{"id":63,"date":"2017-03-23T14:07:35","date_gmt":"2017-03-23T21:07:35","guid":{"rendered":"https:\/\/live-optics-wp.pantheonsite.io\/jpalmer\/?page_id=63"},"modified":"2018-11-15T11:55:27","modified_gmt":"2018-11-15T18:55:27","slug":"lunacy","status":"publish","type":"page","link":"https:\/\/wp.optics.arizona.edu\/jpalmer\/interests\/lunacy\/","title":{"rendered":"Lunacy"},"content":{"rendered":"<p><big>I have a long-term interest in our nearest neighbor, the Moon.\u00a0 Specifically, why is it so much brighter at full moon than on the one or two adjacent evenings waxing or waning?\u00a0\u00a0The Moon still appears full, but is nowhere near as bright.\u00a0 In addition, some full moons are quite a bit brighter than others.<\/big><\/p>\n<p><big>If the moon were a perfectly diffuse (i.e., lambertian) reflector, the ratio of apparent intensity of the full moon to the quarter moon would be pi, but in reality the ratio is more like 10.\u00a0 We know the reason, it&#8217;s the so-called opposition effect, also known as retroreflection.\u00a0 We also are pretty sure of the causes; a fraction of the lunar dust covering the surface came from ejecta from meteorite collisions.\u00a0 When a meteorite strikes the lunar surface with sufficient energy, the ejecta will melt, and\u00a0 then harden into quasi-spherical glassy particles before landing on the surface.\u00a0 If the material has a relatively high index of refraction and the particles are sufficiently small that they are partially transparent, then they act like cats eyes, and reflect a disproportionate amount of radiant power back to the original source, our Sun. The closer we get to the line of sight between the Sun and the Moon, the brighter the Moon appears.\u00a0 We can&#8217;t see full opposition, because we get in the way, casting our big ol&#8217; shadow on the Moon (a lunar eclipse).\u00a0\u00a0<\/big><\/p>\n<p><big>Another important component is shadow-hiding, where shadows cast by finite objects are not seen at opposition. I have not yet found a good model to determine how much brighter.\u00a0 So I have been making measurements for the past several years to attempt to determine the amount of this bizarre behavior. Here is some of the stuff I have come up with.<\/big><\/p>\n<p><u><strong><big><big>Appearance of the full Moon:<\/big><\/big><\/strong><\/u><\/p>\n<p><big>The full moon looks sorta like a flat, uniformly illuminated disc. That would be OK if indeed it were a flat disc.\u00a0 However, we know that it is spherical, and the irradiance (power per unit area) from the sun falls off as we go from the surface directly facing towards the sun around to the terminator.\u00a0 In fact, it falls off as a cosine function.\u00a0 Therefore the brightness (a.k.a. radiance) should likewise fall off as a cosine from the center of the &#8220;disc&#8221; towards the edge.\u00a0 The moon would look brightest at the center, gradually becoming dimmer at the edge of the disc.\u00a0 What&#8217;s wrong with this picture? Is the Moon not a diffuse reflector? Apparently not!<\/big><\/p>\n<p><big><strong><u><big>Sun-Earth-Moon geometry and definitions:<\/big><\/u><\/strong><\/big><\/p>\n<ul>\n<li>Phase angle &#8211;\u00a0The angle between the Sun-moon line and the Earth-moon line.\u00a0 It approaches 0\u00b0 during a lunar eclipse.<\/li>\n<li>Full moon &#8211; The Sun, the Earth and the Moon are all in a line, with the Earth between the sun and the moon. The phase angle is 0\u00b0.<\/li>\n<li>New moon &#8211; The Sun, the Earth and the Moon are all in a line;\u00a0 the moon is between us and the sun and cannot be seen during daylight.\u00a0 The phase angle is 180\u00b0.<\/li>\n<li>Quarter-moon &#8211; we observe that half of the disc is illuminated, that we are halfway between new moon and full moon, but\u00a0 what we see illuminated is only a quarter of the total surface. The phase angle is 90\u00b0.<\/li>\n<li>Crescent &#8211; the illuminated part of the moon is a crescent.\u00a0 The phase is between new and quarter.<\/li>\n<li>Gibbous &#8211; the illuminated part of the moon is lopsided or humped; the phase is between quarter and full.