The following is a long note on guiding and de-rotating. Also discussed is the use of two or more tubes for guiding and imaging on the same mount including considerations of wedge mounting, de-rotating and piggy back camera mounting. The several possible cases are discussed in detail starting with simple cases and going to the more complex. These thoughts are based on experience with my two LX200 telescopes over the past three years but apply to any telescope.
Several ideas for guiding are presented for cases where a camera (imager) is used piggyback and the main tube is used for guiding, the main tube is used for imaging and a separate tube (telescope) fastened to it is used for guiding or some similar combination. This does not preclude using several imagers on several tubes with a guider on another tube all on the same mount and possibly all in use simultaneously. The advantage of using a separate guider is apparent to anyone who has ever used an off axis guider. The separate guider tube can be of fast focal ratio so as to get a bright guide image, its axis can be adjustable and so be moved a bit from that of the imaging tube so as to center the guide star on the guider chip and an independent, stand alone guider chip can be used in its independent operational mode. It is very convenient to have guider and imagers independent. Of course the ST 7/8 offer additional options. This arrangement is not discussed here.
To start, I assume accurate POLAR alignment of the RA axis. I assume that the guider, whatever type is used, finds a star and is working correctly. These assumptions insure that the guide tube is pointed at a star and locked onto it. This is the most straight forward setup and is relatively easy to establish when a reasonably fast guide tube is used. I use a C5 on both my 12" and 10" LX200s. If polar alignment is not excellent, some rotation of the field will take place. (The case of Alt/Azm mounting is considered as well later.) I also assume for this discussion that the atmosphere causes no distortion of the celestial sphere. This assumption is not quite correct, so this issue will be discussed later.
For the ideal case, the declination is fixed and the telescope needs only to track the RA perfectly. There will be no de-rotation required and a de-rotator is not needed. If the telescope moves precisely in RA no guider correction is required either. This situation could happen accidentally but is not typical by any means. Now assume that the telescope does not move precisely in RA. Typically worm drive defects are enough to cause significant RA rate deviations. The LX200 has a fine system for correcting periodic worm drive defects which can reduce them by 10 or 15 to one. For example 50 arc seconds worm induced wobble and be reduced to 3 or so through a training program. Even with a well trained drive a guider will be required to reduce RA wobble to 1 arc second for high precision imaging. This has been done manually in the past and now, with a guider, can be done automatically.
If there is no distortion of the celestial sphere by the atmosphere, then the guider tube can be pointed at a declination or RA other than that of the imaging tube and guiding will still be perfect. This is true since the undistorted celestial sphere in its entirety moves with the same angular rate in RA everywhere. Even with atmospheric distortion, a guider pointed within a few degrees of the imaging tube will provide excellent guiding. Also note that this assumes that the guide tube and the imaging tube are rigidly held with respect to each other. Considerable care must be exercised to insure mechanical rigidity of the tubes.
A counter example can be imagined by suggesting that this scheme will not work with the guide tube pointed at the pole star. And sure enough it will not. So what is wrong? It is this. The effective length of the guider telescope is longer when it is pointed at a guide star that is moving the greatest linear amount for a given angular motion of the celestial sphere. This linear amount is proportional to the sine of the angle from the pole to the declination being guided upon. When pointing the guide telescope at the pole star, the motion of the star is nearly zero and the guider fails. To get the best accuracy with the above conditions, the guider should be pointed to a declination 90 degrees from the pole. Then, the guider will be locked to the celestial sphere where the linear motion of the guide star is great and the guiding will be accurate and all will be well.
Even if the polar alignment is perfect and the guider is perfect there is a complication. That is the atmosphere. Except at the zenith, the atmosphere distorts the position of the stars in the celestial sphere as it is seen from the telescope position. Thus as a star moves from near the horizon to a position higher in the sky and again toward the horizon, its motion is not perfectly regular in angular rate of RA. nor does it maintain exactly the same declination. The amount of this deviation while quite small requires keeping the guide star near the star field being imaged. The LX200 has a first order correction for this effect in its computer. This is why the correct latitude and longitude must be entered into the computer to insure refraction correction that relates exactly to the local horizon and thus insure pointing accuracy. The two tubes do not have to be exactly axial aligned. Usually, imaging is done away from the horizon because of light pollution so the atmospheric distortion is usually not a primary consideration. In summary, polar mounting reduced or eliminates principle tracking defects and is thus very attractive for imaging. Several tubes can be pointed from the same platform and all will function well with a single guider.
A separate but important consideration is the focal length (more importantly, the effective focal length) of the guider tube compared to that of the imaging tube. The guider nominally cannot point more accurately than the angle subtended by one pixel within the guider image. Sophisticated software might improve this resolution in some cases. The focal length of the guider tube should be similar to that of the imaging tube. It is usually recommended that the guider optic be at least 1/2 the focal length of the imager. For example, I use a C5 (fl = 1300) for my 10 inch f6.3 (fl = 1200) and also on my 12 inch f10 (fl = 3000). I feel this is an adequately long guider focal length. Any shorter focal length imaging lens such as might be used on a typical piggy back camera would be guided easily.
Now, what about the Alt/Azm mounting for which celestial field rotation is a problem when imaging. Normally a de-rotator is necessary. The rate of de-rotation is a complex function of the altitude and azimuth to which the telescope is pointed. A table of de-rotation rates is attached at the end of this note. The table is normalized and so the numbers must be multiplied by the local rotation rate which is 15.2 times the cosine of the latitude. Values above 80 degrees are not given since rotation becomes very large near the zenith. The formula for the rate of rotation of the star field is rate = (const.) X cos(latitude) X cos(azimuth) / sin(zenith distance) The rotation goes to zero at azimuth 90 and 270 degrees (due East and due West) and on a line connecting these points. Even on this line the rotation becomes singular (infinite) at the zenith. At the pole the rate is 360 (approx.) degrees per 24 hours. At other points the constant must be calculated for the observers location.
