Reducers, Barlows and spacing: why the factor depends on distance

One equation for both

Celestron’s focal reducer article 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). The Orwell Astronomical Society’s Barlow page gives the Barlow’s amplification as 1 + x ÷ f, where x is the distance from the Barlow lens to the new focal plane and f is the size of its (negative) focal length. Written side by side:

reducer: R = 1 − d ÷ f_reducer    Barlow: M = 1 + d ÷ |f_Barlow|

These are one thin-lens relation, factor = 1 − d ÷ f, with the sign of the lens’s focal length flipped: positive for a reducer, negative for a Barlow (derived from the thin-lens equation). In both, d runs from the lens to the focus your camera or eyepiece sits at, and the new focal length is the native one times the factor. The reducer and Barlow calculator accepts either a factor or these distances.

Why spacing changes the factor

The distance appears in the formula, so moving the camera changes the factor. Celestron’s classic 0.63× reducer has a working distance of 105 mm, measured from its rear threads. Celestron does not publish the lens’s focal length, but if you treat 105 mm as d, the formula implies a focal length near 284 mm, and the factor then moves by about 0.035 for every 10 mm (thin-lens estimate, computed):

Distance to sensor	Factor	C8 focal length	Focal ratio
85 mm	0.70	1423 mm	f/7.0
105 mm (design)	0.63	1280 mm	f/6.3
125 mm	0.56	1137 mm	f/5.6

Treat these as an illustration of the trend. The rear threads are not the lens’s optical center, so the real factors differ, and Celestron adds that an SCT focuses by moving its primary mirror, which changes the back focus and the effective focal length when a reducer is fitted. If you need the exact value, a plate solver such as Astrometry.net returns the image scale of a real frame, and focal length = 206.265 × pixel size ÷ scale.

Barlows: more distance, more magnification

For a Barlow, a 2× factor means the focal plane sits one Barlow focal length behind the lens, since 1 + f ÷ f = 2. OASI gives typical Barlow focal lengths of 75 to 100 mm (negative) and notes that a star diagonal adds a little more than its fitting size to the light path, about 35 mm for 1¼-inch and 60 mm for 2-inch. For a Barlow of −100 mm focal length (an assumed value at the top of OASI’s range), the formula gives (computed):

Barlow to focus	Factor
100 mm (as designed)	2.00×
135 mm (+ 1¼-inch diagonal)	2.35×
160 mm (+ 2-inch diagonal)	2.60×

OASI reports its own case: lengthening the path from about 110 mm to about 130 mm raised a Barlow from 2.0× to around 2.2×, and 1 + 130 ÷ 110 = 2.18 agrees. The second form on the same page, M = 1 ÷ (1 − d ÷ f) with d measured inside the original focus, shows the limit: as the lens moves in toward one focal length from focus, the factor grows without bound and the image can no longer be brought to focus. OASI also cautions that a Barlow designed for 2× may not work well far from that factor.

Designed back focus

The thin-lens formula describes the factor; it says nothing about image quality. Celestron states that every focal reducer works at a specific working distance, typically 55 mm for reducers made for f/7 or f/8 refractors, and that putting the sensor more than a few millimeters beyond it will “degrade image quality”, often as distorted stars near the edge of the field. The 0.7× reducer for the EdgeHD 800 lists an optimal back focus of 105 mm and is optimized for APS-C sensors. Camera makers publish adapter stacks to hit such distances; the ZWO ASI2600 page includes connection diagrams for 55 mm. The back focus guide covers how to add up the spacers.

What a faster focal ratio does

Celestron says a 0.63× reducer makes the image from an 8-inch f/10 SCT about 2.5 times brighter, and (1 ÷ 0.63)² = 2.52 (computed). That brightness is per unit area of the focal plane. The aperture, and so the light collected from an object, does not change; the reducer squeezes an extended object’s image onto 0.63² ≈ 40 % of the area, so each pixel receives about 2.5 times as much (derived from the geometry). The Sheffield optics notes state the same principle from the other side: at a fixed aperture, a longer focal length spreads the light over a larger area.

Two makers mark the limits. Tele Vue lists “faster telescopes show brighter images” as a myth for visual use, since equal apertures at equal magnification give the same image brightness, while agreeing that in photography fast f-ratios mean shorter exposures for extended objects. Diffraction Limited says the fast-f-ratio advantage applies only to distributed sources: a star already smaller than a pixel gets no brighter by shrinking further.

The calculator’s exposure ratio, (N₁ ÷ N₀)², answers one question: how much time the new setup needs to put the same signal from an extended object into each pixel. The price is image scale, since each pixel now covers (1 ÷ R)² more sky.

Worked example: EdgeHD 8 with its 0.7× reducer

Celestron’s table gives the EdgeHD 8 as 2032 mm native and 1422 mm with its 0.7× reducer. With a 3.76 µm APS-C camera (all computed):

	Native f/10	With 0.7× (f/7)
Focal length	2032 mm	1422 mm
Image scale	0.38″ per pixel	0.55″ per pixel
Field (23.49 × 15.70 mm sensor)	39.7′ × 26.6′	56.8′ × 37.9′
Sky area per pixel	1×	2.04×
Signal per pixel, extended object	1×	2.04×

A galaxy delivers the same total light through the same 203 mm aperture in both cases. With the reducer that light lands on fewer pixels, each holding about twice as much, and the frame gains about 43 % in width. The EdgeHD and SCT page lists the reducers for each tube.