Video of Scenery During a Total Eclipse: Luminance and Effects of Solar Limb Darkening E-Mail: Mjtruiz@Gmail.Com

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54 P a p e r Phys. Educ. 54 (2019) 035001 (8pp) iopscience.org/ped 2019 Video of scenery during a total © 2019 IOP Publishing Ltd eclipse: luminance and effects PHEDA7 of solar limb darkening 035001 Michael J Ruiz M J Ruiz Department of Physics, University of North Carolina at Asheville, Asheville, NC 28804, United States of America Video of scenery during a total eclipse: luminance and effects of solar limb darkening E-mail: [email protected]

Printed in the UK Abstract A video of darkening scenery during the 15 min prior to totality of the 21 PED August 2017 total solar eclipse is presented and discussed. The luminances of the scenes at 1 min intervals are obtained by averaging pixel data of stills from 10.1088/1361-6552/ab0311 the video. The experimental results are compared to the theoretical model of uniform radiance from the exposed area of the Sun during the eclipse. The measured values deviate from the model since light from the solar disk 1361-6552 decreases towards the edges of the Sun.

Published The darkening world of a total solar To capture the awe of the darkening world eclipse described by Mark Twain, I shot a video [3] of the local scenery for the 15 min just before the onset 5 The total solar eclipse is one of nature s most spec- ’ of totality during the 21 August 2017 total solar tacular shows in the sky. Mark Twain includes a eclipse. I set my digital video camera on a tripod climactic scene in his novel A Connecticut Yankee 3 and selected manual mode so that the camera in King Arthur s Court. In the story a 19th-century ’ would not automatically adjust for the decreasing Yankee from Hartford, Connecticut suddenly finds available light. My pilot friend Bruce Greene flew himself in England at the time of King Arthur. his girlfriend April and me ninety miles from our Knowing about an eclipse, he escapes being burned hometown in Asheville, North Carolina, USA to at the stake by threatening the court [1]. Greenville, South Carolina, to be in the path of totality. See figure 1 for a series of 15 stills from ‘Go back and tell the King that at that hour I will smother the whole world in the dead the video spaced 1 min apart, arranged from right blackness of midnight; I will blot out the to left and in rows. Since the video frame rate was −1 sun, and he shall never shine again; the 30 frames s , each still has an exposure of 1/30 s. fruits of the earth shall rot for lack of light Be sure to watch the exciting video from which and warmth, and the peoples of the earth the stills came and view the included high resolution image gallery [3]. The video includes a sped- shall famish and die, to the last man! ’ up version as totality approaches. Historically, there was no actual eclipse on For one with no understanding of an eclipse, 21 June, 528 as stated in the novel. Three total the series of images in figure 1 would be quite eclipses over England in the 5th and 6th centuries alarming. Temperature drops accompany the occurred [2] respectively in the years 413, 458, darkening surroundings and animals get con- and 594. fused as to the time of day. In the Mark Twain

Figure 1. A series of stills from a video of the surroundings during a total solar eclipse. The time between each still is 1 min. The last image is at the onset of totality. The author is closest to the camera. A slide show of these photos at high resolution is included in [3]. story, King Arthur readily orders the Yankee to be square meter per steradian, based on the sensitiv- released and to name the terms in order to bring ity of visual receptors in the human eye [5]. The the sunlight back. The negotiations need to take brightness is the nonlinear perception of luminance place in minutes since the duration of totality due to processing in the eye and brain. See figure 2, is typically 2 or 3 min with the longest possible based on a slide by Gordon Kindlmann [6]. eclipsed Sun being 7.5 min [4]. In the next sec- The units are not important since ratios will tion luminance measurements are obtained from be taken throughout this paper. An important the stills and plotted as a function of time. point is that the luminance is a linear function of the radiance. So if one has two identical sources, the luminance doubles as well as the radiance. Radiance, luminance, and brightness However brightness, the perception of luminance, Radiance refers to the total amount of electro is a nonlinear function [7] of the luminance due magnetic radiation with units Watts per square to image processing by the eye and brain. The meter per steradian. Luminance is the weighted eye receptors adjust sensitivity depending on electromagnetic radiation in units lumens per available light through ‘a process called dark

May 2019 2 Phys. Educ. 54 (2019) 035001 Video of scenery during a total eclipse: luminance and effects of solar limb darkening

