Polar alignment, refraction and the King tracking rate
In the second part we saw how drift alignment was influenced by refraction. In this final part we will look at the effects of refraction on the apparent stellar motion, and whether we should align on the true or refracted pole. Finally, we'll summarise what we have learnt about drift alignment.
E.S.King and stellar photography
At the end of the nineteenth century Edward Skinner King was employed as an Observer (later to become Professor) at Harvard Observatory. One of his interests was photographic photometry and to obtain the best results, some means of accurately following a star was required. Remember that this was in the days before autoguiding was possible. He was also using long focal-length refractors and slow photgraphic film, so needed to make longer exposures than we might typically do today. King discussed the problem in detail in a monograph titled "Forms of Images in Stellar Photography". He was principally concerned with the possibility of offsetting the telescope axis in elevation from the pole to counter the effects of refraction and instrumental flexure. He derived a set of analytical expressions for the effects of refraction on the apparent tracking rate and declination drift (and which give answers equivalent to the spreadsheet).
Using his analytical expressions, King investigated the effect of varying the amount of adjustment in elevation and found that it was always possible to adopt a value that compensated for refraction in declination for a given zenith distance. To make use of this, the observer should alter the elevation of the telescope axis for each particular observation. But King remarked that most telescope mounts were not constructed to readily allow accurate adjustments in elevation for each observation, and furthermore that using polar alignment methods typical of that time, the telescope axis would most likely be aligned with the refracted pole: "...If we employ the method of adjustment by photographing stars near the pole, aiming to obtain perfect images at that point, the axis of the instrument will now be directed to the apparent pole as lifted by refraction." Consequently, King went on to show that aligning on the refracted pole was a satisfactory compromise because it did largely correct for refraction in declination except at great zenith distances.
We haven't looked at the effect of polar misalignment or refraction on the tracking rate yet. But the variation in tracking rate caused by refraction was one of King's concerns, and once again he was interested in the possibility of offsetting the telescope's polar axis to correct for it. He wrote: "The natural impression, that a star moves on sidereal time, and that our instruments must be regulated accurately by the sidereal clock, is erroneous. Though it is desirable, as has been said, to maintain a rate which does not change suddenly, that rate is not necessarily sidereal or even constant." The apparent variation in tracking rate is actually quite complex but King concluded that "...In general, the elevation of the axis [to the altitude of the refracted pole] exerts a corrective influence upon refraction north of the equator, and within six hours of the meridian." (A similar effect applies in the southern hemisphere, of course).
True or refracted pole?
King didn't formally prove that using the position of the refracted pole was the best solution but suggested that "...Much may be done graphically toward the solution of this problem." With the assistance of modern digital computers we can now investigate this thoroughly for ourselves. Suppose we adopt the root-mean-square (RMS) variation in the declination drift or hour angle drift for three hours each side of the meridian as a measure of a star's drift over an important part of the sky. We can use the spreadsheet to calculate that RMS value for a range of latitudes and star declinations, and compare the values we obtain for the telescope axis aligned with the true pole or the refracted pole. The results are summarised graphically below, where the columns correspond to a star that has the given altitude when on the meridian and the rows correspond to the given latitude.
The green squares indicate the parts of the sky for which it is always better to align on the refracted pole (i.e. the RMS drift in declination and hour angle is less for the refracted than the true pole for three hours each side of the meridian). For example, at a latitude of 45 degrees north or south, stars of altitude 50 degrees when on the meridian [and consequently with declination of +5 degrees (northern hemisphere) or -5 degrees (southern hemisphere)] drift less if the telescope axis is aligned on the refracted pole.
As we move closer to the equator, there comes a point where aligning on the refracted pole is generally not better. Stars that reach high altitudes on the meridian are spending much of their time free of significant refraction at low latitudes, so offsetting the pole by a large amount will worsen the declination drift for them. This is indicated by the beige squares. But we saw in the second part of this discussion that drift aligning at low latitudes tends to set the telescope mount's axis below the refracted pole, towards the true pole, which is what we want; so drift aligning serves us well here too.
The pink squares correspond to altitudes and latitudes where aligning on the refracted pole gives a worse drift in hour angle than does aligning on the true pole. In fact, the pink squares correspond to declinations south of the equator for the northern hemisphere, or north of the equator for the southern hemisphere; King had commented (as quoted above) that alignment on the refracted pole for the norhern hemisphere was beneficial for the tracking rate only for stars north of the equator. Note that for these low altitudes, it is still better for the declination drift to align on the refracted pole than the true pole, even though the corrective effect is not as great as for stars at higher altitudes.
In general we can say that it is better to align on the refracted pole at mid to high latitudes, and better to align on the true pole at low latitudes. In the mid to low latitudes we may wish to align on one or the other depending on our application.
True pole versus refracted pole - an example
With the aid of the spreadsheet, we can see the effect of adjusting the elevation of the telescope axis to the refracted pole in the following graphs. The red, green and blue lines show the drift for stars of various declinations when the telescope mount is aligned with the refracted pole, whilst the cyan line shows the drift if the telescope mount is aligned on the true pole:
The upper graph shows the drift in declination, including the effects of refraction, for stars with declinations of +20 degrees, +30 degrees and +40 degrees; these correspond to altitudes of +70 degrees, +80 degrees and +90 degrees when on the meridian. The latitude is +40 degrees, but the elevation of the telescope axis is set to the refracted pole at 40 degrees 1 minute 9 seconds. The lower graph shows the rate of drift. The drift in declination doesn't exceed about 0.2 arc minutes (12 arc seconds) for any of the stars for about three hours each side of the meridian.
