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wh48gs
Pooh-Bah

Reged: 03/02/07

Re: Useful info about secondary mirror alignment [Re: Jason D]
#3369179 - 10/03/09 06:51 PM

Quote:

No one is claiming anything “new”. This thread is not meant to introduce new concepts or terminology.

I don't know if anyone is claiming anything, but I do know that something that is not new is being called that. If we step back from talking in terms of focuser/primary axis, this collimation mode is essentially identical to the conventional 1-2-3 step procedure, with #1 being centering the diagonal in the focuser's circle by axially shifting it toward primary, #2 centering the primary's reflection in the diagonal by tilting the diagonal so that focuser's axis "reflected" from it hits the center of the primary, and #3, centering reflection of the diagonal in the primary by tilting the primary's axis toward focuser (so that it nearly coincides with focuser's axis "reflected" from the diagonal).

What is new is that I figured out why I couldn't get primary's axis to fall back onto the focuser's axis (didn't take into account that axial shift of the diagonal cancels most of the primary's reflection displacement in the diagonal). The result is that the magnitude of axial disparity is significantly lower than I thought. It is given by 57.3a^2/SH in degrees, with "a" being the diagonal minor semiaxis, S the diagonal-to-primary and H the diagonal-to-focus separation. For a 12" f/5 Newtonian, with a=1.5", H=10" and S=50", that comes to 0.26 degrees (assuming square focuser).

As usual, it's good for me to hang with you guys.

Vla

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wh48gs
Pooh-Bah

Reged: 03/02/07

Re: Useful info about secondary mirror alignment [Re: sixela]
#3369192 - 10/03/09 07:00 PM

Quote:

If you follow a sane collimation protocol, lack of squareness of the focuser cannot cause axial miscollimation (or do you mean an optical axis not along the tube axis when you say "axial disparity"? That's certainly a "new" term to me, so I'm unsure as to what exactly it means.)

That is true only if the "unsquarness" of the focuser is compensated by the diagonal alone, leaving axes of the primary and the tube nearly coinciding. If these two axes are at an angle, it induces pointing and tracking errors - assuming that the tube/structure axis is aligned with the mechanical axes of the mount. In all, there are four axes other than focuser's that need to be aligned for a perfect alignment.

Vla

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Starman1
Vendor (EyepiecesEtc.com)

Reged: 06/24/03

Loc: Los Angeles
Re: Useful info about secondary mirror alignment [Re: wh48gs]
#3369849 - 10/04/09 01:23 AM

Quote:

Quote:

No one is claiming anything &#8220;new&#8221;. This thread is not meant to introduce new concepts or terminology.

I don't know if anyone is claiming anything, but I do know that something that is not new is being called that. If we step back from talking in terms of focuser/primary axis, this collimation mode is essentially identical to the conventional 1-2-3 step procedure, with #1 being centering the diagonal in the focuser's circle by axially shifting it toward primary, #2 centering the primary's reflection in the diagonal by tilting the diagonal so that focuser's axis "reflected" from it hits the center of the primary, and #3, centering reflection of the diagonal in the primary by tilting the primary's axis toward focuser (so that it nearly coincides with focuser's axis "reflected" from the diagonal).

What is new is that I figured out why I couldn't get primary's axis to fall back onto the focuser's axis (didn't take into account that axial shift of the diagonal cancels most of the primary's reflection displacement in the diagonal). The result is that the magnitude of axial disparity is significantly lower than I thought. It is given by 57.3a^2/SH in degrees, with "a" being the diagonal minor semiaxis, S the diagonal-to-primary and H the diagonal-to-focus separation. For a 12" f/5 Newtonian, with a=1.5", H=10" and S=50", that comes to 0.26 degrees (assuming square focuser).

As usual, it's good for me to hang with you guys.

Vla

The formula I use is that the displacement of the optical axis is equal to the offset in classical offset positioning of the secondary.
As such, you can then compute the angular error between the optical and mechanical axes as the inverse tangent of the offset divided by the distance from primary to secondary.
For a 12.5" f/5, that was about 1/8" offset and about 8' deviation between the optical and mechanical axes.
With a 2.6' resolution on my encoders, that could cause errors in pointing, but the object is always in the center 50% of the field of a 45' field eyepiece if I took the time to align using a reticle eyepiece. I've achieved 0.0 warp factors.
However, I should point out that my focuser is exactly on the side of the UTA, which means the deviation is only in azimuth, which it compensated for in alignment. If the focuser is at other angles, the axial deviation could lead to more problems in pointing.

