PROBLEM 8. The Solar Maximum Mission (SMM) satellite orbits Earth at a height of 560 km. In many computations, the Earth-Sun distance of 1.5 X 10^8 km is used to approximate the distance of SMM from the Sun. What is the maximum relative error of this approximation?
Solution: The distance of SMM from the Sun is contained within the range (Earth-Sun distance +/- (Earth diameter + 560 km)), or (Earth-Sun distance +/- 6930 km). From Problem 5., if 1.5 X 10^8 km is used as the Earth-Sun distance, the absolute error is less than or equal to 2.48 X 10^6 km. 6930 km is much smaller than this error, so that the absolute and relative errors incurred in using 1.5 X 10^8 km as the SMM-Sun distance are the same as those of part (b) of Problem 5. If greater accuracy is required, it will be necessary to use daily "Ephemeris" values such as those in the listing given in Problem 7. In this event, it is still true that since 6930 km < 7 X 10^3 km, the relative error in approximating the SMM-Sun distance with the Earth-Sun distance < (7 X 10^3)/(1.5 X 10^8) = 4.67 X 10^-5, or about 0.005 percent.
The Astronomical Unit (AU), although useful for measuring distances within the solar system, is too small to be convenient for distances to stars. We shall therefore consider two other units of length used by astronomers. The first is called the "light-year."