30 November 2001Trouble at Full Moon
by Art WinfreeThe plan for this week was finish with Curved Lines in the Sky, at least as far as I pursued it in the spirit of unsophisticated thinking about the obvious. But while waiting for November full moon, last week we tried to anticipate its brightness using the astronomer's magnitude scale. This ran into an unforeseen problem that might position any interested party on the brink of Discovering something interesting. Let's deal with that first, as the Full Moon (tonight) won't wait. Curved Lines is deferred yet again, until the next column.The 23 November column contains a link to the challenge of estimating brightness when a chalky sphere is illuminated from the side. I hadn't tried it at the time but did later manage to integrate the trigonometric polynomial that seems pertinent according to "reasonable" assumptions. It comes out wrong. It says the quarter moon (sometimes called half moon: half the disk is lighted a quarter cycle from new or full) should be 1/
as bright as full moon: less than the naïve estimation of 1/2, but not so much less as in fact observed (1/11: see data graph). This little checkup applied to our glib estimation of full moon magnitude seems to have uncovered an interesting flaw in the assumptions. We assumed the full moon would look dimmer than "full" Sun by a geometric factor (its radius in radians, squared: about 52,000, or 11.8 magnitudes) augmented by an absorption factor averaging the grayness of its rocks (if 7-fold, about 2 magnitudes more, for 14 altogether). But it now seems that the putative absorption factor depends peculiarly on the angle between incident and emitted light. Something unexpected is sure to be learned from this.
One way to pursue this riddle might be to find out what mistakes linger in the chosen trigonometric expressions. One way to do that is to look in computer graphics books or on the web for algorithms used to color the image of a sphere. This leads to the name "Phong" and to distinguishing between specular (mirror) components of reflection and "diffuse" components.
For example, play around with Scott Herod's nice graphics applet at http://amath.colorado.edu/faculty/sherod/classes/Color/phong.html. He was kind enough to provision it with new sliders D and T in the bottom right for this column. They allow you to move the light source: see the formula. Maximize distance D and set the specular reflection and ambient lighting to 0. At T=0 the light source is behind so you have the Phong version of full moon. At T=90 degrees the light is in the upper right and shining at right angles to your line of sight as at quarter moon. At T=180 degrees it is behind the sphere so you see no light, as during eclipse of the Sun.
Notice the formula for the diffuse component, the one we care about. It depends on the angle between "the surface" and the incident light, but not on the direction of viewing. It goes to zero where the light strikes the surface horizontally, as at sunset or as at the rim of the full moon. But the full moon does not darken toward its rim. Surely a man on the surface would report very long shadows on the mostly-dark ground --- in a moment the Sun will sink below the lunar horizon --- but strange to say, the ground around him looks to us, viewing practically at a tangent to the average lunar surface, as bright as anyplace else. And try this: set the direction of the light source to 90 degrees as it is one week before or after full or new moon, and see if the image resembles a disk uniformly lit to the right of its terminator, and dark to the left of that sharp line, as the Moon in the sky will look next week (I think). The Moon does not fit the Phong model used for computer graphics, nor the assumptions on which that model is based, which are basically the ones that went into my abortive integral. Could this reveal that the Moon has rocks where Venus and Jupiter have clouds? Photos of "full Jupiter," the only view we ever get from Earth, are dark around the edges, I find in web rummaging today. Or that the Moon has peculiar rocks?
It seems there is something to Discover here, and --- the key point and recurrent theme of this column --- all it took to notice that is to think for a moment about the familiar facts that the full moon is mighty bright compared to a few days later, and that it looks about uniformly bright. Here is a great opportunity for ingenious thinking and testing of speculations by further observation.
Maybe the hills and cliffs of the Moon make it optically different from a smooth "surface": what we really see near the rim is not a surface tilted almost parallel to the incident sunlight and our line of sight, but the faces of cliffs and of rocks on hillsides. But a tennis ball has similar bumpiness on a smaller scale, and if you hold it at arm's length in front of you, illumined only by a lamp over your shoulder, its rim does look darker, unlike full moon. It looks about the same as a ping-pong ball even though the surface roughness is quite different. So this does not seem an adequate explanation. (Nevertheless, it may be good to return to this issue: can the notion of "surface" be useful if not smooth on the scale of light wavelengths, and if not so over areas bigger than the angular resolution of human vision? And if an illumination model indispensibly entails angles relative to a surface, can it be pertinent?)
