       Determining Polar Axis Alignment Accuracy (by Frank Barrett)

Determining Polar Axis Alignment Accuracy
by Frank Barrett
7/6/2008

Abstract: In order to photograph dim celestial objects, long exposures on the order of minutes or
hours are required. To perform this process successfully the mount needs rotate to compensate
for the earthвАЩs rotation. To achieve this, the mountвАЩs rotational axis must be critically aligned
parallel to the earthвАЩs rotational axis. This process is known as polar axis alignment. Two
determine my polar alignment error?вАЭ, and second, вАЬWhat is the required tolerance for alignment
error for a given imaging session?вАЭ

Drift Alignment Overview

Before we examine the details of polar axis alignment it is beneficial to review the drift alignment method. Drift alignment is a very popular method for polar alignment of equatorial mounts particularly when very high accuracy is needed. The method requires that the scope point at a carefully selected reference star. If the star drifts in declination it indicates misalignment in the mountвАЩs rotational axis.

Here is how the procedure works. To measure and adjust the azimuth axis, monitor a star near the intersection of the celestial equator and the meridian. If the star drifts north the mount is pointing too far west. A southern drift indicates the mount is pointing too far east. Likewise, to measure and adjust the altitude axis, monitor a star in the east near the celestial equator. If the star drifts north the mount is pointing too high. A southern drift indicates the mount is pointing too low. In each case the rate of drift is indicative of the magnitude of the error and the adjustment required for correction. Note that the drift direction should be reversed in the southern hemisphere or if an altitude reference star is selected in the west. The procedure is repeated on each axis until no discernable drift is observed or, as this article will show, until the remaining alignment error is within the tolerance required for the imaging session. 

HookвАЩs Equations

The undesirable consequence of a poor alignment is that a guided image may show field rotation centered on the guide star. It is this field rotation that we seek to eliminate by carefully aligning the polar axis of our mount. In an article in the February 1989 issue of the Journal of the British Astronomical Association, Richard Hook derived a number of equations which showed how far a star would drift on an unguided mount and the alignment tolerance required to hold field rotation to within a given limit. We will use HookвАЩs equations to answer our primary questions and then take a closer look into these equations to understand how various factors of the imaging process affect the field rotation. 

Question 1: What is my polar alignment error?

With the drift method we have a means of detecting declination drift for each polar alignment axis. The question here is: вАЬCan I determine the polar axis misalignment given an accurate measure of the declination drift?вАЭ The answer is: Yes! Hook showed that the declination drift was related to the angle of alignment error as follows: Where:
ќіerr¬∞ is the declination drift in degrees
t is the time of drift in minutes
ќі is the declination of the drift star used
qr is the alignment error in radians

Since the drift error is typically very small it is more convenient to express the drift in arc seconds:
(formula2)
Solving for r пБ±пА†gives:
(formula3)
We generally express alignment errors in arc minutes, so to convert radians to arc minutes we arrive at:
(formula4)
So:
(formula4)
Where:
пБ±пА†is the alignment error in arc minutes
err пБ§пА†is the declination drift in arc seconds
t is the time of drift in minutes
пБ§пА†is the declination of the drift star used
For example, given: err пБ§пА†= 20.5вАЭ, t = 10 minutes, and пБ§пА†= 35 degrees:
(formula5)

Measuring the declination drift

The declination drift can be estimated visually using a well-calibrated reticule eyepiece. More accurate measurements can be achieved with camera assisted methods. For example, autoguiding software works by taking images of a reference star and measuring the distance of that star from a reference pixel in the image. Corrections are sent to the mount to reposition the star at the reference point and the process is repeated throughout the duration of the exposure.
Most autoguiding software can operate in a mode where the star offset is reported without any correction made. This is perfect for our purposes since the offset in the declination axis over time is the measurement in which we are interested. The image exposure time should be long enough to average out atmospheric seeing effects or a reasonable estimate can be obtained by averaging the last few measurements. Also note that a requirement of Equation (1) is that the drift needs to be expressed in arc seconds. If the autoguiding software reports the error in pixels the needed conversion is as follows:
(formula5)
Where:
ќіerrпА†is the calculated drift in arc seconds
пБ≥errпА†is the measured drift in pixels
пГђпА†is the pixel size in microns
F is the focal length in mm

Question 2: How do I know when my alignment is вАЬGood EnoughвАЭ?

