# Optics

With CCD detectors being so much smaller than 35 mm film, which optics to use for which objects has to be reconsidered. Some truths still hold, namely that point source detection improves with the square of the aperture while extended source detection improves in proportion to the square of the f ratio (the focal length divided by the aperture).

The choice of lens must depend on the size of the object and possibly on the required angular resolution. So it may be handy to have a table like the one below, listing the field of view and angular resolution for the lenses available.

The mathematics of this is quite simple. You may have to use an image taken with a lens of reliably known focal length to determine the pixel size. Also determine whether the pixels are actually square. Once the pixel size is known you can similarly determine the focal lengths of other lenses. I found that my 2x photo adapter does not always double the focal length.

The relationship between a certain length d on the detector close to the optical axis and the corresponding angle p on the sky is

p = 2 atan[d / (2 f)] ~ d / f

The aperture of the lens together with the wavelength of the light determines the angular resolution of the optics. For 500 nm wavelength (green light) this is

a = 610 nm / D

If your lens arrangement involves a telescope with eyepiece projection then
the focal length is calculated from the primary focal length f_{1}, the
focal length of the eyepiece f_{2} and the distance of the detector
from the eyepiece x:

f = f_{1} [(x/f_{2}) - 1]

If x is more (less) than twice the eyepiece focal length then the eyepiece acts as a focal extender (reducer). The larger x or the shorter the eyepiece focal length the more focal extension you achieve.

If you have a camera with macro lens it may be possible to use it instead of the eyepiece. Usually it will then be a focal reducer, though.

