Moon

Overview

This page has an analysis of photos of the moon that I took with my Nikon P900. The moon photos are cropped so that the moon is tightly enclosed in in a square photo since doing so makes it easier to act as input to sphere2equirect.py in image-utils. Clicking on an image leads to a higher resolution version. The orFiginal photo taken from the camera can be seen by removing the "-crop" from the photo name. Original photos that are not already linked to directly can also be found in this table:

Section Name Original Photo
Moon Maps Full Moon moon-full.jpg
Moon Maps Quarter Moon moon-quarter.jpg
Day Moon Day Moon moon-day.jpg
Moonrise Moonrise moonrise.jpg
Lunar Eclipse Lunar Eclipse lunar-eclipse.jpg



Moon Maps

A nearly full moon taken in North Austin TX (30.37 N 97.76 W) at 2018-10-25 21:29:05 CDT (Austin TX time):

Craters are most visible near the terminator, which in this case is near the upper right side of the moon. Since it's an almost full moon the terminator in the upper right side of the moon is is nearly a half circle, but it's actually half of an ellipse.

This photo is a bit dark and tan possibly due to a low exposure compensation value of -1 to assure that details would not be lost due to over exposure. But it being relatively low altitude at about 18° above the horizon may also explain it.

Since it's close to the horizon atmospheric refraction has flattened it a bit by one part in 362.4, which can be seen by using the formulas on that page for the upper part of the moon, and the lower part of the moon using the altitude given by Stellarium. Because of the flattening this image is not quite square. The calculated flattening is a starting point followed by trial and error.

The above full moon in Stellarium (which shows libration) for the same location and time:

From the above we can see that the full moon had an angular size of 31'42.2" or 0.5284° which is helpful for building an equirectangular map using sphere2equirect.py in image-utils:

sphere2equirect.py -a 0.5284 -b --center-lat 6.68 --center-lon -3.43 --rotate -48.3 --ellipse moon-full-crop.jpg

Which produces the following equirectangular map:

The latitude and longitude of the center point as well as the amount the moon is rotated was determined by trial and error so that it matches a known good equirectangular map of the moon. Such a map can be found in Solar System Scopes's textures.

Since the sphere is identified in the input image by specifying a bounding box the encloses it when working with the moon it's helpful to note that two adjacent sides will be visible and the other two sides will be hidden in darkness. In this case the left and lower sides of the moon are visible and the right and upper sides are hidden in darkness. Consequently "--in-begin-x" and "--in-end-y" can be specified for those sides since they are known and "--in-size" can be estimated.

One way of getting an initial estimate is to use an image editing program such as GIMP to fit a circle to the visible part of the perimeter of the moon.

If sphere2equirect.py is run with "-v" (verbose) it will indicate the range used. That information can be used to crop the images so that they tightly enclose the sphere (moon in this case), produced the images we see here.

In this case the input image is not quite square due to atmospheric refraction flattening the moon a bit (discussed above). Since sphere2equirect.py assumes by default that the input sphere is perfectly spherical the "--ellipse" in this case indicates the entire input should be consumed instead of the largest square that fits within it.

Since equirectangular is a common mapping that can act as input for mapping software it's possible to use G.Projector to convert to other projections. Here's the the above full moon converted to a Mercator projection with a maximum latitude of 65°:

The Mercator projection has some helpful characteristics. In spite of distorting size dramatically it maintains both the aspect ratio (it does not stretch or shrink on one axis) and direction (North is always North). Because of that we can see that the craters that are near the terminator (close to the edge of the moon in the upper right hand corner) are no longer elliptical due to being viewed at an angle. We can also see that the terminator line, which was quite curved in the original image, is a nearly straight vertical line due to the terminator being a great circle that runs nearly North and South.

With the Gnomonic projection the terminator, as well as any other great circle, will be a straight line. It's hard to tell since it's a bit blurry near the poles and since there may be some distortion there, but it looks like that may be the case. Also, lines radiating outward from craters as straight. At 0° North, 30° East and a radius of 65°:

A quarter moon taken in North Austin TX (30.37 N 97.76 W) at 2018-09-02 06:51:28 CDT:

The above quarter moon in Stellarium for the same location and time:

In this case the orientation is off since the moon was so high above the horizon that I didn't bother to have my camera in the "correct" orientation, but it's still reassuring that the terminator matches Stellarium. Also, like with the full moon, the crater shadows are most prominent near the terminator. From the above we can see that the quarter moon had an angular size of 32'5.7" or 0.5349° which is helpful for building an equirectangular map using sphere2equirect.py in image-utils:

sphere2equirect.py -a 0.5349 -b --center-lat 5.9766 --center-lon -6.3281 --rotate -26.46 moon-quarter-crop.jpg