<\/li>\n<li>Waning &#8211; the illuminated part of the moon is getting smaller every day<\/li>\n<li>Waxing &#8211; the illuminated part of the moon is getting larger every day<\/li>\n<\/ul>\n<p><u><strong><big><big>If the Moon were a diffuse reflector<\/big><\/big><\/strong><\/u><big><big>:<\/big><\/big><\/p>\n<p><big>At first glance, it looks like we should receive twice as much light from the full moon as we should receive from the quarter moon, which looks like just half of a sphere.\u00a0 My TA (Heidi Hall) and I did the integrals a few years ago to demonstrate the properties of a theoretical (i.e. lambertian) moon.\u00a0 The ratio between the integral for the full moon to the integral for the quarter moon (90 degree phase angle) is pi and not 2.\u00a0 So the ratios of light received here on Earth between the full\u00a0 moon and the quarter moon would be pi if the Moon were indeed lambertian.\u00a0 The equation for a lambertian sphere illuminated by a distant point source (<\/big><big>Spiro, I.J. and M. Schlessinger, Infrared Technology Fundamentals, Dekker,1989)\u00a0<\/big><big>\u00a0looks like<\/big><\/p>\n<p align=\"center\"><big>I = const<\/big>*<big>R<\/big>*<big>E<\/big>*<big>[sin(PA)+(pi-PA)<\/big>*<big>cos(PA)]<\/big><\/p>\n<p><big>where I is the intensity, R is the reflectance (lambertian), E is the solar irradiance, PA is the phase angle (zero at opposition), and the const involves fixed geometry.\u00a0 This equation predicts the ratio of pi between full and quarter moon.\u00a0 Numerous observations show that this ratio is typically on the order of 10, and I have observed it as high as 15.\u00a0 So now we have two pieces of information that says that the Moon is definitely not lambertian, the ratio and the appearance.<\/big><\/p>\n<p><big>Here are two pictures showing a computer generated lambertian model illuminated with collimated light (as from the sun).\u00a0\u00a0\u00a0 The figure on the left shows the &#8220;quarter&#8221; sphere with the light coming from the left.\u00a0 It us brightest on the left edge, and\u00a0<\/big><big>falls<\/big><big>\u00a0of<\/big><span style=\"font-size: large;\">f at the t<\/span><big>erminator, which divided the light from the dark.\u00a0 The figure on the right is the &#8220;full&#8221; sphere, looking head-on with the light source behind us, clearly\u00a0<\/big><big>showing<\/big><big>\u00a0the darkening as we proceed from the center to the edge.<\/big><\/p>\n<div align=\"right\">\n<table border=\"1\" width=\"100%\" bgcolor=\"#000000\">\n<tbody>\n<tr>\n<td width=\"50%\">\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-121\" src=\"https:\/\/wp.optics.arizona.edu\/jpalmer\/wp-content\/uploads\/sites\/65\/2018\/11\/lambsphquarter.gif\" alt=\"\" width=\"288\" height=\"288\" data-id=\"121\" \/><\/p>\n<\/td>\n<td width=\"50%\">\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-120\" src=\"https:\/\/wp.optics.arizona.edu\/jpalmer\/wp-content\/uploads\/sites\/65\/2018\/11\/lambsphfull.gif\" alt=\"\" width=\"288\" height=\"288\" data-id=\"120\" \/><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p><strong><span style=\"font-size: large;\">For comparison, here are two photos of the moon under the same illumination conditions.<\/span><\/strong><\/p>\n<div align=\"center\">\n<p>&nbsp;<\/p>\n<table border=\"1\" width=\"100%\" bgcolor=\"#000000\">\n<tbody>\n<tr>\n<td width=\"50%\">\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-123\" src=\"https:\/\/wp.optics.arizona.edu\/jpalmer\/wp-content\/uploads\/sites\/65\/2018\/11\/qmoon02r.jpg\" alt=\"\" width=\"186\" height=\"276\" data-id=\"123\" \/><\/p>\n<\/td>\n<td width=\"50%\">\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-122\" src=\"https:\/\/wp.optics.arizona.edu\/jpalmer\/wp-content\/uploads\/sites\/65\/2018\/11\/moonfullclemfr.jpg\" alt=\"\" width=\"217\" height=\"207\" data-id=\"122\" \/><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<p><strong><u><big><big>Retroreflection and how it is obtained:<\/big><\/big><\/u><\/strong><\/p>\n<ul>\n<li><strong><big>Corner cubes<\/big><\/strong><\/li>\n<li><strong><big>Backscatter from transparent spheres<\/big><\/strong><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><big><strong><u><big>Other explanations of the opposition effect:<\/big><\/u><\/strong><\/big><\/p>\n<ul>\n<li><big><strong>Shadow-hiding<\/strong><\/big><\/li>\n<li><big><strong>Interparticle coherent backscatter<\/strong><\/big><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><big><strong><u><big>Measurement methodology:<\/big><\/u><\/strong><\/big><\/p>\n<p>&nbsp;<\/p>\n<p><strong><u><big><big>Corrections to measurements:<\/big><\/big><\/u><\/strong><\/p>\n<p><big>(1)\u00a0<b>Moon-Sun distance.