Estimation of the Eclipse Solar Radius by Flash Spectrum Video Analysis

During the last 50 years several methods have been proposed and applied to measure the solar radius by taking advantage of the favorable conditions occurring during total solar eclipses. As highlighted in Lamy et al. (2015) the occultation happens in space and hence it is free of atmospheric effects. As measurements are performed very close to the internal contact times, when the solar disk is almost entirely occulted by the Moon, the glare of the photosphere is minimal and the level of scattered light becomes negligible. Finally, the Moon provides a reliable and long-term reference, independent of any specific instrumentation. One drawback is that total solar eclipses happen infrequently, they are sometimes marred by unfavorable weather, and they only provide one chance to collect data. The main characteristics of each method are summarized in Table 1.

The idea of using the flash spectrum to time the contacts of total solar eclipses can be traced back to the 1950s, when Kristenson applied it during the 1954 June 30 total solar eclipse (Kristenson 1960) in the context of geodetic investigations to measure the size of the Earth. Kristenson collected a video of the flash spectrum and extracted light curves of photospheric light corresponding to the last Baily's beads before totality and to the first Baily's beads after totality. By looking at the inflection point of these light curves he was able to estimate the time of second and third contact.

A more extensive use of Kristenson's technique aimed at measuring the solar radius was carried out in the 1970s and 1980s by the Japanese Hydrographic Department (Kubo 1993). Once the contact times have been obtained, the solar radius can be estimated by fitting eclipse computations to reproduce them. Kubo et al. observed several eclipses, at times near the centerline and at other times not far from the umbral shadow path limits, and estimated values for the solar radius in the range [95974, 95988], values systematically larger than the standard value of 95963 used for eclipse computations.

A long-running project to measure the solar radius by observing total and annular solar eclipses from the limits of the umbral shadow path (Fiala et al. 1994) is being carried out by the International Occultation Timing Association (IOTA). Their results show tantalizing evidence of fluctuations of the solar radius over time. Positioning close to the limit has the advantage of enhancing all transient phenomena. The main technique they use is to time the disappearance and reappearance of Baily's beads by recording time-stamped videos of filtered telescopic views of the eclipse (Nugent 2007). Based on eclipses observed from the 1990s, IOTA estimated values for the solar radius in the range [95640, 95680] (Dunham et al. 2016).

One of the most comprehensive recent experimental efforts to measure the solar radius is reported in Lamy et al. (2015). By measuring light intensity curves collected with photometers around the time of second and third contact and by simulating them by integrating the limb darkening function (LDF), that study has possibly obtained the most robust estimate of the solar radius S_⊙ so far: 95999 ± 006. The observations were conducted both at the centerline and near the umbral shadow path limits over the course of four eclipses.

Our group attempted contact timing by flash spectrum analysis during the total solar eclipses on 2009 July 22 (China), 2010 July 10 (Cook Islands), and 2012 November 13 (Australia). Several factors, mainly unfavorable weather, prevented the gathering of usable data. Finally, during the 2013 November 3 annular–total solar eclipse (Gabon), under almost perfect sky conditions, we succeeded in collecting a time-stamped low-resolution video of the flash spectrum and we presented the results at the 2014 Solar Eclipse Conference in New Mexico, USA. On all those occasions we were stationed close to the centerline. Appreciating the advantages of collecting data from the limit of the umbral shadow path, we decided to target the southern limit for the 2017 August 21 total solar eclipse.

The main objectives of this work are to present an estimate of the value of the eclipse solar radius during the 2017 August 21 eclipse and more broadly to outline a technique that could be used at future eclipses to build a data set of eclipse solar radius measurements. This technique brings together several building blocks:

• 1.  
  eclipse flash spectrum video analysis
• 2.  
  observing from the limits of the eclipse umbral shadow path
• 3.  
  precise eclipse computations
• 4.  
  light-curve simulations

These individual building blocks have been used in the past (see Table 1) with the aim of measuring the eclipse solar radius but we are not aware of their being used all together. Each building block has a strong aspect and their synergy produces a measuring procedure that can be implemented with modest equipment and can be undertaken by competent amateurs, at least where it concerns the first two building blocks.

In the methodology section we will extensively discuss each of those building blocks and how they come together into a coherent procedure. In the results section we will first present an account of a naked-eye view of the eclipse as seen from the limit of the eclipse shadow path. This description will help with a general understanding of the evolution of the flash spectrum. Then we will provide an initial estimate of the eclipse solar radius by estimating the duration of totality based only on visual inspection of the flash spectrum video. This will provide clear evidence that Auwers' radius (Equation (1)) is notably too small to reproduce the observations. Finally, we will estimate the eclipse solar radius with a more quantitative procedure based on extracting light curves from the flash spectrum video and matching the simulated light curves with the observed curves.

