Detection and solution of alignment errors
Proper alignment is an important factor when it comes to experienced performance. If you want to do astrophotography you really need to align your mount very precisely. Before using your mount, make sure that the azimuthal ring is levelled horizontally. And do your polar alignment before you do the alt-az alignment. A camera is very handy to do your alignment. This instruction relies on using a camera. Make sure that the orientation of your camera fits with the directions of your mount. Slewing to the west must result in stars that move precisely horizontally to the east, slewing to the pole must result in stars that apparently move down.
All equatorial telescope mounts need to be polar aligned, this mount isn't an exception. I will save you from going into the details to do a decent polar alignment, you can retrieve good polar alignment with any method that you like. The only way you can't use for polar alignment is usage of a polar finder, simply because there is no physical polar axis were you can attach it to. Not to mention that polar finders are pretty useless anyway if you want to do serious photography with long exposures at a long focal distance.
A mount that isn't properly polar aligned will show objects that apparently move north or south on the sensor. That is important to understand because we need to distinguish this from speed errors that are caused by improper alt-az alignment. Avoid using stars that are below 20 degrees above the horizon because atmospheric refraction will disturb much of your measurements.
And make sure that your camera stays well oriented with the slew directions, especially if you had to make large alterations to reach polar alignment.
It is sufficient to adapt parameters in the software. (There isn't even a physical polar axis that you can tilt.) There is no need to change the physical orientation of the mount. It can be done to create an algorithm to calculate values from a few astrophotos. I described a similar algorithm already for the alt-az alignment.
An extra requirement for alt-az mounts is that they need to be alt-az aligned as well. Regardless of how well you have done the polar alignment of the mount, a mount that isn't properly alt-az aligned will show deviations in speed, so an object will show apparent motion in east or west directions on the sensor. Before you continue with the following instructions, we assume that you have done a proper polar alignment and that the sensor of your camera is oriented in such a way that the X-direction of pixels correspond with the east-west direction.
After proper polar alignment, this mount still has three degrees of freedom when we want to operate it in equatorial mode. They can be seen as a combination of the following phenomena:
- Rotation in hour angle. The telescope can be mounted with skew in the fork in such manner that there is a difference between the hour angle that telescope points to and the hour angle that mount is set to. We may suppose we previously did a good job on polar alignment, so there is no need to search for an error in azimuth.
- Rotation around the fork axis. To consider this error we assume the mount is at an hour angle of 0 hour. The azimuthal ring of the mount isn't levelled properly, there is difference in the height between north and south.
- Rotation around the horseshoe axis. Again, to consider this error we assume the mount is at an hour angle of 0 hour. And again, the azimuthal ring of the mount isn't levelled properly, now there is difference in the height between east and west.
The result of error 1 is a fixed difference in hour angle, so it can be depicted as a phase shifted graph. To correct for this error we only need to add or subtract this difference from the configured hour angle. The rotations of the fork (2) and horseshoe axis (3) both cause amplitude errors. The resulting amplitude that we retrieve is the result of the summation of both errors. We can correct the mount with the value of this summation. That can be done simply by adding or subtracting this value from the latitude setting because that has an identical influence on the speed.
We can correct in a simple way for both errors by adjusting parameters in the software that drives the motors. We need to determine the size of the phase and amplitude errors. Both errors result in differences of tracking velocity, so you can distinguish them easily from residual polar alignment errors or atmospheric refraction. So we need to find a strategy to measure and calculate the errors. It is best to measure the speed difference were we can expect to find the largest difference. First let's see what happens to the graphs when we have these errors separated.
An error example
It may be rather unlikely to have large errors like these, but we need to get an idea of the consequences of a mount that isn't properly alt-az aligned and how to fix this. The left picture shows a phase shifted graph. The largest difference in angular velocity can be found were the graph crosses the value of one, in this case at hour angles of 48°, 132°, 228° and 312°. These optimal angles to do the measurement are dependent of the latitude of course and are best to be calculated. It is clearly visible that there is very little influence of the phase error on the angular speed at hour angles 0°, 90°, 180°, and 270°.
The right picture shows a graph with differences in amplitude. The right graph is in some respect the opposite of the left graph. Clearly you can see that the differences in angular velocity are the largest at hour angles 0°, 90°, 180°, and 270°. There is hardly any influence visible were the graph crosses the line were the angular velocity is 1.0.
Strategies to measure the phase and amplitude error
The mount at this time has no notion of the true hour angle. That is not a real problem, we simply put our confidence in the mount for now and let it rotate to the optimal hour angles that we calculated for our latitude were we want to take our pictures.
We can determine the alignment errors by taking pictures. Based on a known pixel scale we can measure shift in X-direction over time to find velocity errors. When we know the velocity errors on certain points we can determine the type of error and we can calculate how large the error is.
I shall describe two ways to do these measurements. It is possible to have this programmed automatically so that the mount will slew to the angles were we need to measure and have the mount controller send a pulse to the camera to take the pictures on the right angles and times.
