A Geometry for use in the computation of the topocentric Penumbra and Umbra

Byron Soulsby

Isabella Plains Lunar Observatory Australia

The geometry used for computation of the Umbra for comparison with measured images has been determined and is shown in the diagram here.

For the umbra, the shadow formed by the Earth is a converging cone, whereas for the penumbra it is formed by a diverging cone with the Earth's fainter shadow intersecting the Moon's surface from the time of first penumbral contact and departing at second penumbral contact (see comment below about partial and total eclipses).

Using a modified diagram similar to Figure 2 above, the penumbra edge intersects the surface of the Moon at point G and at point E on the plane passing through the Moon's centre at an angle δ to the vertical AG. The semi-diameter of the Moon is S_c with point G at an angle θ to the plane.

Here, sin θ = EG / S_c and tan δ = AE / AG where AE = S_c Sin θ tan δ

In the modified Figure 1 above,

 then AE = AM – EM

                                    = OM – F₁ – (IM – F_p)

                                                = (x² + y²)^½ – IM – (F₁ – F_p)

we need to find δ and F_p, where F_p     = CH² + EH² / 2EH and EH = CH / tan γ₂

where CH     = EC * sin γ₂

For the umbra Δ = (S_s – π_s) + (Δ'_m - Δ'_s). The relationship between δ and Δ, is tan δ = tan Δ (F₁ – R_p) / (R_p – F₂), and at mid-eclipse 20080816 where the ratio of the Earth's radius (R_p) to that of the Moon (S_c) = 6403/1738 = 3.6842, tan δ = tan Δ 0.2859/ 0.2516 = 1.1363 tan Δ , so at mid-eclipse where Δ = 2.1^o, then tan δ = 0.0417, hence δ = 2.388^o.

As the penumbra edge does not contact the Moon's surface during partial or total phases, hence the value of Fp is only shown from umbral first contact to any fourth contact. In penumbral eclipses where the Moon's distance (OM) plus its semi-diameter (S_c) is greater than the penumbra (F₁) and in this case the penumbral edge only contacts the Moon at first, second, third and fourth contact, hence we need to allow for this geometry in the program ViaX8.exe. Similarly for the umbra, while  OM plus S_c is less than the umbra (F₂) its edge does not contact the Moon's surface and Fi is also not shown in this case.

The above rationalè has been included in Program ViaX8.exe to find the expected F_p and F_i so that later comparison may be made with the measured images of the penumbra and umbra.

The results for 20080816 can be seen here and for 20080221 here.

Produced by the Isabella Plains Lunar Observatory Australia 20081126.