By Dr. Frank Melsheimer, President, DFM Engineering, Inc.
Introduction
Telescopes have used a variety of hardware to provide pointing position information. Some of the old refractors
have beautifully engraved and filled setting circles and verniers allowing a position readout finer than
1 arc minute. Later telescopes have used syncro transmitters and receivers to increase the resolution and
readability of the telescope position. All of these early systems have a fatal flaw however. The pointing
accuracy of even a perfect telescope will always be limited by the earth's atmospheric refraction which
amounts to about 1/2 degree at the horizon.
A modern telescope position readout system consists of electronic encoders read by a digital computer.
Typically, incremental optical encoders are used with a resolution of 1 arc second. The incremental position
is electronically counted within the computer and various corrections are made. Sometimes the encoders
are gear driven and sometimes they are directly mounted onaxis. The onaxis encoders need to have very
high resolution, but don't see the gear errors and backlash of the encoder drive gearing. The gear driven
encoder can have considerably less resolution and cost less than the onaxis encoder.
Many corrections need to be applied to the telescope's raw axis position data. Primarily, the corrections
are of two typescoordinate information corrections (or coordinate transforms), and telescope corrections.
These corrections are mathematically calculation intensive and involve many transcendental floating point
operations. A task that is performed by a modern personal computer very well and very quickly. The software
program that performs these corrections is often referred to as the
"pointing model". It is possible to model the pointing with a geometrical model, numerical model, or a
combination of the two techniques.
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Coordinate Corrections
The positions of celestial objects are cataloged in "mean"
coordinates for a specific date or "Epoch". However the telescope works in today's epoch coordinates. This
means that the "mean"
coordinates must be corrected or transformed from the given epoch to today's epoch. This involves calculating
the effects of precession, mutation, and aberration. The effect of the earth's atmospheric refraction is
usually lumped into the coordinate correction category.
It is very convenient to be able to input coordinate data to the Telescope Control System (TCS) in any
epoch, and it is likewise convenient to have the TCS display in any selected epoch. This means that the
pointing model needs to be able to work the corrections forward and backward.
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Telescope Corrections
Even a "perfect" telescope will need error correction to achieve satisfactory pointing. This is because
the telescope should be polar aligned on the "refracted" (by the earth's atmospheric refraction) celestial
pole and not on the true celestial pole to minimize the field rotation introduced by the atmospheric refraction.
Also, it is not practical to "perfectly"
polar align the telescope in azimuth and in altitude.
The typical telescope pointing errors consist of the following items:
1. Azimuth and altitude misalignments to the celestial pole.
2. Mechanical and optical nonperpendicularities.
3. Encoder scale factors, offsets, and eccentricities.
4. Flexures of the telescope mechanical structure.
5. Flexures of the optical mounts.
The azimuth and altitude alignment accuracy will be set by the mechanisms provided to perform the adjustments
and the time available to make these adjustments. The telescope should be aligned about 1 arc minute above
the true celestial pole to be aligned on the refracted celestial pole. The alignment mechanisms need to
be of sufficient quality to allow adjustments smaller than about 10 arc seconds for a professional telescope.
The mechanical nonperpendicularity between the declination axis and the polar axis is set by the machining
and assembly tolerances of the various parts. Sometimes the flexure of the structure is the controlling
factor. With some telescopes it is possible to measure the mechanical nonperpendicularity and shim or
adjust the declination bearings to reduce the amplitude of this term.
The optical nonperpendicularity (sometimes called optical collimation error) is an error not really appreciated
by many astronomers. This is an error where the optical axis of the primary mirror is not perpendicular
to the declination rotational axis. It introduces large R.A. pointing errors. Sometimes the primary mirror
tilt is changed when adjusting the collimation of the optics. This will change the correction coefficient
and introduce large pointing errors. The primary mirror needs to carefully setup mechanically in its cell
and then not adjusted in tilt (centering adjustments are OK) when the optics are collimated. If pointing
measurements show that the collimation error is larger than desired, then the primary mirror needs to be
adjusted to minimize the nonperpendicularity correction term.
The position encoder scale factors, offsets, and eccentricities may be corrected by introducing the proper
values. The mounting of the encoders needs to be accurate enough to keep the eccentricity errors to less
than 60 arc seconds.