The cyan curve in each graph is for a star at +30 degrees declination but with the telescope axis aligned to the true pole. This shows the drift that is caused solely by refraction. Clearly this is worse than for a star at +30 degrees declination with the axis set to the refracted pole (dark blue line).
Here are the graphs for the drift in hour angle:
The variation in a star's apparent position is typically a bit worse in hour angle than it is in declination in this example. From the top graph we see that a star at +20 degrees has an offset of nearly +0.5 arc minutes (30 arc seconds) relative to its expected position at an hour angle of 21 hours (three hours east of the meridian). Hence it appears to be west (ahead of) its expected location. By three hours west of the meridian, at an hour angle of 3 hours, it has an offset of nearly -0.5 arc minutes, now lying east of (behind) its expected position by the same amount.
The cyan line again shows the curve for the star at +30 degrees declination with the telescope aligned on the true pole. Once again it is clearly worse than for a star at +30 degrees declination with the telescope axis set to the refracted pole (dark blue line).
In principle, we might think it desirable to take advantage of the drift in hour angle as well as the drift in declination for drift alignment, and perhaps achieve alignment on the pole using just one star. In practice this isn't achievable; irregularities in the telescope drive generally create too much uncertainty in the east/west drift measurement, and the lower of the two graphs above now gives us another reason why this won't work: even on the meridian, refraction typically means that the rate of east/west drift isn't zero anyway (though it may be very small).
King tracking rate
It isn't possible to remove the effects of refraction on the tracking rate completely, and the remaining variations in some parts of the sky can be quite large. It would be nice if we could compensate for them automatically. Nowadays we have the advantage of microprocessor-controlled drives - something King could only have dreamed about - so we can do precisely that. Using King's analytical expressions, a drive controller that knows where the telescope is pointing can calculate the appropriate drive rate, similar to the curves in the lower of the two graphs above. And that is exactly what some telescope control units offer: an "adaptive King rate". King also referred to certain situations where the apparent rate is a bit less than sidereal - running slow by one second per hour is typical - and simpler controllers may offer that as a fixed "King rate". Finally, to use King rate tracking, we must make sure that we have adjusted the elevation of the telescope axis to the refracted pole.
Drift alignment: a summary
Here are some drift alignment do's and dont's:
DO level the equatorial head before you start. This isn't essential but it means that azimuth adjustments won't affect elevation adjustments (or vice-versa) so you will only need to do each of the following steps once.
DO make sure that the optical train is not flexing. Mirror flop, bendy focussers etc will ruin your attempts at polar alignment.
DO make the azimuth adjustment first. It isn't significantly affected by refraction since stars are moving parallel to the horizon as they cross the meridian, so you can take time to get this as accurate as you wish.
DO ideally use a star chart or planetarium program to pick a star in the right part of the sky for the elevation adjustment, at an hour angle of 6 hours or 18 hours and about 60 degrees or more declination. However, a star at 30 degrees or so altitude due east or west may serve about as well.
DO try to use a star in the eastern half of the sky for the elevation adjustment if you can. The effects of refraction will decrease as it rises higher in the sky.
DO approach the pole from below for the elevation adjustment when using traditional trial-and-error drift alignment - this is likely to place you closer, and in addition you are raising the axis against gravity which may give fewer problems with backlash on the adjustment.
DON'T expect perfection - there will always be drift in hour angle and declination; it's a consequence of atmospheric refraction, and we need air to breathe.
DON'T fiddle on all night - an important part of drift aligning is knowing when to stop. Remember to take some images before the clouds roll in!
- The altitude of the star affects the elevation that we set the axis to, and the hour angle of the star affects the accuracy with which we can do it.
- If we measure the drift using a camera of some sort (giving us higher accuracy in the measurement), and make just one drift measurement at the 0 hours and 6 or 18 hours positions, we can use the spreadsheet to calculate the direction in which the telescope axis is pointing. We can then offset to the desired position, without requiring time-consuming trial and error.
- We may choose to align on the refracted rather than the true pole to counteract the effects of refraction an a star's path across the sky. We can then make use of the King tracking rate, if available.
How does drift alignment compare with other methods of polar alignment in terms of accuracy and length of time required?
If drift alignment is performed in the traditional way, by observing the drift, tweaking the axis adjustment, re-observing the drift and repeating until the drift is sufficiently small, then a good drift alignment probably takes between half an hour and an hour. A calculated drift alignment using the spreadsheet probably needs half an hour to do well.
A good polar scope may well be as accurate as traditional drift alignment, and certainly faster. Other methods, such as building T-point based pointing models, should yield a similar (or better) accuracy, whilst possibly taking longer. Some telescope controllers offer polar alignment assistance as a model-building option. For super-accuracy, the best methods are probably 'photographic' ones based on imaging stars close to the pole.
Nonetheless, we should keep drift alignment in our arsenal of polar alignment tools because it will always work well, and can be done with just an eyepiece; it doesn't need calibrated setting circles, a computerised mount or even a view of the pole.
King's "Forms of Images in Stellar Photography" was published in Annals of the Astronomical Observatory of Harvard College, volume 41, no.6. It is available via the SAO/NASA Astrophysics Data System.
The size of polar misalignment that is acceptable for particular imaging requirements has been covered in the Journal of the British Astronomical Association, volume 99, no. 1 (1989) by R.N.Hook in a paper titled "Polar axis alignment requirements for astronomical photography", also available via the SAO/NASA ADS.
This website: http://leq.one-arcsec.org was a source of inspiration and contains an excellent set of references.
There are many places on the internet that describe how to do drift alignment, but do be careful with the choice of star for the elevation adjustment.