For this question assume the moon is 110 of its diameter away from the Earth.
The largest apparent size of Mercury that we ever see in the sky is 1/150 the apparent diameter of the Moon. The diameter of Mercury is about 4900 km. What is the closest that the Earth and Mercury approach each other.
To determine the closest distance at which Earth and Mercury approach each other, we need to calculate the distance between them when Mercury appears at its largest apparent size.
Given that the largest apparent size of Mercury is 1/150 the apparent diameter of the Moon, we can calculate the apparent diameter of Mercury by multiplying the apparent diameter of the Moon by 1/150.
The apparent diameter of the Moon is the actual diameter of the Moon (3,474 km) multiplied by the ratio of how far away it is from the Earth to its actual diameter (110). So, the apparent diameter of the Moon is 3,474 km * 110 = 382,140 km.
Now, we can calculate the apparent diameter of Mercury:
Apparent diameter of Mercury = Apparent diameter of the Moon * (1/150)
= 382,140 km * (1/150)
= 2,547.6 km
Knowing the diameter of Mercury is about 4900 km, we can estimate the ratio between the apparent diameter and the actual diameter of Mercury to find the distance between Earth and Mercury when they are closest.
Ratio = Apparent diameter of Mercury / Diameter of Mercury
= 2,547.6 km / 4900 km
= 0.5207
Now, we multiply this ratio by the actual distance between Earth and the Moon (assuming the Moon is 110 times its diameter away from the Earth) to find the closest approach of Earth and Mercury:
Closest approach distance = Ratio * Actual distance between Earth and Moon
= 0.5207 * (110 * Diameter of the Moon)
= 0.5207 * (110 * 3,474 km)
= 0.5207 * 382,140 km
= 199,014 km
Therefore, the closest approach of Earth and Mercury is estimated to be approximately 199,014 km.