Astronomy Lab #1, Fall 00

Exercise 2: Measurements and Unit Conversions


Astronomy involves, among other things, making measurements of distances and angular separations of stars, as well as masses and motions of planets. Therefore it is important that we review some of the characteristics of obtaining and recording meaningful data. A measurement normally incorporates two parts. The first specifies the number (or magnitude) of that which was measured; the second specifies the units in which the measurement was taken. One piece of information is usually not useful without the other. This exercise will provide some experience in working with numbers and units common to astronomy.

A. Dimensions

Units vary with the nature of the measurement. Distances are measured in units such as miles, yards, and inches in the English system, and meters in the metric system. When considering astronomical distances, the quantitites become so great that English and metric units become unwieldy. For instance, the distance between the sun and the earth is approximately 93 million miles. This number can be difficult to manipulate, and it is sometimes more convenient to think of distances as multiples of the earth/sun distance. The astronomical unit (AU) is defined as teh average distance between the earth and sun, and this is equivalent to 93 million miles or 1.5 x 1011 meters.

Another commonly used astronomical measure of distance is the light year (ly). This is the distance light would travel in a vacuum in one year. Light travels at a speed of 3x108 meters/second; so in one year, it can cover a distance of 9.5 x 1015 meters. It is common to measure distances to even our nearest stellar neighbors in tens or hundreds of light years.

Another useful measure of astronomical distance is the parsec (pc). A parsec is the distance an object is from Earth with the object exhibits a heliocentric parallax of one second of arc. (One second of arc is the apparent positional shift of an object relative to the star-field background, and is caused by a 1 AU orbital displacement of the earth.) One can easily observe this effect on a smaller scale by first looking at a nearby object with one eye and then with the other eye. The apparent shift in position against some distant background is called parallax. If one looks at a star from Earth and plots its position in comparison to background stars, and then does the same thing approximately three months later, the star may seem to shift in relation to the background stars. The angular shift is its heliocentric or sun-centered parallax. If it shifts one second of arc, then the star is said to be one parsec distant. In more practical terms, a parsec is equivalent to 3.26 ly or 206,265 AU.

Conversion

The astronomer must often convert from one set of scale units to another. In the practical study of celestial objects, where electronic equipment is commonly used, the equipment often outputs data to strip chart recorders, magnetic tape, or even a computer hard disk. These electrical signals ultimately must be translated into some form that can be related to a meaningful astronomical effect. No one uses analog strip chart recorders any longer, but data is often plotted on a laser printer in a very similar format.

For instance, a strip chart recorder may record an event such as an earthquake or the light intensity-level of a star. The detector of the light, in the case of the star's radiation, bputs out an electrical signal that the recorder translates into movement of a pen. If paper is moving beneath the pen at a constant rate, then one can measure quantitatively how the light varies with time.

For example, suppose we know that the chart paper moves a specific distance of 45 millimeters (mm) in one second. During that second, a marking pen would draw a horizontal line 45 mm long, provided the signal remained at a constant level. Next suppose we find that the star we are observing is a pulsar and that the light intensity pattern on the chart paper looks like the following: 
            _       -       ^       -       _
       ____/ \_____/ \_____/ \_____/ \_____/ \___
where we have calibrated the strip chart recorder so that 
            <========= 45 mm =============>  = 1 second
To determine the period of light-intensity pulsation of the pulsar, i.e., the time interval between consecutive blips, we can say that one second is equivalent to 45 mm in horizontal distance. Thus, 1 sec = 45 mm, and the appropriate conversion factor is:
1 sec / 45 mm

the factor by which length measurements in millimeters are converted to units of time (sec).

