K3's AstroPhotography
"When I consider your heavens, the work of your fingers, the moon and the stars which You have set in place, what is Man that You are mindful of him?" -- Psalm 8:3,4

Equatorial Mount Tracking Errors

Because of Earth rotation longer exposures in astrophotography require tracking of objects. Longer exposure without tracking shows the path of objects on the sky (see Star trails photo).

This web page deals with equatorial mount measuring:

A, Equatorial Mount Tracking Errors (this page)
1, Sources of Mount Tracking Errors
2, Mount Error Measuring
3, Mount Error Analysis
4, RA Motor Speed Error Measurement

B, Modification of GEM1 mount for computer control

C, Tests and Analyses of Homemade GEM1 Guiding System

D, Homemade GEM1 Autoguiding Setup

E, Vixen GP-DX mount + Astromeccanica DA-1 Deluxe dual axis drive

1, Sources of Mount Tracking Errors

For longer exposures it is necessary to use motor driven equatorial mount. Telescope mounted on equatorial mount is rotating in opposite direction of Earth rotation, thus no trails should appear. The word SHOULD is used intentionally, because there are several factors which affect perfect tracking:

  1. The mount should be perfectly aligned with Earth's polar axis. Any deviation of mount's polar axis from Earth's axis causes tracking errors.
  2. Even quality machined mount parts like worms, worm gears, shafts are not absolutely perfect. The parts are machined in micrometer precision at best. We must realise that in astrophotography we require tracking precison up to arcseconds. That means, that e.g. teeth of teeth on perimeter of wheel with diameter of 8cm must be machined with accuracy of hundreeds of nanometers!
    As mount's shaft rotates, any error in its surface and shape and also in worm and worm gear surfaces and shapes causes a periodic bump in tracking. The most observable is so called periodic error of the mount which is caused by inaccuracy of of worm. The period of this error takes one revolution of gear (usually 5-10 minutes for common mounts).
    More expensive mounts has possibility to suppress this error by means of electronics - Periodic Error Correction (PEC). The principle of PEC is based on recording tracking corrections made by observer by star tracking during one period. This tracking corrections are then applied during normal mount use.
  3. Atmospheric refraction causes that stars are not moving exactly according to their calculated trajectories.
  4. Further effects - tripod, scope, focuser and other parts firmness, vibrations, thermal changes agffect the result tracking accuracy.

2, Mount Error Measuring

Mount errors can be simply measured by means of webcam. Here is example of measuring my equatorial mount GEM1 (see my scope) by means of K3CCDTools:

Animated part of screenshot (half size) of K3CCDTools with Reticle ON.

The scope was aimed to Vega star (it was about 43° above horizon).
Capture resolution: 640x480
Scope: 8" F6 OrionOptics Europa Newtonian
Frame size (FOV): 10.27' x 7.70'
Seeing: windy, twinkling stars

"Capture Selected Frames" capture mode was selected with period 1 second (exact period was 1.11s).
I recorded 1321 frames (24min 35s) for further analysis.
Note: Proper camera and scope must be set in Options | Telescope and camera settings to obtain correct results.

I also recorded star's trajectory with motor switched OFF. In this case the star moves from west to east. It is necessary for obtaining RA and DEC axes.

The result shows, that East-West axis is almost identical with reticle's horizontal axis. The deviation is 6.02° (measured in K3CCDTools using mouse).

I processed captured AVI file with "If Lighter" method. It nicely shows the "movement" of the star.

The star trajectory reveals an error caused by bad polar alignment (RA and DEC drifts) and also periodic error.

Then I aligned frames according to Vega star (like if I would want to stack frames for result picture of Vega). When frames are aligned, then the shifts considering the first frame contains data about RA and DEC drifts. Then I used a new function - Export to Drift List (menu Sequence Processing). At first the program asks for angle of West axis (in my case it was 6.02°) and then it exports data into text file.

3, Mount Error Analysis

Then I imported data from the text file to Excel and did the analysis:

Horizontal axis presents time in seconds, vertical axis presents tracking error in arcseconds.
Violet color shows constant increase of DEC error which is caused by inaccurate polar alignment.
Navy (dark blue) color shows more complex RA error. It is composed of linear RA drift, caused by inaccurate polar alignment (blue color) and periodic error (cyan color). The period of periodic error is about 9min 57s.

