MATH PRACTICE PROBLEMS in ASTRONOMY Planetary Data for Solar System Name Radius Of Body Mass Average Orbital Radius (x 10' m) (Kg) (x 10 m) Sun 696 1.991 x 1030 Tiny; such tiny wobbles are the basis for detecting distant planets! Mercury 2.43 3.2 x 1023 0.580 Venus 6.073 4.88 x 1024 1.081 Earth 6.3713 5.979 x 1024 1.4957 Mars 3.38 6.42 x 1023 2.278 Jupiter 69.8 1.901 x 1027 7.781 Saturn 58.2 5.68 x 1026 14.27 Uranus 23.5 8.68 x 1025 28.70 Neptune 22.7 1.03 x 1026 45.00 Pluto (?) 1.15 1.20 x 1022 59.00 1. An asteroid orbits the sun with a mean orbital radius of 3.65 X 10" m. Using earth as a standard (T=365 days), calculate its ratio of R'/T then use that ratio which is universal for everything which orbits the Sun to compute the period of the asteroid. 2. Using the data from the table and earth as a standard, compute the period of Neptune. 3. The moon has a period of 27.3 days and an orbital radius of 3.90 X 10' km. Compute the period of a research satellite which has an orbital radius of 7.1 X 10 km. (The ratio R'/T for masses orbiting Earth is different from masses orbiting the Sun. But since we already have a natural satellite, the moon, we can calculate the ratio for earth satellites.) 4. Geosynchronous orbits enable satellites to remain fixed over one spot on the equator. By definition, they must have a period of 1 day. Using the moon as a standard, compute the orbital radius of such a satellite. 5. Using the table, compute Jupiter's period.6. A satellite is placed into a very high orbit (m of the moon's orbit). Compute the period of the satellite. Universal Gravitation (Note: time must be expressed in seconds when using G). 7. 10. ll. l2. l3. l4. IS. 16. 17. 18. A ball of lead has a mass of 1200ng and is separated from another ball by a distance of 1.00230 m. If the mass of the second ball is 2|.0000Kg, compute the gravitational attraction between them. Using the above table, compute the gravitational attraction between the sun and Saturn. Two identical balls are placed 2.566 m apart. if the gravitational attraction between them is 1.855 x 10''2 N, compute the masses of the balls. One ball has a mass, which is three times the other. [fa distance of 1.450 m separates them and they experience a gravitational attraction of 2.650 x IO'12 N, compute their masses. Calculate the mass of the earth given the fact that the gravitational attraction between the earth and a 2.00 kg mass is 19.60N and the radius of the earth is 6.40 X 10' m. What is the circmnference, C of the Earth? It takes the Earth 24 hours to rotate once. What is the rotational velocity if you were standing on the equator? (velocity = C I 24-hr). What is the orbital velocity of the Earth as it moves around the Sun? Using the speed of light, determine how long it will take light to travel from the Sun to the Earth? How long would it take light to travel from the Earth to the Moon? Again using the speed of light, determine the distance from our Sun to the next star, Proxima Centauri. Consider a 2 Ms neutron star. The mass of a neutron is 1.673(10'2'l g, and 1 Mo - t'.i.0N10"1 g. (i) How many neutrons are in this neutron star? (ii) Assume that the energy released during the gravitational collapse process that led to the formation of 19. this neutron star was 2 MeV per neutron. About 10% of this energy is converted into the kinetic energy of a supernova explosion. Calculate the total kinetic energy output of the supernova in ergs. Note, 1 MeV = L6 \"10"3 ergs. The number of stars in the disk of the Milky Way is about 10", and the shape of the Milky Way's disk can be approximated as a cylinder of radius r = [5 kpc and height h = 1 kpc. (i) Recall that the volume ofa cylinder is given by V = nigh, where it = 3.14, and l kpc=3 .09 x 102' cm. What is the volume of the Milky Way's disk in units of cm"? (ii) What is the density of stars in the Milky Way in units of stars/cm