Question

a) For a solar eclipse to occur, the Moon must be located between the Earth and the Sun. However, both the Earth’s and the Moon’s orbits are not circles but ellipses. The distance between the Earth and the Moon varies between 357,000 and 407,000 km. The distance between the Earth and the Sun varies between 147.1 × 10⁶ and 152.1 × 10⁶ km. Calculate the range of solid angles under which the Moon and the Sun can appear, and decide which orbit is more important to make a solar eclipse total or not anywhere on Earth. (Use these radii: Earth: 6371 km; Moon: 1737 km; and Sun: 696,000 km.)

b) Even if the combination of distances should make a solar eclipse total, it is not always total. Why could that be?

c) Due to tidal friction, the mean Earth-Moon distance increases by 3.8 cm/year. How many more years will total solar eclipses be possible on Earth? (Source of numerical values: NASA Planetary Fact Sheets)

Answer 1

SUN radius = 696.0 e+3 sq.km

Projected area of the SUN disk = $\pi$ r² = $\pi$ *(696.0 e+3)²

Solid angle of sphere is 4$\pi$ sr.

Area of a sphere = 4$\pi$r²  

Solid angle = ds/ r² ; ds is the area and r is distance where the sold angle is formed

Earth at largest distance R = 152.1 e+6 km

smallest distance r = 147.1 e+6 km

Solid angle is larger at smaller distance

SUN solid angle at nearest = $\pi$ *(696.0 e+3)² / (147.1 e+6)² = 70.33 e-6 sr.

at farthest = $\pi$ *(696.0 e+3)² / (152.1 e+6)² = 65.78 e-6 sr.

Moon radius r = 1737 km

Moon distance at the nearest = 357.0 e+3 km

farthest = 407.0 e+3 km

Moon solid angle at the nearest = $\pi$ *(1737)² / (357.0 e+3)² = 74.37 e-6 sr.

farthest = $\pi$ *(1737)² / (407.0 e+3)² = 57.22 e-6 sr.

For a Total solar eclipse to occur The moon solid angle must fall within the SUN solid angle so that whole of the SUN is covered by the Moon disk

for this to happen

Moon must be at the nearest point having a solid angle of 74.37 e-6 sr. earth may be at the nearest are farthest point , SUN disk making a solid angle ranging from 65.78 e-6 sr to 70.33 e-6 sr.

Moon orbit is more important to make a solid angle any where on earth. If Moon is at the farthest from earth we will not have a total solar eclipse anywhere on earth.

B) even of the distance do match Total solar eclipse do not occur always as the orbits of earth and the moon are not in the same plane and the three objects need not be in a straight line to form eclipse.

C) earth moon distance increases by 3.8 cm /yr

For total solar eclipse to form the minimum solid angle of moon is 65.78 e-6 sr.

Max earth-moon distance d is given by

$\pi$ *(1737)² / (d)² = 65.78 e-6

d = 379,601.41289 km

if the moon moves farther by

(379,601.41289 - 357000 ) km then there will be no total solar eclipse

no of years to last = 22601.41289 /3.8 e-6 = 5.95 e+9 yrs