Introduction / Context:
Shadow geometry is central to understanding solar and lunar eclipses. Two principal shadow regions are named: the darker central umbra and the lighter surrounding penumbra. Many test items ask you to identify which description fits the umbra precisely.
Given Data / Assumptions:
- The question mentions the inner portion of the Moon’s shadow and a construction involving tangents.
- We assume basic eclipse geometry with extended light source (the Sun) and an occluding body (Moon or Earth).
Concept / Approach:
The
umbra is the fully shadowed region where the light source is completely obscured. For a solar eclipse, the Moon’s umbra falling on Earth yields totality along its narrow path. Geometrically, the umbral cone is bounded by straight lines that are tangent to both the luminous disk of the Sun and the occluding disk of the Moon, defining the region of complete occlusion. Thus, it is correct to say the umbra is the inner (darkest) portion of the shadow and that it is formed by the external tangents to the two disks.
Step-by-Step Solution:
Recall definitions: umbra (total shadow), penumbra (partial shadow).Visualize tangent constructions creating a conical/umbro-conical region.Confirm both statements correctly describe the umbra.Choose the combined option to capture both perspectives.
Verification / Alternative check:
Astronomy diagrams consistently show the umbra defined by tangent rays grazing the occluding and luminous disks, enclosing the fully dark region.
Why Other Options Are Wrong:
(a) alone or (b) alone: Each is correct but incomplete; the best answer acknowledges both aspects.None of the above: Incorrect because both statements are valid.
Common Pitfalls:
Confusing umbra with penumbra or antumbra; the antumbra occurs when the umbral cone does not reach the observation plane, producing annularity in solar eclipses.
Final Answer:
both (a) and (b)
Discussion & Comments