Focal reducer and Barlow calculator: effective focal length and spacing

Three ways to enter the change

1. Nominal factor. Type the factor printed on the lens (0.8, 0.63, 2) or, for telescopes in our data, pick the official reducer in the field of view calculator. This assumes the lens is used at its designed spacing.
2. Reducer focal length and spacing. If you know the reducer’s own focal length and the distance from it to the sensor, the calculator works out the factor.
3. Barlow focal length and spacing. The same for a Barlow (negative) lens.

The formulas

Celestron’s knowledge base gives the reduced focal length as the original focal length times (1 − distance from the reducer to the focal plane ÷ focal length of the reducer):

R = 1 − d ÷ f_reducer    effective focal length = F × R

For a Barlow lens, the amplification grows with the distance to the new focal plane, as set out by the Orwell Astronomical Society:

M = 1 + d ÷ |f_Barlow|    effective focal length = F × M

Both are thin-lens results. Real reducers and Barlows are multi-element designs corrected for one spacing; moving the sensor changes the factor as the formulas predict, but image quality at the edges usually suffers away from the designed distance. The back focus guide covers how that spacing is built up from adapters.

As an illustration, a reducer lens with a 240 mm focal length placed 85 mm from the sensor gives R = 1 − 85 ÷ 240 = 0.65. For a Barlow with a focal length of −100 mm, the amplification is 1.5× at 50 mm, 2× at 100 mm and 2.5× at 150 mm.

Worked example: Celestron C8 with the f/6.3 reducer

The C8 has a 203.2 mm aperture and 2032 mm focal length in Celestron’s specifications, and the reducer/corrector is a 0.63× design. Celestron’s article gives its working distance as 105 mm from the rear threads.

	Native	With 0.63×
Focal length	2032 mm	1280 mm
Focal ratio	f/10	f/6.3
Image scale, 3.76 µm pixels	0.38″	0.61″
Field, 1-inch IMX533	19.1′ × 19.1′	30.4′ × 30.4′

Focal ratio and exposure

The calculator also reports the exposure ratio (N_new ÷ N_old)². It follows from geometry: for an extended object, the light falling on each square millimetre of the focal plane scales with (aperture ÷ focal length)², that is with 1 ÷ N². With the same camera, reducing the C8 from f/10 to f/6.3 means each pixel collects about 2.5 times as much light per second from a nebula, so it reaches a given signal level in about 40 % of the time.

Tele Vue makes the same distinction in its list of common telescope myths: a fast focal ratio does shorten photographic exposures of extended objects, while visually, equal apertures at equal magnification give the same image brightness whatever the focal ratio.

That is not the whole story. The reducer does not change the aperture, so the total light collected from the object is the same; it is spread over fewer pixels, each covering more sky. For stars, which are points, the comparison is different again. The reducers and Barlows guide works through these cases.

Checking the result

• Use the sampling calculator to see whether the new image scale suits your seeing: reducers often fix over-sampling on long-focal-length telescopes, and Barlows are mainly for planetary imaging, where very fine image scales are wanted.
• Use the field of view calculator to check that the larger field still fits inside the telescope’s corrected image circle, published for many astrographs in the telescope tables.