A while ago in this thread (https://www.cloudyni...in-mirror-slop/) I posted some charts showing the thermal sensitivity of some telescopes along with the equation for computing the thermal sensitivity of a Cassegrain telescope with zero expansion glass and an aluminum structure. I’ll repost them below so that you can review them here. The expression for a Cassegrain was derived using first order optics and based on an educated guess, I stated that the result should be reasonably close to what you might expect from a C14 SCT…and the Edge systems as well.
In that discussion, a question was raised about how much of a difference having non-zero expansion mirrors might actually make and while I hadn’t actually done the calculation, I promised to try to answer that question. It was certainly a valid question that was worthy of a bit more investigation. Unfortunately, it took me longer to get around to it than I expected, but I finally got it done. So, the purpose of this post is to share the results.
One of the things that I did with this computation was to use the first order parameters for the C14 Edge telescope. The first order properties can be derived with good accuracy from published data as well as from direct optical measurements that I’ve made on my own of C14 Edge components (to tighten up the accuracy a bit.) However, because of my work with Celestron, I also happen to have access to proprietary optical design information for the C14 Edge system, which means that I cannot share any of the optical parameters that went into these results. The reason that I took this approach was to ultimately be able to make a direct comparison of the simple models to the C14 Edge as a real product. The good news is that I can describe what I did and how I checked it to reduce (or eliminate) chances of error in the results, which are presented in the table below. (I've had to put the chart in the next post because of the attachment size limits.)
Thermal Sensitivity of a Cassegrain Telescope
The first row of numbers shows the amount of image shift per degree C that you can expect with a Cassegrain telescope using the same 1st order properties as the C14 Edge. This model assumes that the support structure is aluminum and that the mirrors have zero CTE (such as ULE substrate material.) The DOF is the depth of focus in microns (also called the critical focus zone.) The DOF is computed using the Airy criteria, which is given by 2.5 (F/#)^2 at a wavelength of 512 nm. This criterion is a bit less restrictive than the Rayleigh limit, which restricts the P-V wavefront error to be less than ¼ wave over the pupil to maintain a diffraction-limited image. In the chart, the next column labeled C/DOF (C ,) is the temperature change (in C) required to shift the focus by the full DOF. The +/- Tol ( C) is the temperature sensitivity of the system, which is the temperature tolerance required to hold focus within the DOF. This is simply half the value of the previous column. The larger this number, the less sensitive the system is to thermal variations. These results are all shown to two decimal places (which is unnecessarily precise) simply to better compare results.
The second row serves as a cross check. I entered the optical prescription for the Cassegrain system into OSLO (edu) and performed a simple ray trace to confirm the numbers given by the first-order model. You might notice that there is a small difference between the ray trace and the model, which is due to the fact that very small variations in the prescription numbers (like in the 6th place—and beyond) will cause small rounding errors in the result. Ray tracing relies on very small angles and double precision numbers to achieve accuracy so these small variations are hard to avoid without using double precision numbers everywhere—and I didn’t do that. Since this is a straight Cassegrain system, I used paraxial focus to look at the focus shift, which should agree with the model and as you can see, the agreement is nearly perfect.
The third column shows the results for an all-aluminum telescope and it serves as another sanity check. In the previous discussion, it was pointed out that an all-aluminum telescope should be perfectly athermal and that’s correct. I ray-traced this model with the correct CTE numbers to demonstrate that I was able to get the correct result of zero (to within a small rounding error.)
The fourth column gets more interesting because it shows the same Cassegrain telescope with borosilicate mirrors—the same material used in most commercial SCTs. As you can see, using non-zero expansion glass in the telescope does indeed help it to be a little bit less sensitive; but, not by enough to make a meaningful difference in practical terms. Still, increasing the CTE in the mirrors helps a little bit with thermal sensitivity. (NOTE: There are other, more important, reasons to make mirrors out of very low expansion glass so don’t take this as a reason to ignore the benefits of using mirrors with fused silica or ULE substrates!)
Thermal Sensitivity of the C14 Edge Telescope
The data in the last column shows the sensitivity of the C14 Edge telescope and it was the most challenging to compute. OSLO (edu) does not have the capability of introducing thermal variations into a design. So I had to carefully cut and paste full precision numbers between the design program and Excel where I could accurately compute both the change in dimensions using the CTE for all of the aluminum parts and glass types, as well as the change in index due to the dn/dT characteristics of the glass elements. Then I had to carefully cut and paste the results back into the OSLO to do the ray trace. There’s plenty of opportunity to bungle the numbers in this process, so I did the calculation twice, independently of each other, to insure that I got the same results.