Rummaging the web about this mystery I ran across http://www.optics.arizona.edu/Palmer/moon/lunacy.htm: the otherwise unaccountable brightness of the full Moon's perimeter may indicate that lunar soils are peculiarly full of little glass spheres that retro-reflect like a cat's eyes. Such beads apparently formed 400 million years ago and 3-4 billion years ago during heavy meteor bombardment: http://www.cnn.com/2000/NATURE/04/03/moon.glass.enn/
Does this preferentially enhance brightness at the rim of the full disk? How to check? One way would be to grind moon dust into the surface of the tennis ball. I don't have any. Another way might be to substitute the white sandy material, composed of little glass balls from a discarded movie screen or filched from a road crew pouring them into the white paint hopper to put down reflective stripes on asphalt. White sand doesn't work: maybe because the grains are so angular? I tried this afternoon with such spheres as I happen to have: coated a white and a black ping-pong ball with glue, then poured on it an avalanche of 3 mm clear glass spheres. Let it dry, looked at it in a room lit by only a single distant halogen lamp. A basketball might serve better, or glass beads ten times smaller, but what I have doesn't look much like the moon at any phase. Nor does a golf ball; around the rim where sunlight arrives almost tangent to the mean surface, its dimples, like craters, show contrasting shadows and sunlit rear walls. At "full moon" both are dark, on average, around the edges. So is Jupiter.
Well, does a "moon" made of clouds instead of rocks, viz. Venus, change its phases more like the Phong model than like the Moon's data graph? Thus far I found only data covering the phases of Venus with less than half its disk illumined (I guess because the other phases, including "full moon" in particular, are lost in the glare of the Sun). Needed: quantified photos from spacecraft at known distance looking outward toward Venus from positions nearer the Sun. Or in the lab or outdoors in sunlight, some photometer readings on all sides of a sphere coated with each interesting kind of material (against a flat black background).
During the Leonid meteors so much bally-hooed on TV two weeks ago it occurred to me that everyone might be looking in the wrong direction, toward the radiant in Leo rather than toward the setting new moon where maybe one might see splats against hard rock rather than against Earth's air. If such things are going on all the time they might be making new globules of melted glass all the time. Well, in the hour between sunset and moonset I didn't see anything with binoculars, but there were several reports from elsewhere that there was something to see. [LATER: Sky and Telescope magazine, March 2002, shows videotape frames of one such event on page 106.]
A possible project: It would be great to find or make a movie of Moon phases captured in various months from CCD snapshots taken at standard duration, f_stop, and magnification under clear night skies whenever opportunity permits. Each could represent the corresponding 10-hour interval of the 709-hour cycle in a 71-frame movie assembled by dilating ever so slightly and rotating as needed (and maybe masking the background to absolute black) to present a constant appearance except for brightness. I imagine it would dramatically "flash" near full moon, in a way that only certain materials can reproduce. Have you seen such a wonder? I could not find one.
I probably won't get time to cultivate this off-shoot further. If you do, I would be glad to hear of your Adventures in Discovery.
Counting Stars:
Last time there was a second off-shoot concerning the number of stars visible as a function of the least detectable brightness (meaning visible to the Hubble Space Telescope, for example, i.e., not taking into account diurnal and seasonal obstruction of your vision by the bulk of the Earth or by the direction-dependent thickness of its atmosphere.) I gave an "inadequate explanation", partly in desperation and partly to give you opportunity to think and Discover. The desperation stems from time constraints, but there is also a deliberate purpose in leaving loose ends: this column is supposed to represent hand-to-hand encounter with Nature, not calling in air strikes from the Literature. But then I cheated: I found better data on star counts in Sky Catalog 2000.0 proceeded to supplement last week's 4-point plot:
Slope 3/2 is drawn over this log-log plot of a dozen reported star counts (including the four of last week) against the dimmest naked-eye magnitude made visible by telescopes. It would be cheating to immediately ask what astronomers make of this. Our job is to first exploit our own resources.