By means of Equation (1) we know the alignment error based on the declination drift. So how
good is good enough? Hook also answered that question by showing that the maximum tolerable
error is related to the declination of the target, the elapsed time of the exposure, the effective
focal length of the instrument, and the angle between the guide star used and the opposite edge
of the field as follows:
(formula6)
Where:
Qmax ¬∞пА† is the maximum permitted alignment error in degrees
d is the declination of the target in degrees
t is the time of exposure in minutes
F is the focal length in millimeters
D is the angle between the guide star and the opposite edge of the field in degrees
Hook arbitrarily assumed that 30 microns of rotation was tolerable. To allow the rotation to be
more directly addressed we divide by 30 and substitute a factor, пБіпА†which represents the tolerable
field rotation in microns. For convenience we convert from degrees to arc minutes by multiplying
by 60 which gives: Where: is the maximum permitted alignment error in arc minutes is the tolerance for field rotation in microns is the declination of the target in degrees
t is the time of exposure in minutes
F is the focal length in millimeters is the angle between the guide star and the opposite edge of the field in degrees
For example, given: = 3 degrees, F = 655mm, t = 15 minutes, = 35 degrees, and = 9
microns: In other words, if our alignment error is less than about 11.25 arc minutes we should see no field
rotation greater than 9 microns during a 15-minute exposure at 35 degrees declination and the
given setup.
Knowing the maximum permitted alignment error can assist us in our drift alignment. By solving
Equation (1) for the drift rate,(formula9), we can plug in our alignment error and thereby know our
maximum drift rate required to achieve that alignment:
(formula10)
In the example above our maximum required drift rate would then be:
(formula11)

Total Polar Alignment Error

Up until now we have ignored the fact that a single drift alignment effectively only measures the error in one axis. A measurement at the intersection of the meridian and equator indicates azimuth error and a measurement in an eastward or westward direction indicates an altitude error. The calculated value of max q should be viewed in light of these two measurements taken together. In a spherical model one can apply Euclidian geometry as long as the angular distances are small. Since we measure errors typically in small angles of arc minutes this condition applies and we can simply use the Pythagorean Theorem. Therefore:
(formula12)
Where:
total q is the total alignment error in arc minutes
az q is the alignment error due to azimuth error
alt q is the alignment error due to altitude error
If we are striving for an error less than max пБ±пА†such that the error at both axis is identical, i.e.
пБ±пА†= az пБ±пА†= alt пБ±пА†, Equation (5) becomes simply:
(formula13)
Solving forпБ±пА†:
(formula14)
So in our previous example, to achieve max пБ±пА†= 11.25вАЩ:
(formula15)
In other words, we would have to achieve an 8 arc minute (or better) accuracy on both axes to achieve an 11.25вАЩ overall maximum polar alignment error. Plugging this value into Equation (4) reveals that we would need our drift rate to be no more than 1.72 arc seconds per minute in both axes to achieve this alignment tolerance. Alternatively one could measure the error on each axis independently and use Equation (5) to determine the total alignment error.

The Error Angle and Effective Polar Alignment Error

The value total q is only part of the information we need to understand the effects of polar alignment error. The direction of the error, or error angle, is also important. The error angle is defined as the angle between the meridian and the line formed between the mountвАЩs rotational axis (as determined by az q and alt q ) and the celestial pole as visualized in Figure 1:
(figure1)
Figure 1. The Error Angle is measured from the meridian.
Again, since we are dealing with small angular distances, Euclidean geometry can be used and
the error angle, пБ≤пА†, can be calculated as follows:
(formula16)
Why is the error angle important? The fact is that the effective polar alignment error is dependent on where you are pointing in the sky. As an object is imaged over several minutes or hours, the effective polar alignment error is constantly changing since the mount is tracking in right ascension. When you are pointing in the direction of the error angle (or 180 degrees from it), the effects of polar misalignment are minimal. Note also that pointing ¬± 90¬∞ away from the error angle is where you will experience the full effects of the alignment error. We can calculate the effective alignment error, eff q , as:
(formula 17)
Where:
eff q is the effective polar alignment error
total q is the total alignment error calculated by Equation (5)
r is the error angle calculated by Equation (7)
H is the hour angle where the telescope is pointing in degrees