Now, if offset is larger and the secondary-to-primary distance is less, the error grows. As I mentioned, it is probably wise to resurrect the classical offset techniques as scopes get shorter in f/ratio.

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wh48gs
Pooh-Bah

Reged: 03/02/07

Re: Useful info about secondary mirror alignment [Re: Starman1]
#3370217 - 10/04/09 08:53 AM

Quote:

The formula I use is that the displacement of the optical axis is equal to the offset in classical offset positioning of the secondary.

Partial (axial) offset is not quite the same as a full offset, in that it needs larger value to have the diagonal apparently centered in the focuser's view (sinking the diagonal by the offset value has added effect of apparently shifting it toward primary, since the upper edge's apparent horizontal shift is greater). I got that the full offset value is given by (D-A)A/4S and axial offset alone by A^2/4H, where D is the mirror diameter, A the diagonal's minor axis, S the diagonal-to-primary separation and H the diagonal-to-focus separation. This means that the axial alone to full offset ratio is given by SA/(D-A)H. With the usual parameter values, axial shift alone is about 50% greater.

Vla

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Starman1
Vendor (EyepiecesEtc.com)

Reged: 06/24/03

Loc: Los Angeles
Re: Useful info about secondary mirror alignment [Re: wh48gs]
#3370471 - 10/04/09 11:33 AM

Hmm.
As I understand it, in classical offset the secondary is moved along the 45 degree plane of the secondary by an offset amount that is required to bring the opto-mechanical axis coincident with the geometric center of the secondary, and that that results in a centered secondary and even illumination.
Here's a calculator to figure than offset:
http://www.asnsw.com/info/offsetcalc.asp
and
http://users.erols.com/thestewarts/diagonal.html

In the technique that results in tilting the optical path toward the focuser (what has been referred to as the "new model", rightly or wrongly), the *effective* offset amount for the secondary is the same.
However, the actual tilt of the optical axis away from the mechanical axis should be less than the offset on the secondary surface because it doesn't have to tilt sideways as much as the offset because the offset is along a 45 degree line, so the actual tilt of the optical axis in millimeters would be the square root of the quantity of the offset on the secondary surface squared divided by 2.
[If the calculator calculates the offset amount you need to move the secondary away from the focuser, ignore the formula and use this amount as the opposite side of the long triangle in the following paragraph.]
If you then use that offset amount as one side of a long triangle with the distance from secondary to primary as another side, the angular deviation between the optical and mechanical axes can be calculated as the inverse tangent of the lateral tilt in millimeters divided by the secondary to primary distance.
The question is how many minutes of angular deviation between optical and mechanical axes is significant for DSC accuracy? And that is something I can't answer.
The small amount of deviation on my 12.5" f/5 isn't significant. The large amount of deviation on Mike Lockwood's 14.5" f/2.55 scope would be. Where the dividing line is, I don't know.

Vlad would probably argue that if everyone used classical offset collimation this wouldn't be an issue, and he'd be right.
But accuracy in that offset could be hard to achieve and adds an additional dimension to collimation not important to telescopes that are not used with DSCs.

I intend to improve on the instructions for achieving Classical Offset secondary positioning in my next version of the collimation article CN just posted. As telescopes get shorter (and it seems they may be), this will become more critical.

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wh48gs
Pooh-Bah

Reged: 03/02/07

Re: Useful info about secondary mirror alignment [Re: Starman1]
#3370871 - 10/04/09 02:54 PM

Don,

Quote:

As I understand it, in classical offset the secondary is moved along the 45 degree plane of the secondary by an offset amount that is required to bring the opto-mechanical axis coincident with the geometric center of the secondary, and that that results in a centered secondary and even illumination.

The geometric center of the secondary actually slides down bellow the primary's axis, and it is diagonal's apparent center that is brought to coincide with the point of intersection of the primary's and focuser's axis. This centers diagonal both in the focuser's view and with respect to the axial - as well as truncated full-field - converging cone.

The top link gives result identical to that with the formula I posted, and also with the Suiter's approximation (p330 1st ed). He calls it approximation even if it is exact for all practical purposes (the only thing that gets rounded off is the focal ratio number, as f/D, instead of the actual value given as (f minus mirror sagitta)/D).

The other calculator gives slightly lower (<0.1mm) value, but it won't matter.

