There have been a lot of guiding and polar alignment threads here lately, so I thought I would add another one... In this case, I wanted to discuss polar alignment and in particular, what is a tolerable amount of polar alignment error for any given imaging situation. I did a fair amount of research to answer this question for myself. I found the paper by Frank Barrett to be very enlightening. From what I have learned, the minimum tolerable polar alignment error depends on several factors, including:

1) Telescope focal length

2) Angular distance of the guide star from the farthest part of the imaging sensor at the sensor plane

3) Imaging sensor dimensions

4) Sensor pixel dimensions

5) Exposure time

6) Target declination

7) Type of polar alignment error (alt or az)

8) Your tolerance for field rotation

I'll do my best to address each factor in turn, but first, a short discussion of the effects of polar alignment error. There is a common misconception that PA error causes guiding error. This is not correct. Modern guiding systems are perfectly capable of guiding to sub-pixel accuracy, regardless of PA error. What PA error does is introduce field rotation. So, your guiding software may be happily guiding your mount to an accuracy of under 0.25 arc-seconds, yet some stars may show dramatic elongation far in excess of the guiding error. The elongated stars will form concentric trails with the guide star at the center. This is why accurate PA is important. However, as I have learned about guiding, I have found a lack of concrete information about what exactly "good" polar alignment is. So, I embarked to find out. It turns out that there is no perfect answer. In fact, the definitive answer is "it depends" - on the factors listed above.

My experience has been that a little bit of PA error is actually a good thing. I'll explain. Polar alignment error causes the guiding software to require guiding in declination. Perfect PA would mean that no guiding in dec is necessary. Not guiding in dec is a good thing, on the face of it, because backlash in the dec axis of the mount can cause havoc for the guiding software. Additionally, perfect PA means that there will be no field rotation. If the PA is indeed perfect, you should be able to turn guiding in dec off completely. However, perfection is never achieved, and depending on the factors above, it is possible for there to be a very small amount of PA error that will end up causing the guiding software to start hunting in dec, leading to increased guiding error. Turning off dec guiding can be risky because the small but not non-existent PA error may eventually start causing drift in dec. I've found that a very small but noticeable PA error will allow me to set the mount to guide only against the direction of drift in dec. This takes the backlash out of the equation and doesn't carry the risk of turning off dec guiding directly. But how much error is acceptable? Well, like I said earlier, it depends on the factors listed above. I'll discuss each of them. For those of you who are mathematically inclined, it may be helpful to discuss them in terms of their mathematical relationship. I've broken it apart by numerator and denominator to make it easier to follow:

Maximum Acceptable PA Error =

Numerator: 45,000*Maximum Tolerable Field Rotation*cosine(declination)

Denominator: exposure time*focal length* angular distance of the guide star from the farthest part of the imaging sensor

Or, put in terms of the numbers in the list above:

Max PA error = [45,000*8*cosine(6)]/[5*1*2]

Telescope focal length: Being in the denominator, the longer the focal length, the smaller the acceptable PA error. This is fairly intuitive since a long FL will narrow the image scale, causing the star drift from field rotation to span more pixels.

Angular distance of the guide star from the farthest part of the imaging sensor at the sensor plane: When guiding, you are essentially holding the guide star in a fixed position. Any PA error will result in star trails that center around the guide star. The farther away from the guide star, the more pronounced the field rotation. So, to minimize noticeable field rotation on the image, you want to pick a guide star that is as close to the center of the imaging sensor as practical. If you use an OAG, this is a problem. You can't pick a star at the center of the image sensor. However, you can pick a star that is as close to the sensor center axis is possible to reduce the effects of field rotation.

Imaging sensor dimensions: While not part of the equation, this factor is directly related to the distance of the guide star to the farthest part of the sensor. A large image sensor will increase the angular distance of the guide star from the farthest part of the sensor, thereby increasing the visible effects of field rotation at the edges of the sensor.

Sensor pixel dimensions: This is not directly related to the effects of field rotation. However, the dimensions of the sensor pixels influence how much field rotation is tolerable. If you have 8 micron pixels and a maximum field rotation of 3 microns, the field rotation will be confined within the dimensions of a single pixel and will not be noticeable in the final image. So, pixel size directly influences how much PA error is tolerable (8).

Exposure time: This should be obvious. The longer the exposure, the more time the field has to rotate.

Target declination: Field rotation is most pronounced near the poles and drops off dramatically at the celestial equator. This means that you'll need more precise PA for imaging near the poles. The difference can be dramatic.

Let's look at an example. I'll use my imaging rig as a test case. My focal length is 2737mm with the reducer. My imaging sensor (Canon T2i) is 22.3mm x 14.9mm and I use an OAG. This information is necessary to determine the angular distance from the guide star to the farthest part of the sensor, which needs to be in degrees for the above equation. I'll assume that the guide star in the OAG is 10mm from the corner of the imaging sensor. The diagonal of the sensor is 26.8mm. If I add the additional 10mm for the OAG, that puts the far corner of the imaging sensor 36.8mm from the guide star. At my imaging scale, there are 0.021 degrees per millimeter, so the angular distance from the guide star to the farthest part of the imaging sensor is 0.771 degrees.

