Pointing Models

The purpose of the prior chapter is to improve tracking performance during exposure. When the subject of models is introduced we only need to touch upon pointing models, clarify the distinction and move on. Programs such as Maxim’s MaxPoint or the TPoint add-on for TheSkyX reside on a PC and create a model that enables the user to accurately point to objects in the sky and at the same time, report the polar alignment error. The 10Micron mounts have the same ability within their Linux-based controller. These are a step on from the multi-star alignment routines found in many GoTo mount handsets. Here, the model translates a database object position to a set of mount coordinates that account for various mechanical alignment, refractive and time-base errors. In practice, after simply sending the modified slew command to the mount, it then leaves the mount to track on its RA axis and at the sidereal rate.

This is useful but not mandatory for astrophotography: In a permanent setup, model generation is an infrequent event and it is convenient to use a prior model to center objects within several arc seconds, prior to an imaging run. In the case of a portable setup, the time taken generating the model each night is arguably better spent on refining polar alignment, since many acquisition programs now have closed-loop slew features.

These achieve remarkable pointing accuracy in under a minute. In practice, the software slews to the target, exposes, plate-solves, and corrects the mount position to the plate-solved position and repeats the slew command.

If there is backlash in the system another iteration often improves things to a single-digit pixel error. I routinely use this approach in a portable setup using Sequence Generator Pro (which automates the cycle) to point within a few pixels of the target in a single go. If I did use have an extensive pointing model for the pier-mounted scope, I would continue to sync to target at the start of a target sequence or after a meridian flip. This closed-loop system either makes use of the a telescope sync command, or calculates its own offset to issue a slew correction. One thing to take note of; if the mount already has a pointing model, a further sync command on another occasion can cause a hiccup (on account of different atmospheric refraction or a small clock error). To prevent this occurring, some telescope ASCOM drivers have a sync inhibit option, which either blocks the sync command entirely, or instructs the mount to sync outside of the model and not refine it.

Tracking Models

The second type of model is typically deployed within the motor control system of the mount and changes the way the mount moves across the sky. This is a tracking model and is typified by the system in the 10Micron mounts, Paramount’s ProTrack system and the latest AstroPhysics designs. A significant difference between the two model types is a mount using a pointing model tracks with the RA axis only and to eliminate drift during an unguided exposure it is necessary to accurately polar align the mount. A tracking model uses both RA and DEC motors and will effectively eliminate drift and periodic error during tracking. The 10Micron mount employs shaft encoders and a control loop to effectively eradicate all mount mechanical tolerances. The Software Bisque mounts do not have shaft encoders but use a sophisticated model to tackle the residual tracking errors after PEC. In both cases, the model also accounts for mechanical anomalies, such as flexure, in the entire system. It is sometimes claimed that a tracking model can account for less than perfect polar alignment. Even so, it is not good practice to make the tracking model work hard on account of gross polar misalignment in any system, as it places unnecessary demands on the tracking corrections and, even perfect tracking with poor polar alignment, may show up in the form of star rotation in the outer image field over a long exposure.

If the optical configuration is changed, a new model is required to account for the different flexure, collimation and alignment errors. A model is a mathematical equation that relates a set of assumed and actual coordinates for multiple points in the sky and any points in between. The data that is used to calculate the model is generated by slewing the telescope to many different points in the sky and recording the assumed position and the actual position. This is either done by manually aligning a well-defined star on the CCD (using the slew controls) or more conveniently by taking a photograph, plate-solving the center of the image and syncing. To make a robust model, there should be data points on both sides of the meridian and at different altitudes. In this way the model will incorporate the mechanical flexures in the system and misalignments between the motors and the optical axis. To make a successful model it is essential that you address the mechanical, astrometric and the optical properties of the environment with equal care.

Reality Check

Tracking models are very alluring. For some, the ability to make unguided exposures is the holy grail. It is not a panacea for everyone though and it is important to decide whether or not it is entirely necessary or effective in your own particular circumstances. The performance of tracking models is hotly contested and very dependent on your imaging requirements and standards. Please keep in mind that those imaging at short 400-mm focal length have a considerably less demanding requirement than those at 2,000+ mm and that reports of success in less than ideal circumstances may be a consequence of this, imaging at high declination, or a fortunate one-off.