There might be some exceptions, but as I can see the offset is commonly called the increment by which the diagonal moves axially toward primary, and away from the focuser. The actual travel of the diagonal's center is greater by a factor of sq.rt.2.

The secondary shifted only axially toward mirror needs more of a travel in order to have this accomplished. Thus it requires more of a compensatory tilt of the diagonal and primary than what the full offset value would indicate. It seems to me that you are trying to reason about this, but there is no substitute to putting it down on paper, and get the relations from the geometry. Assuming focuser square to the tube axis (or to the plane of the mechanical axis of the mount), the tilt angle is given by by (A^2)/4SH in radians (i.e. tangent), A being the diagonal's minor axis, S diagonal-to-primary and H diagonal-to-focus separation. For a 14.5" f/2.55 with A, S and H, say, 5", 26" and 11", respectively, that would come to 0.022, or 1.25 degrees. Whether it is acceptable or not, depends primarily on the telescope's intended use.

I don't think that the full diagonal offset is a must, especially after I fixed that faulty calculation that made me think that the required primary tilt angle is significantly larger than what it really is. One thing that shouldn't be left to chance is squarness of the focuser, because in the partial offset it translates to the primary tilt. If neglected, it is not too hard to generate up to a few degrees of pointing error.

Vla

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Starman1
Vendor (EyepiecesEtc.com)

Reged: 06/24/03

Loc: Los Angeles
Re: Useful info about secondary mirror alignment [Re: wh48gs]
#3371024 - 10/04/09 04:25 PM

Thanks. I always thought offset, in millimeters, was the amount the secondary had to slide along the plane of its surface.
Since offset is, as you indicate, the amount the secondary has to move away from the focuser and down the tube, then your calculations hold true.
I did graph it on paper, and I thank you for pointing to that as a better way to visualize what's going on. I see the centered diagonal having the opto-mechanical axis hit the secondary below the geometric center of the elliptical surface (toward the end closer to the primary). I also see that if the secondary is slid down along the plane of its surface until the geometric center is coincident with the opto-mechanical axis there will still be a little more secondary outside the cone on the end farthest away from the primary.
As you indicate, the secondary actually has to slide down a little more than the geometric center to get uniform edge illumination all the way around. Fascinating.
Ergo, the tilt of the optical axis away from the mechanical axis is a little more than I calculated.

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sixela
Postmaster

Reged: 12/23/04

Loc: Boechout, Belgium
Re: Useful info about secondary mirror alignment [Re: wh48gs]
#3371043 - 10/04/09 04:38 PM

Quote:

and #3, centering reflection of the diagonal in the primary by tilting the primary's axis toward focuser (so that it nearly coincides with focuser's axis "reflected" from the diagonal).

You've got it slightly wrong. #3 uses a Cheshire, and *perfectly* overlaps the optical axis and focuser axis and their reflections (even though the intercept angle on the secondary is not 90° and the optical axis is not along the tube axis), because you make the reflected optical axis cross the focuser axis half way between the Cheshire ring and the pupil.

Using the reflected secondary's silhouette is a bad idea, and no sane protocol would use it as a reference.

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sixela
Postmaster

Reged: 12/23/04

Loc: Boechout, Belgium
Re: Useful info about secondary mirror alignment [Re: wh48gs]
#3371056 - 10/04/09 04:44 PM

Quote:

If these two axes [the mechanical tube axis and the optical axis] are at an angle, it induces pointing and tracking errors

Yes, if you care about them (and have a GEM with GoTo or use digital setting circles).

But it doesn't introduce axial miscollimation, i.e. aberrations or defocus in the field.

BTW, it doesn't introduce tracking errors on my scope. I have an equatorial platform and its polar axis is quite independent from the axes of the Dobsonian mount and their squareness.

It also doesn't introduce tracking errors on mounts that don't track at all, obviously, and as I star hop and do not use DSCs, it also doesn't introduce pointing errors on my scope either .

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wh48gs
Pooh-Bah

Reged: 03/02/07

Re: Useful info about secondary mirror alignment [Re: Starman1]
#3372227 - 10/05/09 09:47 AM

Don,

It is an unvolved procedure to grasp. The main reason I got into this thread is to make sure the page on Newtonian collimation that I added just recently got it right. Good move for me, since Jason's illustrations and Alexis's comment made me obvious an inconsistency that led me to find and correct an oversight. Seems that we all got something from this (except Alexis, who doesn't sound all that happy ).

