscould be bypassed, obtaining a larger gap: +0.59 arcsec < ∆R < +1.28 arcsec.The lower and upper limit are obtained moving the center of the solar disk in theNorth-South direction till the positions where the eyewitnesses are still valid.

Both the evaluations of the historical eclipses are made without any referenceto the inflection point, and considering the LDF as a heaviside profile. This canenlarge the measured ∆R. It is the main concern with naked eye observations.

6. Conclusions

In this study it is taken into account the potentiality of the observation of eclipsesin defining the luminosity profile of the edge of the Sun. Huge developments inthe method and means have been made, from the consideration of the Baily’sbeads, to the recent mapping of the lunar surface by the satellite Kaguya.

A further improvement takes place in this study, considering the bead as alight curve forged by the LDF and the profile of the lunar valley. A first applica-tion on two beads of the annular eclipse in 15 January 2010, has been describedin this study. We obtained a detailed profile, demonstrating the functionalityof the method. Although it was impossible to observe the inflection point, aconsideration is coming out: the shape of the LDF could enlarge or reduce themeasured solar radius depending on the optical instruments. This could be agood track for the investigation of the enigmatic eyewitnesses of the historicaleclipses. The detection of the LDF, in absolute magnitude in the V bandwidth,is an important future task for considering the effect of the outer regions of thesolar atmosphere for naked eye observations.


Raponi et al.

Moreover the LDF profile is the best source of information for the solar atmo-sphere. Having the LDF profile in different wavelength, the different inflectionpoints position resulting from each wavelength are useful in order to constrainthe models (Thuillier et al., 2011).

Many possible improvements to obtain a detailed profile comparable to thespace astrometry are possible:

• An increased dynamic range of the CCD or CMOS detectors from 8 bitscorresponding to 256 levels to 12 bits, which correspond to 4096 level of de-tectable intensity, is recommended. In this way we can extend the samplingof the luminosity function to regions of the photosphere more internal andluminous than the inflection point, without being saturated before it.

• A right setting of the sensitivity of the optical instruments is necessaryto sample the luminosity function in regions outer and weaker than theinflection point.

• As stated in section 2.3, the profile is highly dependent on the wavelength.One must then observe at specific photometric bands to be able to comparethe results with other measures. The current space mission Picard is a goodopportunity to compare the results. It is therefore recommended observa-tions of the next eclipses at the same wavelengths at which the satellitePicard works (535, 607, 782 nm)6.

• The ratio signal/noise is better for a total eclipse (the eclipse of this studywas annular).

The monitoring of the Sun is attracting increasing interest in recent yearsmainly because the expected maximum of solar activity in cycle 24 is in delayand rather low. It is possible that the Sun slips back into a new grand minimumlike the Maunder one, and this could have interesting implications on Earth’sclimate (Lockwood, 2010).

A variety of observative methods were employed to monitor the profile of thesolar limb. In this contest the IOTA group could give an important contributionthanks to the new approach proposed.

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the measurement of solar diameter and limb darkening function with the eclipse observations

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