NEXA 380 COSMOLOGIES AND WORLDVIEWS
UNITS OF COSMOLOGY

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.)

1. CIRCLES, SPHERES AND ANGLES

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, 360 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 degrees

Be sure to use the same units in the numerator and the denominator of any fraction!
Also useful to remember:
                                  Area of a circle = pi r2
                             surface area of a sphere =4 pi r2
                                Volume of a sphere = 4/3 pi r3
*****

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 r
To find angle in radians, just divide the arc length by r, so complete circle = 2 pi radians.
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2. LENGTHS

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 average distance of the Earth from the Sun is about 93 million miles. This quantity is defined to be 1 Astronomical Unit = 1 AU = 1.5x1011 m.

The distance to a star whose parallax is 1 second of arc is defined to be 1 parsec = 3x1016 m. (Parallax will be discussed in class.)

The distance travelled by light in one year is called a light year and equals about 1016 m, so one parsec is about 3 light years.

3.TIME

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 107 seconds.

4. SPEED

We measure speed in m/s (meters per second). The speed of light is 3x108 m/s.

UNITS CONVERSION.

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 = 1.5x1011 m, and 1 light year is 1016 m, so we can write:

                   1 l.y.
              1 = --------
                   1016 m

                                         1 l.y.
and 1AU = 1.5x1011 m x 1 = 1.5x1011 m x ------
                                         1016 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

1 AU = 1.5x10-5 l.y.

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.

Another example
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 107 s
       t = 1.5x10-5 yr --------- = 1.5 x pi x 102 s
                       1 yr

So it takes about 470 secs, or about 6 minutes.