Polar Alignment using the SBIG STV (minimal drift required)

by Rick Kellogg

8/27/01

rkellog1@twcny.rr.com

 

Introduction

Accurate polar alignment is necessary for long exposure astrophotographs. Even when a scope is perfectly guided, an error in polar alignment can result in field rotation. This paper will provide a method of polar alignment using the SBIG STV that does not require drift alignment except as a final check for accuracy.

Drift alignment is the accepted method of polar alignment by most astrophotographers. I find it tedious, and somewhat frustrating. Usually before I know it, I've spent 90 minutes trying, and I'm still not aligned as good as I would like to be.

The method presented is similar to using a polar alignment scope, but uses images from the SBIG STV autoguider along with precise x,y locations of alignment stars. The accuracy achieved is higher than with a polar alignment scope and is within 2 arc minutes of the celestial pole. I find it relatively easy and reliable, and the whole process can be comfortably completed in 1 hour (about the same time most people claim for drift alignment, even though it takes me a lot longer).

Configuration

This paper assumes a German Equatorial Mount (GEM) and the SBIG STV autoguider working with a guide scope of 300 mm focal length or less. My setup uses a 50 mm aperture, 200 mm focal length finder as the guidescope. This has a 1.35 by 1 degree field of view (FOV) when using the wide image mode. The optional eFinder lens that SBIG offers has a 100 mm focal length with a 2.7 by 2 degree FOV and would be ideal for the proposed polar alignment method. A 200 mm focal length scope is a little more difficult, as Polaris may not be in the FOV when pointed at the pole, depending on orientation of the STV and the date/time of the alignment. When pointed nearly at the pole, I find Polaris in the STV images by rotating the RA axis until I see it. Then I use a finder chart to get my bearings. I usually mount the STV camera head upside down, so that the display has right side up, correct images, without using the flip controls of the STV.

The STV autoguider can take images and display them on the built in TV screen. Using the eFinder mode of operation, a cross-hair can be placed on the screen, and the x,y pixel location of stars determined. When in eFinder mode, I assume that a continuous stream of new images are being displayed, even though some operations, such as determining the x,y position of a star, will stop the taking of new images. Throughout the steps presented below, I assume that after the x,y position is determined, 'NEXT' is pressed and new images start up again.

Polar Alignment Method

The polar alignment method is similar to that used for a polar alignment scope, but achieves higher precision by finding the coordinates of alignment stars on the CCD images and computes both the location of the center of rotation of the RA axis, and the correct position to place one of the alignment stars (using the mounts Altitude/Azimuth controls) such that the celestial pole is located on the center of rotation. The procedure consists of 5 steps described below.

Step 1 - Point the guide scope to DEC 90 degrees, such that when the RA axis is turned, the center of rotation remains in the center portion of the FOV in the STV images. Point the mount toward Polaris to check this. Small changes in DEC may be required. If your guide scope is not square to the RA axis, it may require shimming. The center of rotation needs to be within the center portion of the CCD so that the alignment stars will fit on the image when the center of rotation is perfectly pointed at the pole.

If you have a polar scope, use it to get the mount pointed toward Polaris, otherwise use a low power eyepiece in your main scope or a finder to point the mount toward Polaris.

Set the STV to the eFinder mode, with the image mode = WIDE, exposure = 0.05 second. (To get accurate position readings, Polaris should not be saturating the CCD. If is it, use less exposure time).

After this point, the DEC axis is not changed until alignment is achieved.

Step 2 - Find the x,y location of the center of rotation, as the RA axis is rotated: Using the eFinder mode (image mode = wide, exposure = 0.05), with the counterweight bar horizontal (changes in DEC would be vertical), and the guide scope on the eastern side of the mount, move the cross-hair to determine the x,y location of Polaris. Call this position P1 (p1x, p1y). Now rotate the scope 90 degrees in the RA axis, such that the guide scope is now at the top of the drive (changes in DEC would be horizontal). Find the x,y location of Polaris. Call this position P2 (p2x, p2y).

The center of rotation C (cx,cy) is:

cy = (p1y + p2y)/2 - (p1x - p2x)/2                   (2)

To check this, move the eFinder cross-hair to cx,cy. Use the Alt/Az controls of the mount to move Polaris on top of the cross hair. Now rotate the RA axis and notice that Polaris does not move. We have found the x,y location of the RA axis to within a pixel or so. If Polaris does move repeat this step. It usually only takes one iteration.

Note: any star that stays on the screen throughout the 90 degree rotation can be used to determine the center of rotation. After alignment has been achieved (and Polaris may no longer be in the field of view) this test can be repeated using one of the alignment stars to verify that the center of rotation has not changed.

