I've been giving myself a crash course in optics trying to understand how to set up the optical train for occulation timings. Actually, for the primary system I'm hoping to use, "Scotty's Mighty Mini," the IOTA Yahoo group has been very helpful where I can just build it from the recommended parts and go. But... if I want to use a different primary from the 50mm binocular objective, it pays to be able to calculate the correct spacings for the focal reducer used.

There's a useful primer on the RC Astro site on how to optimally place a focal reducer *if you know its focal length*. But what if you don't? Uhm, that's harder.

Here are a few 1.25" focal reducers and the nominal focal length's I've been able to find.

Model |
Reduction |
Focal Length |
Clear Aperture |
Information Source |

Antares 1.25" | 0.5x | 95mm | 23mm | http://agenaastro.com/ |

GSO 1.25" | 0.5x | 102mm | 22mm | http://agenaastro.com/ |

Orion 1.25" | 0.5x | ??? | ??? | http://telescope.com/ |

OWL 1.25" | 0.5x | 76mm | 26mm | https://groups.yahoo.com/neo/groups/IOTAoccultations |

The ScopeStuff branded focal reducer (based on the pictures on their web site) is really the GSO. But a quick-and-dirty measurement of it's focal length indicates it has a focal length of about 96mm.[1]

The biggest thing here is just how short the focal length of the OWL adapter is. This short length makes it easier to work with for fast systems like the Orion 9x50 finder objective (183mm FL) or the Tasco 10x50 binocular objective.

A focal reducer is designed to be placed a certain distance from the focal plane in order to get the nominal reduction; that distance is called the "working distance" of the reducer and is simply $D=f(1-R)$ where $f$ is the focal length of the reducer and $R$ is the nominal reduction. So, for the OWL reducer, the optimum distance to give the 0.5x reduction, the working distance is 38mm. This is *very* close to the distance in use for Scotty's Mighty Mini design. The flange-to-focal plane distance for the C-mount cameras is 12.5mm, the Cs-mount adapter adds another 5.026mm, and the special 0.5" C-mount to 1.25" adapter adds another 12.7mm giving 30.226mm to the back edge of the reducer mount. I estimate 1-3mm from the back edge of the focal reducer to the optical center. That means the design actually puts the reducer ~5mm short of the "ideal" spacing.

It's worth noting that "ideal" here means the working distance that produces the actual 0.5x reduction factor. Putting it a little closer does three things. First, it insures there is no vignetting by the reducer mounting. Second, it gives us a little wider field of view in that off-axis rays would be the first to be vignetted; having the reducer closer to the focal plane means the original light cone will have had more room converge, so that vignetting will also be reduced. Third, it changes the reduction factor.

## Measuring the OWL Focal Reducer Focal Length

I recently had a chance to measure the focal length of the Tasco 50mm binocular objective for which this system was nominally optimized. Ideally, I should simply have some focuser system and then I can adjust the spacing and measure or, even better, take and image and do a plate fit to get the image scale. What I actually was able to do is put in various combination of spacers and adapters to see where I get my best focus. Since I don't have a wide range of spacers, I'm stuck with limited combinations. In a few cases, I attempted unscrewing an adapter by ~1mm, but that comes with issues rated to tilting the focal plane. The whole assembly tends to droop a little when things are not screwed down tight which means the image shifts and I'm measuring off-axis focus which will naturally be a little worse.

I have the following spacers and adapters available:

- 0.65" C-to-1.25" adapter (16.5mm)
- 1" C-to-1.25" adapter (24.5mm)
- 5mm C-to-Cs adapter
- 0.5" long 1.25" extender (12.5mm)
- 1.5" long 1.25" extender (38.1mm)

The best focus came was when using th 1" C-to-1.25" adapter plus th 1.5" extender + two 0.5" adapters for a total of 62.6mm spacing from the back flange of the focal reducer to the camera flange. The C-mount flange to focal plane distance is another 12.5mm giving a total of 74.6mm plus whatever spacing there is from the back flange of the OWL focal reducer to the optical center. Some crude measurements indicate the spacing to the *geometric* center of the lens is probably 3mm. It's a crude measurement, but I think we're safe in assuming the nominal 76mm FL for the OWL reducer is correct (actually, we *know *the OWL reducer is 76mm, the question is whether or not the reducer bought off eBay is the real deal; the answer is probably yes).

## Measuring the Focal Length of the Tasco 50mm Binocular Objective

To do this, I kept attaching extender tubes to the nosepiece of a Meade DSI Pro II until I could achieve focus when the whole thing was attached to the back of the Mighty Mini in place of the PC164-EX2. I strapped the binocular objective onto an Alt-Az mount and pointed the whole thing at Polaris, then took a 20-second exposure and used MaximDL 6 to do a plate fit. This gave a focal length of 184mm. I've done the same thing with the Orion 9x50 objective, and got 183mm. So for all practical purposes, these two are identical. I'd say using one or the other has no significant performance benefit, but the Tasco is clearly cheaper.

This gives an effective f-number of 3.66 (183mm / 50mm). At that focal ratio, the 2nd Airy disk minimum has a diameter of 4.1μm for 500nm green light (2.23 λF/D). In his analysis, Hristro Pavlov mentions he believes the PC164-EX2 uses the Sony ICX419ALL chip which has 8.6μm x 8.3μm pixels, so we're already good in putting all of the light from a star into a single pixel, maximizing our sensitivity. Assuming we're diffraction limited, which is questionable based on some other tests.

Use of the focal reducer for these finders doesn't do anything to improve the sensitivity of the camera, instead it increases the field of view making it easier to point them adequately. Remember, the field of view is more than 2.4°x3.2° with the reducer, and you're trying to point blindly, no finder, no telrad, no anything to point accurately. It's the increase field of view which is the big win.

## Footnotes

- This was done using the image-object distance measurements. I focused a room light onto a wall using the focal reducer and measured the distance from the focal reducer lens center to the wall at 101mm. The distance from the wall to the light was 1968mm (give or take a few) so the distance to the light from the lens was 1867. This gives

\begin{eqnarray}{1\over f} & = & {1\over l} + {1\over l'} \\ & = & {1\over 1867\,{\rm mm}} + {1\over 101\,{\rm mm}} \\ & = & {1\over 95.8\,{\rm mm}} \end{eqnarray}It's hard to see how I can get a > 5mm error here, the image is severely out-of-focus with that much movement. It seems to me either the information at Agena Astro is incorrect or the GSO variation is large, or maybe they've just changed their focal reducers.