Having direct experience of several dual-axis tracking systems in both permanent and fixed sites and over several years, I have reached a personal conclusion; a mobile setup is not ideal for the purpose of unguided dual-axis tracking. It is not ideal for a number of reasons: In a portable setup it is necessary to generate the model each time you set up. A good model takes time and care to get right and uses up precious imaging time. In comparison, those with a permanent setup can re-use an existing model up until the day they change the imaging hardware. A tracking model relies upon a set of repeatable errors. If not, the tracking is compromised. A tripod support is less rigid than a concrete-mounted pier and made worse if it is on a compliant footing. We have already seen how the smallest movement of a single tripod leg translates into a significant and unexpected drift. In addition, the generally lighter-weight components and exposed nature of a portable rig create their own challenges. I often use the 80–20 engineering paradigm, here with the thought to achieve the 80% with pragmatic polar alignment and a basic tracking model and the remaining 20% with gentle autoguiding to accommodate the residual or variable errors. In theory, this makes a lot of sense, but only if the dual-axis tracking and autoguider systems play together nicely. This is not always the case and something to look out for in your own system. In a permanent setup, the model and hardware can be optimized over time so, even with long focal lengths, unguided operation is an achievable goal. My goal here, however, is to achieve good tracking and if one achieves low RMS errors simply with PEC, accurate polar alignment and autoguiding, happy days. For those who do not have such luck, a mount that features a tracking model may be the answer. To get it to work needs an appreciation of the entire mechanical, software and environment system and how it affects tracking accuracy and repeatability.

fig.1 A screen shot following a TPoint analysis of a pier-mounted Paramount MX and a 10-inch RC telescope. The RMS pointing accuracy is just 4 arc seconds. The list on the right hand side show the various terms and factors that make up the complex modeling equation for pointing (and later on, tracking).

fig.2 Following on from fig.1, the TPoint model in TheSkyX can be used for dual-axis tracking (given there are enough samples). Here, as a precaution, it is autoguiding every 10 seconds using PHD2 and an off-axis guider. The seeing was not great but the RMS tracking error is 0.3 arc seconds, well within the imaging resolution.

Mechanical Model Properties

As we have already realized, a telescope and camera fixed on a mount is a complex mechanical system. There are many physical properties that make it deviate away from perfection. Perfection requires the two motor axes to be perfectly orthogonal, a telescope that is perfectly collimated with the right ascension axis and an optical support system that has no flexibility or play when orientated at different angles. Perfection is an unobtainable goal and the next best thing is to minimize the errors and ensure the imperfections are consistent. In this way they can be measured and corrected for by the model. In practical terms this means that you should collimate the telescope as accurately as possible and ensure that the mount and optical support is rigid. It may be necessary to lock the focus, or in the long term, upgrade the focuser assembly, as these systems often are the largest potential source of flexure.

Another source of mechanical error is the positioning system itself. Gear systems are not perfect and various periodic errors introduce positional errors, possibly up over 30 arc seconds. The better mounts, using periodic error correction, reduce this to 1–4 arc seconds or less. Better still, as in the case of the 10Micron mounts, is to use an optical encoder in a feedback control system and effectively eliminate mount gear variation. Gear variation is otherwise difficult to model as it is a compound of multiple mechanical interfaces and is why tracking models work best with mounts using precision shaft encoders as part of their motor control system. The next best thing is to use a belt-driven precision worm gear. For example, the SuperModel and ProTrack parameters in a TheSky X model correct for numerous mechanical alignment and refraction errors on a Paramount (fig.1, 2).