The telescope structure is not infinitely rigid so there are always deflections. Typically the rotational
deflections or translations causing rotations of the axes are the important flexures. Some of these flexures
don't change with rotation of the polar axle. These symmetrical flexures typically do not introduce pointing
errors. They may introduce image motions due to wind loading, however. The non symmetrical flexures can
be large. For example, the Lick Observatory 3meter telescope has 4 arc minutes of differential flexure
in its fork structure.
Flexure and slop of the telescope optical mounts are problem areas that should be fixed rather than corrected
by the pointing model. Slop (lost motion) is very difficult to predict so it does not model well.
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Telescope Pointing Accuracy
Performed correctly, the conversion between "mean" and "apparent"
coordinates introduces no errors worth consideration. If these coordinate transforms are the only corrections
applied, then the telescope pointing accuracy will be limited by atmospheric refraction.
The correction for atmospheric refraction is a function of zenith distance and atmospheric density (pressure,
temperature and humidity). The correction amounts to about 30 arc minutes at the horizon at sea level conditions.
An approximation of the atmospheric density can be made by correcting for altitude only. This will lead
to an error in the refraction correction of about 1 to 2 arc seconds at a zenith distance of 70 degrees.
Polar misalignment errors can be minimized for pointing or for tracking, but not both. Normally the telescope
is polar aligned on the refracted pole to minimize field rotation during tracking. The resulting altitude
misalignment introduces pointing errors primarily in Declination as a function of hour angle. These errors
can approach 1 arc minute in amplitude.
Optical and mechanical nonperpendicularity errors introduce large pointing errors in Right Ascension
and may exceed 2 arc minutes in amplitude.
With a generalized and nonspecific site pointing model, a corrected telescope will have an RMS (root mean
square or a form of average) pointing error in the 3 arc minute range. The RMS error can be reduced considerably
with site and telescope specific pointing model coefficients. Many professional observatories quote a pointing
error in the few arc second range. However, these values may be the best they have achieved and not necessarily
what they achieve on a routine basis.
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How good is "good enough"?
Even the "best" telescopes will need some form of guiding or correcting the telescope tracking using
optical feedback from a star. If the telescope tracks perfectly, tracking corrections will still be needed
to correct for small refraction effects (seeing for example).
Guiding may be performed with a reticle eyepiece or using some type of electronic detector. In any case,
the guider optical system will have a finite field of view. The guider is used to acquire and center the
object and then to provide tracking corrections.
The pointing accuracy should be sufficient to place the object within the field of view of the guider
system open loop (without optical feedback). Small telescopes tend to have a relatively large field of
view while large telescopes have a very small field of view so the pointing needs to be better for large
telescopes. Typically, it is desirable to have a pointing accuracy (RMS) of about 1/4 of the field of view
of the guider system. This level of pointing is considered "Good".
The guider system will often have a field of view of several arc minutes so the pointing needs to be a
fraction of an arc minute. For moderate size telescopes (0.5 to 1.5 meters) a pointing accuracy of 30 arc
seconds RMS or better is sufficient.
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DFM's Telescope Pointing Accuracy
The DFM Engineering data sheets provide very conservative pointing accuracy with typical values of 30
arc seconds RMS or better stated. The following table shows actual values for a few of the DFM telescopes
installed or serviced lately. The pointing model used by DFM is a geometric model. Typically, the DFM telescopes
have little to no flexure to correct which demonstrates the high stiffness and symmetry of the structure.
The control systems we have retrofitted to many telescopes (mostly Boller & Chivens telescopes) often correct
flexural errors in the 2 arc minute range.
Size 
Institute 
Pointing RMS 
Notes 
16" 
Macalester College 
9.9 arc sec 

20" 
Dr. Ron Kohl 
13.5 arc sec 
Serviced 800 / Installed 691 
24" 
Dickinson College 
9.5 arc sec 

32" 
U. Maryland, Baltimore 
10.0 arc sec 

51" 
USNO, Flagstaff 
11.3 arc sec 

24" 
U. North Carolina 
11.5 arc sec 
Boller & Chivens telescope with DFM Control System 
The following 4 graphs show the various pointing errors for the Dickinson College DFM 24inch telescope.
Notice the largest error is less than 30 arc seconds and this error occurs at a zenith distance of 80
degrees where the refraction model is not very good. The smoothness of the data indicates that further
modeling effort would improve the pointing slightly. The RMS value is less than 10 arc seconds and is considered
to be "excellent".
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Engineering Articles Summary
Steel & Aluminum
Geometry
Deflections
Pointing