Step 3 - With Polaris in the field of view, a longer exposure (10 seconds) is taken and compared to a finder chart to locate 2 stars (1,2) that will be used for alignment (see Figure 1).  The finder chart in Figure 1 shows all of the stars of magnitude 12 or brighter and was created from: http://www.nofs.navy.mil/data/FchPix/cfra2.html

The 2 alignment stars are about magnitude 9-10, and were chosen as they are close to the pole. Notice that the pole lies about 1/3 the distance along a line from 1 to 2.

Adjust the Alt/Az controls of the mount until both alignment stars are visible.

An important timesaver is to make a finder chart, on a transparency, that is scaled the same as the STV screen. This can be placed on the STV screen and rotated/translated to match the screen. This makes it easy to identify the alignment stars.

                 Figure 1 Finder map near the celestial pole.

                              Field is 2 degrees by 2 degrees.

                        Stars of Mag 12 or brighter are shown .

                     Stars labeled 1 and 2 are used for alignment.

 

Step 4 - Use the eFinder cross-hair to determine the x,y location of the 2 alignment stars S1 (s1x, s1y), and S2 (s2x,s2y). The difference D (dx,dy) from S1 to S2 is calculated as:

dy = s2y - s1y                                       (4)

The offset O (ox, oy) from the celestial pole to the first alignment star S1 is calculated as:

oy = (-0.34357)dy + (0.029147)dx                     (6)

(Note: only 2 digits of accuracy are needed for equations 5-6 as the locations are in integer units of pixels from 1-216).

Step 5 - Add the offset O to the center of rotation C to get the target T (tx, ty).

ty = cy + oy                                         (8)

Move the eFinder cross-hair to this target location. Then use the mounts Alt/Az controls to move the alignment star 1 to the cross hair location. You now have moved the center of rotation of the RA axis (C) on top of the celestial pole, and you are aligned.

As a check, do the standard drift test used in drift alignment. (Point the scope to a star toward the south with DEC near 0 degrees. Check for north/south drift. Point the scope to a star to the east, check for north/south drift) . You will find you are aligned, and no adjustment is needed.

The drift test should be done to verify alignment. On one alignment, I was off by quite a bit. This was found by drift testing. When I rechecked (steps 2-5, but using alignment star 1 in place of Polaris in step 2) I quickly found I was way off. Either I made a math error or moved the mount. Once corrected, drift testing showed almost perfect alignment. (Note: the STV has a "Monitor Drive (slow)" function that can be used to show N/S drift on the DEC axis.)

You may be wondering, "if I need to do the drift test as a final check, what's the point?"  The main advantage over a drift alignment is that adjustment of Alt/Az only needs to be done once. You can see what you are adjusting, and you can get accuracy about as good as you can adjust your mount. The drift test just verifies that you are aligned.

Alternate Steps 4 and 5.

If you don't want to have to do multiplication in the field, an alternative graphical method is presented that is an exact analogy to using a polar alignment scope. Create a finder chart transparency that is scaled the same as the STV screen. After step 3, the alignment stars are present on the STV screen. Move the x,y cross-hair to the center of rotation C (it should already be there from Step 2). Place the transparency finder chart on the STV screen matched to the stars. It will be necessary to rotate the chart at some angle to get it to align. Hold this angle and slide the chart over until the celestial pole (on the chart) is aligned with the center C.

Now move the mount's Alt/Az controls to move the stars to align with the chart. After trying this a few times I recommend doing steps 4 and 5, as this alternative is a bit clumsy, and requires accurate scaling of the finder chart.

Accuracy.

If you assume that each location can be determined to within +/- 1 pixel, then the overall accuracy of the pole should be < 4 pixels. In wide mode, each pixel is a 3x3 binned pixel. The size of a 3x3 pixel for a 200 mm scope is ~23 arc sec. Thus 4 pixels is 92 arc sec. I set a goal of 2 arc min, as I don't think I can reliably adjust my mount by increments smaller than this. This accuracy should be readily achievable.

My mount moves 10 degrees in Alt for 3 complete turns. Thus, a rotation of the Alt knob by 1/100 turn (3.6 degrees) results in a change in Alt of 2 arc min. It is doubtful that I could reliably turn the knob by smaller increments.

Test

I conducted a test of this method on 8/22/01. From start to finish, the alignment took about 40 minutes (not counting the drift test to verify the alignment). Then I moved the Alt/Az, and tried again from step 2, and it took about 20 minutes.

To test the alignment accuracy, I did a drift test and drifted about 4-6 arcsec south in 20 minutes when looking east, and <4 arcsec south in 20 minutes looking south.

A 2 arcmin error in the pole location would result in 10 arc sec drift in 20 minutes.

Derivation of Equations 1-8

This section will derive the equations used throughout the paper.

In all calculations a capital letter will be used to represent a point, and lower case letter, followed by x, or y will be used to represent the x,y components of the point. For example, S is a point, and sx, sy are the x and y coordinates of S.