Vla

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wh48gs
Pooh-Bah

Reged: 03/02/07

Re: Useful info about secondary mirror alignment [Re: sixela]
#3372276 - 10/05/09 10:10 AM

Quote:

You've got it slightly wrong. #3 uses a Cheshire, and *perfectly* overlaps the optical axis and focuser axis and their reflections (even though the intercept angle on the secondary is not 90° and the optical axis is not along the tube axis), because you make the reflected optical axis cross the focuser axis half way between the Cheshire ring and the pupil.

How do they cross if they perfectly overlap? Nothing gets really perfect, except on our illustrations. One of the reasons is that actual mirror surfaces, when tilted, do not rotate around the point of their intersections with the axis, but around a point effectivelly displaced from it (particularly diagonal). As a result, any tilt introduces decenter, which makes the surfaces and reflections less than perfectly concentric.

Quote:

Using the reflected secondary's silhouette is a bad idea, and no sane protocol would use it as a reference.

Lots of insanity around: Sidgwick (p204), Thompson (p120), Lecleire (p262), Norton's Star Atlas (p72)...

Vla

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wh48gs
Pooh-Bah

Reged: 03/02/07

Re: Useful info about secondary mirror alignment [Re: sixela]
#3372285 - 10/05/09 10:15 AM

Quote:

It also doesn't introduce tracking errors on mounts that don't track at all, obviously, and as I star hop and do not use DSCs, it also doesn't introduce pointing errors on my scope either.

That's nothing. I have a small Tal Newtonian that never needs collimating at all, because I never use it. Isn't that as good as it gets?

Vla

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Vic Menard
Post Laureate

Reged: 07/21/04

Re: Useful info about secondary mirror alignment [Re: sixela]
#3372358 - 10/05/09 11:02 AM

Quote:

Quote:

If these two axes [the mechanical tube axis and the optical axis] are at an angle, it induces pointing and tracking errors

Yes, if you care about them (and have a GEM with GoTo or use digital setting circles).

Not necessarily. If the plane defined by the two lines (optical axis and OTA axis) is perpendicular to the axis of rotation (declination or altitude), then the axial offset will have no impact on DSC performance.

Even when the plane is skewed, the error is often small enough to fall within the resolution of most DSC displays (+/-0.1-degree). The Argo Navis (used with 8000 or 10000 count encoders) pointing accuracy does have a higher resolution, but the AN can also model the pointing error and correct it on the fly...

That said, I've used an f/4.1 with a Sky Commander and an f/4.0 with a Argo Navis, both aligned using the "New Model" (full offset with centered secondary mirror) and, selecting good alignment pairs, have found the pointing accuracy normally places the selected object within 0.1- or 0.2-degrees from the center of the fov. The worst case scenario is when the selected object is within a few degrees of the zenith--the possible azimuth error can be very large, and still fall within the fov of a widefield eyepiece! Remarkably, once the object has been centered, tracking is largely unimpaired.

FWIW--I think the first time I used the phrase, "the new model", was in 1998 (although back then I suggested mechanically tilting the focuser to ensure a 90-degree intercept). I was getting quite a bit of opposition from those who felt collimation was not accurate unless it was either centered or offset. Choosing a model that was both was probably enough to start some debate--but I didn't stop there, going on to assure those who would follow the procedure that the focuser need not be precisely perpendicular to the OTA, the intercept angle need not be exactly 90-degrees, and the primary mirror need not be perfectly centered in the OTA. The goal, of course, was precise axial (focuser and primary mirror) alignment with an optimal secondary mirror alignment (best available offset and minimal skew).

Considering lightweight, open truss OTAs might have small eccentricities or tilt errors from the lower mirror section to the upper focuser section, and the potential for small geometry errors in the upper focuser section that might impact roundness, squareness, and centration, and then adding that the secondary mirror might not be a true sqrt2 major axis ellipse...it seems prudent to focus on the optical alignment first, and then address the mechanical alignment issues as/if they arise.

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Starman1
Vendor (EyepiecesEtc.com)

Reged: 06/24/03

Loc: Los Angeles
Re: Useful info about secondary mirror alignment [Re: wh48gs]
#3372387 - 10/05/09 11:21 AM

Quote:

Quote:

You've got it slightly wrong. #3 uses a Cheshire, and *perfectly* overlaps the optical axis and focuser axis and their reflections (even though the intercept angle on the secondary is not 90° and the optical axis is not along the tube axis), because you make the reflected optical axis cross the focuser axis half way between the Cheshire ring and the pupil.