D | f | f/D | a | pixel | FOV | foc |
---|---|---|---|---|---|---|

3.0 mm | 6.0 mm | 2.0 | 0.70' | 3.2' | 33° x 25° | end |

10 mm | 28 mm | 2.8 | 0.21' | 0.69' | 7.3° x 5.5° | |

27.8 mm | 50 mm | 1.8 | 4.5" | 23" | 4.1° x 3.1° | 1.5 m |

27.8 mm | 90 mm | 3.2 | 4.5" | 13" | 2.3° x 1.7° | 6 m |

48.2 mm | 135 mm | 2.8 | 2.6" | 8.6" | 1.5° x 1.1° | |

63.5 mm | 400 mm | 6.3 | 2.0" | 2.9" | 31' x 23' | +∞ |

63.5 mm | 770 mm | 12.1 | 2.0" | 1.5" | 16' x 12' | +∞ |

200 mm | 2000 mm | 10.0 | 0.63" | 0.58" | 6.2' x 4.6' | 1.5 |

200 mm | 3500 mm | 17.5 | 0.63" | 0.33" | 3.5' x 2.6' | 1.5 |

D | f | f/D | a | pixel | FOV | foc |
---|---|---|---|---|---|---|

1.8 mm | 3.5 mm | 2.0 | 1.2' | 8.1' x 7.5' | 41° x 29° | |

10 mm | 28 mm | 2.8 | 0.21' | 1.01' x 0.93' | 5.4° x 3.7° | -∞ |

27.8 mm | 50 mm | 1.8 | 4.5" | 34" x 31" | 3.0° x 2.1° | 44 cm |

27.8 mm | 90 mm | 3.2 | 4.5" | 19" x 17" | 1.7° x 1.2° | 1.2 m |

48.2 mm | 135 mm | 2.8 | 2.6" | 13" x 12" | 67' x 47' | |

63.5 mm | 400 mm | 6.3 | 2.0" | 4.2" x 3.9" | 23' x 16' | 20 m |

63.5 mm | 770 mm | 12.1 | 2.0" | 2.2" x 2.0" | 11.7' x 8.1' | 20 m |

200 mm | 2000 mm | 10.0 | 0.63" | 0.85" x 0.78" | 4.5' x 3.1' | +1.8 |

200 mm | 3500 mm | 17.5 | 0.63" | 0.48" x 0.45" | 2.6' x 1.8' | +1.8 |

D | f | f/D | a | pixel | FOV | foc |
---|---|---|---|---|---|---|

3.0 mm | 6.0 mm | 2.0 | 0.70' | 6.4' | 33° x 25° | end |

10 mm | 28 mm | 2.8 | 0.21' | 1.4' | 7.3° x 5.5° | |

27.8 mm | 50 mm | 1.8 | 4.5" | 46" | 4.1° x 3.1° | 1.5 m |

27.8 mm | 90 mm | 3.2 | 4.5" | 26" | 2.3° x 1.7° | 6 m |

48.2 mm | 135 mm | 2.8 | 2.6" | 17" | 1.5° x 1.1° | |

63.5 mm | 400 mm | 6.3 | 2.0" | 5.8" | 31' x 23' | +∞ |

63.5 mm | 770 mm | 12.1 | 2.0" | 3.0" | 16' x 12' | +∞ |

200 mm | 2000 mm | 10.0 | 0.63" | 1.16" | 6.2' x 4.6' | 1.5 |

200 mm | 3500 mm | 17.5 | 0.63" | 0.66" | 3.5' x 2.6' | 1.5 |

- D: Aperture.
- f: Focal length. f = 90 mm, 770 mm and 3500 mm are realised as the next shorter focal length combined with a 2x photo adapter. The actual lengthening factor of the adapter is 1.8x, 1.9x and 1.75x resp.
- f/D: Focal ratio.
- a: Diffraction limit at 500 nm wavelength.
- pixel: Angular size of a pixel.
- FOV: Angular size of the field of view.
- foc: This indicates roughly the focus setting required. This depends on the distance of the detector from the lens and on the focussing mechanism. For the ToUcam Pro this is for the hmecam III camera, for the QuickCam VC the hmecam I camera. For the photo lenses the number is the distance setting as displayed on the lens. For the telescope it is the number of right turns of the focussing knob compared to focus for the 40 mm eyepiece with zenith prism and the spherical abberation of my right eye. -∞ means focus is impossible (beyond the +∞ setting).

One important factor in choosing a lens is angular resolution. The ultimate
limit here is about 1" and is due to the atmosphere. From the table it is
clear that the telescope is ideal for this objective. Its aperture of
200 mm has a diffraction limited resolution better than the atmospheric
limit without overdoing this. A large aperture will simply gather more light,
but not improve resolution. In order that the pixelated images are limited by
the atmosphere or telescope but not by the pixel size, there should be about
three pixels per resolution element (cf. e.g. Kristen Rohlfs, 1986, *Tools
of radio astronomy,* Springer, Berlin, Heidelberg, New York, p. 96, or
http://micro.magnet.fsu.edu/primer/java/digitalimaging/processing/samplefrequency/index.html).
Less means that the pixels undersample and resolution is limited by the pixel
size. More means that the pixels are unnecessarily small and oversample.

The Philips ToUcam Pro VGA on the Celestron 8 with the 2x photo adapter is therefore ideal and longer focal lengths have no benefit. SIF detectors would marginally undersample the image, wherefore it might be worthwhile to use eyepiece projection for even longer focal length.

A more obvious factor in choosing a lens is the field of view. The primary guideline is the angular size of the object to make best use of the few pixels we have in our detector. However, there is a catch here: As you change to longer focal lengths and smaller fields of view, you will detect fewer stars. Although with the larger aperture you catch fainter stars, detectable stars are becoming rarer at a faster rate. If you don't have any star well detected in each frame of a stack, then alignment becomes difficult.

Finally, how fast the lens is, is an important issue. For point sources such as stars, the aperture determines how long it takes to detect a star of given brightness. If this is the only factor, the telescope does best service. Similarly a tele lens will be faster than a standard lens. For extended objects such as nebulae, star clusters and galaxies the f ratio determines how long it takes to detect them. In these terms shorter focal lenghts tend to give better f ratios.

Copyright © 2003 Horst Meyerdierks

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