A video ⁵ of the flash spectrum was recorded ⁶ using a Canon EOS 6D digital camera with a Canon EF 70–200 mm f/4L USM lens. The focus and the exposure were both set to manual and we chose an exposure time of 1/60 s at an ISO setting of 100. A transmission diffraction grating, etched with 235 lines mm^–1, was mounted in front of the lens to separate the eclipse light into its spectral components.

The camera produces compressed video using the AVC/H.264 codec operating on full-range [0, 1] image data in the Y'UV color space with 4:2:0 chroma subsampling. The resulting video stream is stored within a MOV media container. The video has a frame size of 1920 × 1080 pixels and a frame rate of 23.976 fps.

DSLR cameras are not specifically designed for extracting light curves but they are relatively inexpensive and readily available and they have been used successfully for astronomical photometric work (AAVSO 2016). An important aspect to highlight is that the internal CMOS sensor is linear for intensities away from saturation and above dark current noise (Park et al. 2016). These are indeed the characteristics of the video from several tens of seconds before totality to a couple of tens of seconds after totality, which is the period of interest for the video analysis.

However, one other aspect that needs to be accounted for is the presence of a nonlinear gamma adjustment in the color space of the images stored in the video stream. This adjustment originates in the camera's processing of the image data through the sRGB color space prior to their conversion to Y'UV (Canon 2017). The transformations required to undo this adjustment and obtain linearized luminance information are described in Section 3.3.

Watching this video of the flash spectrum is an integral part of reading this article. We invite the reader to watch the video several times—it is only by going back and forth between the details presented in this article and the video that our analysis will fully take shape.

The video of the flash spectrum was recorded during the 2017 August 21 total solar eclipse from a location just south of Vale, Oregon, USA, with WGS84 geodetic coordinates

$$\begin{eqnarray}\begin{array}{rcl}\lambda & = & 117^\circ \,\,13^{\prime} \,\,09\buildrel{\prime\prime}\over{.} 8\,{\rm{W}}\\ \varphi & = & 43^\circ \,\,57^{\prime} \,\,10\buildrel{\prime\prime}\over{.} 9\,{\rm{N}}\\ h & = & 711\,{\rm{m}}.\end{array}\end{eqnarray} \tag{ 2 }$$

This location was very close to the southern limit of the umbral shadow path of the eclipse as Figure 1 shows.

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Positioning very close to the edge of the umbral shadow path enhances all transient phenomena and greatly increases the sensitivity of the duration of totality to the value of the solar radius. Figures 2(a) and (b) show the predicted duration of totality as a function of the solar radius at the observing site and at a site on the centerline experiencing midtotality at the same time. We can see that, very close to the umbral shadow path limits, the duration of totality is noticeably more sensitive to the adopted value of the solar radius S_⊙ than it is at the centerline. If we compare the durations computed assuming the standard value S_⊙ = 95963 and an increased value S_⊙ = 96000, similar to the one found in Lamy et al. (2015), on the centerline the drop in duration is only 1.8 s while at the observing site the drop is 19.3 s. We note that the bend in the plot in Figure 2(a) is not a discretization error (the data are computed every 001) but an actual feature due to the shape of the lunar limb profile.

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Figure 2(c) also shows the distance between the observing site and the umbral shadow path limit as a function of the solar radius. If S_⊙ = 95963, this distance is ∼1200 m, while with an increased value S_⊙ = 96000, it drops to less than 400 m, a substantial change.

The spectrum of solar light is continuous under everyday circumstances. The photons originating from thermonuclear reactions inside the Sun slowly make their way to the surface and radiate away at all wavelengths (being visible to the naked eye between about 400 and 700 nm). While traveling through the solar atmosphere photons at specific wavelengths are absorbed by specific elements like hydrogen, magnesium, sodium, and iron. These absorption phenomena create dark lines in the continuous spectrum and are hallmarks of the presence of those elements in the solar atmosphere.

The spectrum of solar light changes rather dramatically just a handful of seconds before second contact of a total solar eclipse and reverts back to its normal state just a handful of seconds after third contact. Figure 3 displays an example of such a transition during the 2013 November 3 annular–total eclipse. The usual intense continuous spectrum gives way, for some brief precious moments, to a mesmerizing flash spectrum. When the Moon has almost entirely covered the photosphere, the intensity of the solar continuum plummets rapidly before vanishing and the much fainter light emitted by elements in the chromosphere and the inner corona comes into view. The continuous spectrum with dark absorption lines is replaced by a discrete emission spectrum with a few intense arcs and a myriad of fainter ones.

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The emission arcs emitted by the chromosphere are at their brightest just after second contact. As totality progresses and the Moon covers the chromosphere (at least as seen from around the centerline), these arcs fade away and only the faint and ghostly diffuse spectrum of the corona remains visible. The emission arcs come back toward the end of totality and they intensify to become their brightest again just before third contact. Soon afterward the photospheric spectrum reappears, very rapidly brightens up, and hides the flash spectrum from view until the next total solar eclipse occurs. Traditionally, the name "flash spectrum" has been reserved for the highly dynamic spectrum visible for just a few seconds near second and third contact when the Moon has fully covered the photosphere but has not yet occulted the chromosphere. It should be noted though that during intrinsically brief total eclipses (with a magnitude A very close to 1) the flash spectrum might remain in view for most of the duration of totality, if not for the entire duration (this was the case, for example, for the 2013 November 3 annular–total eclipse).