How do they cross if they perfectly overlap? Nothing gets really perfect, except on our illustrations. One of the reasons is that actual mirror surfaces, when tilted, do not rotate around the point of their intersections with the axis, but around a point effectivelly displaced from it (particularly diagonal). As a result, any tilt introduces decenter, which makes the surfaces and reflections less than perfectly concentric.

Quote:

Using the reflected secondary's silhouette is a bad idea, and no sane protocol would use it as a reference.

Lots of insanity around: Sidgwick (p204), Thompson (p120), Lecleire (p262), Norton's Star Atlas (p72)...

Vla

It should be pointed out that the silhouette (reflected) image of the secondary grows more concentric with all other images as the f/ratio gets longer. The illustration that Jason posted is slightly exaggerated for f/6 and longer scopes. Many of the sources you mention were written back when most reflectors were f/8 to f/12.
It might also be pointed out that standard collimation techniques of the '50s and '60s did not always call for even edge-of-field illumination. As late as 1967-1968, books by popular authors such as Sam Brown of Edmund Scientific called for a centered secondary with all images concentric except for the reflected primary, which appeared dropped toward the lower end of the secondary.
The interesting thing is that there are so many different mechanical alignments that can result in good optical collimation. It is when we aim to optimize field illumination and maintain coincidence between optical and mechanical axes that things get more complicated.

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Jason D
Postmaster

Reged: 10/21/06

Loc: California
Re: Useful info about secondary mirror alignment [Re: Vic Menard]
#3372492 - 10/05/09 12:29 PM

Quote:

Not necessarily. If the plane defined by the two lines (optical axis and OTA axis) is perpendicular to the axis of rotation (declination or altitude), then the axial offset will have no impact on DSC performance.

Hi Vic, I believe we discussed this specific point looooong time ago and the end result of the discussion was that the above error can be corrected but it will have to be by the DSC computer. However, if the DSC computer does not have the proper software to make this kind of correction, the error will impact accuracy.
Jason

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Jason D
Postmaster

Reged: 10/21/06

Loc: California
Re: Useful info about secondary mirror alignment [Re: wh48gs]

Quote:

It is an unvolved procedure to grasp. The main reason I got into this thread is to make sure the page on Newtonian collimation that I added just recently got it right. Good move for me, since Jason's illustrations and Alexis's comment made me obvious an inconsistency that led me to find and correct an oversight

Vla,
When I got interested to learn more about the theory of collimation years ago, I made the common mistake of thinking the focuser and primary optical axes have to be at 90 degree angle. Any thing other than 90 degree would introduce an error. Then Vic Menard corrected me.

Focuser and optical axes do not have to be at a right angle.

Here is a thought exercise. You have two pool table setups. In each setup, you strike a billiard ball through a partition with a hole. You do not know what is on the other side of the partition. Each setup has two reflecting walls positioned carefully at different angles. In each case, the billiard ball will retrace its path back to the starting point identically. Even though one setup does not have a "right" angle, you will not be able to tell which setup it is.

Jason

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Vic Menard
Post Laureate

Reged: 07/21/04

Re: Useful info about secondary mirror alignment [Re: Jason D]
#3372694 - 10/05/09 01:55 PM

Quote:

Quote:

Not necessarily. If the plane defined by the two lines (optical axis and OTA axis) is perpendicular to the axis of rotation (declination or altitude), then the axial offset will have no impact on DSC performance.

Hi Vic, I believe we discussed this specific point looooong time ago and the end result of the discussion was that the above error can be corrected but it will have to be by the DSC computer. However, if the DSC computer does not have the proper software to make this kind of correction, the error will impact accuracy.
Jason

I just can't think of any DSC computer that doesn't work that way. The ones I've used employ a basic 2-star alignment procedure. If you choose one star at the zenith (90-degrees elevation) and one at the horizon (0-degrees elevation) and use the telescope optics to align to them...

Of course, it's unlikely you would choose two stars with such widely separated elevations, but then whichever two stars you do choose, once their elevation and azimuth has been correlated to their known right ascension and declination, the software will know where the zenith and the horizon are--even if the telescope horizon (defined by the azimuth bearing) differs from the actual horizon.

