Ad Astra Observatory

Here’s a page that derives Polar Alignment Calculations.

(update 5/2020: the original site is gone, so I found the text on Wayback and am inserting it below. This is by the original author (srobins@…) and is attributed as such.)

I believe that the R/2 term is a little overkill, so I double the eventual “Big Constant” to allow a full pixel of drift in the final answer. That way I can more easily build seeing and FWHM into the final answer.

Original text follows:
This page provides the derivation of the polar alignment error formula

TXF/M=23636.23

where T is integration time in seconds, X is polar alignment error radius in arc-minutes, F is objective focal length in mm, and M is the width of the pixel in microns.

We start with three premises:

1) a ploar alignment error of X arc-minutes results in an error trace circle with a radius of X arc-minutes in one sidereal day.

2) the angle of sky given in arc-seconds covered by one pixel is given by
R=(206265*M)/(1000*F)

3) A displacement of 1 half pixel represents the maximum desired error. Beyond this point, a star looses it’s roundness.

Given 1,2,3 above we can determine the maximum integration time T in seconds given a polar alignment error radius, a focal length, and a pixel size. The maximum time is, of course, the time it takes to traverse R/2.

Therefore:

If the error radius of X arc-minutes traces a path which is

eq0) 2*pi*X

arc minutes in length over a siderial day (approximately 24 hours for convenience), we can say:

a) 1 arc-second=1/60 arc-minute and
b) 1 day=24*3600 seconds.

Applying this to eq0 we get:

eq1) 2*60*pi*X/(24*3600) reducing to
eq2) pi*X/720

arc-seconds per second velocity due to error X.

If T represents integration time in seconds, then we can say

eq3) T*pi*X/720

represents error displacement over time T due to error X.

We know the maximum displacement we can tolerate to be R/2 from the third premise, and from this we can say:

eq4) T*pi*X/720=R/2=206265M/(2*1000F)

Reducing eq4 we get: (note: we substitute 3.1415927 for pi)

eq5) TFX/M=23636.23

with minor alteration of eq5 we get:

eq6) T=23636.23M/FX

eq7) X=23636.23M/FT

and so forth.

I derived the above equation while on vacation in Sedona Arizona, so if there are errors, I’ll just have to blame the red rocks and the spirits therein. I am, however interested in your comments. Send them to srobinso@!mindspring.com

And here’s the text of the original script:


<-- form name="calc">
<-- table>
<-- tbody>
<-- tr>
<-- td colspan="3"><-- big><-- b>Enter Polar Error Data:<-- /b><-- /big><-- /td>
<-- /tr>
<-- tr>
<-- td>1)<-- /td>
<-- td>Enter needed integration time(seconds):<-- /td>
<-- td><-- input name="T" size="12" type="text" /><-- /td>
<-- /tr>
<-- tr>
<-- td>2)<-- /td>
<-- td>Enter objective focal length(mm):<-- /td>
<-- td><-- input name="F" size="12" type="text" /><-- /td>
<-- /tr>
<-- tr>
<-- td>3)<-- /td>
<-- td>Enter pixel size(microns):<-- /td>
<-- td><-- input name="M" size="12" type="text" /><-- /td>
<-- /tr>
<-- tr>
<-- td colspan="3"><-- big><-- b><-- input type="button" value="Compute" /> <-- /b><-- /big><-- /td>
<-- /tr>
<-- tr>
<-- td>4)<-- /td>
<-- td>Error in arc-minutes:<-- /td>
<-- td><-- input name="X" size="12" type="text" /><-- /td>
<-- /tr>
<-- /tbody>
<-- /table>
<-- /form><-- script language="JavaScript">
function calculate(){

var FF=document.calc.F.value;
var TT=document.calc.T.value;
var MM=document.calc.M.value;
document.calc.X.value=23636.23*MM/(TT*FF);

}
<– /script>