Radiance Luminance Brightness

All electromagnetic Weighted for receptor sensitivity Perception: eye/brain processing radiation Luminance Brightness

nonlinear linear

Radiance Luminance

Figure 2. The relationships among radiance, luminance, and brightness adapted with permission from Gordon Kindlmann [6]. Eye image: This "Eye (PSF).png" (https://commons.wikimedia.org/wiki/File:Eye_(PSF).png) has been obtained by the author(s) from the Wikimedia website, where it is stated to have been released into the public domain. It is included within this article on that basis. Brain image: This image has been obtained by the author(s) from the Pixabay website (https://pixabay.com/en/brain-lobes-neurology-human-body-1007686/) where it was made available under a CC0 licence. It is included within this article on that basis. adaptation, which causes the eye to increase its is mapped to a perceived loudness span from 0 to sensitivity in the dark’ [8]. The brain completes 100 decibels (dB). the processing of visual information from the The image processing in a digital camera eye. Digital camera sensors respond to radiance. mirrors human perception of brightness with col- However, in the processing to form a jpg or png our stored in the form of red (R), green (G), and image suitable for viewing on a computer moni- blue (B) pixel values, each ranging from 0 to 255. tor, the raw data is transformed to approximate Microsoft and Hewlett Packard (HP) in 1996 ‘rec- the perception of brightness by the human eye. ognised the requirement for defining a common The next section will discuss how to extract lumi- colour space for use in the embryonic industry’ of nance data from the transformed pixel values in digital cameras [10]. The white point and colour an image taken by a digital camera. ‘as described in Recommendation ITU-R BT.709 (Rec 709), was adopted’ and ‘this colour space was defined as the Standard RGB colour space or Measuring luminance from pixels sRGB’ [10]. It is understood that the pixel values As pointed out in the previous section, humans R, G, and B refer to this standard and they may perceive brightness in a nonlinear fashion. When be written as sR, sG, and sB for emphasis. The the human eye views a scene illuminated by one procedure for obtaining the luminance from pixel light source, two identical light sources, three, values of this standard RGB colour space consists etc., the perception is not twice as bright, three of the three steps outlined below. times as bright, etc. The processing by the eye and brain leads to a perception of brightness which represents a nonlinear compression of luminance. Step 1: finding the average sRGB for the Students will be more familiar with a similar scene compression involving sound, where acoustic First, the average sRGB is found for the still 12 2 intensity ranging from 10− Wm− (threshold image. The original images in figure 1 have 2 2 of hearing) to 10− Wm− (‘disco/rock gig’) [9] dimensions 1920 × 1080 pixels. A 480 × 330

May 2019 3 Phys. Educ. 54 (2019) 035001 M J Ruiz pixel region in the upper left corner was cropped out for the measurements since that portion of the scene was constant throughout, while in other portions people moved around. This cropped section appears in figure 3. A cropped image of direct sunlight on the cement foreground was not possible due to the saturation of the R, G and B pixels at their maximum values of 255 for many of the images during the solar darkening. Therefore, a region of diffuse reflection was chosen. The assump- tion is made that the luminance from the diffuse reflected light is proportional to the luminance Figure 3. Cropped image for luminance analysis of the direct solar light. The cropped image can during the total solar eclipse. be analyzed by a commercial program such as MatLab to determine the average sRGB values of the visible spectrum. Red comes next in sen- for the entire scene. However, a free online pro- sitivity and blue last. Think of equation (1) as gram by Matthias Klein (http://matkl.github.io/ uncompressing the nonlinear sRGB values and average-color/) gives the same results. The online equation (2) as applying the necessary weighting program has the advantage that images can easily factors to arrive at the luminance. The last col- be dragged and dropped into a box to very quickly umn in table 1 lists the luminances relative to the find the average sRGB values [11] with no addi- luminance at time 75 min into the eclipse, a time tional computer code as required by MatLab. The 15 min prior to the onset of totality. results are listed in table 1 as sR, sG, and sB. A plot of the relative luminance as a function of time is provided in figure 4. Totality begins at 90 min and lasts for the duration of about 2 min. Step 2: linearising the normalized sRGB At the end of totality, the luminance begins to values increase as more and more light emerges from the The next step is to normalize each sRGB value by Sun. Note the nice symmetry about the middle of dividing by 255, since pixel values range from 0 the eclipse. In the next section, the empirical data to 255. Then, the normalized sRGB values need will be compared to a simple model based on the to be linearised. The recommended procedure for area of the unexposed Sun. linearising the normalized values sRGB is given by [12] Simplest theoretical model x 0 x 0.040 45 12.92 In the simplest theoretical model the luminance f (x)= x+0.055 2.4 (1) 1.055 0.040 45 < x 1 is calculated by determining the area of the Sun that is unexposed. This model neglects varia- where x is sRnorm, sGnorm, or sBnorm, and the corre tions of luminance due to sunspots, prominences, sponding f (x) values are the linearised Rlin, Glin, and solar limb darkening. Solar limb darkening and Blin® entries in table 1. refers to the fact the light coming from the Sun ‘decreases from the centre of the disc toward the Step 3: obtaining the luminance from the limb’ [13]. In the simple model of a uniform solar linearised RGB values disc, if the unexposed area of the Sun doubles, the To obtain the luminance from the linearised val- radiance doubles as well as the luminance. The model and associated calculations pre- ues Rlin, Glin, and Blin, the following prescription is used [12] sented here represent a simplified version of a publication by Hughes in the Journal of the L = 0.2126Rlin + 0.7152Glin + 0.0722Blin. (2) British Astronomical Association [14]. A similar The largest coefficient is for the green since the treatment appears in a paper by Möllmann and human eye is most sensitive in the green region Vollmer in the European Journal of Physics [15].

