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In what frame(s) of reference are Kepler’s laws valid? Are Kepler’s laws purely descriptive, or do they contain causal information?
A geosynchronous Earth satellite is one that has an orbital period of precisely 1 day. Such orbits are useful for communication and weather observation because the satellite remains above the same point on Earth (provided it orbits in the equatorial plane in the same direction as Earth’s rotation). Calculate the radius of such an orbit based on the data for the moon in [link] .
Calculate the mass of the Sun based on data for Earth’s orbit and compare the value obtained with the Sun’s actual mass.
$1.98\times {\text{10}}^{\text{30}}\phantom{\rule{0.25em}{0ex}}\text{kg}$
Find the mass of Jupiter based on data for the orbit of one of its moons, and compare your result with its actual mass.
Find the ratio of the mass of Jupiter to that of Earth based on data in [link] .
$\frac{{M}_{J}}{{M}_{E}}=\text{316}$
Astronomical observations of our Milky Way galaxy indicate that it has a mass of about $8\text{.}0\times {\text{10}}^{\text{11}}$ solar masses. A star orbiting on the galaxy’s periphery is about $6\text{.}0\times {\text{10}}^{4}$ light years from its center. (a) What should the orbital period of that star be? (b) If its period is $6\text{.}0\times {\text{10}}^{7}$ instead, what is the mass of the galaxy? Such calculations are used to imply the existence of “dark matter” in the universe and have indicated, for example, the existence of very massive black holes at the centers of some galaxies.
Integrated Concepts
Space debris left from old satellites and their launchers is becoming a hazard to other satellites. (a) Calculate the speed of a satellite in an orbit 900 km above Earth’s surface. (b) Suppose a loose rivet is in an orbit of the same radius that intersects the satellite’s orbit at an angle of $\text{90\xba}$ relative to Earth. What is the velocity of the rivet relative to the satellite just before striking it? (c) Given the rivet is 3.00 mm in size, how long will its collision with the satellite last? (d) If its mass is 0.500 g, what is the average force it exerts on the satellite? (e) How much energy in joules is generated by the collision? (The satellite’s velocity does not change appreciably, because its mass is much greater than the rivet’s.)
a) $7\text{.}\text{4}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m/s}$
b) $1\text{.}\text{05}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m/s}$
c) $2\text{.}\text{86}\times {\text{10}}^{-7}\phantom{\rule{0.25em}{0ex}}\text{s}$
d) $1\text{.}\text{84}\times {\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{N}$
e) $2\text{.}\text{76}\times {\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{J}$
Unreasonable Results
(a) Based on Kepler’s laws and information on the orbital characteristics of the Moon, calculate the orbital radius for an Earth satellite having a period of 1.00 h. (b) What is unreasonable about this result? (c) What is unreasonable or inconsistent about the premise of a 1.00 h orbit?
a) $5\text{.}\text{08}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{km}$
b) This radius is unreasonable because it is less than the radius of earth.
c) The premise of a one-hour orbit is inconsistent with the known radius of the earth.
Construct Your Own Problem
On February 14, 2000, the NEAR spacecraft was successfully inserted into orbit around Eros, becoming the first artificial satellite of an asteroid. Construct a problem in which you determine the orbital speed for a satellite near Eros. You will need to find the mass of the asteroid and consider such things as a safe distance for the orbit. Although Eros is not spherical, calculate the acceleration due to gravity on its surface at a point an average distance from its center of mass. Your instructor may also wish to have you calculate the escape velocity from this point on Eros.
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