This is a sort of pendicle to my previous post about the partial solar eclipse during the Battle of Isandlwana in 1879. I wanted to give a better impression of how minimal that 56% eclipse actually was, when compared to the “darkness during daytime” of a total solar eclipse, so I decided I’d plot some comparative light curves.

This involved a couple of simplifying assumptions—I’ve made the moon and sun exactly the same size, and I haven’t simulated the effect of limb darkening. This latter effect is due to the fact that the rim of the solar disc is slightly dimmer than the central portion, but it’s really only relevant during an annular eclipse, or the final moments of a total eclipse or very extreme partial eclipse.

So it was just a matter of calculating the yellow area below, technically known as a lune, for any given degree of offset between the centres of the grey disc and the yellow disc it is partially obscuring.

This turns out to involve a bit of a bonkers formula, but it gets considerably simpler when the two discs are the same diameter.

Knowing that the illuminance from the overhead sun in a clear sky is normally about 100,000 lux, I could then compare the light from the partially obscured sun with a range of other outdoor illuminants.

So I’ve plotted illuminance on the left vertical axis of my diagram above, from 100,000 lux (unobscured direct sunlight) down to 0.1 lux, which is in the ballpark of full moonlight. The scale is logarithmic, because that’s how our eyes and brains interpret light. We perceive equal ratios of illuminance as equal increments—a shift from 1,000 lux to 10,000 lux feels about the same as a shift from 10,000 lux to 100,000 lux. The magnitude scale by which astronomers measure the brightness of stars is also logarithmic—a star of first magnitude is shedding about 2.512 times more light than one of second magnitude, which is putting out 2.512 times more than one of third magnitude, and so on.*

There’s no horizontal timescale, because that varies a bit from eclipse to eclipse. The full width should come in somewhere between two and two-and-a-half hours.

Drawn across my diagram are some approximate thresholds for various kinds of outdoor brightness. A light overcast has very little effect on the day’s illuminance, a dense overcast has a noticeable effect, and if you’re under a really thick cumulonimbus storm cloud you’ll be turning on your car headlights. At which point, the daylight is a thousand times dimmer than bright sunshine.

When the sun sets, we enter civil twilight, which continues until civil dusk, when the sun is six degrees below the horizon. It’s called civil because it has relevance to civil authorities, the organizations that supply services to citizens. During civil twilight, it’s possible to carry on outdoor activities without artificial lighting, so that’s something civil authorities like to know about. You can also still read outdoors, if the print isn’t too small.

But by the time you get down to the brightest full moonlight, at about 0.2 lux, most people will struggle to read anything but a newspaper headline, and will be well on the way to losing colour vision (though it’s not entirely gone).

So that seems like a useful range within which to discuss how dark it gets during an eclipse.

At right of my chart, I’ve marked up the area of the solar disc that’s equivalent to the various illuminants. But once we get down to 100 lux, which would equate to 0.1% of the solar disc, the rough edges of the lunar disc become evident—the remaining solar crescent breaks up into segments, shining between the lunar mountains. This appearance is called Baily’s Beads, in honour of the astronomer Francis Baily. Here’s what they look like:

During a total eclipse, this appearance lasts for just a few seconds before the sun is completely obscured. At which point, it doesn’t get completely dark. Not only do we receive light from the solar corona, but we also have light coming in from the sky beyond the edges of the eclipse shadow, producing a glow of dim daylight along the horizon. So totality bottoms out at around 5 lux, equivalent to the darkest stages of civil twilight, but more than ten times brighter than the full moon.

With that explanation out of the way, let me just repost my graph here, to save you a bit of scrolling:

Notice how rapidly the illuminance falls around the central phase of the total eclipse. It drops more than a hundredfold in the last minute before totality, and leaves us in incontrovertibly dark surroundings.

But look at the broad “wings” of the eclipse curves. For something approaching an hour at start and finish, the illuminance changes only slightly, and very slowly, as the moon creeps across the margins of the solar disc. For most of that period, our pupils can easily adjust so that the light reaching our retinas doesn’t change at all. Anyone with normal pupillary responses, starting from the constricted pupils of full daylight, can easily double their pupil diameter, and therefore compensate for at least a four-fold change in brightness.

The 95% eclipse light curve drops out of this range, but never comes close to matching the final plummet of a total eclipse. Our brains are still happy that the 5,000 lux at the maximum of a 95% eclipse is bright daylight—it certainly never gets to any point we’d remotely consider dark. But the final change is fairly rapid, and it perhaps represents a point at which a lot of people might notice a change in the brightness of the environment. (I looked for, but could not detect, such a change from the recent 90% eclipse in eastern Scotland.)

Now look at the curve corresponding to the 56% eclipse at Isandlwana. It never even dips into “light overcast” territory, and never leaves the “pupillary compensation” zone. Which is why I doubt if it was ever noticeable on the battlefield.

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* Why is there such a weird ratio in the stellar magnitude scale? The number is called Pogson’s ratio, in honour of the nineteenth-century astronomer Norman Pogson, who first suggested it. Astronomers at that time were still using the stellar magnitude scale from Ptolemy’s second-century Almagest star catalogue. But they had discovered, by using filters to reduce the brightness of one star to match that of a dimmer one, that the magnitude scale was operating in a logarithmic way, as I’ve described above. Pogson pointed out that a first-magnitude star was putting out about a hundred times more light than a sixth-magnitude star, and proposed that the magnitude scale be defined according to that ratio. So the ratio of brightness corresponding to a one-magnitude difference is the fifth root of 100—an irrational number equal to 2.511886431…