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Table 1. Pixel averages sRGB, normalized values sRGBnorm, linearised RGBlin, and relative luminance. The calculations are explained in the text. Time (min) sR sG sB sRnorm sGnorm sBnorm Rlin Glin Blin Lrelative 75 176 187 132 0.690 0.733 0.518 0.434 0.497 0.231 1.00 76 170 181 124 0.667 0.710 0.486 0.402 0.462 0.202 0.93 77 164 175 117 0.643 0.686 0.459 0.371 0.429 0.178 0.86 78 158 168 109 0.620 0.659 0.427 0.342 0.392 0.153 0.78 79 151 161 101 0.592 0.631 0.396 0.309 0.356 0.130 0.71 80 142 152 91 0.557 0.596 0.357 0.270 0.314 0.105 0.62 81 134 144 81 0.525 0.565 0.318 0.238 0.279 0.082 0.55 82 121 130 72 0.475 0.510 0.282 0.191 0.223 0.065 0.44 83 110 118 66 0.431 0.463 0.259 0.156 0.181 0.054 0.36 84 97 104 57 0.380 0.408 0.224 0.120 0.138 0.041 0.27 85 83 89 47 0.325 0.349 0.184 0.087 0.100 0.028 0.20 86 67 72 36 0.263 0.282 0.141 0.056 0.065 0.018 0.13 87 50 53 25 0.196 0.208 0.098 0.032 0.036 0.008 0.07 88 29 31 14 0.114 0.122 0.055 0.012 0.014 0.004 0.03 89 9 10 6 0.035 0.039 0.024 0.003 0.003 0.002 0.01 90 0 0 0 0.000 0.000 0.000 0.000 0.000 0.000 0.00 91 0 0 0 0.000 0.000 0.000 0.000 0.000 0.000 0.00 92 0 0 0 0.000 0.000 0.000 0.000 0.000 0.000 0.00 93 12 11 6 0.047 0.043 0.024 0.004 0.003 0.002 0.01 94 32 34 17 0.125 0.133 0.067 0.014 0.016 0.006 0.03 95 52 56 31 0.204 0.220 0.122 0.034 0.040 0.014 0.08 96 70 76 45 0.275 0.298 0.176 0.061 0.072 0.026 0.14 97 86 93 55 0.337 0.365 0.216 0.093 0.109 0.038 0.22 98 101 108 66 0.396 0.424 0.259 0.130 0.150 0.054 0.30 99 114 122 77 0.447 0.478 0.302 0.168 0.195 0.074 0.39

1.00

0.80

0.60

0.40 Relative luminance

0.20

Totality 0.00 75 80 85 90 95 100 Minutes from beginning of eclipse

Figure 4. A plot of the relative luminance of a scene during a total eclipse where the reference luminance is at 75 min, a time 15 min prior to the onset of totality. In our model presented below, the Sun and Moon use of the law of cosines. Therefore, the model are taken to have the same apparent size in the presented below is more easily accessible for sky and the mathematics is simpler, avoiding the introductory students. However, it is important to