The periodic error in detail. The graph is biased to show both positive and negative error values. The peek values are +32/-29 arcseconds. It is quite a lot for doing long exposure astrophotography. On the other hand the graph also shows flat parts, where shorter exposures are successful. Also with this mount it is possible to take deep-sky photos (look here).

Click the graph to see detail graph.

Animation enables to compare two periods. The comparison is not totally exact (because of not perfect timing during videocapture and because of wind), but it allows to create an imagination the situation.

Note: Video capture timing in K3CCDTools (from version was reworked. Now it has no cumulative error and timing error is less than 50ms.

4, RA Motor Speed Error Measurement

During my measurement of mount errors I noticed that RA drift is usually to the East side, so I started to suspect RA motor speed. That's why I did several measurements of my RA axis drive. I measured the mount with telescope loaded during day time.

Place mouse pointer above image to see labels.
Click the image to see higher resolution image.
The driving unit of my GEM1 consists of stepper motor and 2 brass wheels. Both wheels have 31 teeths, so the gearing ratio is 1:1. The second wheel drives the axis with worm.

At first I measured the period of my worm axis. I stick a strip from electrical insulating tape to the stepper motor wheel for better reading wheel position.
The period was about 9min57s. Then I calculated the number of teeths of the main RA axis teeth wheel:
The Earth's sidereal day is:
sid_day = 23h56min04.09074sec = 86164.09074sec
Number of teeth = Round(sid_day / motor_period) = Round(86164s/597s) = 144 teeth

The measurement of a single period is rather inaccurate, so I choose another method for measuring.
I removed drive cover to better see brass wheels and I signed one tooth of wheel with a pen. Then I measured more periods of rotation to be more precise. I also made use of my motor controller, which have possibility to use 8X speed. The factor 8 is accurate, because a binary counter is used as frequence divider. Here are results of my measurements:
Number of period Measured time [min:sec] Worm wheel period [sec]
12 14:55.47 596.980000
36 44:45.37 596.748889
48 59:40.32 596.720000
60 74:35.42 596.722667

Worm wheel period was calculated according to the equation:
worm_period = 8* (time / number_of_periods)

The error caused by inaccurate read out the brass wheel is maximum 5° (less than 1/2 tooth). Error caused by a man reaction time could be +/- 0.3sec. So the result in 60 periods measuring could be 5°/60=0.08333°= 1/4320 of period (i.e. about 0.14sec). The reaction time error should not affect result more than 8*0.3sec/60=0.04s.

If we exclude the first table row (probably higher error) we can get average value:

worm_period = 596.7305sec (+/- 0.18sec)

It means that "mount's sidereal day" is:

mount_sid_day = 596.7305sec * 144 = 85929.2sec

The period of accurate drive system should be:

accurate_worm_period = sid_day / 144 = 86164sec/144 = 598.3611sec

My measurements confirm my suspicions - RA drive is faster than it should be. The inaccuracy of drive speed is 0.273%, which is rather high error. The maximum inaccuracy of my measurement was 0.03% (=0.18/596.7305).
The crystal oscillator unit has much higher accuracy, so my conclusion is, that inproper crystal frequency or inproper division ratio of binary counter were used.

Now we can calculate the drift in RA caused by inaccurate RA drive speed. The speed of star at celestial equator is:

accurate_speed = 360°/sid_day = 360°/86164sec = 0.0041781°/sec = 15.0411 arcsec/sec

The speed of telescope:

scope_speed = 360°/moun_sid_day = 360°/85929.2sec = 0.0041895°/sec = 15.0822 arcsec/sec

The star on equator is moving in telescope by speed which is equal to difference of above speeds:

scope_star_drift = 15.0822arcsec/sec - 15.0411arcsec/sec = 0.0411 arcsec/sec = 2.466arcsec/min

My measurements showed me, that my mount has also (apart from periodic error) constant drift in RA, which is ~2.5arcsec per minute.

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Last Update: 07.10.2002