Figure 4(a) shows the origin of the main emission lines in the flash spectrum. Electronic transitions of various elements, in the neutral state or in ion form, emit photons at specific wavelengths. Prominent are the emission arcs of the Balmer series of hydrogen (Hα, Hβ, Hγ, ...) and some emission arcs from helium, the magnesium triplet, and the sodium doublet. The forest of faint arcs is due to a host of elements like iron, calcium, barium, titanium, and chromium. Superposed on the discrete spectrum is the very faint, almost ghostly circular diffuse spectrum of the inner corona.

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To provide a deeper explanation of the structure of the flash spectrum we need to make some assumptions on the distribution of elements in the chromosphere. We wish to express the caveat that we are not solar physicists and we do not claim to be depicting the most detailed available knowledge. We just want to give an approximate idea of what could be going on in the solar chromosphere to have more tools to understand the flash spectrum. Based on investigations of the chromosphere by Mitchell (1947) we imagine the distribution of elements as in Figure 4(b). We start by pointing out that the location where the photosphere ends and the chromosphere begins is not well defined as it is not precisely known where the top of the chromosphere lies. The Sun is not a solid body and the top of the chromosphere is turbulent and a place of occasional large prominences. Nevertheless, for simplicity's sake we provide some approximate indications of where the light-emitting elements are located. Hydrogen may be present, in probably abundant density, up to a height of ∼12,000 km above the surface of the photosphere; helium, up to ∼8000 km; magnesium, up to ∼3000 km; sodium, up to ∼1500 km; and iron and a host of other elements, up to a height of ∼1000 km. Given these heights, we can infer that, at second contact for example, a wider arc of light will be emitted by hydrogen, extending outward by at least 12,000 km, than by iron, extending outward by only 1000 km. We do not make any hypotheses on the density distribution of specific elements within their respective layers; it is probably inhomogeneous.

From geometrical reasoning we expect to see long arcs at the emission wavelengths of hydrogen: 656.3 nm (Hα, red), 486.1 nm (Hβ, aqua), and 434.0 nm (Hγ, blue). Almost as long is the arc at the wavelength of helium: 587.6 nm (yellow). We expect shorter arcs in correspondence to the magnesium triplet—516.7, 517.3, and 518.4 nm (green)—and to the sodium doublet: 589.0 and 589.6 nm (yellow). Iron and other elements emit a forest of much shorter arcs at a wide range of wavelengths. We need to note that, due to the nature of the electronic transitions involved, not all emission lines have the same strength—some are very intense, like Hα and the helium line at 587.6 nm.

Armed with this knowledge we can try to explain the naked-eye appearance of the chromosphere (refer to the still image at the bottom of Figure 3). As most of the chromosphere above ∼3000 km is dominated by the emission of hydrogen and helium, when we combine the respective red, yellow, aqua, and blue emission lines, we get that deep purple–red color that is so characteristic of many images of the chromosphere. However, much closer to the photosphere, besides the emissions of hydrogen, helium, magnesium, and sodium, we have a myriad of emission lines by iron and other elements. These lines are discrete but they are very densely distributed across most of the visible spectrum. The combined effect of all these emission lines is to generate a whitish light very close in aspect to the light emitted by the photosphere. This light is not to be confused with the remnants of the photosphere.

That part of the chromosphere closest to the photosphere has a whitish color that is distinctly different from the pinkish-purple of the majority of the chromosphere lying higher up. In the bottom frame of Figure 3 we point out these subtle differences in the visible chromosphere—some areas are intense and almost white and others are less intense and more purple–red in color. The corresponding spectrum clearly explains the origin of such differences in color. Although they do not seem to be in common use, we have come across the terms fluctosphere (Wedemeyer-Böhm et al. 2009) and mesosphere (Sigismondi et al. 2012) being used to describe this lowest-lying layer that is the source of the myriad of faint emission lines. Because the fluctosphere/mesosphere emits a quasi-continuum of white light, albeit with fainter intensity than the photosphere, it is troublesome to disentangle the location of the very edge of the photosphere from the mesosphere in white-light images.

The flash spectrum comes to the rescue as it allows, at least in principle, separation of the light coming from the photosphere, the fluctosphere, the rest of the chromosphere, and the corona, as indicated in Figure 4(c). In practice, this statement might seem quite overoptimistic. But it can at least be said that, for the same imaging resolution, the flash spectrum provides a far greater level of detail than the corresponding white-light image (see Figure 3).

The eclipse computational model relies on the latest ephemerides and accounts in a very precise way for all the complexity of the lunar topography and of the orientation of the celestial and terrestrial reference systems via robust algorithms and procedures.