The rate of rotation is seriously large, so that only very short exposures are possible. Depending on your desire for perfection, exposures of only a few seconds up to a minute are possible without serious rotation errors. If a de-rotator is working correctly is will take care of all the calculations and turn the imaging surface at the correct rate as long as it knows where the telescope is pointing. I am told that the de-rotator on the 16" LX200 does this operation very well. ( I cannot give personal confirmation since I have not use one.) The telescope must be leveled and the correct location entered into the computer so that the telescope knows exactly where the pole star is. This is because the de-rotation is calculated on the basis of the known declination and azimuth.. This implies that the Alt/Azm setup must have the same precision as that required for polar alignment.
There are severe limits to the use of a de-rotator. The guide star must be in the field of the de-rotator and be de-rotated with the field being imaged. An off axis guider will work. But off axis guiding is already difficult and with the guider rotating as well may require a triple jointed neck. The ST 7/8 cameras solve this problem by mounting the guider chip next to the imaging chip Thus it rotates with the field and will correctly guide when using a de-rotator. This does not solve the problem of field rotation in any other tubes or piggy back imagers that may be on the same platform.
If a series of short exposures is satisfactory, a single chip could be used as a part time guider. Appropriate software would be required and short exposures would be necessarily acceptable. But single long exposures have better signal to noise ratios. So using a chip to share functions as a guider and imager does not seem like a good idea. The shift and accumulate method used by SBIG is a good, if partial solution to this problem. There are also software image processing techniques that might be applied to combine a series of rotated images. If the guider points the telescope to the correct RA. and declination and tracks a guide star accurately and the telescope computer knows the value of the coordinates, it can calculate the correct rate of de-rotation. The guider tube and the imaging tube need only be aligned to the accuracy and with the rigidity required for the polar guiding case. But conversely, they need to be set at least with this accuracy and not less. Considerations for atmospheric distortions and cetera are the same for either setup.
Neither Polar nor Alt/Azm setups are much simpler, each than the other, to achieve the same accuracy. In one case you need the wedge and in the other the de-rotator. One case is a well known solution the other, currently, somewhat unknown. It is my opinion that for a permanently mounted telescope the polar mount is certainly the simplest and the least problematic. That is because it is a simple, well understood and versatile solution to imaging guiding. For a moveable telescope that has to be reset for each imaging session I still think the polar wedge solution is the best. Alignment, leveling and the like are not that much more demanding for polar as for Alt/Azm setup I have chosen not to use a de-rotator and have opted for a wedge for both my permanent and trailer mounted telescopes. Thus the following opinions my be somewhat biased.
The de-rotation is limited to the main tube for which the de-rotator is designed. The piggyback image is not de-rotated. Use of a separate guider telescope is not possible. De-rotation does not work well near the azimuth because the rate of rotation becomes very large. The clearest part of the sky, at least 30 degrees wide, is lost to imaging because of mechanical problems of the de-rotator clearing the fork. Of course the pole region is lost to some polar mounted telescopes for the same reason. But the polar region is not of as much interest and it is at the zenith that the sky is the clearest and imaging gets the best results. The de-rotator is just another complex mechanical mechanism with bearings, motor, computer software and the like which can fail to work precisely.
My conclusion is that a de-rotator should be the system of last resort for imaging. Since this was written, a de-rotator for the LX200 telescopes has become available. I have chosen to pass on it for some of the above reasons and others mentioned below. There is considerable added extension to the rear of the telescope (about 75 mm). The attachment uses the standard Schmidt thread which is of small diameter (internal tube opening of about 34 mm) which vignettes 35 mm format. It would be fine for CCD chip sizes. For imaging that requires guiding, it must be done with an off axis guider that rotates with the imager or with the ST 7/8 two chip imager. In the case of the LX200 series, the PEC does not work in Alt /Asm mode. The angle of exclusion at the zenith is stated to be as large as 40 degrees from the zenith limited by the physical arrangement of the tube and or fork. .
Those interested in the field rotation problem when using a telescope in Alt/Azm mounting might find the following calculations interesting. I have calculated the rotation rates using the formulas given in Meeus. The reference is Meeus, "Astronomical Algorithms", Chapter 13. Using the concept of the Parallactic Angle this reference explains rotation. The discussion is quite brief and thus not a clear as it might be.
A convenient formula which is very easy to tabulate is:
Angular rate of rotation = (a constant) X cos (azimuth angle) / cos (altitude angle)
The constant is the angular rate of rotation of the earth times the cosine (local latitude). For my location, 43 degrees, the constant is about 11.1 degrees per hour. One must be careful with this equation since there are several singularities. i.e. points where the cosine goes to zero. The singularity causes a line of zero rotation going from 90 to 270 degrees azimuth, which is due East and West. This line intersects the zenith but at the same time the values of rotation at the zenith are infinite since the cosine at zenith is zero as well. The tables, calculated with MATLAB, are attached at the end of this note. The table is for altitudes up to 80 degrees. Above 80 degrees, the rotation gets very large. Conclusions from the tables follow.
The rotation rate is smallest pointing East or West and largest pointing North or South for a given altitude. When pointing at an altitude of 60 degrees, the rate of rotation can get to be 2 times normal (the constant in the equation above). At 80 degrees, the rate can go to 6 times normal drift rate. That is why, I believe, de-rotators are not generally used at pointing angles closer to the zenith than about 20 degrees.
The following table is for rotation rates below 80 degrees altitude. The values are for 43 degree latitude. Note that the values get very large at 80 degrees and above that angle they get larger still.
Below is a graphical representation of the data given above.
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