In this case, using paraxial focus makes no sense, so I used the monochromatic minimum RMS spot size to determine the best focus position at 587.56 nm. The reason that I couldn’t use the polychromatic solution is that OSLO (edu) won’t easily handle the change in dispersion with temperature for the refractive components. This restriction will not affect the result in any meaningful way.
As you can see the temperature sensitivity of the C14 Edge is very close to the sensitivity of the equivalent straight Cassegrain with borosilicate mirrors (within about 1%.) This result shows that holding the temperature to within about +/- 0.5 degrees will insure that any Cassegrain or SCT with the same first order properties will remain in focus as estimated by the original model.
These numbers show how much the focus will change relative to the sensor over a temperature change of one degree C. They also show the temperature sensitivity of the telescope, which is a measure of how much the temperature must change before the telescope goes out of focus under ideal conditions. Remember that this computation relates to the difference between two identical telescopes at different temperatures while both are in thermal equilibrium. That means that when the ambient temperature changes, it may take a while before all of the components are again at equilibrium. In general, the external aluminum tube and mirror cells will change temperature more rapidly than the internal mirrors simply because the external parts are exposed directly to the outside sky and ambient air. That means that the focus shift may not be a simple linear model during rapid temperature shifts. It is certainly possible to model focus shift characteristics due to dynamic thermal changes, but that’s beyond the scope of this study.
Of course, the amount of focus drift will be altered if part of the telescope structure is electrically heated for any reason. If you heat the tube, that will alter the temperature change of the structure; however, it’s important to recognize that nothing should be heated whenever the temperature-dew point spread is larger than about 5-6 degrees C. Heat generates thermals and it is better to turn off all heat when there is no danger of dew. Remember that these results show the effect of temperature change on the telescope without any heaters running.
Atmospheric seeing conditions will also affect the sensitivity of a telescope to changes in temperature. This is because poor seeing effectively increases the blur diameter of the integrated star image on the sensor. When that happens, the size of a star image is driven more by changes in seeing than by changes in focus so it takes more of a focus shift to be noticeable. The last three rows of the chart show the effect of seeing on the C14. When the seeing blur diameter is on the order of an arc-second, the C14 will have a temperature sensitivity tolerance of +/- 0.62C; however, that tolerance rapidly grows to +/- 2.5C when the seeing deteriorates to a 2 arc-second blur size. Many folks report that they see very little focus change over the course of an imaging session with their SCT and this is most likely the reason. Combine mediocre seeing with a stable environment and even a C14 system will appear to produce well focused images without a lot of refocusing. Smaller SCTs will be even less sensitive under similar conditions. Of course this is true for most telescopes under similar conditions.
Understand that the sensitivity of most computer controlled automated focusing systems will also vary with the quality of the seeing conditions. This is true for V-curve focusing as well for astigmatic focusing methods. Since the data used for focusing is typically taken with relatively short exposures, most of these methods can achieve and hold focus to minimize star size well within the limits of the long-term local seeing conditions.
The important take away (and the point that I was trying to make previously) is that SCTs are quite sensitive to thermal variation relative to other telescope types—as shown in the chart. When the seeing is good, a C14 is very sensitive to temperature variations—and it is particularly noticeable through large changes in the ambient temperature during an evening. I used to record the temperature every night when I operated my telescope locally and it was not uncommon to see change rates of nearly 4C/hr as shown in the attached chart. Under these conditions, I learned that my C14s had to be refocused at least every 10-15 minutes to achieve good results, which roughly matches the predicted thermal sensitivity of the system. Since my typical exposure time is 20 minutes, I use a focusing system that holds focus in real-time while the shutter is open. Remember that smaller telescopes will be less sensitive than the C14 that was used as an example here. As we've seen, the results for the simple model are reasonably close to the real telescope so the model can be used to estimate the sensitivity of smaller scopes with reasonable accuracy.
There is no doubt that some users may be able to run for hours with very little focus change; but, that’s only going to occur under relatively stable thermal conditions and/or under relatively poor seeing conditions.
Edited by jhayes_tucson, 10 May 2018 - 11:55 AM.