A Point Without Shadows

I have run into a significant problem I did not expect.

Flying in the Humuʻula Saddle
Flying the drone in the Humuʻula Saddle region on the way to work
I love to fly the drone in the Humuʻula Saddle in the morning and afternoon. It is an area that I find endlessly fascinating, with spectacular scenery in a dramatic landscape.

Even better, I need only stop off on the way to or from work at the summit to find time to fly here. Leave for work early, discharge a few flight batteries, join the rest of the crew for breakfast at Hale Pōhaku.

Just after dawn or just before sunset the light accentuates this dramatic landscape, intensifying the colors, the low sunlight angle creating shadows that reveals the stark terrain.

The low sunlight angle causes some problems as well. One I expected… The drone camera does not deal well with glare. Take an image pointing near the Sun and the image is often ruined by the glare. This is discussed in some of the review videos I watched before I bought the drone. Easy to avoid, just point away from the Sun before you start that panorama sequence.

The second major issue I did not expect… Many of the images feature a very bright spot directly opposite the Sun in the sky. This spot is intense, creating a peculiar feature in the photograph that I find distracting.

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The Opposition Effect

It was pretty obvious, an odd bright spot in the trees below that followed the helicopter. Having educated myself on quite a few optical phenomena I knew exactly what it was I was seeing, and made a point of taking a few photographs.

Opposition Effect
A bright spot in the Puna rainforest caused by the opposition effect
The mechanism for this bright spot is remarkably simple… No shadows.

Called the opposition surge, Seeliger effect or shadow hiding this simple optical phenomena occurs when looking at rough or irregular surfaces that are directly away from the light source, usually the Sun. On an irregular surface some parts will shadow other parts, resulting in an apparent darkening of the overall surface. When looking at that part of the surface directly away from the light source no shadows are seen, making that region appear brighter.

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No Moon in the Shadow

I had planned to go up and photograph the tail end of tonight’s eclipse as the full Moon rose just inside the mountain shadow… No luck.

Thanks to tropical storm Niala passing south of the island the summit is totally weathered in. Fog, rain and no shadows…

Summit Fog and Rain
Current view from MastCam of a foggy, rainy summit. No eclipse to be seen tonight.

The Earth’s Shadow… Twice

One of the more sublime sights seen from Mauna Kea is the shadow of the mountain rising through the mist and haze at sunset. One of the more sublime sights in the heavens is the Earth’s shadow crossing the face of the Moon, a total lunar eclipse. It is possible to combine these two phenomena if the timing is right, the Earth’s shadow seen twice.

Eclipse in the Mauna Shadow
The eclipsed moon rising in the shadow of Mauna Kea, February 20, 2008, photo by Alex Mukensnable, used with permission
The moment of totality in a lunar eclipse occurs when the full Moon is directly opposite the Sun in the sky. By simple geometry this same anti-solar point is where the tip of the mountain’s shadow will be projected for an observer standing near the summit of the same mountain. If the eclipse is in progress at sunset, and you are standing on the summit of a suitably prominent mountain, you will see the Earth’s shadow both in the sky and obscuring the Moon.

Back on Feb 20th, 2008 the timing was right. A friend of mine, Alex Mukensnable, noted the timing and set up to catch the eclipsed Moon rising in the shadow. The result was a great set of photos. The photo is nice enough as a still, be he did more than that, he shot the event as a timelapse and assembled a video of the rising Moon.

There are several possible variations depending on the timing with this sort of event… If the Moon was at the height of totality rising it would also be right at the tip of the mountain shadow as it rises. This is a relatively rare event as the timing requirements are tight. Unfortunately it would also be quite dim, darkened by the shadow, and not easily seen as it rose.

As lunar eclipses are long events, taking several hours to complete, the likelihood of the Moon being in at least the partial eclipse phase at sunset is fairly good. Still a rare event, but not extraordinarily so. Thus for a single site, the summit of Mauna Kea this even happens in both 2008 and 2015.

As the Earth’s shadow is about 2.6° across at lunar orbit, the closest a partially eclipsed Moon will be seen from the the very tip of the shadow is about 1.3°. This is a bit less than three lunar diameters. Of course these numbers will vary a few percent depending in the distance to the Moon which changes as it makes its elliptical orbit.

The Moon moves slowly across the sky from west to east, thus before totality the Moon would be above the tip of the mountain shadow. After totality, with the eclipse ending, the eclipsed Moon will be in the shadow. Placing the Moon in the shadow also makes it easier to see, the bright crescent a better contrast to the dark shadow. This is the case for the 2008 eclipse captured by Alex.

