Polarization of Light - linearly polarised light

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Divide the AFoV, 50°, by the magnification, 48X, to yield a TFoV of 50/48, or about 1.0417°. You can multiply the result in degrees by 60 to convert it to arcminutes, which are used more commonly to refer to an eyepiece field of view. A 1.0417° field is 62.5 arc-minutes, which is abbreviated 62.5’.

The AFoV method suffices for a quick-and-dirty calculation of TFoV and is good enough for most purposes, but it is not accurate enough for critical work.

Because they represent a spherical surface (the sky) as a plane (the view in the eyepiece), all eyepieces must introduce some distortion. That means that even if the AFoV is specified accurately, it cannot easily be translated to an accurate TFoV. For example, Tele Vue Panoptic eyepieces have a nominal AFoV of 68°, which is accurate insofar as it goes. But the distortion present in a Panoptic eyepiece means that if you measure the TFoV accurately you’ll find that it corresponds to an undistorted AFoV of less than 65°. (We’re not knocking Tele Vue or Panoptics here; Tele Vue is a first-rate optical maker, and Panoptics are superb, world-class eyepieces. Distortion is simply an optical fact of life.)

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Because the actual TFoV is determined solely by the focal length of the telescope and the field-stop diameter of the eyepiece, it is possible to calculate very accurate TFoV figures if you have an accurate focal length for your scope and have the field-stop diameter for the eyepiece in question. Unfortunately, Tele Vue is the only eyepiece maker that routinely publishes field-stop diameters for its eyepieces. For some eyepieces, you can use calipers to measure the field-stop diameter yourself. However, the field stop of many eyepieces is inside the eyepiece rather than exposed, which means you have to disassemble the eyepiece to measure its field stop, which is not a recommended procedure.

Divide the focal length of the scope, 1,200mm, by the focal length of the eyepiece, 25mm, to determine that that combination provides 48X magnification.

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The quickest way to determine TFoV is to divide the Apparent Field of View (AFoV) of the eyepiece by the magnification provided by that eyepiece in a given scope. For example, if your scope has a focal length of 1,200mm and you use a 25mm Plössl eyepiece with an AFoV of 50°, you can calculate the TFoV as follows:

If you have a Dobsonian or other alt-az mount, simply center the star in the eyepiece field and start your stopwatch. The human eye is very, very good at centering objects. If you start with the star centered in the eyepiece field, by definition it will cross a half-diameter of the eyepiece field as it drifts to the edge. You can use the same calculation described previously and simply double the result to determine the TFoV of the eyepiece.

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If you have an alt-az scope but prefer to use a full-diameter drift test, first do a half-diameter drift test described in the preceding bullet point. Watch where the star leaves the field of view, and then restart the test with the star entering the view at the opposite point on the clock face. (We prefer to use half-diameter drift tests for alt-az scopes; we can do twice as many tests in the same time and average the results.)

If the star you use for testing is exactly on the celestial equator (declination 0), this factor drops out because the cosine of 0 is 1. If the star is not on the celestial equator, its apparent motion is slower. The larger the absolute value of the declination, the slower the apparent motion. A star located at exactly +90° or -90° declination has no apparent motion at all because it is exactly on the pole of the axis around which Earth rotates. Polaris, for example, is located at declination +89°15', and it requires 24 hours to trace a 1.5° circle around the pole.

Any combination of telescope and eyepiece has a specific true field of view (TFoV), which is determined solely by the focal length of the telescope and the field stop diameter of the eyepiece. TFoV quantifies the amount of sky visible in a particular scope with a particular eyepiece. For example, if a particular telescope/eyepiece combination provides a 1° TFoV, two stars that are separated by exactly 1° will just fit into the eyepiece field, with each star on opposite edges of the field.

It is important to know the TFoV of your eyepieces in your scope and the TFoV of your finder because you use the true field to match the stars that are visible in the eyepiece to those on your charts and to plan and execute star hops [Hack #21]. There are several methods, of varying accuracy, to determine TFoV.

