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.
How does refraction affect the apparent postion and motion of a star?
Here is a slightly modified version of a diagram from the previous page, showing the celestial equator at a mid-latitude, looking due west in the northern hemisphere, or due east in the southern hemisphere:
The celestial equator is shown by the black line. The blue dots represent a star on the celestial equator at two different times. Refraction raises the star to some new position, shown by the red dots. When the star is lower, it is raised by a greater amount. The increase in apparent altitude also affects the apparent declination, shown by the red arrows. But that's not all - the apparent hour angle of the star is affected too. The orange and green lines show the shift in apparent hour angle at the two different locations. Most importantly, notice how the shift at lower altitude "d2" is greater than the shift at higher altitude "d1". Consequently, the distance between the two red dots must be shorter than the distance between the two blue dots, so the star must appear to move a shorter distance in a given time than it actually does. Therefore it appears to move more slowly than the sidereal rate.
Note that this is not universally true: refraction raises and flattens the apparent path of stars below the pole and this contributes to an increase in their apparent angular speed round the pole. There is a complex interplay between this effect and the one illustrated above, but for some range of hour angle beneath the pole, refracted stars appear to move faster than the sidereal rate. But circumpolar objects are better viewed above the pole, at which point their apparent rate of motion is sidereal or slower. So generally we will encounter cases where stars appear to move at a reduced rate.
Historical study of stellar refraction: E.S.King and stellar photography
Edward Skinner King was interested in the problem of stellar refraction when he was employed as an Observer (later to become Professor) at Harvard Observatory at the end of the nineteenth century. 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.
The variation in tracking rate caused by refraction was also 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 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. For alignment on the true pole, the upward drift in declination is seen to be largest near the horizon (at the 6 hour/18 hour points) and decreases towards the meridian (the 0 hour/24 hour point), as expected:
The upper graph shows the drift in apparent 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 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 when adjusted to the refracted pole. The lower graph shows the rate of drift. The rate of drift is negative as the star climbs away from the horizon and decreasing refraction lowers it towards its true position; the opposite effect applies as the star approaches the western horizon.
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.
Even though the apparent hour angle of the star is ahead of its true hour angle in the eastern half of the sky, the drift rate is always negative, i.e. the refracted star appears to move more slowly than the sidereal rate. This is because it rises early as a result of refraction, but still crosses the meridian at the same time as the true star, since refraction is perpendicular to the star's path at that point and cannot affect its hour angle. Consequently, once it has risen, it must appear to travel more slowly across the sky than the true star.
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).
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 corrections to the drive rate, similar to the curves in the lower of the two graphs above. This is exactly what some telescope control units offer: an "adaptive King rate". King gives two expressions for the tracking rate, one assuming that the mount is aligned on the true pole and one for alignment on the refracted pole. It is advisable to check which formula is used by a given telescope controller and then adjust the mount accordingly.
King also showed that the variation from sidereal rate for a star on the celestial equator was independent of latitude. The variation is at a minimum on the meridian, being 1 second per hour slow on the sidereal rate. At 45 degrees latitude, the actual rate is within 10% of this value for an equatorial star for slightly longer than an hour each side of the meridian. Telescope controllers that offer a fixed "King rate" are likely to be set to this value. This rate also applies at the zenith for mounts that are adjusted to the true pole rather than the refracted pole.
Is the refracted pole really better?
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." Of course he was aware that a telescope close to the earth's equator is best adjusted to the true pole, stating that "...for an equatorial station, no shifting of the axis is advisable, as it would introduce an error in declination where none exists." 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 earth's equator, there comes a point where aligning on the refracted pole is generally not better. At low latitudes, stars rise roughly vertically so refraction does not affect their apparent declination, but it does have an increasingly large effect on the apparent position of the pole. Offsetting the mount's polar axis by a large amount will worsen declination drift in this case. This is indicated by the beige squares. But we saw on the previous page that drift aligning with a star at a higher altitude than the pole (as would be the case near the equator) tends to set the telescope mount's axis below the refracted pole, towards the true pole, which is what we want; so traditional drift aligning serves us well here.
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 very low latitudes. In the mid to low latitudes we may wish to align on one or the other depending on our application.
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).
DO make sure that the optical train is not flexing. Mirror flop, bendy focusers 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 use the polar drift method at least for the elevation adjustment if you have a view of the pole. This is quicker and more accurate than the standard declination drift method above the horizon.
DO use a star in the eastern half of the sky for the elevation adjustment if you can, when using the standard method. The effects of refraction will decrease as it rises higher in the sky. Avoid anything much below about 30 degrees altitude where the effect of refraction becomes more severe.
DO use a star in the same region of sky each time for the elevation adjustment when using the standard method, if you are iterating between azimuth and elevation adjustments. Otherwise you will find a different elevation adjustment each time you do it.
DO approach the pole from below for the elevation adjustment - you are raising the polar axis against gravity which may give fewer problems with backlash.
DON'T expect perfection - there will always be drift in hour angle and declination as a consequence of atmospheric refraction - and we need the atmosphere 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!
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 might take an hour or so. A calculated drift alignment using the spreadsheet needs at least half an hour to do well.
A good polar scope may be good enough in many cases, and certainly faster than drift alignment. Other methods, such as building T-point based pointing models, should yield a similar (or better) accuracy than drift alignment, whilst possibly taking longer. Some telescope controllers offer polar alignment assistance as a model-building option. There are also a number of software aids available that make good polar alignment very straightforward.
Nonetheless, we should keep the standard drift alignment method 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 "A Manual of Celestial Photography" can be found in the HathiTrust digital library.
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.
I am grateful to Nathan Towne for a detailed discussion of polar drift which prompted me to add the web page describing it. You can find his analysis from an alternative viewpoint here.