NOTE: This post documents our first attempts to measure the solar radius during solar eclipses. We have improved on that initial methodology, and details on its latest incarnation can be found here: https://iopscience.iop.org/article/10.3847/1538-4365/ac1279/meta
Timing of the Internal Contacts of the 2013 Nov 03 Total Solar Eclipse by Flash Spectrum Analysis
by Luca Quaglia, Konstantinos Emmanouilidis and John Irwin
1) Gather experimental data to support the improvement of accurate predictions of local circumstances.
2) Assess if the measurement of internal contact times is a viable technique to infer the irradiation-less
semidiameter of the Sun at unit distance .
CURRENT ACCURACY OF THE PREDICTION OF LOCAL CIRCUMSTANCES
The total solar eclipse was observed under pristine sky conditions in Benguié IV (Gabon) at the following WGS84 coordinates (obtained by averaging GPS readings, estimated accuracy < 10 m):
To compute the prediction of local circumstances for the observing location, we have used two excellent and accurate eclipse prediction applications [Xavier Jubier’s Eclipse Maestro  (Source 1) and Fred Bruenjes’ Eclipse Orchestrator  (Source 2)] and also relied on some very detailed computations by John Irwin (Source 3). Source 1 and Source 2 follow the classical procedure of computing nominal contact times from Besselian elements and then calculating lunar limb corrections. Source 3 uses an innovative 3D geometric shadow projection algorithm which directly provides lunar limb corrected contact times. Table 1 shows the computed lunar limb corrected internal contact times and eclipse duration for the observation location according to the different sources.
The three sources use slightly different data sets to derive lunar limb profiles: Source 1 and Source 2 use NASA LRO/LOLA RDR and JAXA Kaguya data and Source 3 NASA LRO/LOLA LDEM 128 (version 2014-09-15) data. Source 1 and Source 3 use a lunar limb profile calculated for the lunar range and libration values at the observing place which accurately reproduces the topocentric perspective view of the lunar surface. Source 2 has a database of lunar limb profiles at discrete lunar libration (L and B) values. Once the lunar libration values for the observing place are computed, the lunar limb profiles for the closest values of lunar libration are extracted from the database, interpolated and the resulting lunar limb profile used to compute the lunar limb corrections. The lunar limb profile used in Source 3 has also a very fine angular discretisation providing altitude values over the reference lunar profile every 0.02° around the limb, while the profiles used in Source 1 provide altitude values every 0.05° and the ones used in Source 2 every 0.2°. From the data in Table 1 we can estimate the internal accuracy of local circumstances predictions to be of the order of 0.1-0.2s.
The high resolution lunar limb profile used by Source 3 was created using a simple highest-point binning scheme. It is known that this process tends to overestimate the limb height, particularly in regions of the profile containing rapidly changing elevations. However, an asymptotic analysis of profiles with a wide range of resolutions has shown that the mean excess elevation is only 14m (0.008″) for a profile spacing of 0.02°. This excess elevation corresponds to a change of contact time of about 0.03 s. Therefore, given also that the profile was created for the specific lunar range and libration angles appropriate for the observation point given above, we can confidently say that the profile is robust enough to be used in making contact time predictions which aim for an accuracy of 0.1 s. Links to limb profile data for this eclipse can be found in the URLs section at the end of this post. In contrast, for a profile with 0.2° spacing (as commonly used), the mean excess elevation is some 158m (0.088″). This can lead to contact time errors as large as 0.25 s unless the profile creation process is amended to reduce or eliminate biases introduced by the binning operation.
SENSITIVITY TO THE SOLAR RADIUS OF THE PREDICTIONS OF LOCAL CIRCUMSTANCES
The value of the irradiation-less semidiameter of the Sun at unit distance is one of the fundamental constants that goes into the computation of Besselian elements. The standard value used in eclipse computations is the one given by Auwers in 1890: = 959.63″ . Source 3 uses a solar radius of 696000 km corresponding to = 959.64″ (the 0.01″ difference with the Auwer’s value only shifts contact times by less than 0.03 s in the case of this eclipse). It has been shown  that the value of might slightly fluctuate over time (of the order of 0.05-0.15″) . The usual technique used at solar eclipses to detect this phenomenon is to time the appearance and disappearance of Baily’s beads from the grazing region of the totality zone. But because a change in directly affects the internal contact times, accurately measuring internal contact times could be another viable way of measuring the radius of the Sun at total solar eclipses. To have a feel for the accuracy needed to detect variations in , we computed by how much the nominal internal contact times would shift if was different from the standard Auwers’ value. Figure 1 shows the result of this computation: on the x axis we have the difference (expressed in arcseconds) and on the y axis we have the shift in internal contact times (expressed in seconds) compared to the ones computed with .