Which produces the following equirectangular map:

Due to libration the longitude of the center for the quarter moon has moved about 2.9° to the West compared to the full moon allowing us to see a bit more of the left side of the moon. We can see that there is a spot on the West side that is 65 pixels from the edge for the full moon, and 85 pixels from the edge for the quarter moon which works out to a shift of acos(1 - 85/(2462/2)) - acos(1 - 65/(2436/2)) = 2.61°. Subtracting two pixels from 65 and calculating for 63 gives 2.91°, so that amount of shift seems about right. Comparing to the Stellarium images after rotating so that the equator is horizontal the spot gives acos(1 - 34/1024) - acos(1 - 25/1004) = 1.99°, which is a bit off, but in the right direction. I'd like to understand this discrepancy better since Stellarium seems to calculate libration correctly (longitude, latitude and diurnal). The full and quarter Stellarium images used.

It's an interesting exercise to try to generate the full moon equirectangular map with the quarter moon longitude and then adjusting the "begin" and "end" options to compensate so that the center of the map is the same pixel. The result is a significantly distorted map.

Libration

To illustrate libration with maps the full moon equirectangular map can be generated on top of the quarter moon equirectangular map with the "--multi" option. First "moon-quarter-crop-er.jpg" is copied to "moon-combined-crop-er.jpg", then it is resized to match "moon-full-crop-er.jpg" and then the command that produced "moon-full-crop-er.jpg" previously can be run with the "--multi" option:

sphere2equirect.py -a 0.5284 -b --center-lat 6.68 --center-lon -3.43 --rotate -48.3 --ellipse --multi -o moon-combined-crop-er.jpg moon-full-crop.jpg

Which produces the full moon equirectangular map with a sliver of the quarter moon equirectangular map on the left:

Which doesn't show us much since it's so distorted that close to the edge, but it still gives us a way of visualizing how libration reveals different parts of the moon. It's also reassuring that the light and dark spots line up in some places.

Angular Size

Both the original uncropped full moon and quarter moon photos were taken with maximum optical zoom was used which, as can be seen in the EXIF (properties that are stored in JPEG images) information, is the equivalent of 2000 mm for 35 mm film:

exif:FocalLength: 3570/10
exif:FocalLengthIn35mmFilm: 2000

Following the reasoning on the sun page, but using a more precise version of SPP:

SPP = 3.7558e-06

Also from the sun page is this formula for calculating the angular size, which will be used for the following table:

angular_size = atan(SPP * ((4608 / 2) - left_pixel)) + atan(SPP * (right_pixel - (4608 / 2)))

The previously X ranges and X sizes (horizontal is better than vertical due to atmospheric refraction) can be used to calculate the angular size and compare to Stellarium:

Image Link Left Pixel Right Pixel Image Angular Size Image Error Stellarium Angular Size Stellarium Altitude Stellarium Screenshot Link
Full 1123 3559 0.5242° -0.7949% 0.5284° 18.4914° Full
Quarter 1636 4098 0.5298° -0.9534% 0.5349° 75.4146° Quarter

It seems from both this page and the sun page that image angular size is systematically low, maybe by about -0.75%. If we were to correct for that it would be much closer. The quarter moon is 1.23% bigger than the full moon, so with that correction the error would be small compared to the measured difference.

Dark Side

Since the dark side of the moon blends into the night sky an observer unaware of the moon's spherical shape might think that the moon is entirely the light side of the moon. Such an observer might also think that the different phases of the moon are due to the moon changing shape. Fortunately in some cases it's possible to observe the dark side of moon directly. In the case of a waning crescent moon just before sunrise or a waxing crescent moon just after sunset it is particularly easy to see the dark side of the moon for two reasons. Since only a small fraction of the moon is light there is little light to compete with the dark side. Also, the phase of the Earth seen from the moon is the opposite of the phase of the moon seen from Earth, so the dark side of the moon lit by a nearly full Earth when it is a crescent. Such lighting is known as Earthshine. When the moon is a crescent it's sometimes possible to see the dark side of the moon with the naked eye.

When using a camera the dark side of the moon can be photographed by adjusting the exposure, which may result in the light side being overexposed. For the P900 this can be done by with the "exposure compensation" setting. Here are three photos of a crescent moon that I took with three different exposure compensation settings around 2018-12-10 18:48:53 CST (the "exif:DateTime" for the photo is an hour later since the camera had not been adjusted for standard time).

