EXERCISES FOR ASTRONOMY AND MATHS ENTHUSIASTS
1 · The Moon and the Sun seem to be the same size, that is the reason why total and annular eclipses happen. If both Sun and Moon’s sizes are known, can we know at which distance is each from the Earth?
2 · At what distance must be the Moon from the Earth for a total and an annular eclipse to occur? Does it depend on Earth-Sun distance?
Answers
The Earth is an elliptical (almost circular) orbit around the Sun, just like the Moon around the Earth. The maximum and minimum distances from the Earth to the Sun are 147 and 152 million kilometers, respectively, and the Sun has a diameter of 1,391,400 km.
For the Moon, the minimum and maximum distances to the Earth are 356,500 and 406,700 km, and it has a diameter of 3474.2 km.
With these numbers, we can calculate the angular sizes (the apparent size we see in the sky measured in grades) using the following math relation:
Angular size = tangent (diameter / distance)
If we replace the values mentioned above, we find that the angular sizes are: -Maximum size of the Sun (when it is closer) = 32.54 minutes of an arch (0.54°).
Minimum size of the Sun (when it is further) = 31.47 minutes of an arch (0.52°).
Maximum size of the Moon = 33.50 minutes of arch (0.56°).
Minimum size of the Moon = 29.37 minutes of arch (0.49°).
We can see that if the Moon is at a minimum distance from the Earth, it doesn’t matter at what distance is the Earth from the Sun if the conditions form an eclipse are given, the total eclipse is seen. Whereas if the Moon is at its furthest distance from the Earth, a total eclipse would never happen, only annular (and partial), no matter the distance Earth-Sun.
For any other middle point, it is necessary to analyze in detail the relative distances of the three bodies.
3 · How much longer will it take to stop producing eclipses?
In order to never experience an eclipse again, we have to imagine ourselves in the worst possible scenario. Even when the Moon is the closest to Earth, and the Earth the farthest from the Sun, the Moon looks smaller than the Sun.
Let’s suppose that both of them don’t change their sizes, that the orbit of the Earth remains constant, and the Moon is the only one moving away.
In this case, the minimum size of the Sun is 31.47 minutes of arc, and to have no more eclipses the rule is that the Moon, in its maximum size, must be lower than that.
Then: Angular size = 31.47 = 0.54° =
tangent (diameter / distance).
So, how do we know what size the Moon is (diameter)?
At this distance, the Moon won’t be able to eclipse the
Arctangent (0.54°) = 3474.2 kmdistance
0.0091 = 3474.2 kmdistance
0.0091 = 3474.2 kmdistance = 379.530 km
