When will the last total solar eclipse occur?

Oct 29, 2017

The last total solar eclipse visible from Earth will occur in about 558 million years time.

Explanation:

Solar eclipses occur because the angular diameter of the Moon can be greater than the angular diameter of the Sun. This means that when the Sun, Moon and Earth are in alignment, the Moon's disc can completely cover the Sun's disc.

The Moon is moving away from the Earth at a rate of about 3.8cm per year. Eventually the Moon's angular diameter will always be smaller than the Sun's angular diameter and total solar eclipses will no longer occur.

Let's do the calculations. I will use angular radius rather than angular diameter for convenience. I will also round some of the values as accuracy is not too important.

The Sun has its smallest angular radius when it is at aphelion. The distance of the Earth from the Sun at aphelion is 152,000,000km. The radius of the Sun is 700,000km. The angular radius of the Sun at aphelion is:

$\frac{700 , 000}{152 , 000 , 000} = 0.0046$ radians or ${0.26}^{\circ}$

The Moon has its largest angular radius at perigee. The distance from the Earth to the Moon at perigee is 356,400km. The radius of the Moon is 1,737km. The angular radius of the Moon at perigee is:

$\frac{1 , 737}{356 , 400} = 0.0048$ radians or ${0.28}^{\circ}$

Clearly total eclipses are possible at present.

Now the next thing is to calculate the perigee distance which make the maximum angular radius of the Moon equal to the minimum angular radius of the Sun. This distance is:

$\frac{1 , 737}{0.0046} = 377 , 600 k m$

The distance the Moon needs to move away from the Earth is then:

$377 , 600 - 356 , 400 = 21 , 200 k m$

Given that the distance the Moon moves away from the Earth in a year is $3.8 c m = 0.000038 k m$. The the time it will take for the Moon to be at that distance is:

$\frac{21 , 200}{0.000038} = 558 , 000 , 000 y e a r s$

So, in about 558 million years time, no more total solar eclipses will be visible from Earth.