In the case of a shaft encoder system, the rotational angle of both axis are reported to fractions of an arc second. This information is used to send corrections to the motors to align the system. There are some challenges though: The encoder system has the same resolution as a good autoguider, without the seeing noise, at about 0.1 arc seconds. Unlike an autoguider, the position feedback is almost immediate and without noise. It still has to be integrated into a control loop that moves a motor quickly and without overshoot. The dynamics of the system do not make this a trivial task, especially if the control loop has to additionally cope with external dither and autoguider inputs too. In classical dynamic control theory, designing a responsive yet stable system with nested feedback loops is a challenge.

Astrometric Model Properties

Obviously stars appear to rotate as the Earth spins on its axis and we know we need the precise time to locate them on their endless cycle. In addition to this, the tilt of the Earth’s axis, in astronomical terms, changes quickly (rotation and nutation) and on top of this, stars are not fixed in their relationship to one another as they circle around overhead (proper motion). As such, aligning reality with a catalog’s static star database involves a number of different considerations and adjustments. The most obvious of these is the telescope’s position on the Earth’s surface, in terms of altitude, longitude and latitude as well as the precise time. Less obvious is ensuring that the coordinate system used by the catalog and the telescope mount are aligned. This last point causes some confusion and requires an understanding of Epochs and conver-sions: Catalogs have millions of star references and it is a mammoth task to update them. The result is they are static and standardized to a certain historical point in time (Epoch). In use, a star coordinate can be adjusted by a software calculation to any other point in time. Many applications including the common plate-solving programs use catalogs that define star astrometry in the J2000 Epoch. This defines the position of an object in RA and DEC as it was on midnight of 1st January 2000. Conversely, JNow is the RA and DEC coordinate at the current point in time. The difference can be calculated between the two providing you know your precise position and the precise time of the observation.

Telescope mount systems assume either JNow or J2000 coordinates and conversions happen between software and hardware as required, providing that the different parts of the system are well defined. Maxim DL for instance can work in either J2000 or JNow and in the 10Micron controller, the software works solely in JNow coordinates. If J2000 coordinates are to be used with a mount that assumes JNow, either the driver or the mount software must make a conversion. If not, it will introduce errors into the model, measured today, in the region of a few arc minutes.

Many amateur telescopes use JNow (also called local topocentric coordinates). More sophisticated telescopes use one of the standard reference systems established by professional astronomers of which the most common is the Julian Epoch 2000 (J2000). These instruments require corrections for precession, nutation, aberration, etcetera to adjust the coordinates from the standard system to the pointing direction for the current time and location.

fig.3 The above screen capture is from a piece of donationware by the late Per Frejvall of Stockholm. His utility automates the data acquisition for the 10Micron mount model-building function. Interfacing to PinPoint and Maxim DL, it can acquire 25 sync points in about 10 minutes. With the right attention to detail in the imaging system setup, the resulting model effectively tracks unguided in both axis, without any apparent star elongation.

Time-Bases

Plate solving is the most expedient method to compare the apparent and actual position of sync points with which to calculate a pointing or tracking model. Plate solving is carried out in the PC. Establishing the coordinate pairs using a PC, rather than using the mount’s hand controller to align stars, introduces a further source of error: Consider two identical telescope mounts whose clocks are one hour apart. If each is directed to the same RA and DEC coordinate, they will point to very different parts of the sky. If they are asked to point to the same Alt and Az coordinate, however, they will be aligned. Similarly, any calculations in the PC and the mount itself must assume the same precise time if there are any RA/DEC–Alt/Az coordinate conversions used in the internal calculations. The Earth rotates at 15 arc seconds per second and if the mount thinks it is midnight and the ASCOM driver and PC clock thinks it is 23:59:59, well, you get the picture. There are often several time-bases in play within the computer and the mount hardware. Ideally, they all must be perfectly accurate. That is not easy when one introduces a PC into the mix. They are not designed for this purpose and can drift several seconds over a week. A mount on the other hand, typically has a temperature-compensated crystal at its heart. In reality, there will be an absolute and relative time-base error that, depending on the what is going on at the time, will have a different impact: If both time-bases track perfectly but are 1 second out, the RA pointing accuracy will be out by 15 arc seconds but the tracking and model building process will usually work perfectly, accounting for the offset in the first sync. If there is a difference between the time-bases used for determining the actual and theoretical positions during model building, it translates to an error in the tracking model, especially if the time difference changes during the model building process. If the time-base used for tracking drifts, or is updated during an exposure, the tracking will similarly drift or jump in RA. The latter can happen if the mount (or PC) has been set up to update its time-base from a GPS source.