Notice that Equation (8) holds the key as to why the drift alignment method works. We are able to measure drift and adjust each axis independently as a consequence of this equation. When we measure azimuth error we point at the intersection of the meridian and equator. At this location the hour angle is zero. Equation (8) reduces to:
(formula18)
Likewise, when measuring drift in the east or west for altitude error, the hour angle is пВ±пА†90пВ∞пА†.
Equation (8) reduces to:
(formula19)

Implications of Alignment Error Tolerance

Equation (3) carries with it some far-reaching implications that should be examined carefully. We will examine each variable in turn and look at the effect that variable has on alignment error and vice versa.

Focal Length

Focal length, F , is a factor in the denominator of Equation (3). Therefore, as focal length increases, the polar alignment tolerance decreases, assuming all other factors remain constant. This effect is predominantly due to the image scale reduction at longer focal lengths. The star trail created by the field rotation will occupy more pixels or film grains. To hold пБіпА†constant, therefore, will demand a more accurate polar alignment.

We might then ask, вАЬGiven the measured alignment error, what is the maximum focal length instrument I can use to keep field rotation within tolerance?вАЭ To answer this, we solve Equation (3) for F :

(formula20)

For example, with the parameters: alignment error = 10 arc minutes, field rotation = 9 microns, exposure time = 15 minutes, guide star angle = 3 degrees, target = 35 degrees, the calculated maximum focal length is 737 millimeters.

Guide Star Angle

Guide star angle,пБД, is a factor in the denominator of Equation (3). Therefore, as the guide star angle increases, the polar alignment tolerance decreases, assuming all other factors remain constant. Clearly, if you are guiding with a guide scope, try to use a guide star as close as possible to the center of your imaging camera.

Note that if you are guiding through the same optic as the imaging device and if the guider to imaging device is rigidly fixed (as is the case with dual sensor cameras and some off-axis guiders), the product FпБДпА†will remain constant. This is because in those configurations the guide star selection is rigidly bounded and the worst case angle is inversely affected by the focal length.

For short focal lengths the angle is larger, and inversely, the angle is smaller for larger focal lengths.

To measure the guide star angle (for any configuration), try locating a star on the guider and note a star in an opposing corner of the imaging camera. If these stars can be identified in an atlas or planetarium program and their coordinates derived, the angular distance can be deduced by the following equation: 

(formula 21)

Where:
пБДпА†is the angle of separation between the two stars
1 пБ§пА†is the declination of the first star
2 пБ§пА†is the declination of the second star
1 пБ°пА†is the right ascension of the first star
2 пБ°пА†is the right ascension of the second star

Also remember when determining the guide star angle to take into consideration any planned cropping of the image. If the image is to be cropped (perhaps to remove some spherical aberrations) the guide star angle can be reduced accordingly, giving some relief to the polar alignment tolerance.

The relevant question here is, вАЬGiven the measured alignment error, what is the maximum guide star angle I can use to keep field rotation within tolerance?вАЭ To answer this, we solve Equation (3) for пБД:

(formula22)

For example, with the parameters: alignment error = 10 arc minutes, field rotation = 9 microns, exposure time = 15 minutes, focal length = 655 mm, target = 35 degrees, the calculated maximum guide star angle is 3.38 degrees.

Exposure Time

Exposure time, t , is a factor in the denominator of Equation (3). Therefore, as exposure time increases, the polar alignment tolerance decreases, assuming all other factors remain constant. This is intuitive since the longer we expose, the longer our star trails will become due to field rotation.

We might ask, вАЬWhat is the longest exposure time I can make before field rotation is noticeable?вАЭ To answer this we solve Equation (3) for t :

(formula23)

For example, with the parameters: alignment error = 10 arc minutes, field rotation = 9 microns,
guide star angle = 3 degrees, focal length = 655 mm, target = 35 degrees, the calculated
maximum exposure time is 16.9 minutes.

Field Rotation

Field Rotation, пБіпА†, is a factor in the numerator of Equation (3). Therefore, as we allow for more field rotation the polar alignment tolerance increases. Another way of looking at this is that if we require very small star trails due to field rotation we will require a very small polar alignment error.