Of course you could reverse all this timing and watch the event at moonset and sunrise. If the eclipse was just starting at dawn it would again place the eclipse in the mountain shadow.

Another important point to remember is that the shape of the mountain’s shadow has little to do with the shape of the mountain. The shadow will always be a neatly conical form due to the effects of projection.

What brings this event back to the fore is that the timing will soon be correct to see this same event again. The total lunar eclipse of Sept 27th, 2015 will be a bit of a dud for Hawaii, most of the eclipse already over as the Sun sets and the Moon rises over the islands. However, this event will feature very similar timing to the 2008 eclipse. The Moon will still be in partial eclipse when it rises. As it rises a short time after full Moon it will again be deep in the shadow of Mauna Kea as it comes over the horizon.

You know where you will find me on the evening of the 27th. Now I just need some clear weather that day.

Lahaina Noon

Living south of the Tropic of Cancer we get to experience an interesting phenomena that folks outside the tropics will not see. There are two days each year when the Sun passes directly overhead. In the islands this event is called Lahaina Noon.

Spring Lahaina Noon for 2013
Location  Date Time
Hilo May 18th 12:17pm
Waimea May 19th 12:20pm
Kahului May 24th 12:23pm
Honolulu May 26th 12:30pm
Lihue May 30th 12:36pm

Lahaina noon occurs twice each year as the Sun appears to move northwards with the spring and again as it moves southwards in the fall. For the islands of the Hawaiian archipelago the first day is between May 16th and May 31st. The second Lahaina Noon will be between July 10th and July 25th.

The date on which this event occurs each year depends on your exact latitude, the further north the later in the spring it will occur. Thus the day for Lahaina noon will vary by eight days from Hilo to Honolulu, and another five to Lihue. As you approach the Tropic of Cancer at 23°26’N Lahaina Noon will occur closer to the summer solstice. The date will also slip a little due to the out of sync nature of our seasons and our calendar. This is the reason we insert a leap year into the calendar every four years.

This year Lahaina Noon will occur on May 18th for residents living in Hilo, or May 26th for Honolulu. It is also important to remember that the Sun is not directly overhead at 12:00 exactly. As the islands lie west of the center of the time zone, true local noon occurs up to half an hour after 12:00.

The Symmetry of Mountain Shadows

The sunset view from the summit of Mauna Kea is truly spectacular. From the summit you are usually above the clouds, watching the sun sink into a cloud layer thousands of feet below. The colors are intense, the deep blue sky, the red cinder and the gleaming telescope domes. This spectacle draws tourists from across the globe, trekking up the mountain just in time to witness sunset.

The Shadow of Mauna Kea
The shadow of Mauna Kea rising through the haze at sunset with the pink of the Belt of Venus above
One part of this spectacle is the enigmatic shadow that rises through the eastern haze, a beautiful pyramid of darkness that stretches to the distant horizon. A serene and yet awesome sight, the shadow reaches for infinity through the pastel shades of the Belt of Venus above the blue-grey shadow of the Earth itself. The shape is a perfect pyramid, with a symmetry not expected in a natural phenomena.

Oddly enough, it seems that the actual shape of the mountain is not that important in the creation of such a triangular shadow. The shadow will show that beautiful shape regardless of the mountain’s profile. Even a flat topped mountain will have a shadow that converges to a point at the top. This contradicts our experience, where common shadows match the shape of the casting object. We expect a shadow to portray the object.

A mountain shadow is different, the shadow is elongated to a great distance by the scales involved, in this situation the geometry dictates a different result. The secret to the shape of the shadow is that it is driven by the effects of perspective, with the shadow reaching to a vanishing point in the far distance. In 1979 the problem of the mountain shadow shape was mathematically modeled by William Livingston and David Lynch. They showed that regardless of the mountain’s profile a conical shadow would be perceived by a viewer near the summit. The proportions may differ depending on the profile of the mountain, but the conical shape would remain.

In the case of Mauna Kea, the effect is not obvious, the mountain does have a fairly symmetrical shape with steep sides. A viewer might not recognize the fact that the projected shadow does not match the shape of the volcano. An astute observer may notice a discrepancy, Mauna Kea is notably rounded at the summit, yet the shadow possesses a sharp apex.

I was completely unaware of this until it was pointed out to me a few days ago by Dean Ketelsen when I posted the Mauna Kea mountain shadow image. I suspect Dean has had many opportunities to see this phenomena from atop Kitt Peak, a flat topped mountain that casts a conical shadow.