Chapter 13, Problem 51EAP

Be sure to show all calculations clearly and state your final answers in complete sentences.

1. Lost in the Glare. How hard would it be for an alien astronomer io detect the light from planets in our solar system compared to the light from the Sun itself?
2. Calculate the fraction of the total emitted sunlight that reaches Earth. [Hint: Imagine a sphere around the Sun the size of Earth's orbit (area = 4mz²], then calculate the fraction of that area taken up by the disk of Earth [area = irrjL,rtiil. Earth’s reflectivity is 29%.) b. Earth refleas 29% of the Sun’s light. Based on this fact and your answer to part a, calculate what fraction of total sunlight is refleaed by Earth. [Hint: Your answer will simply be the overall fraction of all the Sun’s light that is reflected by Earth.) c. Would detecting Jupiter be easier or harder than detecting Earth? Comment on whether you think Jupiter's larger size or greater distance has a stronger effect on its detectability. You may neglect any difference in reflectivity between Earth and Jupiter.
3. Transit of 'lTES-1. The planet TrES-1, orbiting a distant star, has been detected by both the transit and die Doppler technique, so we can calculate its density and get an idea of what kind of planet it is.
a. Using the method of Mathematical Insight 13.3. calculate the radius of the transiting planet. The planetary transits block 2% of the star’s light. The star ’1YES-1 has a radius of about 85% of our Sun's radius, b. The mass of the plana is approximately 0.75 times the mass of Jupiter, and Jupiter's mass is about 1.9 X IO ^{2 7} kilograms. Calculate the average density of the planet. Give your answer in grams per cubic centimeter Compare this density to the average densities of Saturn (0.7 g/cm ³) and Earth (5.5 g/cm ³). Is the planet terrestrial or Jovian in nature? [Hint: To find the volume of the plana, use the formula for the volume of a sphere: V — JttH. Be careful with unit conversions.)

4. Planet Around Si Pegasi. The star 51 Pegasi has about the same mass as our Sun. A planet discovered orbiting it has an orbital period of 4.23 days. The mass of the planet is estimated to be 0.6 times the mass of Jupiter. Use Kepler's third law to find the planet's average distance (semimajor axis) from its star. [Hint: Because the mass of 51 Pegasi is about the same as the mass of our Sun, you can use Kepler's third law in its original form, - aⁱ[Section 3.3). Be sure to convert the period into years before using this equation.)
5. Identical Planets? Imagine two planets orbiting a star with orbits edge-on to the Earth. The peak Doppler shift for each is 50 m/s, but one has a period of 3 days and the other has a period of 300 days. Calculate the two minimum masses and say which, if either, is larger. [Hint: See Mathematical Insight 13.2.)
6. Finding Orbit Sizes. The Doppler method allows us to find a planet’s semimajor axis using just the orbital period and the star’s mass (Mathematical Insight 13.1).
7. Imagine that a new planet is discovered orbiting a 2Ms_unstar with a period of 5 days. What i3 its semimajor axis?
8. Another planet is discovered orbiting a 0.5Af_Sun star with a period of 100 days. What is its semimajor axis?
9. One Bom Every Minute? It’s possible to make a rough estimate of how often planetary systems form by making some basic assumptions. For example, if you assume that the stars we see have been bom at random times over the last 10 billion years, then the rate of star formation is simply the number of stars we see divided by 10 billion years. The fraction of stars with detected extrasolar planets is at least 5%, so this factor can be multiplied in to find the approximate rate of formation of planetary systems.
a. Using these assumptions, estimate how often a planetary system forms in our galaxy. (Our galaxy contains at least 100 billion stare.) b. How often does a planetary system form somewhere in the observable universe, which contains at least 100 billion galaxies? c. Write a few sentences describing your reaction to your results. Do you think the calculations are realistic? Are the rates larger or smaller than you expected?

10. Habitable Planet Around 51 Pegasi? The star 51 Pegasi is approximately as bright as our Sun and has a planet that orbits at a distance of only 0.052 AU.
a. Suppose the planet reflects 15% of the incoming sunlight. Using Mathematical Insight 10.1, calculate its "no greenhouse” average temperature. How does this temperature compare to that of Earth? b. Repeat part a, but assume that the planet is covered in bright clouds that reflect 80% of the incoming sunlight.

11. Based on your answers to parts a and h, do you think it is likely that the conditions on this planet are conducive to life? Explain.