ALGEBRA

PROBLEM 4.

PROBLEM 4. a. The Space Telescope will be able to see stars and galaxies whose brightness is only 1/50 of the faintest objects now observable using ground-based telescopes. The brightness of a point source such as a star varies inversely as the square of its distance from the observer. How much farther into the universe will the space telescope be able to see compared to ground-based telescopes?

Solution: Let dG be the distance from Earth of the faintest object visible to a ground-based telescope, and let BG be the brightness of this object. Let d be the distance of an object of brightness 1/50 BG. Since brightness varies inversely as the square of distance, BG = k/dG^2 and (1/50)BG =k/ d^2 . Then k/BG = d^2/50, so d^2=50dG^2 or d = 7.1 dG.

The Space Telescope will see about seven times farther.

b. Because of the time it takes for light to travel from distant stars and galaxies, we see them as they were some time agočthe photons that reach us from an object that is 1 parsec away were actually emitted 3.26 years ago (see Chapter 2, Problem 12). The best ground-based telescopes can see objects about 10^9 parsecs from our solar system. How long ago were the photons emitted that we now see when we observe such an object?

Solution: Since 10^9 parsecs = 3.26 x 10^9 light-years, the photons were emitted 3.26 x 10^9 years ago.

c. When the Space Telescope begins its observations, how far back in time will it see stars and galaxies?

Solution: Since it will see 7.1 times farther, it will see photons that were emitted 7.1 x 3.26 x 10^9 = 2.2 x 10^10 years ago. (If, as suggested by cosmological theory, the age of the universe is between 10 and 20 billion years, the space telescope should enable us to see stars and galaxies in the earliest stages of formation.)

Pioneer 10 was launched on 3 March 1972. It outlived and outperformed the fondest dreams of its creators. Designed to last at least 21 months, it has continued well beyond the accomplishment of its mission. On 25 April 1983, its distance from Earth equaled that of Pluto, and the following June it crossed Neptune's orbit and left the solar system. (Although Pluto is normally the outermost planet in the solar system, it has a highly eccentric orbit and will be closer to the Sun than Neptune will be for the next seventeen years.) To add to its record of endurance, most of Pioneer lO's instruments are still working, and Earth-based tracking stations were still receiving signals bearing information about the behavior of the Sun's extended atmosphere as of this writing.

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