Modeling of the lunar topography is based on the gridded elevation data sets SLDEM-256 and LDEM-128 originating from measurements by the Lunar Orbiter Laser Altimeter (LOLA) on board the Lunar Reconnaissance Orbiter (NASA 2021). The former data set is a special reduction of the LOLA observations in conjunction with photogrammetric data obtained from the Kaguya mission, which resulted in a more accurate and reliable digital elevation model (Barker et al. 2016). But it is available only for latitudes within 60° of the lunar equator; therefore we have used the latter data set to cover latitudes outside this range. These data are referenced to the mean Earth/polar axis (ME) lunar orientation frame as defined for JPL's DE421 planetary and lunar ephemerides (JPL 2021). Additionally, the data reference the center of mass of the Moon so that they can be used directly with the lunar ephemeris.

However, we have used the later DE430 ephemeris to provide better accuracy for the positions of the Sun, Moon, and Earth. We also used the lunar orientation data contained in this ephemeris but note that there is effectively only a few meters' difference between the ME orientations provided by DE421 and DE430 (Archinal et al. 2018), which is negligible for our purposes. We also note that updated ephemerides released by JPL since DE430 do not provide any data for the ME frame, only for the principal axis (PA) frame. The difference between the PA and ME frames is typically around 850 m on the surface of the Moon and cannot be neglected in computing reliable lunar limb profiles.

For Earth's orientation we used the IAU-2006 model, comprising IAU-2006 precession, IAU-2006A adjusted nutation, and IAU-2006/IAU-2000A sidereal time (Petit & Luzum 2010). Combined with measured Earth orientation parameters for UT1–UTC, polar motion, and the celestial pole offsets (IERS 2021), this model provides the orientation of the ITRS relative to the GCRS to better than 1 mas (<0.1 m on the ground). Therefore, given accurate coordinates for the observation site, this modeling ensures accurate topocentric positioning of the Sun and Moon.

The prediction of apparent topocentric positions follows a fully relativistic procedure involving adjustments for (a) light time to the body in question, (b) gravitational bending of the light path by the mass of the Sun, and (c) planetary aberration due to the barycentric velocity of the observer. The latter is used in preference to stellar aberration as it produces, in particular, a more accurate prediction for the distance of the Moon and therefore a better estimate of the apparent lunar semidiameter (Irwin 2019).

Our computational model does not depend, like the traditional method, on Besselian elements (Urban & Seidelmann 2012) to compute nominal predictions and then on applying limb corrections (Herald 1983) to account for the Moon's topography. Instead, it performs its computations directly with the solar limb and the lunar limb profile to find times of contact and other related quantities. No approximating adjustments are involved with these computations.

The way the solar and lunar limbs move with respect to each other determines how the flash spectrum evolves around the times of second and third contact. To assess these dynamics for the 2017 August 21 total solar eclipse as seen from our observing site we have computed accurate time-dependent polynomials providing the apparent topocentric semidiameters of the Sun and Moon and the relative positions of their centers of mass. The topocentric lunar limb profile is also computed. All these data can be derived from the eclipse computational model.

The polynomials are centered at the time

$$\begin{eqnarray}&&{T}_{0}={17}^{{\rm{h}}}\,{25}^{{\rm{m}}}\,55\mathop{.}\limits^{{\rm{s}}}1\,\mathrm{UTC}\end{eqnarray} \tag{ 3 }$$

which corresponds to the time of mid-eclipse at the coordinates given in Equation (2) and S_⊙ = 96000. All polynomials are of the form

$$\begin{eqnarray}&&{c}_{0}+{c}_{1}\,t+{c}_{2}\,{t}^{2}+{c}_{3}\,{t}^{3}\end{eqnarray} \tag{ 4 }$$

where the coefficients c_i are given in Table 2 and t is the time in minutes from the reference time T₀. The polynomial for the apparent topocentric solar semidiameter corresponds to S_⊙ = 96000. For any other solar radius within ∼10'' of this value the polynomial can be scaled by the dimensionless factor

$$\begin{eqnarray}&&\left(\displaystyle \frac{{\text{}}{S}_{\odot }}{960^{\prime\prime} }\right)\end{eqnarray} \tag{ 5 }$$

to give the corresponding semidiameter to better than 0.1 mas. The apparent lunar semidiameter corresponds to a lunar radius of 1738.091 km. The right-handed reference frame for the (x, y) coordinates is the same as that for the lunar limb, with the y-axis pointing in the direction of the lunar north pole.

The lunar limb profile for an observer at the coordinates given at Equation (2) is presented in Figure 5. The mountains and valleys on the lunar limb, despite having a height of just 2''–3'' (compared with the radius of the lunar datum of around 976''), have a great impact on determining the times of second and third contact. The contact angles of second and third contact are always at the bottom of a lunar valley. These two valleys, together with nearby valleys, are the cause of the remarkable Baily's beads seen on the lead-up to second contact and just after third contact. In correspondence to these Baily's beads we can see bands of photospheric continuum in the flash spectrum.