Center of rotation

Figure 2 shows the x,y location of the star used to determine the center of rotation C (Step 2 above). Shown in Figure 2 is the first location of the star (P1), with the RA axis such that the counterweight bar is horizontal, and the scope on the east of the mount. Also shown is the location of the star (P2) after rotating the RA axis 90 degrees, such that the scope is now at the top. The figure also shows the point in the field of view of the CCD about which the RA axis is rotating (C).

We want to calculate C, from the x,y observations of P1 and P2. From Figure 2, for a 90 degree rotation, we have:

p2y =  (p1x - cx) + cy

Solving for cx, cy we get equations 1 and 2:

cx = (p1x + p2x)/2 + (p1y - p2y)/2                   (1)

cy = (p1y + p2y)/2 - (p1x - p2x)/2                   (2)

Coordinates of alignment stars

The reference used to create the finder map also returns the data of the stars. The data for the 2 alignment stars are:

S2 Mag 9.1, RA 09:46:20.4636, DEC 89:34:09.949

These are shown in Figure 3. Figure 3 is the same as Figure 1, except with a magnitude limit of 10 instead of 12, and I have added X, and Y axes, along with a vector D (from S1 to S2) and a vector O (from the origin to S1).

We want to be able to calculate O from observations of x,y positions of S1 and S2.

The vector D (dx,dy) from S1 to S2 is by definition:

dy = s2y - s1y                                       (4)

                                                                              Figure 2

                                        P1 is location of Star with RA axis horizontal, scope on east

                                               P2 is location of Star with RA vertical, scope on top

                                                                  C is the center of rotation

                                           Figure 3. Finder map with X,Y axes.

                                          Alignment Stars S1, S2, and vector D

                                 (from S1 to S2) and vector O (from the pole to S1)

The X and Y axes are in units of degrees. First we compute 90 degrees - DEC to get the angle distance (AD), away from the pole. Then we convert RA to angle (A) by multiplying by 15 degrees/hour. Finally the polar coordinates are converted to rectangular the x,y coordinates as x = (AD)cos(A), y = (AD)sin(A).

Thus the locations for S1, S2 are:

   = -0.359399 x, 0.237113 y

The value of D = S2-S1 becomes:

dy =   0.237113  - (-0.147656) =  0.384769 y

Note: in polar form, D is:

0.65528 at angle 144.0428 degrees

The vector O has the same coordinates as S1. That is:

oy = -0.147656 y

We are going to need to both multiply and divide vectors.

The formulas for multiplication and division of 2 vectors and are given below:

Division of vectors:

If O (ox, oy) is divided by D (dx,dy), and the resulting vector is O_D = O/D then:

o_dx =(oxdx + oydy)/(dxdx + dydy)

o_dy =(oydx - oxdy)/(dxdx + dydy)

Multiplication of vectors:

If Q (qx, qy) is divided by R (rx,ry), and the resulting vector is QR = (Q)(R) then:

qrx = qxrx - qyry

qry = qyrx + qxry

We compute the vector O divided by the vector D. using division by vectors above:

     = 0.029147 y

We are almost done. Now to convert x,y observations of S1, S2 to O, we compute D = S2-S1, and multiply by O_D above to get O.

Using the formula for multiplication of 2 vectors above, we get equations 5-6 from step 4.

oy = (-0.34357)dy + (0.029147)dx                     (6)

Once the offset O is calculated, it is added to the center of rotation C to get the point (T) where S1 should be placed to get the celestial pole to be over the center of rotation. This results in equations 7-8.

ty = cy + oy                                         (8)

Example.

The mount is pointed toward Polaris, DEC set to 90 degrees, the STV is placed into eFinder mode, image mode-wide, exposure = 0.05 second. When the mount is rotated 90 degrees about RA axis, Polaris stays on screen.

Step 1 complete.

Images taken for Polaris at 2 RA angles corresponding to counterweight bar horizontal, scope on eastern side of mount (1), and rotated 90 degrees such that scope is at top of mount (2) give x,y positions of:

   = 100x, 115y

The center of rotation C (cx,cy) is:

Step 2 complete.

Step 3 moves the 2 alignment stars on screen with the Alt/Az controls. Polaris may move off the FOV.

Step 4 determines the location of the 2 alignment stars:

dy =  s2y - s2x = 153 -  72 =  81 y

Offset O calculated as:

Target T calculated as:

ty = cy + oy = 96 -29.8 =  66.2 =  66 y

The cross-hair is moved to 112 x, 66 y, and the Alt/Az controls of the mount are used to move the alignment star S1:

Step 5 and alignment complete. Now do a drift test to verify alignment.

Send comments to: rkellog1@twcny.rr.com