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This question was posed to myself and Reg Dunkley, both retired meteorologists, by Victoria Centre member David Lee during an astro-café. Like many avid skywatchers in the mid-latitudes, he was frustrated at the seemingly high frequency of cloud in the twilight arch.
Fig 1. Pairing of Venus and Jupiter 01:10 UT November 26, 2019, taken from Ogden Point, Victoria B.C. by David Lee. Venus is about 4.8 deg altitude.
The one-word answer is "foreshortening", but I suspect that many readers would prefer some diagrams and explanation to help them really grasp the phenomenon.
Fig 2. A foreshortening example familiar to all readers. The spacing between the poles is constant, but the apparent gaps become shorter as our fore-ward viewing angle decreases.
Fig 3. This schematic turns Fig 2. on its side, and paints alternating gaps with cloud and clear.
It demonstrates how the gaps in regularly spaced cloud streets can appear large overhead but compress towards the horizon.
Fig 4. The lowest section of Fig 3. zoomed-in depicts how cloud seems to take up an increasing amount of sky closer to the horizon.
Thankfully we do not live on a flat planet; the inevitable distant cloud would always stripe the lowest part of the sky. I find it instructive to examine the geometry to scale. Readers can access Fig 5. at https://www.desmos.com/calculator/vxdlpyod8h and adjust the values to see changes in real-time. Clicking on a line will pop-up the X and Y values (in km). For simplicity, cirrus cloud is taken to lie 10km up.
Fig 5. A schematic of the geometry, where the lower green section is the solid Earth (radius 6371 km); horizontal blue line is the horizon; diagonal purple line is a user-changeable way to measure the altitude; the orange and black segments are 1/4 degree arcs placed 10km above the surface. Note that the word "altitude" in this article refers to the angle above the horizon in degrees.
Despite the astronomer's knowledge of the size and scale of the Earth and sky, their animal instinct and tactile experience of nearby nature can lead them to misjudge angles and distances of farther things. For example, while standing on a hilltop you might figure you can see 30 or 40 km (assuming no prominent mountain ranges), and subconsciously put distant clouds at that range. On occasion we see public reports of UFOs post-sunset because their bright orange vapour trails have no visible craft at their leading edge. As the diagram shows, objects near the horizon and 10km high are actually 300km away!
How does the angular size of the gaps change with altitude? The diagram shows strips of cloud and gaps 28km wide, a 1/4 degree spacing as seen from the centre of the Earth.
Table 1. Horizontal distance, altitude above horizon, and angular spread of regularly spaced 28 km strips at a height of 10km, ignoring refraction.
Distance Altitude Spread0 km 90.0 deg27.8 19.6 70.455.7 9.9 9.783.5 6.5 3.4111.4 4.6 1.9139.2 3.5 1.1167.0 2.7 0.8194.8 2.1 0.6222.7 1.6 0.5250.5 1.2 0.4278.3 0.8 0.4306.1 0.5 0.3334.0 0.2 0.3361.8 -0.04 0.2The gaps near the horizon subtend angles barely larger than half a lunar diameter. Compounding the problem, increasing refraction at lower altitudes through the greater path length of air "pulls up" objects from beyond the geometric horizon. The very lowest part of the sky can contain cloud from 400km away.
For the photographer, what does this 400 km distance mean? When interpreting a satellite image, wondering anxiously if the advancing cloud will scuttle the celestial display, it helps to know that one degree of latitude is about 111km. At 50 deg N, this is 1.6deg of longitude. Quick, approximate math means that if a deck of cirrus cloud is less than 4 degrees of latitude or 6 degrees of longitude away, it will show up.
Fig 6. GOES-W Satellite image at 02 UTC, 20191126, with thanks to Michael Skarupa at Environment Canada for retrieving it. This is 1 hour after David Lee's image from Fig 1, so the two cannot be exactly compared. The main gap below Venus is about 2.4 degrees altitude, a linear distance of about 180km or 1.5 degrees of latitude. Allowing for some eastward motion of the cloud between the two times, this corresponds to the orange edge in the satellite image, just west of Vancouver Island.
Despite the cloud, take heart! Decades of experience have shown us that the leading edge of a cirrus shield can suddenly thin, dissipate, or form gaps just as often as jet vapour trails (contrails) make things worse. If we are lucky, then just a touch of cloud adds "atmosphere" to the scene. The relevant truism here is that in the end, the most persistent skywatchers are the ones most rewarded.
In summary, foreshortening and refraction visually compress hundreds of kilometres of atmosphere into the lowest 4 degrees of sky. Compared to large gaps seen overhead, the horizon is indeed more likely to be obscured by clouds.
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