<\/b>\u00a0 Our orbit around the sun is elliptical, and we are closest to the sun on January 6.\u00a0 The total effect is a change in the solar &#8220;constant&#8221; of +\/- 3.5%.\u00a0 This correction is easy to make based on Ephemeris data.<\/big><\/p>\n<p><big>(2)\u00a0<b>Earth-Moon distance:<\/b>\u00a0 Ephemeris data shows the semi-diameter of the moon as a function of time.\u00a0 On an angular scale, the half-angle varies from about 14.6 arc-min (Earth-moon distance a maximum) to just over 17 arc-min\u00a0 (Earth-moon distance a minimum).\u00a0 The solid angle of the Moon as seen from Earth is pi*[sin(half-angle)]^2, and thus varies from about 0.0000574 steradians to 0.0000771 steradians.\u00a0 The irradiance (W\/m<sup>2<\/sup>)\u00a0 received at the Earth is equal to the radiance (W\/m<sup>2<\/sup>-sr) of the moon multiplied by its solid angle, and thus the total difference between the &#8220;biggest&#8221; moon and the &#8220;smallest&#8221; is about 34%. This important effect is\u00a0<\/big><big>also easily correctible.<\/big><\/p>\n<p><big>(3)\u00a0<b>Earth atmospheric transmission<\/b>:\u00a0 Reliable corrections for the transmission of our atmosphere are quite difficult, requiring precise knowledge of the atmospheric constituents and their distribution along the line of sight.\u00a0 I have used LOWTRAN7 to do atmospheric transmission calculations to determine the transmission as a function of elevation angle for several different water vapor contents.\u00a0 When surface RH data is available, I can select one of several nominal transmission values.<\/big><\/p>\n<p><u><strong><big><big>Measurement results:<\/big><\/big><\/strong><\/u><\/p>\n<p><big><big><u><strong>Eclipse measurements:<\/strong><\/u><\/big><\/big><\/p>\n<p><big>The smallest phase angle that we can observe\u00a0 without our shadowing the Moon is 1.52 degrees. If the opposition surge is sharply peaked, it is conceivable that the peak irradiance received from the Moon may be implied from observations made during the penumbral phase of an eclipse.\u00a0<\/big><\/p>\n<p><big><u><strong><big>How to deconvolve the Earth&#8217;s shadow:<\/big><\/strong><\/u><\/big><\/p>\n<p><big><u><strong><big>Other refinements:<\/big><\/strong><\/u><\/big><\/p>\n<p><u><strong><big><big>Other sources of data:<\/big><\/big><\/strong><\/u><\/p>\n<p><big>The Clementine mission took many images of the Moon, some of which included parts of the surface at opposition.\u00a0 I&#8217;ve only begun to look at and sort out this data. The images are full of structure, and clever averaging will be needed to get a meaningful profile.<\/big><\/p>\n<p><u><strong><big><big>Resources and links:<\/big><\/big><\/strong><\/u><\/p>\n<p><strong><span style=\"font-size: large;\">The starting points for any serious lunar eclipse observers are\u00a0<\/span><\/strong><span style=\"font-size: large;\">\u00a0<\/span><span style=\"color: #008000; font-size: large;\">sunearth.gsfc.nasa.gov\/eclipse\/eclipse.html<\/span><span style=\"font-size: large;\">\u00a0and\u00a0<\/span><big><a href=\"http:\/\/aa.usno.navy.mil\/AA\/data\/docs\/UpcomingEclipses.html\">http:\/\/aa.usno.navy.mil\/AA\/data\/docs\/UpcomingEclipses.html<\/a>\u00a0.<\/big><\/p>\n<p>&nbsp;<\/p>\n<p><u><strong><big><big>Observation opportunities (past and future)<\/big><\/big><\/strong><\/u><b><big><big>:<\/big><\/big><\/b><\/p>\n<p><big>A recent total lunar eclipse was 21 January 2000, and it was a humdinger.\u00a0 The moon was nearly overhead (75 degree elevation angle) when the moon left our shadow at 0024 on January 21.\u00a0 With relatively low humidity and all, it was a bright one.\u00a0 I measured a peak illuminance of 0.92 lux on a horizontal surface.\u00a0 Remember all that hoopla about how bright the December 1999 full moon was going to be?\u00a0 Here in Arizona, two out of the four factors were favorable.\u00a0 It was partly cloudy, indicating a moist atmosphere and poorer transmission than usual.\u00a0 The moon was at its closest approach to Earth, and we were closest to the sun.