May 2019 5 Phys. Educ. 54 (2019) 035001 M J Ruiz point out that the simple model ‘corresponds to a A duration of totality of zero’ [15], when the moon precisely covers the Sun for an instant. The actual relative apparent size of the Moon is typically slightly larger than 1. See Möllmann and Vollmer 1 [14] for all general cases of apparent lunar sizes that describe all types of solar eclipses: partial, BCDEθ F annular, and total. α 1− α Our model appears in figure 5, where the discs 2 2 Solar of the Sun and Moon as they appear in the sky Sun centre Moon are taken to be the same size, each with radius 1. The goal is to calculate the area of the exposed Sun that forms the crescent ABGCA. Students will be surprised that this area is not hard to calcu- G late using elementary geometry and trigonometry. α The plan is to first find the area ACGFA of the covered Sun. Then, subtract this area from Figure 5. The Moon covering the Sun during an the area π 12 of the Sun in our units where the eclipse where the covered solar disc is the region radius of the· Sun is 1. The area ACGFA of the ACGFA. The exposed Sun is the region ABGCA. covered Sun is twice the area of the chord sec- The parameter α ranges from 0 to 2 from the time the Moon just begins to cover the Sun to the time the Moon tion AEGFA. This chord section is equal to the completely covers the Sun. angular pie section ADGFA minus the triangular region ADGEA. The area of the angular pie section ADGFA is the start of the eclipse. At our location, totality was reached at t = 90 min. Therefore 2 2θ AADGFA = π1 = θ, (3) t (min) 2π α = . (8) 45 min where θ is measured in radians. The area of the triangular section ADGEA is two times the area The parameter α is related to the relevant trig of the triangle ADEA functions as follows α 2 α 1 (9a) A = 2 cosθsinθ = cosθsinθ. (4) cosθ = 1 = − ADGEA · 2 − 2 2 As noted above, the chord section AEGFA = √4α α2 sinθ = 1 cos2θ = − . (9b) ADGFA − ADGEA, − 2 AAEGFA = AADGFA AADGEA = θ cosθsinθ. With these trig relations, equation (7) becomes − − (5) 1 2 α 2 α The area of the covered Sun is twice the area A = π 2cos− − + − 4α α2, exposed − 2 2 − given in equation (5) (10) A = A = 2A = 2θ 2cosθsinθ. where α is related to time by equation (8). covered ACGFA AEGFA − (6) Figure 6 is a plotï of theò relativeï ò luminance Finally, the area of the exposed Sun (Aexposed) is starting at 75 min, i.e. 15 min before the onset of 2 π 1 Acovered, totality. The relative luminances are referenced to · − the luminance at 75 min for both the experimental A = π 2θ + 2cosθsinθ. (7) exposed − data and for the theoretical curve. The theoretical A remaining challenge is to express equation (7) plot during the 15 min prior to totality shown in terms of the parameter α which is proportional in figure 6 is surprisingly linear. Equation (10) to the time as the Moon moves across the Sun. does not appear to be linear in any obvious way. When the eclipse begins, α = 0; when the eclipse However, if the derivative is taken to find the reaches the onset of totality, α = 2. Let t = 0 at slope, the result is

May 2019 6 Phys. Educ. 54 (2019) 035001 Video of scenery during a total eclipse: luminance and effects of solar limb darkening

1.00

0.80

Theory (based on area of exposed sun) 0.60

Relative luminance 0.40 Data (showing solar limb darkening)

0.20

0.00 75 78 81 84 87 90 Minutes from beginning of eclipse

Figure 6. Theoretical (blue) and experimental (red) plots of relative luminance for the last 15 min before the onset of totality (90 min).

dA The limb darkening evident in figure 6 is due exposed = 4α α2. (11) dα − − to integrative effects of wavelengths and crescent solar areas. A similar pair of plots is found in the From t = 75 min to t = 90 min, α ranges from paper by M llmann and Vollmer [15] for the 29 75/45 = 1.67 to 90/45 = 2.00. The slope given ö March 2006 total solar eclipse with a path over by equation (11) ranges, respectively, from −1.97 southern Turkey. M llmann and Vollmer measured to −2.00, very close to being constant. ö light with a digital light meter. More advanced The measured values for the luminances models incorporate limb darkening [19, 20]. drift below the theoretical curve due to solar limb darkening, i.e. the regions towards the edges (the limb) of the Sun are darker. When light is measured from the center of the solar disc (defined Conclusion as 0°), the light emerges from ‘the hottest part of This paper studies the darkening of a scene during the photosphere. But when we examine the limb, a total solar eclipse using elementary mathemat- the relatively darker portion of the solar disc, we ics. The decreasing luminances are obtained from receive radiation coming from the higher layers camera pixel data. The theoretical model assumes of the solar photosphere’ [16]. Limb darkening is uniform radiance from all regions of the Sun and wavelength dependent. The ratio of radiation at does not require any use of calculus. Therefore, the very oblique angle of 87° coming from the the results can be presented to introductory stu- higher layers of the photosphere compared to the dents with a knowledge of geometry and trigo- direct path at 0° emerging from the lower pho- nometry. The analysis is rich in interdisciplinary tosphere ranges from about 0.2 for 400 nm (blue connections involving astronomy (eclipses), extreme of the visible spectrum) to about 0.4 for physics (radiance), biology (luminance), psy- 700 nm (red extreme of the visible spectrum) [17]. chology (perception of brightness), and computer It is interesting to note that Sir Arthur Eddington science (RGB pixel color). The deviation from proposed an ‘approximation of solar limb darken- experiment and theory reveals the subtle concept ing for light integrated over all wavelengths’ [18] of solar limb darkening in astronomy. For stu- that predicts the relative radiation dropping from dents in elementary grades and for the general the 1.00 value at 0° to 0.40 at 90°. public, the teacher can show the video [3] and the

