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The Focus Position Problem.
One project at the observatory is a spreadsheet containing focus numbers for each
eyepiece and various accessories. At first glance one might expect a linear
relationship between the focus number and the position of the focal plane. That is
not correct. This means we must use the telscope to measure the actual
focus number for each combination of accessories. Simply using a different adapter
ring will change the focus number. If we can find a way to relate the focus number
to the position of the focal plane of the telescope, it should be possible to
simplify the creation of focus settings for new equipment.
I believe we can measure the location of the focal plane of an eyepiece fairly easily.
Other accessories like 1.25 adapters, the variable density filter and the ARE should
have a single length that can be added to the focal plane of an eyepiece to determine
the focal plane position of a combination of items in use. I am not sure about barlows,
they may not have a single effective length for all eyepieces.
The problem is to relate the focus number to the position of the focal plane of the
telescope. After reviewing a number of possible ways, I think the easiest is to
create a mathematical model of the telescope.
The Solution
The focus number appears to track the position of the secondary mirror, which has a
hyperbolic shape that will move the focal plane more than the movement of the mirror
would alone.
The image to the right is a slice through the center of the telescope.
We are looking at the path of a single ray of light, from a point source at infinite
distance, located at the center of the field of the telescope. This is a common
simplification when analyzing optical behavior, and is the same thing as looking at
a star.
The large curved
blue line represents the primary mirror, and the two smaller blue lines represent the
secondary mirror in each of its limit positions. The dark red line shows the center of
the telescope. The green lines, A and B represent an incoming light ray, and its
reflection from the primary. The orange line C shows the path of that ray when the
secondary mirror is as close as possible to the primary. The yellow line is the path
with the secondary as far as possible from the primary. Note that the focal plane
position, where lines C and D cross the center line, moves more than the mirror.
First Steps
Lines A and B are the same for any position of the secondary, so we solve it one time
and use it later. Our goal is to find where our selected ray intersects the secondary.
Any ray from an infinitely distant source at the center of the field should be
acceptable, I chose one near the outer edge of the mirror.
The proper way to find the slope of ray B is to find a line tangent to the surface
of the mirror, but for a hyperbola that takes calculas that is beyond me. I believe
I have another way. Find two points on the surface of the primary, equidistant from
the location where ray A strikes the mirror. Now find the slope of this line, and
the angle of a line perpendicular to it. Ray B will leave the primary mirror the
same distance in degrees, on the opposite side of this perpendicular line as Ray A
strikes it. (Note: The curve and angles are exaggerated in the image.)
The hard part
I think the intersection of ray B and the secondary will be linear in relation to the
finder position number. At worst, I may have to iterate to adjust for the curve of the
secondary when computing focuser number from desired focal plane location. The next
part of the project will be to relate the focuser position number to the actual distance
between the primary and secondary mirrors. This may have to be determined with a tape
measure. We only need to consider the actual range of secondary position.
The complicated part of the project happens when ray B hits the secondary. Its surface is
hyperbolic and is the place where the non-linearity in focus position originates. I think
this should be the same problem as the reflection of A to B. The position on the secondary
will be based on where ray B hits the surface. Once we have the slope of the ray when it
leaves the secondary, the focuser distance can be computed as the place where this line
crosses the center line of the telescope.
Final Steps
The final steps in this project will be to produce two equations, or algorithms that
relate focus number to position of the focal plane, and position of the focal plane to
the focus number. I expect that once I have a model of the telescope, the rest will
be algebra.
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