When determining a good value to use for field rotation two rules of thumb can be applied. Some astrophotographers strive to limit the rotation to 1/3rd the minimum star size produced by their optics. So if your smallest stars are about 30 microns you might want to limit the worst case rotation to 10 microns. Another solution is to use the pixel size of your camera so that the rotation is no more than a single pixel.

Clearly, calculating the field rotation is the critical factor of this whole exercise since it is the very variable we would like to keep to an absolute minimum. Simply solve Equation (3) for пБіпА†:

(formula24)

For example, with the parameters: alignment error = 10 arc minutes, exposure time = 15 minutes, guide star angle = 3 degrees, focal length = 655 mm, target = 35 degrees, the calculated field rotation star trail is 8 microns.

Declination of the Target

Declination,пБ§пА†, is a factor in the numerator of Equation (3). Therefore, as we image closer and closer to the celestial pole we will need a smaller and smaller alignment error. This may not seem intuitive at first. We know that star trails in untracked images are shorter near the pole than they are near the equator. Why, then, do we require more alignment accuracy near the pole? The answer is quite simple. For field rotation to occur there must be a correction in both declination and right ascension. The prevailing myth is that during drift alignment there is no drift in right ascension. This is incorrect. The drift in right ascension when measured near the equator is very small and is usually overwhelmed by the periodic error of the right ascension worm gear, but it does exist in an imperfectly aligned mount and must exist for field rotation to occur. Suppose, for a moment, that the guiding corrections were only in the declination axis. The image would move laterally with, perhaps, some side to side movement for periodic error, but no overall rotation would occur. However, this is simply not the case in an imperfectly polar aligned mount. Over time a small drift in right ascension will cause field rotation.

The situation is unique near the pole. Near the pole the correction required for right ascension is more significant than is needed at the equator given the same polar alignment error and will result in more field rotation when guiding. Why? At the equator the movement due to correction is predominately linear. The declination error is greater than the right ascension error. Near the pole the movement is more circular and the correction in right ascension is more significant. It is this increase in right ascension correction that causes the rotation to be more severe at the polar regions. The illustration in Figure 2 may make this clearer. Here position A represents the starвАЩs starting position, B is where the misaligned mount would point after some period, and C is where the mount should be pointing if it were perfectly aligned. Note also that for illustration purposes the alignment error near the equator is exaggerated; still you can see that the correction in right ascension is relatively small. This correction amount will decrease significantly as the error angle is decreased.

The implication here is clear; if we are going to image at declinations near the celestial poles we will require very tight alignment tolerances. The other alternative is to attempt to image unguided since star trailing near the pole should not be too offensive and may fall below our tolerance, пБіпА†.

So how do we calculate the maximum declination given our measured alignment error? The answer requires us to solve Equation (3) for пБ§пА†:

(formula 25)

For example, with the parameters: alignment error = 10 arc minutes, exposure time = 15 minutes, guide star angle = 3 degrees, focal length = 655 mm, field rotation = 9 microns, the calculated maximum declination is 64.1 degrees.

(figure 2)

Figure 2. Field rotation is more severe near the poles due to more significant corrections in R.A.

Conclusions

With the mathematical model disclosed here, we have at our disposal the means to determine the polar alignment error of our equatorially mounted instruments. Armed with this information, we can also determine if the alignment error is within acceptable tolerances for the imaging session.

It is hoped that this narrative will take some of the mystery out of the polar alignment procedure. The information presented here can be used as an important aid to the normal drift alignment procedure. Having the ability to quantify the polar alignment error may reduce the time for accurate alignment by making the axes adjustments in a more deterministic, data-driven fashion.

The implications of the various factors affecting alignment error tolerance provide the astrophotographer with intelligent alternatives. If the measured alignment error is insufficient given the calculated tolerance for error the choice can be made to:

- Reduce the exposure time
- Find an image target at a lower declination
- Choose a guide star closer to the target
- Allow for more field rotation
- Image with a shorter focal length
- Adjust the mount to improve the polar alignment.

–Ю–њ—Г–±–ї–Є–Ї–Њ–≤–∞–љ–Њ: 31-07-2012