The Argo Navis manual refers to this alignment error as a "collimation error". In an Alt/Az configuration, it's "...the non-perpendicularity between the pointing axis and the Alt axis." They further note that this error results in a "...left-right shift in the sky that is constant for all Altitudes" and "...there will be an area around the pole (zenith) of the scope that the scope cannot point." This only happens when the optical axis is offset (skewed) away from the perpendicular plane--if it's offset and parallel to the plane, or falls inside the plane, the error will be compensated for when the DSCs are initialized (and the area around the pole will be accessible).

(The Argo Navis also uses an altitude reference (usually 90-degrees), but this is only an estimate and is referenced to the telescope (not the zenith). With the horizon stop turned on, it's possible to have the Argo Navis tell you that the object you have selected is below the horizon, when in fact, you can manually override the GoTo feature and view the object in the telescope (a common occurrence at the WSP when observing Eta Carinae or Alpha Centauri).)

Edited by Vic Menard (10/05/09 02:00 PM)

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Jason D
Postmaster

Reged: 10/21/06

Loc: California
Re: Useful info about secondary mirror alignment [Re: Vic Menard]
#3372825 - 10/05/09 03:05 PM

Quote:

I just can't think of any DSC computer that doesn't work that way. The ones I've used employ a basic 2-star alignment procedure. If you choose one star at the zenith (90-degrees elevation) and one at the horizon (0-degrees elevation) and use the telescope optics to align to them...

I wonder if low cost DSC computers such as Orion utilize such a correction in their software.
Mechanical/optical axes misalignment does not alter the angular distance between stars. As long as the “actual” angular distance between the two alignment stars matches the database, then it is a matter of simple math to translate, back-and-forth, between the actual DSC coordinates and the database coordinates.
If the angular distance is not the same then the discrepancy is caused by other reason(s) than optical/mechanical axes misalignment.
Jason

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Vic Menard
Post Laureate

Reged: 07/21/04

Re: Useful info about secondary mirror alignment [Re: Jason D]
#3372932 - 10/05/09 03:57 PM

Quote:

...As long as the “actual” angular distance between the two alignment stars matches the database, then it is a matter of simple math to translate, back-and-forth, between the actual DSC coordinates and the database coordinates.

Exactly--although that's not exactly how the DSCs are initialized. Generally speaking, for best precision, the alignment stars are opposite each other on a line drawn through the zenith (180-degree change in azimuth), and are thirty or so degrees different in elevation, not too close to the zenith, not too close to the horizon. That's enough for the DSC computer to do the coordinate conversion.

Quote:

If the angular distance is not the same then the discrepancy is caused by other reason(s) than optical/mechanical axes misalignment.

But that's not the case. Given a scope with the offset optical axis pointing one degree above the mechanical axis, and using the mechanical axis as the reference, a star on the horizon (centered in the eyepiece) would be minus one degree elevation and a star at the zenith would be 89-degrees elevation. The difference is 90-degrees and the computer easily makes the adjustment. But if the optical axis is offset one degree to the side, then the star on the horizon is still at zero degrees elevation, but the star at the zenith cannot be pointed to! You can raise the scope 90-degrees (mechanically) to the zenith, but optically the scope will be pointing 89-degrees--one degree away from the zenith and 89-degrees from the starting azimuth (as viewed in the eyepiece). This requires fairly complicated coordinate conversion remodeling, along with the fact that there's a 2-degree diameter hole at the zenith that is inaccessible to the optical axis.

Of course, the actual offset difference is typically much smaller than a whole degree and, since the coordinate conversion is modeled on stars a good distance from the horizon and zenith, the pointing accuracy (especially away from the zenith) is usually quite good.

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sixela
Postmaster

Reged: 12/23/04

Loc: Boechout, Belgium
Re: Useful info about secondary mirror alignment [Re: wh48gs]
#3373159 - 10/05/09 05:55 PM

Quote:

How do they cross if they perfectly overlap?
[µ/quote]
I'm talking about the theory. You make the axes cross at the focal plane and at the primary mirror's centre, and then they overlap everywhere (and "cross" everywhere).

Quote:

As a result, any tilt introduces decenter, which makes the surfaces and reflections less than perfectly concentric.

In real life, you'll find many procedures are iterative. But of course, once you're within the proper tolerances, you stop, rather than take an infinite number of extra steps .

Quote:

Lots of insanity around: Sidgwick (p204), Thompson (p120), Lecleire (p262), Norton's Star Atlas (p72)...

No argument from me. I'd bet they didn't try it on a large f/3.66 Newt, though.

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