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The actual focal lengths of both the telescope and eyepiece often differ 5% or more from nominal stated value because of normal manufacturing variances. For example, a telescope with a stated focal length of 1,200mm may actually be 1,173mm (or 1,234mm), and an eyepiece labeled 10mm may in fact be 9.7mm (or 10.4mm).

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The specified AFoV is often an approximation. An eyepiece with a nominal 55° AFoV may actually provide as little as 50° or as much as 60°. The nominal AFoV of premium eyepieces tends to be reasonably accurate; that of cheap Chinese eyepieces tends to be inaccurate and very optimistic. Also, the true AFoV of different focal length eyepieces in a series with nominally identical AFoVs may differ significantly from one focal length to another.

When we test a new finder, we simply point it at a bright star when we first set up the scope and then go about our business. If the star is anywhere near the celestial equator, we know about how long the half-diameter drift test should take, based on the published FOV of the finder. For example, if the finder supposedly has a 6° field, we know that our half-diameter drift test should take about 12 minutes (the 3° half-diameter at about four minutes per degree). We center the star in the crosshairs and start our stopwatch. About 10 minutes later, we wander back to the scope and watch as the star drifts out of the field of view. We jot down the elapsed time and which star we used, and do the calculations later at our convenience.

If you have or can get accurate values for your scope’s focal length and your eyepieces’ field-stop diameters, you can calculate an accurate TFoV. To do so, divide the eyepiece field-stop diameter in mm by the focal length of the scope and multiply the result by (360/2φ) or about 57.2958. (There are 2φ radians in a full circle of 360°. Dividing 360° by 2φ yields about 57.2958° per radian.) For example, if you use a 27mm Tele Vue Panoptic eyepiece with a 30.5mm field stop in a scope of 1,255mm focal length, the TFoV is (30.5/1,255)*(360/2φ), or about 1.39°. Multiplying that result by 60 yields the TFoV in arcminutes, about 83.5’.

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Using a “slower” star without a correction factor would overstate the TFoV of an eyepiece. (The star takes longer to drift across the field, so the TFoV appears larger than it really is.) But you can use any star for drift testing if you calculate its declination correction factor. To do so, convert the star’s declination to a decimal value and determine the cosine of that value. For example, if you use Antares, convert its declination of -25°26’ to -25.433°. The cosine of -25.433° is about 0.9031.

Let’s assume that you’ve drift tested using Antares, which took exactly 4:43 or 283 seconds to drift across the diameter of the eyepiece field. What is the TFoV of that eyepiece in arcminutes?

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where T is the elapsed time in seconds and Cos(Dec) is the cosine of the declination in decimal degrees of the star you used for testing. That’s fine as far as it goes, but it does require a bit more explanation. Let’s look at each of these terms:

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You must be certain that the star drifts across the actual full diameter of the field of view, rather than across a chord.

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Up to this point, we’ve considered only Earth’s rotation on its axis, which is the basis of Solar time. But to calculate stellar motion properly, we have to use sidereal time (star time), which is slightly different from Solar time. The difference exists because as Earth rotates on its axis, it also orbits Sol. Over the course of one year, Earth makes one full additional rotation in its orbit, and this must be taken into account.

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The factor 0.2507 is the constant that converts seconds of time to the TFoV in arcseconds. The earth rotates on its axis 360° in 24 hours, which translates to 15° per hour, 1° every four minutes, 1/4° per minute, and 1/240° per second. There are 60 arcminutes per degree, so Earth rotates at 60/240, or 0.25 arcminutes per second. (This is known as the Solar Time Constant.) So why the extra decimal places?

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If you know the focal length of your other scopes, you can determine their true fields of view without drift testing simply by using the field-stop diameter calculation described in the preceding section.

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For perfectly accurate results, the star you use must be located exactly on the celestial equator [Hack #17], at declination 0°0’0”. This is true because a star at other than exactly 0 declination has a different apparent drift rate. In practice, any star within ±5° or so of declination 0 yields sufficiently accurate results for all but the most critical work. If Orion is visible, use Mintaka (the belt star on the Bellatrix side), which is at declination about 0°18’. If Orion isn’t up, other good choices are Porrima (α-Virginis) at about 1°29', Sadalmelik (γ-Aquarii) at about -0°17', and σ-Serpens at about 1°00’. (You can use any star for precise drift testing if you apply a declination correction factor, described in the next section.)

