During a solar eclipse, the lansdcape will slowly dim until it is nearly complete darkness along the path of totality. other observers wil see te landscape dim a bit but then brighten to normal intensity. If you didn;t know that an eclipse was going on you might not even notice the dimming, mistaking it for a cloud passing across the sun. The geometric condition for this dimming have to do with the area of exposed solar surface and how this changes as the disk of the moon passes across it. Below is a simple mathematical model for ambient light dimming that you can put to the test the next time a solar eclipse passes over your geographic location.

I have reanalyzed the geometry and defined it in terms of the center-to-center distance, L, between the sun and moon, and their respective radii Rs and Rm as the figure of the upper half-plane of the intersection shows, with the yellow area on the left representing the disk of the sun and the white area on the right the disk of the moon. This problem was previously considered in 2000 by British astronomer David Hughes who used the distance defined by the segment FE, which he called alpha, but L = 1+M-a. The figure shows the moon overlapping the disk of the sun in a lens-shaped zone whose upper half is represented by the area AFDE.

The basic idea is that we want to compute the area of the lunar arc cap AFD by computing the area of the sector BAF and subtracting the triangle BAE from the sector area. That leaves the area of the cap as the left-over area. We perform the same calculation for the solar sector CAE and subtract the triangle CAD from this. The resulting area of the full lens-shaped overlap region is then

Occulting Area = 2x(AreaAFD + AreaAED).

Because of the geometry, the resulting area should only depend on the center-to-center separation and the radii of the sun and moon. You should not have to specify any angles as part of the final calculation. In the following we will use degree measure for all angles.

The area of the sector of a circle is just A = (Theta/360)piR^{2} so that gives us the first two relationships:

To simplify the problem, we are only interested in the fraction of the full sun disk that is illuminated. The full sun has an area of pi Rs^{2}, so we divide Am and As by pi Rs^{2} , and if we define Rs=1.0 and M = Rm/Rs we get:

Although M is fixed by the solar-lunar ratio, we seem to have two angular variables *alpha* and *theta* that we also have to specify. We can reduce the number of variables because the geometry gives a relationship between these two angles because they share a common segment length given by h.

so that the EQ-1 for A can be written entirely in terms of the center-to-center distance, L, and moon-to-sun disk ratio M = Rm/Rs. This is different than the equation used by Hughes, which uses the width of the lens (the distance between the lunar and solar limbs) segment FDE=a as the parameter, which is defined as L = 1+M-a.

During a typical total solar eclipse lasting 4 minutes, we can define L as

L = 1900 – 900*(T/240) arcseconds where T is the elapsed time from First Contact in seconds. Since L is in units of the current solar diameter (1900 asec) we have

EQ 3) L = 1 – T/480.

If we program EQ 1, 2, and 3 into an excel spreadsheet we get the following plot for the April 8, 2024 eclipse.

First Contact occurs at 16:40 UT and Fourth Contact occurs at 19:57 UT so the full duration is 197 minutes. During this time L varies from -(1+M) to +(1+M). For the April 8, 2024 eclipse we have the magnitude M = 1.0566, so L varies from -2.0566 (t=0) to +2.0566 (t=197m). As the moon approaches the full 4-minute overlap of the solar disk between L=-0.05 and L=+0.05 (t =97m to t=102m), we reach full eclipse.

We can re-express this in terms of the landscape lighting. The human eye is sensitive to a logarithmic variation in brightness, which astronomers have developed into a ‘scale of magnitudes’. Each magnitude represents the minimum change in brightness that the human eye can discern and is equivalent to a factor change by 2.51-times. The full-disk solar brightness is equal to -26.5m, full moon illumination is -18.0m on this scale. The disk brightness, S, is proportional to the exposed solar disk area, where E is the solar surface emission in watts/m^{2} due to the Planck distribution for the solar temperature of T=5770 k. This results in the formula:

m = -26.5 – 2.5log10(F)

where F is the fraction of the full disk exposed and is equal to Equation 1. For a sun disk where 90% has been eclipsed, f=0.10 and the dimming is only 2.5log(1/10) = 2.5m. How this translates into how humans perceive ambient lighting is complicated.

The concept of a Just Noticeable Difference is an active research area in psychophysics. In assessing heaviness, for example, the difference between two stimuli of 10 and 11 grams could be detected, but we would not be able to detect the difference between 100 and 101 grams. As the magnitude of the stimuli grow, we need a larger actual difference for detection. **The percentage of change remains constant in general**. To detect the difference in heaviness, one stimulus would have to be approximately 2 percent heavier than the other; otherwise, we will not be able to spot the difference. Psychologists **refer to the percentages that describe the JND as Weber fractions**, named after Ernst Weber (1795-1878), a German physiologist whose pioneering research on sensation had a great impact on psychological studies. For example, humans require a 4.8% change in loudness to detect a change; **a 7.9% change in brightness is necessary**. These values will differ from one person to the next, and from one occasion to the next. However, they do represent generally accurate values.

The minimum perceivable light intensity change is sometimes stated to be 1%, corresponding to +5.0^{m}, but for the Weber Fraction a 7.8% change is required in brightness corresponding to only -2.5log(0.078) = +2.7m. This is compounded by whether the observer is told beforehand that a change is about to happen. If they are not informed, this threshold magnitude dimming could be several magnitudes higher and perhaps closer to the +5.0^{m} value.