Moon shadow - Approximation of clock and incidence angles

Oct 22, 2026

Uploaded by Adam

Moon shadow defines the geometry of lunar shadow using the clock angle and the normal incidence angle, and relates incidence to shadow percentage. It outlines how to approximate the clock angle over time by combining known elevations of the Moon and Sun with a spherical-trigonometry model. The procedure includes adjusting observer-related parameters, solving for intermediate angles via cosine-rule equations, and computing later-time angles and the resulting clock angle.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages3

Reading Time8 min

AccessFree to download

Lunar illumination

Approximating the clock angle

Adjusting s using p, zs and T (optional)

Calculating i from m, s and I

Calculating T′

Calculating zs′ from p, s and T′

Calculating zm′ from p, m, I and T′

Calculating C′ from zm′, i and zs′

Moon shadow The Moon shadow can be described using two angles, the clock angle, c, and the normal incidence angle, j. The \clock angle", c, indicates the angle between zenith and clockwise direction towards Figure 1: Upper limb elevation. the sun, where 0 indicates Sun above Moon, 90 indicates Sun to the right etc. The \incidence angle", indicates angle between Moon and Sun. The normal incidence angle, j, is reported in degrees, so that-90 gives 100% shadow, 0 gives 50% and 90 gives 0% shadow. Lunar illumination The lunar illumination, b, is given by b = 1+si2n(j) 􀀁 100% . Approximating the clock angle As the Moon moves over the sky, the apparent direction towards the Sun (clock angle) changes as the observer turns. The Moon shadow clock angle as a function of time can be approximated if you at one time know the Moon and Sun elevation, Moon clock angle and the position of the observer, assuming that Moon and Sun positions do not change. The algorithm for approximating the elevation of the Sun or Moon at a given time, given max and min elevations is given in another document. Note that zs is complementary to the Sun elevation, eSun so that zs = ~~􀀙~~2 􀀀eSun . Similar goes for the the Moon zenith angle, zm. Figure 2: Spherical trigonometry The known variables at a time, C , p, T , zs and zm are marked in red and blue, the varying arcs are dashed. We assume that m, s, i, p and I are constant, where s can be estimated from p and the maximum Sun elevation, emax. 􀀙 zsmin = 2 􀀀 emaxs = p + zsmin: Note that the Sun may have maximum elevation to the North at low latitudes (check the azimuth) in which case s = p 􀀀 zsmin. We may estimate m through a similar approach by using the maximum Moon elevation instead of the maximum Sun elevation. The angle I can be estimated by comparing the time di􀀋erence between maximum Sun and Moon elevations and scaled so that a 6 hour di􀀋erence matches an I angle of 􀀙=2 . The time angle T is zero at noon and 􀀙 at midnight. Let us denote the angles at the later time with an asterix, for instance T 􀀃 is the time angle atthe later time. Assuming that we know p, zs, zm and the clock angle C at time angle T , we would like to approximate the clock angle at a later time, C 􀀃 . Adjusting s using p, zs and T (optional) You may want to 􀀌ne-adjust s so that it is consistent with p and zs at T. From 􀀌gure 2, using the cosine rule from spherical trigonometry gives, cos(zs) = cos(s) cos(p) + sin(s) sin(p) cos(T ): The problem here is that the limited information allows several geometrical solutions, a property that manifests itself through the appearence of both sin(s) and cos(s) . To solve this equation for s, we use \Weierstrass t-substitution" and introduce t so that, t = tan( s2) 2t sin(s) = 1 + t2 1 􀀀 t2 cos(s) = 1 + t2 which gives us 1 􀀀 t2 2t cos(zs) = 1 + t2 cos(p) + 1 + t2 sin(p) cos(T ) sin(p) cos(T ) 􀀆 qsin2 (p) cos2 (T ) 􀀀 cos2 (zs) + cos2 (P ) st = ~~ ~~ = tan( ) (cos(zs) + cos(P )) 2 yielding several solutions for s as mentioned above. Pick the one closest to your original estimate. This correction procedure can also be repeated for zm. Calculating i from m, s and I From 􀀌gure 2, using the cosine rule from spherical trigonometry gives, cos(i) = cos(m) cos(s) + sin(m) sin(s) cos(I) which can be solved for i. Calculating T 􀀃 The new solar time angle is calculated by assuming that the earth spins around its axis once a day. The solar time angle is zero at noon, and 􀀙 at midnight. Calculating zs􀀃 from p, s and T 􀀃 From 􀀌gure 2, using the cosine rule from spherical trigonometry gives, cos(zs􀀃 ) = cos(s) cos(p) + sin(s) sin(p) cos(T􀀃 ) which can be solved for zs􀀃 . Calculating zm􀀃 from p, m, I and T 􀀃 From 􀀌gure 2, using the cosine rule from spherical trigonometry gives, cos(zm􀀃 ) = cos(m) cos(p) + sin(m) sin(p) cos(I + T􀀃 ) which can be solved for zm􀀃 . Calculating C 􀀃 from zm􀀃 , i and zs􀀃 From 􀀌gure 2, using the

FAQs

What do the clock angle and normal incidence angle represent?

The clock angle c describes the direction from the zenith toward the Sun, while the normal incidence angle j is the angle between the Moon and Sun that governs shadow reduction through the given illumination relation.

How is the lunar illumination b related to the incidence angle?

Lunar illumination b is defined by b = 1 + sin²(j) with percentages derived from the model, including cases like -90 giving full shadow and 90 giving no shadow.

How do you approximate the Moon shadow clock angle at a later time?

Use the spherical-trigonometry setup to compute parameter adjustments (optionally refine s), solve for intermediate angles at the later solar time angle T′, then derive the later clock angle C′ from the cosine-rule relation involving i and zm′.

Oct 22, 2026

Uploaded by Adam