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The Solar year is 365.2425 days long, which is 525,949.2 minutes or 31,556,952 seconds. A full circle is 360°, or 21,600 arcminutes. The correction for sidereal time therefore adds (21,600 arcminutes / 31,556,952 seconds) = 0.00068447675+ arcminutes/second to the apparent drift speed of a star located on the celestial equator. (This figure is only a very close approximation because Earth’s orbital speed around the Sun varies according to its orbital position; when Earth is near Sol, it moves faster than when it is more distant.) Adding the ~0.0006845 correction to the Solar time constant 0.25 and rounding the result gives us 0.2507.

It’s surprising how much the actual FOV of a finder may differ from published specifications. We saw one finder with a nominal FOV of 6.5° that turned out to be more like 5.8°. Conversely, we remember another finder with a supposed 5° FOV that actually had a 5.5° FOV. That amount of difference can be significant, particularly if you are a dedicated star hopper, so it’s worth testing your own finders to determine their actual fields of view.

Once you determine an accurate TFoV for a particular eyepiece in one scope, you can back-convert that TFoV to a field stop diameter for that eyepiece by multiplying the TFoV in decimal degrees by the focal length of the scope in mm and then dividing by (180/φ) or about 57.2958. For example, to back-convert the 64.1 arcminute field we determined using a scope with a focal length of 1,255mm, convert the 64.1 arcminute field to 1.07° and use the formula 1.07 * (1,255/57.2958) = 23.44mm.

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But the timing and calculations are the easy part. The first time you actually drift test an eyepiece, you’ll learn that the hard part is getting the star to drift across the full diameter of the eyepiece field instead of a chord. If you try to do this by trial and error, you’ll waste a lot of time and become quite frustrated. We know. We’ve watched people do it. However, there are some easy solutions:

Sometimes the best way to be sure of something is just to go and look at it. That’s the basis of the drift-testing method of determining TFoV. Drift testing is simple in concept. You place a star on one edge of the field of view of an eyepiece and then use a stopwatch to measure how long it takes to drift all the way to the opposite edge of the field of view. Because stellar motion (actually, Earth’s motion) is known very precisely, it’s easy to convert the elapsed time to an accurate TFoV.

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If you have a polar-aligned equatorial mount, center the chosen star in the eyepiece field, turn off your drive motor(s), and use your RA slowmotion control to put the star just outside the eastern edge of the eyepiece field. As the star appears in the field, start your stopwatch. Stop it just as the star exits the eyepiece field.

These are the usual instructions for drift testing. Place the chosen star just outside of the field of view of the eyepiece. When the star appears at the edge of the field of view, start your stopwatch. Just as the star disappears on the other side of the field of view, stop the stop watch and record the elapsed time. Calculate the TFoV in arcminutes using the formula:

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It’s interesting to compare the results of this method with the AFoV method. Assuming a telescope focal length of 1,255mm and a 27mm Panoptic with a 68° AFoV, the AFoV method yields a true field of about 1.46°. Calculating based on the 27mm Panoptic field stop diameter of 30.5mm yields a true field of view of only 1.39°, which specifies accurately the amount of sky visible with that telescope/eyepiece combination. Back-converting the accurate TFoV to determine a “real” AFoV gives us about 64.72° rather than the nominal 68°. Again, this is not a knock on Tele Vue or Panoptic eyepieces; the AFoV in Panoptics simply “looks” wider than it really is.

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Most astronomers use drift testing only for eyepieces, but it’s just as accurate a means of determining the true field of view of optical finder scopes. Of course, finders have much wider fields of view than eyepieces, so it may take half an hour or more to do a full-diameter drift test on a finder, which you really don’t need to do. Finders have nicely centered crosshairs, so it’s trivially easy to do an accurate half-diameter drift test. If you center the star in the crosshairs, there’s no question that the star crossed the exact center of the field.

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