I'll assume a 600-second exposure, which is 10-minutes (the equation needs minutes), and a target declination of 45 degrees. The equation needs the declination in radians, so 45 degrees converts to 0.785 radians. Finally, I would like the effects of field rotation to be some fraction of a pixel - let's say an overly challenging 0.25 of a pixel. My pixels are 4.3 microns, so that is a tolerable rotation of 1.075 microns. The equation needs the tolerable field rotation in arc-seconds, so that convert to about 0.075 arc-seconds.

If I crunch the numbers, I get a maximum tolerable PA error of 1.62 arc-minutes. To illustrate how important the angular distance of the guide star to the farthest part of the image sensor is, if I were not using an OAG and was instead guiding on a star at the center of the image sensor, my maximum tolerable PA error would be 5.7 arc-minutes.

We can also examine the effect of declination. My above example is for 45 degrees. If I want to image an object within 1 degree of the north pole, the numbers change dramatically. For my OAG setup, I would get a maximum tolerable PA error of only 0.4 arc-minutes. If I move to the equator, I get a more luxurious 2.3 arc-minutes of tolerable error.

If I adjust my expectations for tolerable field rotation to something a bit more realistic, things get more tolerant of PA error. I can typically guide to an RMS error of less than 0.4 arc-seconds. That's a little over 1 pixel at my image scale. If I change my tolerable field rotation to 1 pixel, I can accept a PA error of 6.5 arc-minutes at 45 degrees dec using my OAG. If I could guide using a guide star at the center of the image sensor, the tolerable PA error skyrockets to 23 arc-minutes.

The astute among you will probably note that I have not yet discussed factor #7 above. The calculations I have done so far determine the total tolerable PA error. That total error is the vector sum of the error in alt and the error in az. Those of us with computerized mounts capable of displaying PA error know that these mounts report error in both alt and az, but do not report the total error. To calculate the total error, we can assume that since the angles are very small, the geometry is Euclidian, so the errors add by the Pythagorean theorem:

total error = square root(alterror^2 + azerror^2)

A simpler approach is to just determine the maximum tolerable PA error for each axis. You can do this in your head by dividing the total maximum PA error calculated above by the square root of 2, or more simply, multiplying by 0.7. So, for my last example where my total tolerable PA error was 6.5 arc-minutes, the maximum tolerable error in each axis would be 6.5*0.7 or about 4.5 arc-minutes. This means that if the mount reports errors in alt and az that arebothless than 4.5 arc-minutes, I will not see noticeable field rotation.

This type of calculation can be very useful when planning an imaging session. For example, let's say that I want to image the horsehead nebula with my 300mm telephoto using the same imaging camera piggyback on my scope. With this setup, I can guide through the scope on Altinak and put Altinak at the center of my image. For a 600-second exposure, I would only need to get my total PA error to be less than 25 arc-minutes, or the error in each axis to be less than about 18 arc-minutes. There would be no need to spend a bunch of time drift-aligning and I could do a simple one-time pass with ASPA and easily get the PA error less than this tolerance on the first try.

On the other hand, if I want to image M81 (much closer to the pole) at prime focus using the OAG, I find that I need to bring my per-axis PA error to below about 2.3 arc-minutes to successfully mitigate field rotation for the same 600-second exposure.

As I have gone through this exercise, I have learned that PA accuracy in less important than I thought. I have found myself spending a whole lot of time getting my PA to less than an arc-minute on each axis when in actual fact, such precision is utterly unnecessary even for my very long FL scope and relatively long exposures for most targets. Now, I can easily calculate what PA accuracy I need for my imaging session and stop when I get it within the calculated tolerances.

I'm sure that for some folks, the math can be a bit daunting and the concept a bit difficult to visualize. To make things easier, I have created the attached simple spreadsheet. Enter your imaging rig parameters into the Input Quantities section and the spreadsheet does the rest. It will tell you both the total maximum tolerable alignment error and the tolerable alignment error per axis. These results are reported in the Maximum Tolerable Alignment Error section. The spreadsheet reports both the tolerable PA error if the guide star in at the center of the sensor as well as if the guide star is at one extreme corner of the sensor. If you use an OAG, you can enter an offset in the Image Quantities section and look at what the spreadsheet reports for "guide star in corner". The reported values in the Maximum Tolerable Alignment Error section are for the declination that you enter in the Input Quantities section. If you want to know the error for any declination, the spreadsheet also produces a chart showing each tolerable error for each declination.

https://drive.google.com/open?id=0B1...MWM&authuser=0

If you have Excel, or a spreadsheet program capable of reading Excel files, it is probably best to download the spreadsheet. It seems that Google Sheets only allows 2 data series in the charts so the per-axis alignment curves in the chart don't show when opened in Google Sheets.

Tim