In astronomy and cosmology, we need to measure times, distances, and angles. Because of the vast dimensions of space, units are used which are not used in our day-to-day lives. (Material marked with *** is not absolutely essential.)

Our measurement of angles dates back to the Babylonians, who used multiples of 60, and divisors of 60 (like 12) as the base of their numerical system (just as our monetary system is based on the number 10, and multiples of 10 like 100). Thus we measure angles in degrees, 360ø making a full circle. Degrees are divided into 60 minutes of arc, and each minute is divided into 60 seconds of arc. (The ressemblance to hours, minutes and seconds is NOT coincidental!) Physicists also use a measure of angle called a radian, defined below. To relate these units, recall that the circumference of a circle is pi times its diameter, or 2 pi times its radius, where pi is the number 3.14159.....

1 degree = 1° = 60 minutes of arc =60´1 minute =1´ = 60 seconds of arc =60"

Complete circle = 360 degrees

To measure the distance between two objects when the angle between them is known, we use the idea of proportion. The length around the circumference of the circle is proportional to the angle. So

length angle -------------- = ------------ circumference 360 degreesBe sure to use the same units in the numerator and the denominator of any fraction!

Also useful to remember:

Area of a circle = pi r^{2}surface area of a sphere =4 pi r^{2}Volume of a sphere = 4/3 pi r^{3}*****

1 radian is the angle such that the fraction of the circumference marked off is equal to the radius of the circle.

For a complete circle, arc = circumference = 2 pi rTo find angle in radians, just divide the arc length by r, so complete circle = 2 pi radians.

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The primary unit of length in science is the meter (abbreviated m) A meter is slightly more than three feet. An inch is exactly 2.54 cm = 2.54 centimeters, or 2.54 hundredths of a meter. Because lengths in the universe are so large, we'll want to use some larger units.

The primary unit of time is the second. We also use years, and it is useful to note that 1 year is almost
exactly pi x 10^{7} seconds.

We measure speed in m/s (meters per second). The speed of light is 3x10^{8} m/s.

Converting between units is just multiplying by the number 1, written in an odd way. For instance,
suppose we wanted to write the distance from the Earth to the sun in light years. We already know
this distance is 1 AU = 1.5x10^{11} m, and 1 light year is 10^{16} m,
so we can write:

1 l.y. 1 = -------- 10^{16}m 1 l.y. and 1AU = 1.5x10^{11}m x 1 = 1.5x10^{11}m x ------ 10^{16}m

Now the unit "m" appears in both the numerator and the denominator, and so it cancels out. To divide the numbers which are powers of 10, we just subtract the powers, 11 - 16 = -5, so

Note that I chose to write the number 1 with l.y. on the top and m on the bottom so that the m would cancel. Writing it the other way up would be correct but not useful in this problem.

How long does it take a for the Sun's radiation to reach the Earth?

The speed of light is most usefully written as 1 ly/yr. Then using the result above, and the fact that

distance time = -------- speed we have 1.5x10^{-5}ly t = ------------- = 1.5x10^{-5}yr. 1 ly/yr Now we do the units conversion. pi x 10^{7}s t = 1.5x10^{-5}yr --------- = 1.5 x pi x 10^{2}s 1 yrSo it takes about 470 secs, or about 6 minutes.