The Sun has mass \(M_{\odot}=1.9910^{30} \mathrm{~kg}\), radius \(R_{\odot}=7.010^{8} \mathrm{~m}\) and rotates on itself (approximately) in 30 days. Using the data in Table 11.1, determine:

1. the acceleration of gravity on the surface of the Sun, \(g_{S}\);

2. the fraction of the total mass of the solar system that is concentrated in the Sun.

3. the total angular momentum \(L_{\odot}\) of the Sun. Note: Use the relation: \(L_{\odot}=J \omega\), where \(J=\) \(2 / 5 M_{\odot} R_{\odot}^{2}\) is the momentum of inertia of a sphere.

4. the total angular momentum \(L_{S S}\) of the Solar System, using the data in the table. Jupiter and Saturn contribute \(85 \%\) of the angular momentum.

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Table 11.1 Today's data related to Kepler's third law for the planets of the Solar System. Mass is expressed in terms of the Earth's mass, (MT = 5.97 x 1024 kg). The Astronomical Unit (A.U.) corresponds to 1.5 10 km. The distance from the Sun is indicative of the semi-major axis of the ellipse, a. The ratio T2/a3 is in the units used for T and a. The eccentricity of the elliptical orbit is defined in Fig. 11.10. All orbits are nearly circular except for Mercury and Pluto. The inclination of the orbit shows that the planes in which the orbits occur are practically parallel, except for Pluto. Also for these reasons, Pluto in 2006 was downgraded by the International Astronomical Union to a dwarf planet Planet Eccentricity Inclination M/MT Sun Period T/a distance (years) () (A.U.) Mercury 0.055 0.387 0.241 1.002 0.206 7.01 Venus 0.815 0.723 0.615 1.001 0.007 3.39 Earth 1.00 1.000 1.000 1.000 0.017 0 Mars 0.107 1.520 1.880 1.006 0.093 1.85 Jupiter 318 5.200 11.9 1.007 0.048 1.31 Saturn 95.2 9.540 29.4 0.996 0.056 2.49 Uranus 14.5 19.20 84.0 0.997 0.047 0.77 Neptune 17.1 30.100 165.0 0.998 0.009 1.77 Pluto 0.002 39.440 248.0 1.003 0.249 17.2