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The position of the solar limb with respect to the lunar limb at second and third contact is presented in Figure 6. Totality starts and ends at the bottom of two valleys in the lunar southern polar region. These two valleys are very close to each other and are separated by a narrow and less deep double valley (indicated by stars in the diagrams). The solar limb passes just below the bottom of this double valley during totality. These three valleys generate some interesting features in the flash spectrum. We expect the photospheric continuum to disappear in the valley labeled C₂ and to reappear in the valley labeled C₃. The photospheric continuum does not shine in the intermediate double valley but the presence of this double valley is seen in the flash spectrum, as we will show later on.

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We can also look at the mutual movement of the solar and lunar limbs on the lead-up to second contact and just after third contact. These dynamics are presented in Figure 7. In this figure we indicate (with the tags B1, B2, and B3) the valleys generating the last three Baily's beads before C₂ and the first three Baily's beads after C₃. About 30 s before C₂ we expect to see three beads. Soon afterward the bead B3 should vanish. By 20 s before C₂ the bead B2 should have also almost vanished. The last inconspicuous Baily's beads will remain for another 20 s, fading away very slowly until the onset of totality. After third contact Baily's beads should reappear at a noticeably faster rate. After 10 s there should already be three beads of moderate size. The photosphere will brighten up into a small arc by 30 s after C₃.

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The computations support the broad evolution of the flash spectrum. We see thin bands of photospheric continuum shrinking and disappearing or reappearing and widening up in correspondence to these Baily's beads. The dynamical simulation especially shows how the last bead before C₂ lingers for so long and fades away so slowly.

Light curves for the last and first Baily's beads can be extracted from the video of the flash spectrum. These observed light curves provide a quantitative measurement that we can attempt to model.

To simulate the light curves we need to determine the area of the photosphere that is still uncovered by the lunar limb. Figure 8 depicts the situation. At any time we can determine the mutual position of the lunar and solar limbs and then the shape of the area ${ \mathcal R }$ over which the computation is performed.

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The light intensity of the photosphere is not uniform but decreases from the center of the solar disk to its limb. The function describing the change in intensity is called the LDF. Its functional form, according to Hestroffer & Magnan (1998), is provided by Equation (6), where ϕ is the angular distance from the Sun's center, Σ_⊙ is the (topocentric) solar semidiameter, and λ is the wavelength measured in micrometers. Figure 9 shows a plot of the LDF for three specific wavelengths—of the Hα line, the main He line, and the Hβ line.

$$\begin{eqnarray}\begin{array}{rcl}{\mathrm{LDF}}_{\mathrm{HM}98}(\phi ,\lambda ) & = & {\left[1-{\left(\displaystyle \frac{\sin \,\phi }{\sin \,{{\rm{\Sigma }}}_{\odot }}\right)}^{2}\right]}^{\tfrac{1}{2}\alpha (\lambda )}\\ \alpha (\lambda ) & = & -0.023+\displaystyle \frac{0.292}{\lambda }.\end{array}\end{eqnarray} \tag{ 6 }$$

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The numerical integration is set up and performed in Wolfram Mathematica (Wolfram Research, Inc. 2018). It uses the time polynomials, computed previously, describing the apparent topocentric lunar and solar semidiameters and the (x, y) coordinates of the Sun's center of mass (it is also the origin of the LDF). By fixing a specific time t and combining those polynomials and the functional form of the LDF, we can model the light intensity originating from the uncovered area ${ \mathcal R }(t)$ of the photosphere by computing the integral

$$\begin{eqnarray}&&I(t)={\iint }_{{ \mathcal R }(t)}{\mathrm{LDF}}_{\mathrm{HM}98}\,d{ \mathcal R }.\end{eqnarray} \tag{ 7 }$$

Integrations are performed separately for the light curve corresponding to the C₂ valley and to the C₃ valley. The computations are done for a set of times broadly spanning the lead-up to totality, totality, and afterward, at 1 s intervals.

We would like to highlight the fact that the limb of the photosphere is considered to be sharp in these calculations. This is not physically true but all eclipse computations make that implicit assumption.

The progression of the eclipse at the limit of the umbral shadow path is markedly different from the progression of the eclipse near the centerline. We would thus like to give an account of the experience of totality from the limit because, besides its being interesting in itself, we are only aware of a few such descriptions in the literature and it will offer some useful insights into interpreting the dynamics of the flash spectrum.

The experience that most eclipse chasers (almost always stationed close to the centerline) have of the last minute before second contact is of a rather rapid and accelerating sequence of phenomena—the level of ambient light decreases faster and faster, the lunar shadow materializes more and more noticeably and comes rushing toward the observer, the photospheric crescent shrinks and breaks into rapidly vanishing Baily's beads, often one last bright point of light shimmers past the lunar edge at the same time as the solar corona fully coming into view, and then totality begins. The preceding dramatic and rapid progression of events is followed by the rather calm, albeit spectacular, phase of totality, which evolves far more subtly.

