A scotch mount without tangent error
In 1974, George Haig has proposed an original design to make a cheap and simple equatorial mount (School Science Review Septembre 1974). This design is now widely used by amateur astronomers (see Phillip S. Harrington's book Star Ware : Plans for a "Scotch Mount" Camera-Tracking Platform).
The tangent error of the original barn door tracker designed by George Haig limits the duration of astro-photographies to about 10 minutes pauses, even with a 50 mm focal lens (eq. 24x36). As time lapses, the tracking is getting worse, after 1 hour, the deviation is about 0.3 degrees, and 2.4 degrees after 2 hours. The use of an uncorrected scotch mount is not recommended for focal lenses greater than 200 mm.
This article is proposing a way to easily remove the tangent error by inserting a special cam between the threaded rod and the top board.
Already fabricated scotch mount can be fixed with this cam.
in these sketches, the green circle is the pivot axis that shall point toward the celestial polar axis
the blue board is the top board that bears the camera or the telescope
the grey board is the mount base, firmly attached to the tripod
the yellow bar is the threaded rod
the green rectangle is the drive handle
the marron/orange rectangles are the studs
Several other designs exist to reduce the error, such as the single or two arms isoscele barn doors (studied by Dave Trott, for more information see his site The Double Arm Barn Door Drive), but their fabrication is complex and out of the original idea of George Haig to make something easy to fabricate. If you curve the threaded rod, it is possible to completely remove the error, but some unstability is added in the mount because of the curvature, and curving the rod without damaging the threads can be problematic if you choose a large diameter rod.
The french astronomer André Hamon has presented a solution to the tangent error in summer 1953, long before George Haig described his tangent barn door. His article has been published in the french magazine Astronomie of the Société Astronomique de France and later translated in the june 1978 issue of “Sky & Telescope”. Since then, the idea of A. Hamon has been ignored… until Gerard Cutting presented his plastic cam in the Stellafane convention of 1986 (ref. Sky & Telescope nov 1986, pages 530-531), then ignored again.
The original design of A. Hamon is based on two shaped cams, one driven by the rod, the other one being on the drive arm. The preparation of two different pieces is not difficult, but yet more complex than one, and their proper alignment is an additional concern.
Left : original design of A. Hamon (1953), and improvement of S. Dodson (top corner). Right : Gerard Cutting design (1986)
The threaded rod is pushing the top board at a constant linear speed. But the top board should move at a constant angular speed, and this angular speed does not match the speed of the rod : the rod is too slow. Some geometric and trigonometric analyse quickly drives to the difference between the speeds and you can calculate a way to compensate the difference. In fact, this problem is similar to the one the engineers had to solve to design spur gears in the 19th century. And the solution is the same : an involute of a base circle.
The curve of the corrective cam I propose is therefore the involute of the barn door base circle, which radius is d. The equation is as follows :
This curve can be drawn on any worksheet software, such as Open Office Calc, of Microsoft Office Excel.
d = ε / tan ω : is the base radius of the scotch mount (distance between the mount pivot and the rod axis on the base board)
θ = an angle, in radians, that varies from 0 to ωN, where N is the total number of threads of your bolt
ε = thread pitch
ω = rotation speed of Earth, in rad/minute = 0,004375 rad/min
x,y : coordinates of one point on the involute curve in the Oxy axis system
Nota : d and ε have the same length unit (if ε is in mm, d will be in mm, and so on)
The easiest way to make the cam is by printing the following picture on a transparent paper AT THE RIGHT SCALE !
The equation shows that the curve is proportional to the thread picth ε. the unity curve is plotted on the picture below (for ε = 1). You just have to print the picture at the proper scale ratio.
cliquer sur l'image pour la télécharger (zip)
The picture is a square with side length equal to 65 times the thread picth :
Redraw the red curve to the piece of brass, steel or hard wood in which you want to cut the corrective cam, then carefully grind of the extra material until you reach the perfect fit between the printed curve and your cam. The corrective cam is made of the yellow part of the picture (note that the blue side is the corrective cam of the mount arm of A. Hamon).
The vertical lines are showing the tracking duration, it is not necessary to cut a too long piece, just allow one centimeter extra length for confort on each side, and below. Install the cam on the top board of your mount, making sure that the "zero" corner of the cam is exactly at the base radius distance from the barn door axis.
Now you have corrected your tracker, you shall make sure that the polar alignment is good, but this is another story !
The accuracy of the grinding needs to be good but not that precise. At the contact point, the angle between the threaded rod axis and the corrective piece surface is always a right angle. For a rod with 1 mm thread pitch, a defect of of ½ mm will cause an error of about 0.13 degree. With some care, it is possible to get a better accuracy that will further reduce the error.
The second cause of error is the positionning of the corrective piece relative to the pivot axis of the barn door. An error of 1 mm will cause an angular error of 1 arc.min after 150 minutes of tracking, very neglictible.
Though this design is theoretically "error free", the fabrication tolerances induce some tracking errors. But their magnitude is similar to one can expect from a far more complex design such as the double arm barn door.
See my page - written in french - here.
Well, the corrective cam is a very simple way of fixing the tangent error of the orginal Haig mount. There is also another simple solution, it is the curved rod. But, you will have to curve the bolt at the proper radius of curvature without damaging the threads. This design is however well known and prooven in wide-field astrophotography. See the "barn door" page of Peter Kennett's web site.
In a letter to Arvind Paranjpye, dated 8th July 1996, George Y. Haig, the inventor of the so called Scotch Mount explained in a note why he chose to call the mount 'The Scotch Mount'.
I first made the door-hinge mount for the use in the winter of 1972 - 73. Since it was neither a German nor an English equatorial, it could have been described as a Scottish equatorial; however, the word "Scotch" rather than "Scottish" seemed to me to convey a hint of that comic parsimony (allegedly associated with Scotsmen) which I felt was appropriate for such a cheap and simple device. Then of course there is a brand of Scotch whisky made by Haig (not a relation of mine, unfortunately!) so the epithet is also an oblique (perhaps even obscure) allusion to my own surname.
Source : OBSErvatory in EducaTION
Pierre Charpentier, Charles Rydel, Société Astronomique de France