EBK LIFE IN THE UNIVERSE
4th Edition
ISBN: 8220101460417
Author: SHOSTAK
Publisher: YUZU
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Chapter 11, Problem 59IF
To determine
The center of mass of a planet.
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A spherical, nonrotating planet has a radius R and a uniform density ρ throughout its volume. Suppose a narrow tunnel were drilled through the planet along one of its diameters, as shown in the figure above, in which a small ball of mass m could move freely under the influence of gravity. Let r be the distance of the ball from the center of the planet.
a. Show that the magnitude of the force on the ball at a distance r < R from the center of the planet is given by F = –Cr, where C = 4/3 πGρm.
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The ball is dropped into the tunnel from rest at point P at the planet's surface.
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Show that the period of orbit for two masses, m1 and m2 , in circular orbits of radii r1 and r2 , respectively, about their common center-of-mass, is given by T = 2π √(r3 / (G(m1 + m2))) where r = r1 + r2 . (Hint: The masses orbit at radii r1 and r2 , respectively where r = r1 + r2 . Use the expression for the center-of-mass to relate the two radii and note that the two masses must have equal but opposite momenta. Start with the relationship of the period to the circumference and speed of orbit for one of the masses. Use the result of the previous problem using momenta in the expressions for the kinetic energy.)
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Chapter 11 Solutions
EBK LIFE IN THE UNIVERSE
Ch. 11 - Prob. 1RQCh. 11 - Prob. 2RQCh. 11 - Prob. 3RQCh. 11 - How do habitable zones differ among stars of...Ch. 11 - Briefly describe the conditions under which...Ch. 11 - Why are extrasolar planets hard to detect...Ch. 11 - Briefly describe the astrometric, Doppler, and...Ch. 11 - Briefly summarize the planetary properties we can...Ch. 11 - Why does the Doppler method generally allow us to...Ch. 11 - How does the transit method tell us planetary...
Ch. 11 - How do the orbits of known extrasolar planets...Ch. 11 - Summarize the key features shown in Figure 11.20,...Ch. 11 - According to current statistics, how common arc...Ch. 11 - What types of worlds seem most likely to support...Ch. 11 - How might a stars habitable zone be wider than we...Ch. 11 - How might future imagery and spectroscopy allow us...Ch. 11 - Prob. 17RQCh. 11 - Prob. 18RQCh. 11 - What is the HertzsprungRussell diagram? How does a...Ch. 11 - Prob. 20RQCh. 11 - Date: February 16, 2025. Headline: Astronomers...Ch. 11 - Prob. 22TYUCh. 11 - Date: June 19, 2028. Headline: Spectrum Reveals...Ch. 11 - Date: November 7, 2020. Headline: New Images Show...Ch. 11 - Date: November 7, 2050. Headline: New Images Show...Ch. 11 - Date: July 20, 2020. Headline: Giant Planet Found...Ch. 11 - Date: September 15, 2045. Headline: Sun-Like Star...Ch. 11 - Prob. 28TYUCh. 11 - Date: December 13, 2033. Headline: Orphan Planet...Ch. 11 - Prob. 30TYUCh. 11 - Prob. 31TYUCh. 11 - Prob. 32TYUCh. 11 - Which method could detect a planet in an orbit...Ch. 11 - To determine a planets average density, we can use...Ch. 11 - Based on the model types shown in Figure 11.20, a...Ch. 11 - According to current statistics, about what...Ch. 11 - The term super-Earth means a planet that is (a)...Ch. 11 - Our best hope for determining that life exists on...Ch. 11 - Jupiter has had an important effect on life on...Ch. 11 - Prob. 40TYUCh. 11 - Prob. 41POSCh. 11 - Unanswered Questions. As discussed in this...Ch. 11 - Explaining the Doppler Method. Explain how the...Ch. 11 - Explaining the Transit Method. Explain how the...Ch. 11 - Comparing Methods. What are the strengths and...Ch. 11 - Super-Earth. Youve discovered a super-Earth...Ch. 11 - Stars with Habitable Planets. Based on what youve...Ch. 11 - Are Earth-Like Planets Common? Based on what you...Ch. 11 - Prob. 50IFCh. 11 - Science Fiction Planet. Choose one fictional...Ch. 11 - Number of Stars with Habitable Planets. Assume...Ch. 11 - Prob. 54IFCh. 11 - Finding Orbit Sizes. The Doppler method allows us...Ch. 11 - Finding a Planetary Mass. Using the Doppler...Ch. 11 - Transit of TrES-1. The planet TrES-1, orbiting a...Ch. 11 - The Doppler Formula. The amount of Doppler shift...Ch. 11 - Prob. 59IFCh. 11 - Future Mission. Imagine that a wealthy benefactor...Ch. 11 - Is It Worth It? Thanks to rapidly advancing...Ch. 11 - Prob. 62IFCh. 11 - Extrasolar Planet Mission. Learn about a proposed...
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- Astronomical observations of our Milky Way galaxy indicate that it has a mass of about 8.0 • 1011 solar masses. A star orbiting on the galaxy’s periphery is about 6.0 • 104 light years from its center. a) What should the orbital period of that star be in years? b) If its period is 6.0 • 107 years instead, what is the mass of the galaxy in solar masses? 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.arrow_forwardSuppose we are given the values for the two masses and their initial velocities along with the final velocity of B as follows: ??=1.2;??,??=5.7???=2.7; ??,??=2.3??????,??=4.1? What is the final velocity (speed and direction) of mass A?arrow_forwardOur solar system is roughly 2.2 x 1020 m away from the center of the Milky Way galaxy, and the system is moving at roughly 231.4 km/s around the galaxy's center. Since most of the galaxy's mass is near its center (and we are on an outer arm of this spiral galaxy), let's model the galaxy has a spherical mass distribution (like a single, giant star that our system is orbiting around). What is the mass of the galaxy (according to our rough, spherical model)? Obviously, this will be a VERY big answer, and so enter in your answer to the order of 1040 kg. In other words, calculate the answer, and then divide by 1040 and then enter in the result. BTW - by assuming that all mass in the galaxy is made up of stars that are about the same mass as our sun, it isn't too hard to then estimate how many stars are in the galaxy!). As an another aside, some measurements and observations that we have taken in Astronomy suggests that in reality, stars only make up a fraction of the total massarrow_forward
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