From the limit this familiar experience is upended and turned into a rather alien experience. We would like to describe what we saw from observing the 2017 August 21 total solar eclipse from just a few hundred meters inside the southern limit of the umbral shadow path. A couple of minutes before second contact, the progression of the eclipse started diverging from the common picture usually reported by seasoned observers. The ambient light level decreased organically without noticeable acceleration—it was as if the entire sky, almost without anyone noticing, had taken a deep purplish-blue tinge. Instead of an acceleration of transient phenomena, the exact opposite occurred—a deceleration of transient phenomena. The mad rush of the breaking of the photospheric crescent into rapidly vanishing Baily's beads that happened over 10–15 s at the centerline got extended to over a minute or more. From our observing site we observed for 30–35 s, with the naked eye and without eclipse glasses, three large Baily's beads slowly evolve, shrink, and vanish on the lead-up to second contact. Initially they were as bright as Venus. Then they were only as bright as Jupiter. Then they became less and less bright. Two Baily's beads evolved and vanished at an average rate but the last one lingered on for 10–15 s after the other ones had vanished. During this gentle and subtle evolution of Baily's beads, the most striking and totally unexpected feature we observed was that the whole corona had already fully come into view—not just the inner corona but also the outer corona extending out to several solar radii. We were busy setting up the camera so we do not know exactly when the full corona appeared but it was at least 35–40 s before second contact (when we left the camera running by itself and we were just observing). The view of the slowly evolving Baily's beads on the backdrop of the full solar corona was absolutely mesmerizing and enthralling. The last Baily's bead did not vanish with a bang in a diamond ring effect. It just faded away very slowly. There was no noticeable brightening of the solar corona after the disappearance of the last Baily's bead. The full corona was just there all along. After around 10–15 s, during which the photosphere was totally extinguished, new Baily's beads came back into view, more rapidly than the old beads had faded away before second contact. Again, no noticeable change was detectable in the brightness of the corona. After another 10–15 s of unfiltered naked-eye observation we averted our gaze and put on eclipse glasses for eye safety.

As shown in Figure 6, the very different experience of totality from the edge is due to the peculiar way the lunar and solar limbs are traveling with respect to each other. They stay very close to each other and they move almost tangentially to each other.

We would like to highlight something that will seem so trivial to many eclipse chasers but will illuminate understanding of the experience of a total eclipse from the edge of the umbral shadow path. We know that the solar corona is there at all times, but the brightness of the sky is almost always so intense that it hides it away from view. When the Moon completely occults the photosphere, the brightness of the sky becomes lower than the brightness of the solar corona and the corona becomes visible. We usually define the period when the photosphere is completely out of view (photospheric extinction) as totality and approximately identify it with the period when the solar corona is fully visible. Near the centerline this is basically the case as the solar corona only fully emerges into view a couple of seconds before second contact and almost entirely disappears out of view a few seconds after third contact. The photosphere is too intense at all other times to allow a clear view of the corona. We can try to artificially cover the reemerging photosphere just after third contact and still see some traces of the corona but that is dangerous and unsafe. In contrast, very near the edges of the umbral shadow path, this is not really true anymore. As we have described, by observing from just a few hundred meters within the southern limit, despite the period of photospheric extinction being only less than about 15 s long, the solar corona (including the faint outer corona) was fully visible for about 50–60 s. That means that the brightness of the sky had dropped below the brightness of the outer corona for far longer than the mere period of totality. We feel that we should coin a new term, coronality, alongside totality. Coronality is the period when the brightness of the sky falls below the brightness of the outer corona. Totality is the period when the Moon completely occults the photosphere. On the centerline (and almost anywhere else in the umbral shadow path), the two terms are almost synonymous. Very close to the edge, the two terms part ways and coronality becomes substantially longer than totality.

Our first estimate of the eclipse solar radius is based on visually inspecting the flash spectrum video to assess the duration of photospheric extinction. From that estimate we can then infer the value of the solar radius by reading it off Figure 2(a).

This is not our main methodology (see the next subsection) but we present this simple procedure as a way of showing that Auwer's value (Equation (1)) can be ruled out as being too small to be compatible with observations, even without having to perform complex numerical integrations.

Inspection of the video shows the usual features of the flash spectrum, as Figure 10 depicts. The resolution is not as high as what can be achieved with still images (see Figure 4, for example) but nevertheless the main spectral signatures present in the eclipse light can be clearly seen.

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The video shows both the photospheric light of three Baily's beads fading away before second contact and the photospheric light of three Baily's beads reappearing after third contact. We expect to see this evolution based on the dynamical simulations depicted in Figure 7. In that figure the valleys generating these Baily's beads are indicated. By 20 s before second contact, there is basically only light coming from the very last Baily's beads and that light fades away very slowly. The simulation confirms this slow fading away. The video then shows Baily's beads reappearing at a faster rate after third contact.

