# Umbra, penumbra and antumbra

I'm trying to understand following schema of the wikipedia page here. Image subtitle is "Umbra, penumbra, and antumbra, for a lunar eclipse".

Distances and sizes are not to scale.

If distance Earth-Sun is about 150 Gm (gigameters) and Sun diameter is 1.4 Gm, $\arctan 1.4/150 = 0.57º$ (degrees). Thus, the Sun is seen from Earth with an angle of approx. 0.57º (0.01 radians). The Blue and red lines in schema should be near to parallel.

That means the point where umbra and antumbra touch is (being 12 Mm the approx diameter of Earth) at a distance of about $\frac{150}{1.4\times0.012}=1.3 \mathrm{Gm}$ from Earth.

As distance Earth-Moon is only 0.400 Gm, it seems impossible that Moon was never in the antumbra zone of the Earth.

I'm doing a basic error somewhere, but I do not know where.

• As you say, it’s not to scale. Not at all. What is your question exactly? – chirlu May 17 '17 at 11:07

## 2 Answers

As distance Earth-Moon is only 0.400 Gm, it seems impossible that Moon was never in the antumbra zone of the Earth.

That is correct. You were reading yet another poorly written wikipedia page. The information on that page isn't quite incorrect, but it isn't quite correct, either. There is no such thing as an annular lunar eclipse.

That page would have been much more meaningful had it used the Moon's shadow as the example. The Moon's umbra trails from 368000 km to 381000 km behind the Moon, depending on how close the Moon is to the Sun. This is very similar to the distance between the center of the Moon and the closest point on the surface of the Earth to the Moon, which varies from 362000 km to 400000 km.

When the Moon is directly between the Earth and the Sun, the resulting solar eclipse can be total or annular. When the Earth is directly between the Moon and the Sun, the resulting lunar eclipse is always total.

The calculations seem reasonable. The conclusion is wrong.

The Earths antumbra starts 1.3 Gm from Earth. At less than 1.3 Gm there is umbra, penumbra and non-shadow. The moon is 0.4 Gm from Earth.

$$0.4 < 1.3$$

Hence it is impossible for the moon to be in the Earth's antumbra.

However, the Moon's antumbra forms at almost exactly one lunar distance. We notice that the moon and the sun are almost exactly the same size in the sky. When the moon is further from the Earth, an annular eclipse can form, as a small part of the Earth will then be in the moon's antumbra.

I've made a scale model of the sun-earth-moon shadows, at https://ggbm.at/CbbvmBFq

• Thanks for your answer, and very nice model. The scheme subtitle in wiki page is "... for a lunar eclipse" (and position of astros agrees, "Sun-Earth-Moon"). Thus, it seems we are talking about Earth umbra-antumbra and Moon traveling on them. As the second case, Moon in the antumbra of Earth is not possible, I do not understand the meaning of the wiki schema. – pasaba por aqui May 17 '17 at 17:44