Ultimately, one needs to decide which device tracks time most accurately (normally the mount) and then set that up directly via a GPS unit or Network Time Server using the network time protocol (NTP) and ensure any other time-bases are synced to this. This sets a baseline for the initial pointing accuracy. During the process of modeling, which may last all night, the time-bases usually free-run or at most, they are synced to the master in-between exposures. This ensures there are no discontinuities in the time-base that can be misinterpreted by the model calculations.

In practice, I set the time-base to the GPS clock or use an NTP utility, such as Dimension 4, at the beginning of the imaging session to improve the initial pointing accuracy. This is not as essential as one might think, since if the time-base is wrong, a single image sync offset will correct any offset. If there are any updates to the tracking time-base (or to the environmental parameters either) make sure these occur in-between exposures to preventing streaking during the exposures. (A simple time update may cause a sudden tracking correction and for that reason some applications low-pass filter environmental and time updates to make them less obvious.)

Plate-solving accuracy is the key to effective automated modeling. It is not perfect, however, and false matches do occur, especially if the pixel scale is incorrect. By way of example PinPoint commonly uses the GSCII catalog (a compilation of catalogs from different Epochs, converted to J2000) and reports the position of the image center in RA/DEC coordinates in J2000. If you are using Maxim DL to acquire images, the pixel scale is reported in the FITS header of the image and is calculated from the Site and Optics settings for focal length and instrument in Maxim’s settings dialog. If you have several telescope and field-flattener combinations, as I do, save specific Maxim DL configurations and load the correct one for the imaging session. After measuring the position at each sample point, it is important to review the individual pointing errors and delete the obvious outliers. The SuperModel option in TheSkyX does this automatically and other applications progressively delete the worst offenders above a threshold error value between the sample and the model.

fig.4 This screen capture of ModelMaker for the 10Micron mount shows the reported sync point errors in the model and their direction. This model was generated in less than ideal conditions or settings and required a re-run after going through the checklist.

Optical Model Properties

The last major consideration is the refraction caused by the atmosphere. There is no image shift at the zenith but progressively more towards the horizon. The ultra-precise polar alignment finders forget to mention that the apparent position of the North Celestial Pole and its actual position are different! Refraction effects can be modeled comparatively easily. Many PC applications (planetariums and applications such as MaxPoint) make allowances for refraction when they issue a slew command. This helps with pointing accuracy but does nothing to improve tracking. When the mount controller calculates the effect of atmospheric refraction on a star’s apparent position, it not only improves pointing accuracy but makes it easier to model the remaining mechanical attributes more accurately and improve tracking accuracy.

To calculate the correct amount of atmospheric bending, we need the altitude of the object (coordinates), the ambient pressure and the ambient temperature, both measured on-site. The formula below, devised from Gar-finkel in the 1960s for calculating the refraction based on the true altitude is accurate to 4 arc seconds (assuming 10°C and 101 kPa):

At 40° altitude, an unguided 5 minute exposure can have up to 3 arc second apparent drift in RA. For other temperatures and pressures (in °C and kPa) it is multiplied by the following factor:

This formula is used in many applications and is in ASCOM’s astrometric function library. It makes a big difference; at 25° it is 130 arc seconds and even at 40°, the shift is 73 arc seconds. Compared to a 5 arc second RMS (root mean square) target error, this is significant as a star changes its altitude during tracking. The refractive index relates to the air density and depth, which in turn is determined by air pressure, temperature and altitude. One could theoretically calculate it all in the model, along with all the mechanical considerations. Mathematically speaking, if the parameters are well understood and the refraction model is robust, it is better to deploy this model up front and load the refraction parameters into the mount before measuring points for a model. Refraction parameters can be entered directly into the mount via the handset or PC software. Refraction is particularly sensitive to temperature changes and the refraction increases as the temperature drops. Some ASCOM drivers and utilities also allow for the automatic updating of these parameters at the start of the run or during tracking. Others use the standardized text format files produced by some electronic weather stations. Paramount mounts have an internal temperature sensor which tracks ambient trends and the latest ASCOM environmental definitions will likely lead to full parameter update from external sensors in due course.