During totality there are some interesting characteristics that can be noted. The emission arcs from hydrogen and helium stay almost unchanged, just slightly rotating to a different position. The most interesting observation is that the light from the magnesium triplet never disappears either. The same is also true for the faint light from the sodium doublet, next to the helium arc. The beads that can be seen during totality at the wavelength of the magnesium triplet indicate that the limb of the Moon was not able to travel too far past the limb of the photosphere. We know from Mitchell (1947) that magnesium is present from the Sun's surface up to ∼3000 km. The bulk of the density is probably at a lower height. Let us take a height of 1000 km, assume a solar radius S_⊙ = 960'', and simulate the dynamics of this locus of points 1000 km higher than the photosphere. Figure 11 depicts this situation. Despite the limb of the photosphere gliding under the double valley indicated by the stars, the lunar limb is never able to occult the light generated by the magnesium triplet. Throughout totality we can see an evolution of the intensity of these beads of light from the magnesium triplet. Initially the light floods the C₂ valley and a bit of the other two, and then the light starts flooding the C₃ valley and diminishes in intensity in the C₂ valley.

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Now that we understand some of the features seen in the flash spectrum video, we can focus on our main task—estimating the duration of totality. We need to detect when the photospheric continuum disappears and when it reappears in order to estimate the duration of totality, that is to say, of complete photospheric extinction. Instead of looking at a whole frame, we can focus on a small region of the spectrum and inspect the video frame by frame. We select the area around the magnesium triplet as it is feature-rich. To make the signal more contrasted, the video frames were first converted to gray scale and then the colors were negated. Figure 12 presents a sequence of zoomed-in images of that spectral region. The video was not recorded with a UTC time signal, so we have chosen one frame situated more or less at midtotality and referenced temporally all other frames with respect to this one using the known frame rate:

$$\begin{eqnarray}&&t={t}_{0}+(n-{n}_{0})/23.976\end{eqnarray} \tag{ 8 }$$

where n is the frame number of a generic frame, n₀ is the frame number of the reference frame, and t₀ is the (unknown) UTC corresponding to the reference frame. We should note that Equation (8) makes the implicit assumption that no frames were dropped, which is reasonable as we used a fast SD card and we did not employ the fastest frame rate available on the camera.

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The magnesium triplet is clearly visible toward the top of every image. In line with the C₂ valley we can see the photospheric continuum in frames (a) and (f) of Figure 12. The faint darker beads that we can see to the right of the magnesium triplet are emission lines from the fluctosphere. If you refer back to Figure 4, you can identify those as emissions from chromium and iron. We want to look between those emission lines and try to see when the photospheric continuum vanishes. The blue arrows indicate what to look at. We can estimate that the photospheric extinction lasts for about 13 s, possibly slightly less.

The photospheric continuum is present in frames (a) and (f) of Figure 12 and it is absent in frames (c) and (d). The duration of totality is between these two cases, 9 s and 17 s, with a possible value of 13 s. The solar radius S_⊙ compatible with a duration of 13 s is about 960''. This estimate has elements of uncertainty due to the presence of a background of faint emission from the fluctosphere but one thing is certain: the standard radius (Equation (1)) is wildly incompatible with the duration of totality that can be extracted from the flash spectrum video. If S_⊙ = 95963, photospheric extinction should have lasted for 32.6 s at the observing site, which is clearly not the case.

Our first result is that visual inspection of the flash spectrum video supports an eclipse solar radius S_⊙ of around 960''. By interpreting the flash spectrum video in light of the very precise eclipse computations performed here, we can provide the following estimate of the solar radius based on the range of durations given above. A duration of 9 s gives S_⊙ = 96009 and a duration of 17 s gives S_⊙ = 95993. Therefore, our estimate of the eclipse solar radius obtained from visual analysis can be given as

$$\begin{eqnarray}&&{S}_{\odot }=(960\buildrel{\prime\prime}\over{.} 01\pm 0\buildrel{\prime\prime}\over{.} 08).\end{eqnarray} \tag{ 9 }$$

The flash spectrum video contains sufficient data to allow us to conduct a more quantitative analysis to estimate the value of the eclipse solar radius. The idea is to extract light curves for the last Baily's bead before C₂ and for the first Baily's beads after C₃. The main spectral region chosen to perform the extraction in is around 580 nm and it is indicated in Figure 13 by the white boxes. The region just to the left of the helium arc is reasonably quiet with respect to the fluctosphere, as can be assessed by looking at the high-resolution spectrum in Figure 4.