With 8 more data points added this looks lots more interesting than it seemed last week. It then seemed necessary that the cumulative count (expressed as a log to the same base as used for magnitudes) would rise as slope 3/2 against magnitude. This is nearly linear as expected on log-log coordinates, but the slope is more like 1.2 than 1.5 . It seems that counts fall short of theory (or magnitude is unexpectedly dim) in a simple regular way that depends on distance as a power law: the deficit of stars is proportional to (limiting magnitude)0.3, which would be (maximum distance)0.6 were our simple model correct. (Notice this is not somethingdistance, as might be expected if light traverses an obstacle course in getting to us.) So let's consider possible meanings:
1) Our idea of a 3/2 power law came from imagining Sun-like stars uniformly distributed in volume. What changes if there are two or three or a continuum of kinds of star, also uniformly distributed in volume, but with intrinsic absolute Magnitudes different from the Sun's? I think nothing of importance, but this is something you can explore in a model.The point here is not really about stars and magnitudes, which I assume that astronomers have completely figured out: it is that asking even childishly simple obvious questions about observations available to everyone seldom fails to lead to surprises. The surprises do not have to be facts new to collective and cumulative human experience: it is good enough if they lead the solo inquirer to notice personal deficiencies of concept (inconsistencies, ambiguities, absences) and to correct them. And "good enough" becomes "better" if one of these little kindlings serves to start a more substantial fire.2) Are we Discovering that stars actually thin out with distance from us? Are we in the middle of a dense patch like a globular cluster? I doubt it, but how to check?
[LATER: Astronomers tell me we are in the middle of a thick patch, in that the Sun is about in the galactic plane, and stars thin out exponentially away from that center plane, with scale height about 1000 light years.]3) Or that at magnitude 12 we are looking beyond the plane of the galaxy and so not picking up as many new stars as expected? Pancake half-thickness out where we live is said to be about 1000 light years; you should be able estimate the distance corresponding to 27+12 magnitudes diminution in proportion to distance2. Does it get outside? [LATER: it is not really a cutoff so much as an exponential thinning: see (2). This may well be the main reason why we get about 3 times more stars per magnitude rather than 4 times as expected under the assumption of uniform density.]
4) In principle, this is the sort of evidence one would look for when suspecting 3d space is positively curved: volume increases slower than radius3. But in a logarithmic way?? And to posit noticeable curvature within the confines of a mere galaxy seems too ridiculous: it would amount to a hell of a strong gravitational pull.
5) What if star light attenuates faster than 1/distance2? What if it additionally attenuates like e-distance ? Could there be such density of dust in interstellar space (inside the galaxy) as to fog out remote beacons, and would uniform fog uniformly alter the slope on this log-log plot? A quick-and-dirty spreadsheet or algebraic model will answer this for you. Or look at it this way: Suppose stars are uniformly scattered in volume around us, and that the outermost are most dimmed by some intervening fog. The heavy straight line through the dozen data above is not what we would expect of a uniform fog that gives standard stars at range r less brightness in proportion to exp(-r/L) 1/r2 , where L indicates a distance sufficient to extinguish visible starlight e-fold, about 1 magnitude. In the xy log coordinates of the graph above (prove this to yourself) that would be not x = constant + 2/3 y + about 0.13 y, as the graph suggests, but rather x = constant + 2/3 y + something times 100(1/5 y/3). With y ranging from 2 to 16, and "something" fitted as about 0.015 to go through the extremities of the line of data, these two versions of the third term, the interesting and informative departure from default expectation, are still not very similar. They both get from one endpoint to the other, but the exponential one does it along a conspicuous arch not seen in these data. If the data were somehow biased, and "better" data might traverse such an arch between the same endpoints, can you figure out what distance L would be implicit? I think it would be a couple hundred light years. Could there be that much dust and gas fogging interstellar space? I doubt it, but maybe such a fog is one component of the sought-for explanation.
6) What if it becomes increasingly difficult to reliably census more remote stars in the glare of all those closer, i.e., what if the stars themselves are the fog? Again, my model says "no"; what about yours?
