Why do the Sun and Moon appear to be the Same Size in the Sky?

Why do the Sun and Moon appear to be the Same Size in the Sky?

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The Sun and Moon’s physical diameters are obviously different from each other, so what causes them to appear in the same size? It’s their ANGULAR DIAMETER that’s coincidentally almost the same. The angular diameter or apparent size of an object is its apparent diameter as seen by an observer in a specific point. Intuitively, the angular diameter of an object changes as the observer’s distance changes.

How do we derive the equation for the angular diameter?

Figure 1: Angular Diameter

With θ_d as the angular diameter, it’s easy to see that the tangent of the angular radius (half of the angular diameter) is equal to the actual radius (D/2) over the distance.

Solving for θ_d:

Therefore the equation for the angular diameter is:

Now we can use this to calculate for the angular diameters of the Sun and Moon.
Note that the distances used were the average distance, since orbits are elliptical. The Sun’s corona is excluded in the diameter.

For the Sun:                                                  For the Moon:

D = 1 392 684 km                                        D = 3474 km
l = 1.496 x 10⁸ km                                        l = 384 405 km

θ_d = 0° 32’ 0.18’’                                          θ_d = 0° 31’ 4.07”

 

As we can see, their angular diameters are nearly the same with the moon being a little bit smaller, that’s why when perfectly in line during a solar eclipse in a given position, an annular or total eclipse could be seen; annular when the moon is a little bit farther and a total eclipse when the moon’s distance is enough to make its angular diameter equal or a little greater than the Sun’s. 

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By: Lanz Anthonee Lagman
       BS-Astronomy Student