Effective Model Making

As an example, the tracking model for a 10Micron mount can deliver unguided 10-minute exposures, with an aspect ratio of less than 15%, even with modest polar misalignment. It is not a free ride; with all that has been said, generating an accurate tracking model requires attention to detail and optimization.

The following are the basics prerequisites. In no particular order:

• ensure the mechanical system is stable

• careful balance on both axis (reduces flexure)

• tighten the RA and DEC clutches

• set accurate location and altitude in the mount

• set correct epoch ASCOM driver settings

• set and synchronize time between the plate-solving system (PC) and mount with an accurate time source

• set air temp for refraction calculation

• set air pressure for refraction calculation (in the case of the 10Micron mount you should either enter the air pressure at sea level and have enable auto altitude in the mount or supply the surface air pressure and disable the auto altitude function)

• enable refraction

• set correct focal length in Maxim DL in Settings, Site and Optics for plate solving

• check camera sensor pixel pitch is set (modern astro CCDs communicate this automatically)

fig.5 This screen capture from TheSkyX is the equivalent to the one in fig.3. Sample placement can be automated and constrained by horizon, altitude and either evenly-spaced or randomized. The sample order minimizes the number of meridian flips and all samples are considered in the determination of the polar alignment and model parameters.

In the case of the 10Micron modeling software, you should additionally check the following:

• mount is set to treat ‘syncs’ to refine model

• set the PC clock with a NTP server or GPS

• one-time sync the mount clock to the PC clock (or use the Clock Sync utility)

• set exposure, binning and mount settle times for reliable plate-solving

• remember to clear the existing model in the mount before beginning a new model, or after polar adjustment

• enable dual tracking in the mount

If you are solely using the handset for model making, the Epoch parameters take care of themselves and it is only necessary to make sure that time, location (say with an external GPS) and environment parameters are set in the mount. You also need to check that the sync setting is set to “sync refines”, rather than treat the sync as an offset, so that all your patient work adds further data points. (Before Per passed away, he added a facility for continual environmental updates to the 10Micron model.)

fig.6 This screen capture shows the tracking graph from Maxim DL (with the guider output disabled) for the imperfect model in fig.4. In context, the vertical axis is just ± 2 arc seconds. A very slight drift in RA can be seen over the full exposure. The actual exposures were perfectly usable, however, as seeing conditions were the limiting factor.

Running and Checking the Model

The first part of a 10Micron model is usually a three-point synchronization to determine the assumed celestial pole position. This can be done manually with an application like Maxim DL or a model making utility. The accuracy of these are fundamental to the final model accuracy and require special attention. These three points should be about 120 degrees apart in azimuth, about 40 degrees in altitude and avoid points on the meridian. It is good practice to use a longer exposure, say 15 seconds, no binning and a generous settling time to ensure an accurate plate-solve. It is important to check the errors for these first three points using the virtual handset or model making utility and redo them until they have single digit arc second errors. This may involve going through the checklist again and updating refraction settings.

The initial physical alignment of the mount requires a position that ideally is within the field of view of the camera for the plate-solve to work. Use your favorite method to polar align. One alternative is to slew to and center a named star, using the Alt and Az knobs, before starting the three-point calibration. In the case of the 10Micron mount, the polar alignment error report includes turn instructions for the Altitude and Azimuth adjustment knobs. If you follow the recommendations, turn the knobs by the required amount and re-run Model Maker (remembering to clear the model first) and you should be within 5 arc minutes. This may not be sufficiently accurate for unguided operation and may require further calibration. In a permanent setup, there is no reason why one should not aim for sub arc minute polar alignment error.

