During a total solar eclipse, a group of students on the Earth observes the phases of the eclipse. They notice that the sky becomes darker, and they can see stars appearing. Based on this observation, how does the positioning of the Moon and the Sun relate to the concepts of umbra and penumbra during this event?

When the Moon passes directly between the Sun and the Earth, it casts a very dark cone called the umbra, and the students inside this cone see the Sun completely blocked, so the sky darkens and stars become visible. Outside the umbra but still within the Moon’s outer shadow, the penumbra, the Sun is only partly covered, making the eclipse appear partial and the sky only slightly darker. The students are located in the path of totality, where the Earth’s surface lies inside the umbra, so they experience the darkest phase and can see stars. In contrast, observers outside this path see only the penumbral shading, with the Sun still largely bright. Thus the darkness and star visibility directly show that the Moon’s umbra is covering the Sun while the penumbra creates a gradual dimming around it.