PROBLEM 9. The light-year is the distance traveled by light during one Earth year. To three significant digits, the speed of light is 3.00 x 10^5 km/s. Find the length of the light-year in km and in AU.
Solution:1 Earth year = 365.25 days = 365.25 x 24 x 60 x 60 seconds. In one year, light travels 3.00 x 10^5 x 365.25 x 24 x 60 x 60 km = 9.47 x 10^12 km.
To express this distance in AU,
1 light-year = 9.47 x 1O^I2 km x 1 AU/(1.50 x lO^8km)
= 6.31 x 10^4 AU.
The "parsec" is the astronomical unit of distance that relates to observational measurements. In order to define this unit, we must consider the fact that when we observe the heavens, we have no direct perception of depth or distance. A useful model developed to portray the heavens is the celestial sphere. In this model, Earth is surrounded by an imaginary sphere with infinite radius. A coordinate system, similar to latitude and longitude, is imposed on the celestial sphere by projecting Earth's rotation axis on the sphere to identify the celestial north pole (CNP) and celestial south pole (CSP) as shown in Fig. 2.1. Since the radius of the celestial sphere is infinite, all parallel lines point to the same spot on the sphere, and so every line parallel to Earth's rotation axis also points to the celestial north and south poles.
Every star or celestial object can now have its position identified by the ordered pair of angles in the previous example. Because Earth rotates with respect to the celestial sphere, the time of observation must also be known in order to use the coordinate system. Differences in the positions of two objects on the celestial sphere are expressed in terms of the angle subtended at Earth by the arc joining these points. As Earth revolves around the Sun, very distant stars show no discernible changes in position, but closer stars will show apparent motion with respect to the celestial sphere when viewed from different points in Earth's orbit, as shown in Fig. 2.3. This apparent motion is called "parallactic motion", and the change in position is called the "parallax angle". In this context, 1 parsec is defined as the distance at which the radius of Earth's orbit subtends an angle measuring 1 arc-second (see Fig. 2.4).