Now we measure the distance in millimeters between the blips of the pulsar. To convert this measurement into time, simply multiply by the conversion factor. If the horizontal displacement between consecutive peaks were 9 mm, then
9 mm x 1 sec/45 mm \ 1/5 sec = 0.2 sec

Notice how the mm unit in the numerator cancels that in the denominator, leaving only seconds as the final unit.
For practice in converting distances to time, three examples are provided. The scaling line at the left designates the length equivalent of the time that is given adjacent and to the right of it. To the right in example #1, there is a recording of sunspot numbers from which time intervals are to be determined. Examples #2 and #3 are recorded-time variations in light intensities of a Cepheid variable star and a pulsar, respectively. Convert the scaling line lengths and the corresponding time equivalents from the beginning to the end of the scaling lines into appropriate conversion factors. Then use the procedure discussed previously to determine periods of variation in sunspot maxima, the Cepheid variable, and the pulsar variable.

RECORD YOUR RESULTS ON THE ANSWER SHEET AT THE END OF THIS LAB EXERCISE. 1.
50 years = ___________mm
How many years lapse between consecutive sunspot maxima? _______________

2.
5 days = _____________mm

3.
1 sec = _____________mm
What is the period of this pulsar? __________ seconds

Not everything with which an astronomer works has to do with measuring time between two events. Quite often it is the size of an object or the separation between two objects that interests the astronomer. These apparent distances in the sky are measured in units of angle, which are degrees, arc minutes, and arc seconds. If one remembers that a circle circumscribes 360 degrees, then it will ber apparent that our horizon from north to south spans one-half of a circle or 180 degrees of arc. Smaller units include 60 minutes of arc in one degree and 60 seconds of arc in one minute of arc.

For reference, our moon and sun both have angular diameters of about 1/2 degree. In terms of minutes (') or seconds (") of arc, respectively, the diameters are
1/2 degree x 60'/degree = 30 minutes of arc

and
30' x 60"/1' = 1800 seconds of arc

Astronomers, when studying a photograph of an object, often want to know its angular size in the sky. The size of the photo will change, depending on the amount of magnification used to take the photograph, but if the photographer gives us a scaling factor, we can apply the same technique used for the time-measurement conversions.

For example, using the scaling factors to the left, find the angular distance or diameters of the respective objects to the right.

4.
1 degree = _____________ mm
The angular distance between stars 21 and 28 = _____________ degrees

5.
150 seconds of arc = ______________mm
The largest angular diameter of a Virgo nebula = _______________ seconds of arc.

6.
1 degree of arc = __________________mm
The diameter of the moon = ___________ degree.

One also should know how to convert units from one form to another. The metric system is particularly convenient because it is based on a decimal system. For example,
1 meter = 10 decimeters
= 100 or 102 centimeters (cm)
= 1000 or 103 millimeters (mm)
= 0.001 or 10-3 kilometers (km)
= 10,000,000,000 or 1010 Angstroms ()

To convert from one distance to another, one must move the decimal point the correct number of places to the left or right. Using the list of definitions on the page preceding the answer sheet, convert the following:
7. 254 meters = _________ centimeters (cm)
= ___________millimeters (mm)
= ___________Angstroms ()
= ___________kilometers (km)

One is often required to convert some of the various units that are specific to astronomy. The ones most frequently needed are:
1 parsec (pc) = 3.26 ly = 206,265 AU = 3.086 x 1016 m

For example, our nearest star, excluding our own sun, is Alpha Centauri, and its distance from us is 4.6 ly. In other words, its distance is:
< 4.6 ly x 1 pc/3.26 ly = 1.41 pc

or
1.41 pc x 206,265 AU/1 pc = 290,834 AU

or
290,834 AU x 3.08 x 1013 km / 206,265 AU = 4.34 x 1013km

Using conversions similar to those above, answer the following questions.
8. The star Arturus is a red gian located in teh constellation Bootes, and it is the sixth brightest star in the sky. It is about 36 ly from us. How far is it in parsecs (pc)? Kilometers (km)?


9.Saturn is about 1427 million km from the sun. What is the distance in astronomical units (AU)? Light years (ly)? Parsecs (pc)?


last updated August 28, 2000