Normal exposure, which is exposure compensation 0. Only a hint of the dark side is visible:

Brighter exposure, which is exposure compensation 1. The dark side can be seen, but without detail:

Brightest exposure, which is exposure compensation 2. The dark side can be seen with some details:

The above crescent moon in Stellarium for the same location and time:

Day Moon

The moon in the day is the same thing as the moon at night, but with blue light added. Here's a gibbous day moon at 2018-10-25 21:29:05 CDT (Austin TX time):

And the Stellarium screenshot for the same time and location:

In the image there is a large magenta square surrounding the sea of Serenity and two much smaller red squares pointed to by arrows at the lower part of the image. For the magenta square here's a comparison to the same area for the full moon in the first image.

Our eyes adapt to the prevailing color bias of a particular scene, but the above makes it clear that the day moon is much bluer than the night moon.

The small red squares are from inside a crater shadow, and from the sky next to the moon. A comparison of their contents:

It's very close to the same shade of blue which suggests that the color of the crater shadow is black, as it is at night, with the blue light found in the sky added.

Here's a photo taken from flight 2126 from Houston to Austin TX at 2018-12-28 11:12:38 CST:

Due to the atmosphere the distant land is bluer than the close land. The same phenomena can be seen with mountain ranges (follow the link to see links to great examples).

Sphere Lighting

During the day, as seen in the previous section, it's possible to compare the way the moon is lit by the sun to other spheres and see that it is similar.

Here's a composite photo of the moon (left) next to a 5 cm yellow ball (right) taken in Austin TX at 2019-01-28 08:35:16 CST:

The above composite image in Stellarium for the same location and time:

Since the depth of field is constraining with the P900, even with the f-number set to f/8 (the highest supported value), the composite image was created from two images so that each sphere could be in focus. This is the image with the moon in focus at 2019-01-28 08:35:05 CST (Austin TX time):

The above was followed by this image with the ball in focus taken 21 seconds later at 2019-01-28 08:35:26 CST (Austin TX time):

For the image with the moon in focus the center of the moon is approximately at X=1901 and the center of the ball is at X=2984. From the EXIF data the 35 mm equivalent focal length is 380 mm. Combining this information the ball is 1.23° to the right of the moon:

SPP = ((sqrt(36**2 + 24**2)/2)/380)/(sqrt(4608**2 + 3456**2)/2) = 1.9767e-05
atan(SPP*(2304 - 1901)) + atan(SPP*(2984 - 2304)) = 1.23°

So the phase angle of of the ball is approximately 1.23° fuller, which is hard to notice.

The photo with the ball in focus seems is rotated rotated clockwise 2.1° relative to the photo here the moon is in focus. The rotation can be seen in the composite image with careful examination. If the 2.1° was removed by rotation the composite image would be even closer to matching.

Here's a photo of the moon next to a sphere on top of a flagpole taken in Austin TX at 2019-12-19 09:52:30 CST:

And the Stellarium screenshot for the same time and location:

It should be possible to get better results by using a camera with a smaller aperture, a larger farther ball and a tripod (instead of placing it on a slightly uneven surface), but the the above is sufficient.

There is nothing special about the time and location that the pictures were taken. Here's a video with similar results.

This strongly suggests that both the moon and the ball are both spheres similarly lit bit a distant light source (the sun). The light rays are close to parallel.

Moonrise

Like the the sun the moon rises and sets. Here's a picture just after moon rose over the ocean in Jacknsonville FL (30.19 N 81.36 W) at 2018-12-25 21:07:02 EST:

The above moonrise in Stellarium for the same location and time:

It's noticeably compressed due to atmospheric refraction due to being so close to the horizon (the cropped image is square). Since anomalous refraction normally occurs within three meters of the water and since this photo was taken approximately five meters above the water, and since the entire line of sight, other than the very bottom of the moon, is entirely over three meters, this strongly suggests that standard atmospheric refraction is in effect.

Here's a non-zoomed picture taken nine minutes later at 21:16:58 where it's possible to see that the top of the moon is about 4 moon diameters over the horizon, which corresponds to about 2.1° over the horizon. This is the expected distances given timeanddate.com's prediction that the moon would rise at 21:02.

Lunar Eclipse

Here's a photo of the lunar eclipse taken 2019-01-20 23:30:01 CST (Austin TX time):

It's a bit blurry since it was hard to photograph due to it being much dimmer than the moon normally is. The shutter was open for a full second at ISO 1600. The eclipse happened exactly as predicted.

The above lunar eclipse in Stellarium for the same location and time:

There's a video of the lunar eclipse by Soundly that I recommend. Here's a screenshot that I took of it:

The red arc was added by me (click here to see the original screenshot without the red arc). It indicates the curve expected by the umbra portion of Earth's shadow, which is approximately 8/3 (2.67) times the diameter of the moon. It seems to roughly line up with the shadow seen. Click here for more information on the math behind the 8/3 calculation.

Conclusion

The moon is clearly a large (tiny bit oblate) sphere which can be mapped and also predicted with heliocentric software such as Stellarium.