## Forensic physics puzzle.

These days we are always rightly suspicious of anything on the web (the misinformation highway). Fake news stories, and faked photographs proliferate. Photo manipulation software makes fake pictures difficult to detect. While writing the rainbow puzzle for the last issue I searched for a picture of a rainbow against a sky, inclouding the moon. I found one, and only one such picture. Is it real, or faked? A little knowledge of physics could help us determine that. This picture is a nice forensic physics puzzle.

Are there clues in this picture to help us determine whether it is a fake? Assuming this is a genuine photo, estimate how many days before full moon it was taken. Do this by three independent methods.

Do not confuse this with a moonbow, which are usually so faint as to appear colorless, and are seen on the opposite side of the sky from the moon. Lunar halos are another phenomena entirely, and are centered on the moon. The web has many pictures of both of these.

Reader feedback is appreciated. Email dsimanek@lhup.edu. If you have a favorite physics puzzle that is not well known, not easily found on the web, or in the many published physics problem books, send it along. Include your answer, if you have one. Original ideas will be credited. I especially like puzzles with unexpected answers, that can be solved with insightful and simple arguments, preferably with minimal mathematics.

## Answer to the forensic physics puzzle

There are three pieces of evidence in this picture that agree with each other and confirm that this is a real picture, or one faked by a very skilled and knowledgable person who understood celestial geometry.

(1) The moon subtends an angle of 0.5° in the sky. The primary rainbow colors from violet to red have arcs from 40° to 42° a difference of 4 moon widths. The moon size and the color spread of the rainbow are consistent with that.

(2) The moon's terminator (the boundary between its illuminated and dark surface) is on the side toward the anti-solar point at the center of the rainbow, as it should be between first quarter and full moon, and is symmetric with the rainbow's radial line.

(3) The moon's synodic period is 29.53 days. Relative to the sun (and relative to the anti-solar point) it moves 360°/29.53 days = 12.2°/day. The rainbow's radius is 42°. So the moon would take 42/12.2 = 3.44 days to move from the edge of the rainbow to its center. Therefore since the moon appears so near the arc of the rainbow, just under 40° from its center, this picture must have been taken about 3 days before full moon. At full moon the moon would be at the rainbow's center, and usually just below the horizon. But, thanks to atmospheric refraction, the moon can sometimes be seen just rising as the sun is just setting and both are still above the horizon. In the rare case where there would be a rainbow at that time, the full moon would be at the center of the rainbow. I would like to see that! Send me the photo when you see it.

Furthermore, now that we have concluded that the picture is genuine, we can reasonably assume that this was taken in the northern hemisphere at mid-latitude. Of course that does depend on the assumption that the camera was held as it normally is—not tilted with respect to the horizon, but the distant hills confirm that. It also assumes that no one has flipped an image L/R that was taken in the southern hemisphere.

Based on the size of the moon's image we can conclude that the picture width is about 31°. So it was taken with a "normal" camera lens (these usually have horizontal width of 30° to 40°) with only modest cropping.

• Donald Simanek, 2017.