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The measurement of light curves from the video is based on extracting a sequence of consecutive frames and on analyzing them to gather light intensity information. Frames are extracted using the OpenCV library, ⁷ which imports and transforms video data into the required sRGB color space. Given the triplet of values {R', G', B'} for each pixel, with components expressed in the range [0, 1], the linearized luminance Y is obtained from the transformation:

$$\begin{eqnarray}&&Y=0.2126\,{\rm{\Gamma }}(R^{\prime} )+0.7152\,{\rm{\Gamma }}(G^{\prime} )+0.0722\,{\rm{\Gamma }}(B^{\prime} )\end{eqnarray} \tag{ 10 }$$

where

$$\begin{eqnarray}{\rm{\Gamma }}(x)=\left\{\begin{array}{ll}\displaystyle \frac{x}{12.92} & x\leqslant 0.04045\\ {\left(\displaystyle \frac{x+0.055}{1.055}\right)}^{2.4} & x\gt 0.04045.\end{array}\right.\end{eqnarray} \tag{ 11 }$$

The operation Γ(x) undoes the gamma adjustment and linearizes the RGB components. Both Equations (10) and (11) are defined in the international standard defining the sRGB color space (IEC 1999).

For each frame we then compute the average linearized luminance of the pixels in the extraction region (size 11 × 11 pixels). The resulting light curve can be seen in Figure 14 (left plot). Timing is obtained by using the same relationship in Equation (8).

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The intensity of the light vanishing at the bottom of the C₂ valley decreases almost linearly and then flattens out. Correspondingly, the intensity of the light reappearing at the bottom of the C₃ valley is initially almost flat and then starts increasing also almost linearly. We observe that the two light curves do not exactly reach zero intensity, even at midtotality. We think that there might be two reasons for such behavior. First, the flash spectrum mostly collects the eclipse light but there is some, albeit very minimal, diffuse ambient light that is probably emitted at all frequencies. So we would expect some background intensity equally affecting all light curves. Second, we know that the light from the fluctosphere is in line with the photospheric continuum. Even if we have chosen a quiet area of the flash spectrum in that regard, there are probably always traces from faint emissions from the fluctosphere. The very gentle sloping of the flat parts of both light curves might be a sign of that. Nevertheless, the bulk of the intensity in the linear sections must be due to the photospheric continuum.

Figure 14 shows the experimental and simulated light curves by assuming a solar radius S_⊙ = 95995. To be able to compare the simulated and observed light curves, we need to make some adjustments. The time t₀ in the light curves is arbitrary (Equation (8)) so we can apply a suitable global time translation to the observed light curves to make their timescale compatible with the UTC timescale of the computations:

$$\begin{eqnarray}&&{t}_{0}\to {t}_{0}+\tau .\end{eqnarray} \tag{ 12 }$$

The LDF is normalized to 1, so we can apply a suitable scale factor a to the computed intensities to bring them on the same intensity scale the light curves are expressed in. Afterward, we also add a global shift to the computed intensity to account for the fact that the observed light curves do not go to zero during totality (due to the background diffuse light and faint emission from the fluctosphere we pick up alongside the photospheric continuum). We apply a simple linear transformation to the simulated intensity I, where a and b are the fitted parameters:

$$\begin{eqnarray}&&I\to a\,I+b.\end{eqnarray} \tag{ 13 }$$

These parameters are kept constant to transform all simulations at the same wavelength, so as to allow a fair comparison. We should also note that, despite the light curves at second and third contact being independent, we are able to apply the same adjustment to both of them.

The results of the modeling of the light curves at λ ≈ 580 nm are presented in Figure 15. Four values of the solar radius S_⊙ are taken into consideration: 95985, 95990, 95995, and 96000. The best simulation of the observed light curves is obtained for S_⊙ = 95995. The simulation is very good in the linear sections of the light curves. Small discrepancies can be seen very close to the interval of photospheric extinction. This is not surprising as the LDF ends sharply. We can take the simulations with 95990 and 96000 as providing uncertainty bounds for the value S_⊙ = 95995. Therefore, our estimate of the eclipse solar radius obtained from light-curve simulation can be given as

$$\begin{eqnarray}&&{S}_{\odot }=(959\buildrel{\prime\prime}\over{.} 95\pm 0\buildrel{\prime\prime}\over{.} 05)\end{eqnarray} \tag{ 14 }$$

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To test if a dependency of the solar radius S_⊙ on the wavelength could be detected, we extracted light intensity curves from two other spectral regions—one to the left of the Hβ arc at λ ≈ 480 nm, and one to the left of the Hα arc at λ ≈ 640 nm (see Figure 13). These light curves and their simulations are displayed in Figure 16 together with the light curves at λ ≈ 580 nm we have already presented. The time shift τ (Equation (12)) is the same in all three cases. The bias b (Equation (13)) is very close in all three examples. The scale factor a (Equation (13)) had to be adjusted at different wavelengths because the camera sensor is less sensitive in the blue and in the red. In Figure 16 we can see that a solar radius of about S_⊙ = 95995 better fits the observations at these wavelengths. However, we do not consider the minor shift in the best-fit solar radius at different wavelengths to be significant relative to the uncertainty on each estimated radius.

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