Written by Toshimi Taki

Revision A: January 12, 2002

Revision B: April 3, 2004

Revision A: January 12, 2002

Revision B: April 3, 2004

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I found errors in TAKI.BAS recently. Please make the following

corrections in the BASIC program.

Wrong: 760 Y(1,0)=Y(1,0)+SIN(F)*COS(H)*(Z1/G)

Correct: 760 Y(1,0)=Y(1,0)+SIN(F)*

Wrong: 795 Y(1,1)=Y(1,1)-SIN(F)*COS(H)*(Z1/G)

Correct: 795 Y(1,1)=Y(1,1)-SIN(F)*

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Palm Pilot version is developed by David Ditch and you can download it from his website.

Since then, the algorism has been widely used by amateurs and some of the advanced amateurs are using the algorism to control Dobsonian telescopes. Mel Bartel has developed such kind of control system and he presents his software in his home page.

Because of the limited space of the magazine, the detailed equations behind the algorithm were omitted. So I would like to present all the detailed equations here. I also present a software to find mount error. I hope this will be helpful to understand the algorithm.

If you interested in the matrix method which I used to develop the equations, see "Matrix Method for Coordinates Transformation" in my home page.

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The relationship between telescope coordinate and equatorial coordinate is expressed as follows using direction cosines of both coordinates.

Transformation from equatorial coordinate to telescope coordinate,

.... equation (1)

Transformation from telescope coordinate to equatorial coodinate,

.... equation (2)

Where,

: Direction cosine of an object in telescope coordinate system .... equation (3)

: Direction cosine of an object in equatorial coordinate system .... equation (4)

, : Transformation matrix and its inverse matrix

: Time

: Initial time

: Horizontal angle of an object

: Elevation of an object

: Right Ascension of an object

: Declination of an object

For reference star 1

Observed time :

Telescope coordinate : ,

Equatorial coordinate : ,

For reference star 2

Observed time :

Telescope coordinate : ,

Equatorial coordinate : ,

.... equation (5)

.... equation (6)

.... equation (7)

.... equation (8)

.... equation (9)

.... equation (10)

.... equation (11)

: True horizontal angle

: True elevation

: Apparent horizontal angle

: Apparent elevation

: Mount error angle between horizontal axis and vertical axis

: Mount error angle between vertical axis and telescope optical axis

: Mount error angle, zero point shift

The direction cosine of true telescope angles can be expressed using apparent telescope angles as follows.

Since mount error angles, and are very small, we can use the following approximation. Note that and are in radian.

Equation (12) is simplified as follows.

.... equation (13)

Again, mount error angles, and are very small, we can use the following approximation.

, , ,

Apparent telescope angles can be calculated from true telescope angles as follows.

.... equation (14)

(1) Point your telescope to two known stars (call them reference stars) and measure the apparent telescope angles.

(2) Using equation (12), calcuate direction cosines of true telescope angles for the 2 reference stars.

(3) Using equations (7) and (9), calculate direction cosines of equatorial coordinates for the 2 reference stars.

(4) Using equations (10) and (11), calculate the additional directions cosines.

(5) Apply equation (5) and calculate the transformation matrix, [T]. Note that the transformation matrix is for the direction cosine of true telescope angles.

(1) Using equations (4) and (1) with equatorial coordinate of a known object, calculate direction cosine of true telescope angles of the object. Then, apply equation (14) to find direction cosine of apparent telescope angles of the object. The apparent angles is calculated from the direction cosine.

(2) Point your telescope to the apparent angle and you will find the object in the field of your telecope.

(1) Measure apparent telescope angle of unknown object and record the observed time.

(2) Calculate direction cosine of the true telescope angles using equation (12).

(3) Calculate direction cosine of equatorial coordinate using equation (2). Then, equatorial coordinate of the unknown object is calculated using equation (4).

Click here to download "mount_error_b.xls".

[2] Toshimi Taki, "New Method for Pointing of Alt-azimuth Mount Telescope," Hoshi-no-Techo, Volume 35, 1987, pp.94-97. (in Japanese)

[3] Toshimi Taki, "Improvement of Pointing Method of Alt-azimuth Mount Telescope," Hoshi-no-Techo, Volume 39, 1988, pp.94-100. (in Japanese)

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- Eorrors in equations (12) to (14) are corrected.

- "mount_error.xls" is revised using coorected equations.

- "mount_error_b.xls" is revised.