# Focal Ratio

## Overview

The ratio of a telescope's aperture to focal length is known as its focal ratio. The smaller this ratio, the "shorter" or "faster" the telescope is said to be. The higher this ratio, the "longer" or "slower" the telescope is said to be. Phrases like "fast scope" or "long focal ratio" or "telescope speed" all refer back to its focal ratio number.

The focal ratio itself isn't as important as the telescope's aperture or its focal length, but it does play a key role in describing the various aberrations that can be present when looking through the telescope.

## Focal Ratio Effects

### Focal Ratio and Astrophotography

The "speed" nomenclature of the focal ratio stems from photography and is analogous to the F-stop setting on a camera lens. A small F-stop number means a faster shutter speed can be used, while a large F-stop number means a longer shutter speed needs to be used.

This same concept applies to astrophotography. The faster the telescope is (e.g. the shorter its focal ratio), the less time you need to expose your camera in order to get a given signal level on the sensor. The slower the telescope (e.g. the longer its focal ratio), the longer you need to expose your camera to get the same signal.

This relationship between focal ratio and exposure time is based on the square of the focal ratio. An F/10 focal ratio requires four times the exposure time of an F/5 focal ratio to get the same signal.

This relationship also holds true across apertures. A 16" aperture at F/10 will still require four times the exposure length to get the same signal as even a 4" aperture at F/5. While this sounds counter-intuitive that a significantly smaller aperture can still require 4x shorter exposure, it's important to note that focal ratio is always relative to both the aperture and focal length of the telescope. F/10 always means that light will travel 10x farther than the size of the aperture, while at F/5 it means light will always travel only 5x farther than the size of the aperture. This means, relative to the aperture, F/10 will always spread the light out 4x more than at F/5, thus the amount of light that falls on a given pixel on a sensor is always 4x weaker.

So what role does aperture play in astrophotography? It allows for greater image scale a given focal ratio. Compare a 16" F/5 telscope against a 4" F/5 telescope. Each scope has the same focal ratio, so they will require the same exposure length to get the same level of signal. However, the 16" F/5 scope has a focal length that is 4x longer than the 4" F/5 scope, thus it can produce 4x the image scale. If the 4" scope tried to match that same image scale, a 4x barlow would be required and it would make the effective focal ratio of the 4" scope F/20, so it would require 8x the exposure length as the 16" scope.

It should be noted that the focal ratio isn't the *cause* of the increased or decreased exposure time (ultimately it's the aperture and how long of a focal length the light collected by the aperture has to travel), but it's merely a good descriptor for how much more quickly one scope will acquire good signal than another.

### Focal Ratio and Visual Astronomy

When it comes to visual astronomy, the focal ratio doesn't have any direct effects like focal length does on magnification, or aperture does on brightness, but it does affect how well an eyepiece performs.

A short focal ratio means light rays converge to the focal plane more steeply than a long focal ratio. This steep angle of convergence must be straightened out or re-bent by the eyepiece in order to form a magnified, focused image. However, this process of refracting the light from the telescope can distort it, which can cause an aberration known as astigmatism to manifest near the edges of the field of view. The steeper this converging angle, the more aggressively it needs to be refracted, and thus the more extreme the aberration becomes. To eliminate this aberration requires eyepieces with more lens elements and elements with more complex shapes, which drives up cost and weight.

Thus cheaper eyepieces often work well on long focal ratio telescopes (F/10 and longer), but increasingly poorly on shorter focal ratio telescopes. Only the most expensive "premium" eyepieces are designed to work well with short focal ratio telescopes.

### Focal Ratio and Refracting Telescopes

Refracting telescopes with short focal ratios can suffer from chromatic aberration without the use of additional, low dispersion glass elements in the objective lens assembly. An achromatic refractor often requires a long focal ratio in order to minimize the effects of chromatic aberration, while an apochromatic refractor can get away with shorter focal ratios since it uses special glass that does a better job of preventing light from splitting up into its constituent colors than standard achromats do. Some apochromatic refractors also make use a 3-element objective assembly to further shorten the focal ratio without causing too much chromatic aberration.

### Focal Ratio and Parabolic Newtonian Reflectors

The parabolic mirror in a Newtonian reflector produces an aberration known as coma, which gets worse the shorter the focal ratio of the mirror gets. This relationship is based on the cube of the focal ratio. An F/5 mirror produces 8x the coma as an F/10 mirror. Even an F/4 mirror produces nearly twice the coma as an F/5 mirror.

### Focal Ratio and Spherical Newtonian Reflectors

Newtonian reflectors that use spherical mirrors are especially sensitive to focal ratio. The shorter the focal ratio, the more pronounced the spherical mirror's inability to focus light to a single point becomes.

Using the formula, `e = 22 D / F^3` where `e` is the wavefront error, `D` is the diameter of the telescope's aperture in inches, and `F` is the focal ratio of the telescope, you can see that the wave front error is based on the cube of the focal ratio. This means as the focal ratio goes down, the wave front error goes up by the cube.

Examples:

• 5" F/5 telescope produces 0.88 waves of spherical aberration.
• 5" F/10 telescope produces just 0.11 waves of spherical aberration.
• 10" F/10 telescope produces 0.22 waves of spherical aberration (as you can see, spherical mirrors require you to increase focal ratio when increasing aperture, in order to keep the wavefront error down)

For reference, barely passable diffraction-limited optics is 0.25 waves, while excellent optics are 0.1 waves. Thus the 0.88 waves of spherical aberration from 5" F/5 telescope is very bad, but the 8x smaller spherical aberration of a 5" F/10 scope is acceptable.

Note that spherical aberration is in addition to other errors present on the mirror, so just because a 5" spherical mirror might only have 0.11 waves of spherical aberration, that doesn't mean the total wavefront error is only 0.11 waves. That said, at F/10, the spherical aberration is effectively too insignificant to matter.