7) Anything else? Most likely I just perpetrated some embarrassingly simple algebraic error. But maybe not, this time; this just ostensibly the sort of surprise we hope to stumble upon as scientists. Such unopened doors superficially all look the same and it is easy to pass them by with glib "explanations", but if we resist that temptation some lead to unexpected Discovery. For example, a scrupulous investigator would realize that failure to observe the expected 3/2 power relation might just subvert the whole idea that stars are sun-like objects distributed in 3d space. If you thought they were points of diverse brightness, all stuck on a relatively nearby ceiling, you would not be at all surprised to find an arbitrary law of abundances of the various brightnesses. In this view, there are no vast spaces and if there were they would have to be air (spatial extension being a property of matter, nothingness not existing in itself) so you could not see more than 100 miles or so into them anyway. The Sun is unique in heavens (of relatively modest extent), the stars are something else altogether, and only a crazy person would suppose they are other suns clearly seen while inconceivably remote, and only a charlatan would continue so even after the expected 3/2 power law fails.
[Added LATER:
see 28 December column for specifics about "completely figured out"]Addenda re prior columns:
Look here about seeing Venus with naked eye in the daytime (16 November column).
Three notes on the proffered rough criterion for visibility of a point source like a star in the daylight sky: did you think of these on 16 November?
1) Once enough magnification is applied so the source is no longer a mere "point", further increasing magnification to further thin out the background no longer helps: it now dims both disk and background equally and so should no longer affect visibility if contrast is the decisive factor.2) With full moon 14 magnitudes dimmer than the Sun, the moonlit sky is as much dimmer than daylight sky. So should we suppose that threshold magnitude -5 for naked-eye visibility in daylight implies magnitude 9 under full moonlight? I bet only stars of magnitude 2 or brighter will show up tonight (but I have to post this column before looking. What do you see?). Magnitude 9 differs from 2 by an unmistakable 7 magnitudes.
3) I suppose (1) is reminding us that there is another criterion: there must also be enough light entering our pupil to register with our rod cells. So replace "9" by the reported dimmest visible to naked eye on a totally dark background: 6. There remains a problem: given even as little sky glow as daytime magnitude -5 per arc minute (our effective pixel) dimmed in switching to moonlight by 14 magnitudes to magnitude 9 per arc minute, we can no longer see those magnitude 6 stars (or anyway at least the magnitude 4 stars that we can really see on a clear night) but now only maybe magnitude 2. That dim moonglow cost us at least 2 magnitudes. Can you figure out why?
And here are some belated attempts to write clearly:Last week's "How many stars are visible?" was anything but clear. What I meant is "with optical help" and "if none are obscured by Earth, its atmosphere, or other obstacles such as interstellar dust clouds."
Similarly "2.51 times more stars per magnitude," despite its quantitative disguise, is a totally unclear answer. It is clear that enhancing sensitivity to detect stars a magnitude dimmer is equivalent to detecting them (same kind of stars) sqrt(2.51) times further away if energy is transmitted without loss in expanding spheres. This means the boundary of that enlarged sphere has 2.51 times the former area. But that is 2.51 times as many stars only if we consider the compared volumes to be spherical shells both of the same radial thickness. So a better answer to the riddle "inadequately explained" might be that skipping a magnitude fattens the sphere by sqrt(2.51) times the prior radius, not by a fixed amount, so the new shells added on each time are geometrically similar and thicker and increase in volume by factor sqrt(2.51)3 at each step. The sum of all the nested shells very nearly does likewise, differing only by the constant omission of some inner-most sphere. This is why we expected a 3/2 power law in cumulative star count vs magnitude limit (and might have been content that Nature agrees, had I not gotten more data, above: so now we get to Discover something, perhaps including the existence of matter thinly occupying interstellar space.)
And the phrase last week, "half that ratio of f_stop change," also sounds precise but proves to be ambiguous: does "change" means clicks or printed numbers? In terms of light intensity or film exposure two clicks of the shutter speed ring (printed number doubling per click) gets you two factors of two, and so does two clicks of the f_stop ring (printed number doubling in two clicks but area doubling per click ).
Interpreting the Curvature of Straight Lines (written 8 November with 16 November intent) will finally appear next week (7 December), after these three weeks of detour to exploit Discovery opportunities latent in Alan MacRobert's 9 November "Stellar Magnitudes" article.
Copyright 2001 by A.T.Winfree. All rights reserved. Used by permission.