\u00a0 But the phase angle was several degrees.\u00a0 My measurements showed it was, shall we just say &#8220;average, no big deal.&#8221;\u00a0 The 21 January 2000 full moon\u00a0 knocked the socks off the December 1999 full moon!\u00a0<\/big><\/p>\n<p><big>The next eclipse was in July 2000, and observations were a bit tougher.\u00a0 We saw the penumbral phase just before sunrise.<\/big><\/p>\n<p><big>There was a\u00a0<\/big><big>total eclipse of the Moon on 9 January 2001.\u00a0 None of the eclipse phases were visible in Arizona.\u00a0 We also had a great deal of cloud cover, rendering measurements impossible.<\/big><\/p>\n<p><big>There was a partial lunar eclipse on July 5, 2001.\u00a0 I was making music in Italy with no radiometric instrumentation.\u00a0<\/big><\/p>\n<p>The\u00a0<big>penumbral lunar eclipse that occurred on\u00a0 December 30, 2001 was a total bust due to heavy cloud cover.\u00a0 Too bad, I had my Meade ETX90 telescope fitted with a new eyepiece and my Canon digital camera.<\/big><\/p>\n<p><big>A penumbral lunar eclipse\u00a0on 26 May 2002.\u00a0 It was totally cloudy here in Tucson.\u00a0<\/big><\/p>\n<p><big>Another penumbral lunar eclipse\u00a0occurred on 24 June 2002, but the peak\u00a0<\/big><big>was<\/big><big>\u00a012 hours away in China.\u00a0 Here the phase angles were far too large for maximum intensity.\u00a0<\/big><\/p>\n<p><big>Future lunar eclipses will be as follows:<\/big><\/p>\n<ul>\n<li><big>2002 November 19-20 &#8211; penumbral lunar eclipse\u00a0<\/big><\/li>\n<li><big>2003 May 16 &#8211; total lunar eclipse\u00a0<\/big><\/li>\n<li><big>2003 November 8-9 &#8211; total lunar eclipse<\/big><\/li>\n<li><span style=\"font-size: large;\">2004 May 5 &#8211;\u00a0<\/span><big>\u00a0total lunar eclipse &#8211; good if you live in Eastern Europe<\/big><\/li>\n<li><span style=\"font-size: large;\">2004 October 28 &#8211;\u00a0<\/span><big>\u00a0total lunar eclipse &#8211; Because of the strong phase function, I have predicted that the luminous intensity would have its maximum sometime before the moon completely exits the penumbra.\u00a0 Sufficient data was taken between times U4 and P4 to demonstrate, and first look at the data shows the prediction to be true.\u00a0 Now I have to work out the convolution of a uniform circular source (sun) obscured by circular obscuration of about three times that size (earth) to see the function for a lambertian reflector.\u00a0 I might end up doing it graphically using my CAD program.<\/big><\/li>\n<li><big>2005 April 24 &#8211; penumbral lunar eclipse\u00a0<\/big><\/li>\n<li><span style=\"font-size: large;\">2005 October 17 &#8211;\u00a0<\/span><big>\u00a0partial lunar eclipse<\/big><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: large;\">Last revision:\u00a0 3 November 2004.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>I have a long-term interest in our nearest neighbor, the Moon.\u00a0 Specifically, why is it so much brighter at full moon than on the one or two adjacent evenings waxing or waning?\u00a0\u00a0The Moon still appears full, but is nowhere near as bright.\u00a0 In addition, some full moons are quite a bit brighter than others. If the moon were a perfectly<\/p>\n","protected":false},"author":3,"featured_media":0,"parent":12,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-63","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/wp.optics.arizona.edu\/jpalmer\/wp-json\/wp\/v2\/pages\/63","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp.optics.arizona.edu\/jpalmer\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/wp.optics.arizona.edu\/jpalmer\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/wp.optics.arizona.edu\/jpalmer\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/wp.optics.arizona.edu\/jpalmer\/wp-json\/wp\/v2\/comments?post=63"}],"version-history":[{"count":4,"href":"https:\/\/wp.optics.arizona.edu\/jpalmer\/wp-json\/wp\/v2\/pages\/63\/revisions"}],"predecessor-version":[{"id":125,"href":"https:\/\/wp.optics.arizona.edu\/jpalmer\/wp-json\/wp\/v2\/pages\/63\/revisions\/125"}],"up":[{"embeddable":true,"href":"https:\/\/wp.optics.arizona.edu\/jpalmer\/wp-json\/wp\/v2\/pages\/12"}],"wp:attachment":[{"href":"https:\/\/wp.optics.arizona.edu\/jpalmer\/wp-json\/wp\/v2\/media?parent=63"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}