The first way is a simple but effective method to measure the phase and amplitude errors by taking three pictures. One picture can be taken at an optimal calculated hour angle to minimize the influence of the amplitude error. We have the mount tracking at sidereal speed while taking pictures of course. The integration time can be short, a second or so, anyhow long enough to do proper measurement of the star and short enough to avoid saturation of that star. We use plate solving to find the true hour angle and compensate the mount configuration for that by adapting an offset value for hour angle. Next we slew the mount to a position a bit before 90° (or 270°) and have it tracking at sidereal speed again. Now we take two pictures. We take the first picture just before an hour angle of 90° (or 270°), when the mount is 7.5 minutes of arc (30 seconds in time) before the 90° position. We just have the mount tracking for 1 minute in time before taking the second picture. Based on the pictures we can verify if our correction of the hour angle that we did earlier was right. And we can search for differences in X-positions of the pixels for the same star. We can neglect deviations in Y because with a well oriented camera, these deviations must be the result of a residual polar error or atmospheric refraction. (Actually you shouldn't see deviations in the Y-direction.) Based on the difference in X-pixel position and the interval time we can calculate the speed difference and we can calculate an equivalent value for latitude that causes such a difference. With that value we can compensate for the amplitude error. Now we have a fully aligned mount.
The second - more complex - way is based on 3 sets of 2 pictures. This way makes it possible to correct for alignment errors without the need of plate solving. We take sets of pictures just as the previous method described for the hour angle of 90°. We simply slew the mount to a position a little before the optimized position and have the mount tracking at sidereal speed. OK, the mount slewed to a position of which it expects to be just before the first optimal hour angle, it isn't aligned yet.) For every set, we take the first picture at 7.5 minutes of arc before an optimal hour angle and the second picture one minute in time later. During this interval time we expect that the mount tracked over 15 minutes of arc but that is not our concern. We keep equal integration times for each set of pictures, the stars only need to be bright enough to have a good measurement of the X and Y value of the pixels. (Don't use saturated stars for your measurement.) We do take sets of pictures like here described for a pair of calculated optimal hour angles to measure the phase error. We also take a set of pictures at 90° (or 270°) to measure the amplitude error. Now we measure the shifts in X direction of the pictures for each set. First we need to compare the two sets that we took at the optimized angles in order to compensate for the influence of the amplitude error. Now we retrieve the phase error. With this result and the differences of the X values of the pixels in the pictures that we too k at 90° we can calculate the amplitude error.
The graph with the magnifying glasses show us an example were we have the combination of both errors that we showed in the previous chapter. The magnifying glass on top (90°) is were we take two pictures to measure for the amplitude error. Both methods need to measure here. The magnifying glasses at the 1.0 velocity line mark the spots were we take pictures at the calculated optimized hour angles to determine the phase error. The first method requires only one picture on one of these spots, the second requires a total of four pictures.
How to solve for the influence of residual errors
The next graph shows you enlarged what is behind the magnifying glass at the left. In detail, this is what happens when we take a set of integrations at that position. We have the mount to calculate the optimized hour angle. In this example for a latitude of 52°, the left optimized hour angle is 48.4°. We have the mount slew to the position that it considers as this calculated hour angle. In our example we have put an error of 2°, so the actual position were it stops is 50.4°. The mount stops just before the calculated position, and it starts tracking at sidereal speed. It takes a first, short exposure picture 30 seconds in time before it reaches the calculated position. Precisely 60 seconds after the start of the first exposure it takes the second exposure. There was a deviation in angular velocity that caused stars in the field of view to be shifted in the X-direction. We can measure this difference, but a small part of this deviation was caused by the amplitude error. With a measurement like this alone we cannot determine which part of the measured value would have been caused by the phase error or amplitude error. But we can measure precisely how much of the velocity error was contributed by the phase error by doing another measurement, symmetrically, at equal distance in hour angle at the right side from the top of the graph.
The next picture shows us a similar situation as above. Now we see the details of the right magnifying glass. Again we see two pictures taken at the calculated optimized position. If the mount only suffered from a phase error, the measurement would result in exactly the same velocity error, somehow proportional to the difference in hour angle that the mount holds for true and the reality of our setup. In our example we see that the phase error combined with the amplitude error leads to a slightly different value as measured at the left side. Now we can do some math and combine these results to get rid of the influence of the amplitude error in order to retrieve a pure value of the phase error.
The last picture shows us what is under the magnifying glass at 90°. This is the spot were we do our measurement to find the amplitude error. Like the previous measurements, there will be some influence of the other error, in this case the phase error. That is not going to cause us trouble, because we determined that value by combination of the two measurements before. And also here, we have the mount taking the pictures for us. Now it takes them just before and just after passing the position that it considers as 90° in hour angle. We count how many pixels in X a star was shifted in the picture and based on the pixel scale we can do our math to find a value that we can use to compensate for the velocity deviation. That is enough. There is no need for us to find out which of the fork axis, the horseshoe axis or the azimuthal ring are a bit off and what they contribute. The value that we measured can simply be used as an offset for latitude, because a difference in latitude causes exactly the same behaviour. Position control of the axes is a matter of synchronisation and proper use of encoders.