Observing Planets beyond Our Solar System NASA is considering a project called Planet Imager that would give astronomers the ability to see details on planets orbiting other stars. Using the same principle as the Very Large Array (see Section 36.7), Planet Imager will use an array of infrared telescopes spread over thousands of kilometers of space. (Visible light would give even better resolution. Unfortunately, at visible wavelengths, stars are so bright that a planet would be lost in the glare. This is less of a problem at infrared wavelengths.)
(a) If Planet Imager has an effective diameter of 6000 km and observes infrared radiation at a wavelength of 10µm, what is the greatest distance at which it would be able to observe details as small as 250 km across (about the size of the greater Los Angeles area) on a planet? Give your answer in light-years (see Appendix E).
(b) For comparison, consider the resolution of a single infrared telescope in space that has a diameter of 1.0m and that observes 10-/µm radiation. What is the size of the smallest details that such a telescope could resolve at the distance of the nearest star to the sun, Proxima Centauri, which is 4.22 light-years distant? How does this compare to the diameter of the earth (1.27 x 104km)? To the average distance from the earth to the sun (1.50 x 108km)? Would a single telescope of this kind be able to detect the presence of a planet like the earth, in an orbit the size of the earth's orbit, around any other star? Explain.
(c) Suppose Planet Imager is used to observe a planet orbiting the star 70 Virginias, which is 59 light-years from our solar system. A planet (though not an earthlike one) has in fact been detected orbiting this star, not by imaging it directly but by observing the slight ''wobble'' of the star as both it and the planet orbit their common center of mass. What is the size of the smallest details that Planet Imager could hope to resolve on the planet of 70 Virginias? How does this compare to the diameter of the planet, assumed to be comparable to that of Jupiter (1.38 x 105km)? (Although the planet of 70 Virginias is thought to be at least 6.6 times more massive than Jupiter, its radius is probably not too different from that of Jupiter. The reason is that such large planets are thought to be composed primarily of gases, not rocky material, and hence can be greatly compressed by the mutual gravitational attraction of different parts of the planet.)