The computation shows that internal contact times are very sensitive to the value of : a 0.1″ change in solar semidiameter (at unit distance) causes a shift of around 0.3 s in contact times.
If we could refine the algorithm to compute local circumstances so to reach 0.1 s accuracy and if we could experimentally measure internal contact times with the same accuracy, we could, at least in principle, rather precisely determine the actual value of the irradiation-less semidiameter of the Sun at the time of a total solar eclipse.
AN EXPERIMENTAL TECHNIQUE TO ACCURATELY TIME INTERNAL CONTACTS
The internal contacts of a total solar eclipse are defined as the instants of photospheric disappearance and reappearance. By measuring the time evolution of the light intensity coming from the last (first) Baily’s bead and also a bit after (before) disappearance (reappearance), the moment of photospheric extinction could in principle be determined quite accurately. Usually at the time of second (third) contact, the last (first) Baily’s bead disappears (reappears) at the location of a valley of the lunar limb. Figure 2 shows (in a simplified way) the different light sources contributing to the light intensity at that location. There is the bright photosphere, a fainter thin transition layer, the mesosphere , then the much fainter chromosphere which transitions into the inner corona.
In order to disentangle the contribution of the different light sources, it has been proposed by Kristenson , by Kubo  and more recently by Koutchmy’s team  to record the time evolution of the eclipse flash spectrum in order to determine the internal contact times. The video recording needs to be accurately time stamped in UTC time. Figure 3 shows an example of spectrum of the solar light just before second contact. The spectrum of the light coming from the last two Baily’s beads (double diamond ring) is continuum and contributes to the flash spectrum with two bright multicoloured lines. Superposed to this spectrum (to both bright lines), there is the spectrum of the mesosphere which is formed by a forest of faint emission lines in the shape of tiny arcs. The spectrum of the chromosphere is discrete and formed by well defined long arcs. The spectrum of the corona is diffuse and very faint compared to the other three.
The idea of the method is to record a time stamped video of the flash spectrum and to observe the evolution of the different spectra. Second contact will be heralded by the disappearance of the photospheric continuum and third contact by its reappearance. The advantage of using the flash spectrum is that it separates the different light sources contributions.
The experimental setup used to record a time stamped video of the flash spectrum of the 2013 November 03 total solar eclipse is shown in Figure 4 in schematic form. The spectrum of the solar light is created by a transmission diffraction grating film (500 lines/mm) and imaged by a video camera (Canon MV600i) fitted with a wide angle lens. The analogue video signal is then sent to a video time inserter (Kiwi OSD) which accurately time stamps every video frame with UTC time. The time stamped video is then recorded by another video camera (Canon MV600i).
Using an analogue video signal was essential to ensure the utmost accuracy in the time stamping. The first video camera can output (through an analogue AV port) the captured signal with almost no delay and this signal can immediately be processed by the video time inserter. The Kiwi OSD acquires its time base from the 1 peep-per-second (1 PPS) time signal from a GPS unit (Garmin 18X) and can timestamp each frame with UTC time with an accuracy of around 1 ms. Digital video signal are often outputted with unavoidable time delays, due to internal processing, that can amount to 50-100 ms or even more. Figure 5 shows an example of a time stamped frame extracted from the eclipse flash spectrum video. The video is very low resolution but it allows detecting the main chromospheric emission lines and a continuum/diffuse spectrum which is a superposition of the photospheric, mesospheric and inner coronal spectra.
