Question

Table 13.1 Solar system data (in SI units and relative to Earth) Orbit eccentricity Mass Equatorial radius semimajor axis period (a^) (years) 30 Sun 2.0 X 10 3.3 × 10 Mercury 3.30 X 1023 Venus 4.87 X 1024 Earth Mars Jupiter 1.90 x 1027318 Saturn 5.68 × 1026 95.2 Uranus 8.68 X 1014.5 Neptune 1.02 x 102617.1 Pluto 2.440 ×106 6.052 X 106 6.378 X 106 3.396 × 106 5.79×1010 1.082 x 1011 1.496 × 1011 2.279 ×1011 11.2 7.783 x 101 1.427 ×1012 2.871×1012 4.498 X 1012 6 0.206 0.007 0.017 0.09 0.05 0.05 0.05 0.06 0.38 0.39 7.60 X 10 1.94 × 10 3.16 × 10 5.94× 10 3.74 X 10 9.29 × 10 2.65 × 10 5.20 x 109 164.8 7.82 X 10 247.9 0.24 0.95 0.72 0.62 5.97 × 1024 6.42 × 1023 0.11 0.53 1.88 11.86 29.45 84.02 7.149 × 9 x 10 5.20 6.027 × 107 2.556 X 10 2.476 × 107 9.45 4.01 3.88 9.54 9 30.1 1.31 x 1022 0.002 1.151×106 0.18 5906×1012 39.5 0.25 Moon 7.3 X 1022 0.012 1.737 X 106 0.27 3.84 X 108 0.0026 0.055 2.36 X 10 0.075 The elliptical orbits of the planets and the Moon are specified by their semimajor axis a (half the major axis) and eccentricity e; see Figure 13.7. With the exception of Mercury and Pluto, the eccentricity is small and so the orbits are close to being circular We can use the table above to answer. A black hole has a mass of 4*10A6 suns A) How big is the Schwartzschild radius for the black hole? Answer using both meters and astronomic unit (AU) B) Based on the answer from (A), is it realistic that a star may be located in a circular orbit around the black hole at a distance of 5 AU? C) If so, how big os the orbital velocity? D) i) What is the mechanic energy of the system (the black hole and the star - mass 5 suns)? j) Show that this energy can be determined without knowing the orbital velocity when the system is closed and the orbit is circular k) If we calculaehe tangental yelocity from the result i j), does that fit the answer in (C)? Call this velocity V_0 1) If this star instead had had an elliptical orbit with the exact same mechanical energy calculated in (i), what is the maximum distance from the black hole that the star could obtain (in AU)?

Answer 1