Answer to Question #201543 in Astronomy | Astrophysics for Majedali

Question

A total solar eclipse occurs when the Moon moves between the Earth and the Sun and completely

blocks out the Sun. This phenomenon is very spectacular and attracts people from all cultures.

However, total solar eclipses can also take place on other planets of the Solar System.

Determine for each of the following moons if they can create a total solar eclipse on their planet.

Moon I Radius I Distance to Planet I Planet I Distance to the Sun

Phobos I11 km I 9376 km I Mars I 228 × 10^6 km

Callisto I 2410 km I1.883 × 10^6 km | Jupiter 779 × 10^6 km

Titan | 2574 km |1.222 × 10^6 km | Saturn |1433 × 10^6 km

Oberon | 761 km | 0.584 × 10^6 km | Uranus | 2875 × 10^6 km

Note: The radius of the Sun is 696 × 10^3 km.

Answer 1

The total eclipse may occur when the angular size of the moon is greater or equal to the angular size of the Sun. We may compare the angular radii

1) the angular radius of Phobos from Mars is $\alpha = \dfrac{R}{d} = \dfrac{11}{9376} =1.2\cdot10^{-3}\,\mathrm{rad}.$ The angular radius of the Sun from Mars is $\beta = \dfrac{R_{\odot}}{r} = \dfrac{696\cdot10^3}{228\cdot10^6} = 3\cdot10^{-3}\,\mathrm{rad}.$

The angular radius of Phobos is smaller, so the total eclipse can't occur.

2) the angular radius of Callisto from Jupiter is $\alpha = \dfrac{R}{d} = \dfrac{2410}{1.833\cdot10^6} =1.3\cdot10^{-3}\,\mathrm{rad}.$ The angular radius of the Sun from Jupiter is $\beta = \dfrac{R_{\odot}}{r} = \dfrac{696\cdot10^3}{779\cdot10^6} = 8.9\cdot10^{-4}\,\mathrm{rad}.$ The angular radius of Callisto is greater, so the total eclipse may occur.

3) the angular radius of Titan from Saturn is $\alpha = \dfrac{R}{d} = \dfrac{2574}{1.222\cdot10^6} = 2.1\cdot10^{-3}\,\mathrm{rad}.$ The angular radius of the Sun from Saturn is $\beta = \dfrac{R_{\odot}}{r} = \dfrac{696\cdot10^3}{1433\cdot10^6} = 4.9\cdot10^{-4}\,\mathrm{rad}.$ The angular radius of Titan is greater, so the total eclipse may occur.

4) the angular radius of Oberon from Uranus is $\alpha = \dfrac{R}{d} = \dfrac{761}{0.584\cdot10^6} = 1.3\cdot10^{-3}\,\mathrm{rad}.$ The angular radius of the Sun from Uranus is $\beta = \dfrac{R_{\odot}}{r} = \dfrac{696\cdot10^3}{2875\cdot10^6} = 2.4\cdot10^{-4}\,\mathrm{rad}.$ The angular radius of Oberon is greater, so the total eclipse may occur.