ECLIPSE FLASH SPECTRUM VIDEO DATA ANALYSIS
The flash spectrum video was analysed by choosing small areas in each frame, integrating the intensity of the pixels in each area and obtaining light intensity curves as a function of time. The areas chosen are in between chromospheric lines in order to eliminate the chromospheric contribution to the light intensity curve. The coronal contribution is almost constant and will appear as a constant background intensity in the curves. Due to the movement of the lunar limb, the photospheric spectrum will change in intensity depending on the time the frame was taken. Very close to the solar limb the brightness of the photosphere falls considerably (a phenomenon known as ”limb darkening”) and consequently we expect to see, in the light curves around second and third contact, a change in derivative in the light curve intensity. The forest of mesospheric emission lines cannot be resolved in the frame shown in Figure 5 and distinguished from the photospheric spectrum. However, the photospheric spectrum is very bright while the mesospheric spectrum is much fainter. Even in the presence of limb darkening, we do not expect to see a contribution from the mesosphere in the light curves because the mesosphere is very much fainter. We have taken the time when the light curves go flat for the first and last time as signalling the occurrence of second and third contact, respectively. This is just a working definition to analyse the low resolution flash spectrum video. We need to note that very accurate spectroscopic studies have highlighted that the solar limb is not razor sharp, so its precise location remains somewhat uncertain  . The excellent software LiMovie  was used to extract the light curves from the flash spectrum video. Figure 6 shows the result of this analysis: the light intensity curves around the internal contact times for different spectral regions. We need to stress that the video is very low resolution so the measurement is likely affected by a 0.2-0.3 s uncertainty.
In Table 2 the results of these measurements are compared with theoretical computations from the three sources of predictions mentioned previously.
The difference between observed and predicted internal contact times is striking. The measured time of second contact is around 1 second later than predicted while the measured time of third contact is around 1 second earlier. Hence the measured duration of totality is about 2 seconds shorter than theory predicts using the standard solar radius.
ESTIMATION OF THE SOLAR RADIUS
We can estimate the value of the solar radius compatible with the measured internal contact times by adjusting the value of in the computations so as to reproduce the empirical values. Table 3 shows the results of a parametric study by John Irwin using very precise and detailed computations of the local circumstances. The contact times and duration were calculated using a higher internal resolution and the output was rounded to the nearest tenth of a second.
Table 3 enables us to see that the value of the solar radius compatible with the empirical contact times should be in the range 959.90-960.00″. A specific analysis of second and third contact yields the following values:
Despite the low resolution of the flash spectrum video and the consequent uncertainty in the determination of , the measurement of contact times supports a value of the solar radius Σ substantially greater than the traditional Auwers’ value which is still the standard one in use nowadays.
The internal contact times of a total solar eclipse were measured and compared with theoretical calculations. A noticeable discrepancy between measured and predicted internal contact times has become apparent. Because most other sources of uncertainty in the predictions (especially the uncertainty due to the topography of the lunar limb) have been drastically reduced in recent years, the source of the discrepancy needs to be found in the assumed value of the solar radius. The observation supports a value of the solar radius significantly larger than the standard value: .
The next iteration of this experiment is to vastly improve the resolution of the flash spectrum video while keeping the same UTC time stamping accuracy . We are confident that with this improvement, reaching a 0.1 second accuracy in the measurement of internal contact times is achievable. We are aiming to benefit from the developments of DSLR technology and to use the high resolution video recording capabilities of modern digital cameras. Besides the higher resolution, these cameras are capable of shooting videos at a very high frame rate which should help us to increase the time resolution. Moreover, by collecting several flash spectrum videos from various locations, we can reduce the uncertainty in the derived value of the solar radius by taking advantage of statistical averaging.
We would like to thank Xavier Jubier and Fred Bruenjes for providing their lunar limb data, Alessandro Pessi and John Tilley for useful technical discussions and Michael Seeborger-Weichselbaum, for being a great travel companion during the very eventful eclipse expedition to Gabon.
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 H. Kristenson. “The Precise Timing of the Total Solar Eclipse June 30, 1954”. In: Arkiv for Astronomi 2.29 (1960)
 Y. Kubo. “Position and Radius of the Sun Determined by Solar Eclipses in Combination with Lunar Occultations”. In: Publications of the Astronomical Society of Japan 45 (1993), p. 819
 K. Miyashita. LiMovie. 2014. URL: http://www.astro-limovie.info
 A. Raponi et al. “Solar Limb Darkening Function and Solar Diameter with Eclipse Observations”. In: EAS Publications Series 55 (2012), p. 389
 F. Suits. Methods for Timestamping Analog and Digital Video. 2014. URL: http://www.altazinitiative.org
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