After the initial calibration any subsequent syncs are treated as “refine points”. As more refine points are collected in the model, it becomes more sophisticated with potentially more equation terms in the model to map the theoretical and actual coordinates. Theoretically the error between reality and the model should decrease with more points but that is not always the case; it all depends upon the data. The mount alignment report displays how many terms it is using in its modeling equation and the error between each measurement point and the model. More points or terms do not always equate to better tracking accuracy. Errors do happen and it is better to have fewer, good points than many points with large errors. The better systems report the error of each sync point and have the ability to delete the worst point and resubmit the remainder of the data points to generate a new model. This is usually done to achieve a target RMS error. A rule of thumb is that no point should be over 10 arc seconds and an RMS error of 5 arc seconds or lower is a good result. A combination of a low RMS and a low term count implies a simpler and “smoother” model, with fewer anomalies.

The Software Bisque mounts are supplied with TheSkyX and their modeling software TPoint®. The intelligence for the mount and the modeling software reside on the same PC, bypassing time conflicts. The TPoint add-on is highly integrated with TheSkyX, making use of its planetarium, plate-solving and camera functions. Although these mounts are built to exacting standards the current Paramount models do not employ shaft encoders and so their residual tracking errors (after PEC) are higher than the 10Micron mounts. TPoint is very easy to use, however, and automates the image acquisition, plate solving, filtering and modeling terms (fig.1, 5). In practice my Paramount MX closes-in on the 10Micron accuracy with approximately double the number of sync points. The Model Maker software tops out at 100 points, whereas TPoint can exceed 500. TPoint requires about 20 sync points to nail Polar alignment error and 50 suffices for very accurate pointing. Each point has equal weighting in the model calculation and hence there is no need to make special efforts to define polar alignment error at the beginning of the process. The TPoint software also realizes that it obtains better results with fewer meridian flips during calibration. It can create random or regularly spaced sample points over the imaging horizon and organizes them into an optimum sequence. Although the tracking model accounts for atmospheric refraction, flexure and a multitude of mechanical errors, a TPoint requires more points for accurate unguided tracking. Moving along, and ideally after acquiring a few hundred data points, their Super Model feature automatically deletes outlier points and adjusts and adds model terms to reduce the RMS pointing error further and to generate a tracking model (ProTrackTM) for unguided exposures. The Paramount has negligible backlash and stiction and guides well via ST4 or pulse guiding interfaces. I typically run unguided with focal lengths less than 500 mm and use gentle autoguiding with the longer refractors and the 250 mm f/8 RCT.

One can check the tracking performance of any mount by calibrating and running your autoguider application, disabling the guider outputs and viewing its tracking graph. To distinguish tracking errors from seeing, use an exposure time of 5 seconds or more. Tracking performance is still affected by the mount’s mechanical properties. Non-linear effects, such as backlash and stiction have to be addressed by the motor control feedback system. This is most likely to have an effect on the accuracy of the DEC axis tracking.

fig.7 This is a 10-minute unguided exposure with a 924-mm focal length refractor fitted to a Paramount MX, using TheSkyX TPoint modelling and ProTrack. The eccentricity of 10% is very difficult to detect and indistinguishable from the guided exposure.

Summary

Tracking models (and encoders) appear to be a simple panacea for those who dislike autoguiding. They are not, since the system requires considerable diligence for it to be effective. In particular, a portable setup still requires optimized balance, alignment, live environment parameters, system rigidity and imaging time to generate an effective tracking model. Ground stability is not guaranteed and wet and dry seasons may require a new model. To my mind, a quality mount (polar aligned to 2 arc minutes or better) that is responsive to guider commands and whose motor axis exhibit little backlash is consistently easier to set up and a more effective use of a clear night. In a permanent setup, it is worth spending an entire night or more refining a multi-point model to improve tracking performance. Although some may shudder at the heresy, with a reasonable tracking model autoguiding is easier to implement, using a long exposure and a low aggression setting, to correct tracking drift and the repercussions of atmospheric refraction changes.