May 2019 7 Phys. Educ. 54 (2019) 035001 M J Ruiz images of figure 1. Students of all ages will find [8] Goldstein E B 2010 Sensation and Perception the video and associated images fascinating. 8th edn (Belmont, CA: Wadsworth, Cengage Learning) pp 52–3 [9] Bishop C 2016 Physics: Medical Physics Acknowledgments AQA A-Level Year 2 (London: HarperCollins) p 16 The author would like to thank his pilot friend [10] Tooms M S 2016 Colour Reproduction in Bruce Greene for flying the author to Greenville, Electronic Imaging Systems: Photography, South Carolina, USA to view the 21 August 2017 Television, Cinematography (Chichester: total solar eclipse. The author would also like to Wiley) p 440 [11] Klein M Get average color of image (http:// thank the reviewer for helpful comments. Finally, matkl.github.io/average-color/ (Accessed: the author has confirmed that any identifiable par- 6 November 2018)) ticipants in this study have given their consent for [12] Dubois E 2010 The Structure and Properties publication. of Color Spaces and the Representation of Color Images (San Rafael, CA: Morgan & Received 15 October 2018, in final form 6 November 2018 Claypool Publishers) pp 62–7 Accepted for publication 30 January 2019 [13] Sánchez-Bajo F, Vaquero J M and Rubio https://doi.org/10.1088/1361-6552/ab0311 Montero M P 2002 Measuring solar limb-darkening with modest equipment Eur. J. Phys. 23 323–30 References [14] Hughes D W 2000 Brightness during a solar [1] Twain M 1889 A Yankee at the Court of eclipse J. Br. Astron. Assoc. 110 203–5 King Arthur (London: Chatto & Windus, [15] Möllmann K-P and Vollmer M 2006 Measurements Piccadilly) p 51 (Later editions have the title and predictions of the illuminance during a A Connecticut Yankee in King Arthur’s Court) solar eclipse Eur. J. Phys. 27 1299–314 [2] Espenak F 2014 Eclipse predictions by Fred [16] Bhatnagar A and Livingston W 2005 Espenak (NASA’s GSFC), NASA Eclipse Fundamentals of Solar Astronomy Web Site (https://eclipse.gsfc.nasa.gov/ (Singapore: World Scientific) pp 94–5 SEcat5/SEcatalog.html (Accessed: 3 [17] Gray D F 2005 The Observation and Analysis of November 2018)) Stellar Photospheres 3rd edn (Cambridge: [3] Ruiz M J 2017 Video: solar eclipse darkness Cambridge University Press) p 174 (http://mjtruiz.com/ped/eclipse/) [18] Carroll B W and Ostlie D A 2014 An [4] Seeds M A and Backman D E 2010 Astronomy: Introduction to Modern Astrophysics 2nd edn the Solar System and Beyond 6th edn (Harlow: Pearson) p 299 (Belmont, CA: Brooks/Cole Cengage [19] Vollmer M 2009 Sonnen- und Mondfinsternisse: Learning) p 35 Beobachtungen Messungen und quantitative [5] Bohren C F and Clothiaux E E 2006 Modelle Praxis Nat. Phys. 58 38–44 Fundamentals of Atmospheric Radiation: an [20] Lee S W and Lee S 2012 Illuminance during Introduction with 400 Problems (Weinheim: a solar eclipse with limb darkening: a Wiley) pp 200–1 mathematical model J. Korean Astron. Soc. [6] Kindlmann G, Reinhard E and Creem S 45 111–7 2002 Face-based luminance matching for perceptual colormap generation Presented at IEEE Visualization 2002 (Boston, MA, USA, Michael J Ruiz is professor of physics 27 October–1 November 2002) (see slide at the University of North Carolina at 5 terminology in the online HMTL version Asheville (UNCA), USA. He received at www.cs.utah.edu/~gk/papers/vis02/talk/ his PhD in theoretical physics from the slide005.html (Accessed: 15 October 2018)) University of Maryland, USA. His [7] Poynton C and Funt B 2014 Perceptual innovative courses with a strong online uniformity in digital image representation component aimed at general students and display Color Res. Appl. 39 6–15 have been featured on CNN.

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