Proper Positioning of a Focal
Reducer
on a RitcheyChrétien Cassegrain Telescope
Russell Croman
Introduction
The RitcheyChrétien optical system provides superior
optical performance over a very wide field of view compared to other, more
easily manufactured designs. While focusing an RC telescope can be done in
the usual way using a drawtube focuser at the back of the telescope, many
current designs also employ an electronicallycontrolled secondary mirror
positioning system for fine focus. This enables automatic focus when doing
electronic imaging. Also, the standard focuser can then be replaced by a
more rigid system to directly attach instruments to the telescope,
eliminating flexure from the system.
A consequence of the RicheyChrétien design is that
the spacing between the primary and secondary mirror must be kept very
near the value intended by the optician who figured the mirrors. If this
spacing is allowed to deviate much more than a few millimeters,
significant deterioration of performance will result due to spherical
aberration.
The method of focusing by moving the secondary mirror
is thus at odds with maintaining the performance of the system. The
solution is to locate the instrument, be it a CCD camera, an eyepiece or
what have you, close to the focal point of the telescope with the
secondary mirror at its proper distance from the primary. The secondary
focuser can then be moved only for fine adjustment of the focus.
Adding a focal reducer into the mix complicates
matters. With the reducer in place, where is the new focal point of the
system? How does one determine the correct position of the reducer such
that the system will be nearly in focus without having to move the
secondary mirror from its optimum position?
This paper is an analysis of how to do this.
Focal Ratio and the Angle of the Light Cone
To start the analysis of how to properly position the
focal reducer, we need the concept of the “light cone” and how it relates
to the telescope’s focal ratio. A telescope receives a circular section of
the incident light, and focuses it to a point at some distance, the
effective focal length, from the front aperture. Incoming light rays are
formed into a converging cone, with the apex being at the focal point. A
simplified diagram illustrates this:
The angle of the light cone for a telescope of a
given focal length and aperture is easily found by trigonometry:
where A is the aperture diameter, and F
is the focal length. This can be rewritten in terms of the focal ratio,
:
For example, an f/10 telescope will have a light cone
that converges at an angle of 2.86^{o}.
The Reduction Factor and Working Distance
The amount of reduction (R) produced by a
focal reducer is a function of the focal length (F_{R}) of
the reducer and its distance (D) from the camera’s focal plane:
For example the AstroPhysics 0.75x focal reducer has
a focal length of 700mm. Thus the correct working distance to yield a
0.75x reduction is
Note: the AP web page for the 0.75x reducer states
that the distance from the edge of the 2” adapter should be about 65mm for
0.75x reduction. However, measurement of the reducer and 2” adapter
combination yields a distance of about 60mm from the back of the optics to
the back edge of the 2” adapter. Thus, according to the above computation,
an additional 115mm is needed to achieve 0.75x. Using a 65mm spacing from
the adapter to the focal plane will yield approximately 0.82x reduction.
Focal Reduction Steepens the Light Cone
The action of the focal reducer is to cause the
incoming light rays to converge at a greater angle than they were before
entering the reducer. In other words, it steepens the angle of the light
cone.
The new angle is simply a function of the new focal
ratio, which is the product of the telescope’s native focal ratio and the
reduction factor:
To avoid needing to move the telescope’s secondary
mirror to achieve focus, the focal reducer must be placed at a position
such that its light cone intersects with the native light cone of the
telescope. In other words, the diameter of the new light cone must equal
that of the original light cone at the position of the focal reducer. This
diameter is labeled X in the above diagram. D is the
reducer’s working distance, and Y is the distance from the reducer
to the original native focal point of the telescope.
We can write an equation for X for each light
cone. First, for the focal reducer’s light cone:
Then for the native light cone:
As stated above, X must equal X_{R}
for proper positioning. Thus
or
Recalling that and, the above reduces to
The focal reducer must be placed a distance Y
in front of the native focal point of the telescope. This distance is
simply the ratio of the working distance of the reducer to the reduction
factor.
For the AP 0.75x reducer at a 175mm working distance,
Y is approximately 233mm, or about 9.2 inches. On my 14” f/10
telescope, the native focal point is only 8.54” behind the back plate.
Thus the reducer optics must be located ahead of the back plate,
inside the telescope! Fortunately, the AP 0.75x reducer is designed such
that this will be the case if it is mounted directly to the back plate
using RC Optical Systems’ fixedinstruments adapter.
Actual Working Distance
In practice it is difficult to achieve the exact
working distance needed to yield the specified reduction factor. A more
practical approach may be to try to get close, and then use the achieved
working distance in the above formulation to compute the proper reducer
position. Various spacers included with RC Optical Systems’
fixedinstruments adapter, for example, can then be used to try to achieve
this position.
Recalling that the reduction factor is just
, the above equation for Y can be rewritten as:
For example, here is the calculation of the working
distance for my setup:
Then:
or about 9.33”.
Thus the reducer optics must be 9.33” in front of the
native focal point of the telescope, or about 0.79” in front of the back
plate.
The distance from the front flange of the reducer to
the center of the reducer’s optics is about 1.25”. The fixedinstrument
adapter plate adds about 0.30”. So I need an additional 0.16” (4mm) spacer
in front of the reducer (between the reducer and the fixedinstrument
adapter plate) to get the proper positioning.
The working distance just happens to be almost
exactly what is needed for 0.75x reduction:
Conclusions
The above analysis provides the formulas needed to
compute the correct position of a focal reducer relative to the native
focal point of a RicheyChrétien Cassegrain telescope. Using these
formulas, one can determine for a particular telescope and instrument
setup how to position the reducer such that the system can reach focus
with only minor adjustment of the secondary mirror position. The
performance of the system will thereby be optimized.
John Smith has developed an excellent spreadsheet for
performing the calculations involved in properly positioning a focal
reducer. It appears at the following page, along with handy tables of the
space introduced by various camera elements:
