The field of view (FoV) of an eyepiece is the diameter of the area of sky that is visible when looking through that eyepiece. In general, the field of view of any telescope is very small, usually less than 1°. This is why most telescopes require the use of a smaller “finderscope” to get them pointed at the right spot. While the field of view can be calculated based on parameters of the telescope and eyepiece, it is easier and more instructive to measure the FoV with actual observations. Your telescope’s clock drive slowly turns the telescope to the west to counter balance Earth’s rotation. Without a clock drive, or with the clock drive turned off, any object visible in the eyepiece will drift out of view in a short while, due to Earth’s rotation. Since the rate the Earth rotates is known (one rotation or 360° in 24 hours), you can use this “drift method” to determine the FoV of your eyepiece by timing how long it takes an object to drift across the eyepiece’s FoV with the clock drive off—the transit time. What you will do in this lab is to move a star to the eastern edge of your eyepiece, turn off the clock drive, and time how long it takes to drift across and out of your field of view. This time that you measure will be the transit time.
With a bit of calculating one can arrive at a formula for the FoV from the transit time. Here’s the background to get the formula. Since the Earth turns through 360° in 24 hours, it is turning at a rate of 15° per hour, or 15°/hour. (360 divided by 24 is 15.) In one minute of time, the Earth turns through 15°/60 = 0.25°/minute, or 0.25° per 60 seconds. Therefore, if you time how long it takes an object to drift across the field of view (in seconds), then multiply by 0.25°/60 seconds (multiply by 0.25, then divide by 60), you get the field of view in degrees. So, since 0.25°/60 seconds = 1/240 °/sec, finally (almost finally):
FoV (in degrees) =transit time (in seconds) ÷ 240
Most small angles are measured in arc minutes and arc seconds instead of fractions of a degree. So you will want to know the field of view in arc minutes. Since there are 60 arc minutes in one degree, 0.25° (from above) is the same as 15 arc minutes, or 15’ (just like 0.25 hour is 15 minutes of time). To get the FoV in arc minutes, we replace the 0.25°/60 s with 15’/60 s or 1’/4 s. In other words, we simply divide the transit time by 4:
FoV (in arc minutes) =transit time (in seconds) ÷ 4.
(Note that the transit time in the formulas above is in seconds of time, not arc seconds – another one of those places where astronomy can be quite confusing!)
The two formulas above for FoV are strictly valid only for stars on the celestial equator (with declination = 0°). Stars or other objects not on the celestial equator will take longer to drift across the eyepiece’s field; since they are closer to the poles than stars on the celestial equator, they make smaller circles around the pole. The extreme example of this is Polaris, the North Star. It is so close to the north celestial pole that its position barely changes at all over the course of a night. For many low-power eyepieces, it would never drift out of the field of view since it makes a very tiny circle around the true north celestial pole. To correct for stars off the celestial equator requires use of a bit of trigonometry, specifically the cosine of the declination. Rather than crunching through the details of spherical geometry, let’s just write down the result. For stars not on the celestial equator:
FoV (in arc minutes) = transit time (in seconds) x cos(dec) ÷ 4.
In this formula, “cos(dec)” is the cosine of the star’s declination. (For you purists, this formula is really no different than the first two formulas and can be used even for stars on the celestial equator. The declination of a star on the equator is 0°, and cos(0°) = 1, so the last formula gives the same answer as the first two.) If a star is not exactly on the celestial equator, but is within 5° or 10° of the celestial equator, you can use the simpler formulas without introducing any major errors.
Here’s a couple of examples. Suppose you observe that the star Mintaka (the star in Orion’s Belt that is right on the celestial equator) takes 96 seconds to drift across a particular eyepiece’s field. That eyepiece’s FoV is:
FoV (in degrees) = 96 ÷ 240 = 0.40°,
or, in arc minutes:
FoV (in arc minutes) = 96 ÷ 4 = 24’.
For an example off the celestial equator, suppose you used a different eyepiece to observe the star Schedar in Cassiopeia (with a declination of 57°), and it took 115 seconds to go across the field of view. Calculating that cos(57°) = 0.5446,
FoV = 115 x cos(57°) ÷ 4 = 115 x 0.5446 ÷ 4= 16’.
Materials
Telescope and eyepieces, stop watch, calculator (preferably one with sin, cos and tan), star finder (optional)
Procedure
| Read through this entire write up carefully before coming to lab. |
| Set up and align the telescope per the instructions given in Lab 1. |
| Identify a naked-eye star within 10° of the celestial equator. One of the stars in the belt of Orion or the constellation Aquilla would do nicely. |
| With your telescope’s clock drive ON and your lowest power eyepiece installed, center the star in the field of view. Then use your hand controller to move the star just out of the eastern edge of your eyepiece. |
| Turn the clock drive OFF and start the stop watch as soon as the star reappears in the eyepiece. (It’s probably easiest to have one of you work the drive and the one looking through the eyepiece work the stop watch.) |
| Time how long it takes the star to drift out of view. |
| Record this time, the transit time. Note that your stopwatch records the time in minutes, seconds, and hundredths of seconds. You will need to record this as seconds (out to the hundredths place). |
| Install your medium-power eyepiece and repeat the timing to obtain the transit time for this eyepiece. Record your results. |
| Now install the high-power eyepiece and repeat the process. |
| With your high-power eyepiece installed, center on a star with a declination between 30° and 60° and measure the transit time as before. |
| Record this star’s declination. |
| Your data and observations for this lab should be the names of the two stars you observed, the declination of the second star, and the four timings. |
| When you are done making measurements, use your timings to calculate the FoV for each eyepiece (in arc minutes). |
| Also do a separate calculation for the FoV of the high-power eyepiece based on the measurement of the star with a declination between 30° and 60°. |
These measurements will be useful throughout the semester, so record the FoV of each eyepiece in something you will bring to lab each week. (For the high-power eyepiece, use the FoV calculated for the star near the celestial equator.) For your report, make a table of your measurements and results, showing examples of your calculations. Also answer these questions:
1. Do you see any qualitative relationship between the FoV and magnification of the different eyepieces? If so, explain.
2. For your low-power eyepiece, what is the FoV measured in degrees? (Remember there are 60 arc minutes in 1 degree.)
3. How well does the FoV of the high-power eyepiece calculated using the star off the celestial equator compare with the FoV of the same eyepiece calculated using the star near the celestial equator? (They should be the same, since the FoV is a property of the telescope & eyepiece alone.)