Magnification, Surface Brightness, & Exit Pupil
1. Basic Calculations
To calculate the magnification of your telescope and eyepiece divide the focal length of the telescope by the focal length of the eyepiece. The smaller the eyepiece’s focal length the higher the magnification for the same telescope.
The focal ratio is the focal length of the telescope divided by the aperture. The lower this number is the “faster” the telescope is considered. For the same focal length and magnification, a faster telescope will have a brighter image. This is also equivalently a telescope with a larger aperture for the same magnification or essentially a larger telescope. The larger the telescope the brighter the image for the same magnification.
The exit pupil is the diameter of the tube of light exiting the eyepiece and entering your eye. It is calculated by dividing the focal length of the eyepiece used by the focal ratio of the telescope. The best exit pupil for deep sky objects tends to be around 2-3mm.
The equivalent aperture is the telescope aperture needed to make an image appear the same surface brightness as the naked eye at a given magnification.
2. Telescopes and Point Sources
Telescopes are typically explained as big light buckets collecting light from space and making dim objects brighter not bigger. This is correct only for objects that can be treated like point sources such as stars. A point source is one where all the light emitted from the source is assumed to originate from a singular point of zero size. For all stars in the sky, their size in relation to their distance makes them so small in apparent angular size that they can be treated as points.
As you would expect point sources appear dimmer as you increase your distance from them due to the inverse square law. Every time you move twice as far away from a point source the amount of light drops off by a factor of 4 as the area the light is spread is 2 x 2 = 4 times more.
Since stars are point sources when viewing them through a telescope (assuming you are not approaching the diffraction limit), they appear the same brightness at low and high magnifications.
Inverse Square Law for Point Sources
3. Distance and Magnification
Not all objects that are viewed with telescopes are point sources. The majority of interesting objects are not point sources but rather diffuse sources that have their light emitted over an area rather than a point. These types of objects will have apparent magnitudes listed in star charts that take all of their light, bring it to a point, and report the brightness of the point. In practice, this makes a brightness or magnitude reported in this form for diffuse objects rather useless without knowledge of the size of the object.
For example, in common bright suburban skies (Bortle 7 - 8) the dimmest naked-eye stars are between 4th and 5th magnitude. The Andromeda Galaxy however has a magnitude of 3.44 and can not be seen naked eye in a Bortle 7 or 8 sky in even the best of conditions. This is due to the light being spread out over an area the size of a couple of full moons, making any one part of the galaxy much dimmer than a star of equivalent magnitude. This is what allows a telescope like the James Webb Space Telescope (JWST) to take an image of a bright planet like Jupiter but not an equivalently bright star. A bright star being a point source will saturate the sensor but Jupiter’s light spread out over the area is takes up on the sensor lowers the amount of light each pixel receives.
Surface brightness, the amount of light coming from an area of an object is a better way to understand how bright the object actually is. There are countless objects in the sky that are larger than the full moon but are invisible to the naked eye due to their low surface brightness.
JWST Image of Jupiter
4. Diffuse Sources and Surface Brightness
As stated in section 2, point sources that are farther away appear dimmer due to the drop off in light due to the inverse square law. Diffuse objects have their equivalent point source brightnesses decrease with distance but not their surface brightness. A very easy way to see this in practice is to look at a wall at a distance and then up close. The wall is the same brightness from both viewpoints.
You may have heard that in a few billion years that the Milky Way will collide with the Andromeda galaxy and Andromeda will take up a large portion of the sky. This is true but the sky will unfortunately not be ablaze with another galaxy in the sky. As Andromeda gets closer it will get larger but also equivalently dimmer maintaining the same surface brightness. For example, if Andromeda was half as far away, it as a whole becomes 4 times brighter due to the inverse square law, but simultaneously takes up 4 times as much area. The result is that Andromeda retains the same surface brightness and still requires a dark sky to see. In the image (right on desktop, above on mobile) of Andromeda and the Milky Way colliding, Andromeda is no brighter than the Milky Way which requires a dark sky to see.
Artist rendering of an Andromeda and Milky Way Collision
5. Exit Pupil and Ability to See
As stated in section 1, the exit pupil is the width of the beam of light entering your eye from the eyepiece. Light from this beam can only enter your eye if your pupil (the black part of your eye where light enters) is larger or equal to the size of the exit pupil.
In bright conditions, the pupil shrinks down to 2 mm and when fully dark adapted can be up to 7 or 8mm. The size of the pupil in the dark peaks during one’s teenage years and shrinks down to 5 mm 60 to 65. Due to the average person having a maximum pupil size of around 7 mm, this is often defined as the maximum useful exit pupil as a beam of light over this size does not enter the eye.
Since no more light can enter the naked eye or eye while viewing a telescope eyepiece than 7 mm, the view in a telescope of a diffuse object is never brighter than the view seen with the naked eye. The image is only magnified with the surface brightness the same as the naked eye or dimmer.
If telescopes do not make diffuse objects brighter how come we can see them with a telescope but not with the naked eye? It is due to the way our eyes work. An object with the same surface brightness made larger is easier to see. The galaxies at 1 x and 10 x (right on dekstop, above on mobile) have the same surface brightness. The 10x is easy to see while the 1 x is barely visible.
Point sources at magnifications not near the diffraction limit are the same brightness as the light from the point is not spread when still resolved.
Effect of size on Visibility
Change in pupil size
6. How to Use Exit Pupils
M 6 in a 10” telescope left and in a 20” telescope right
M 31 at 20x (2.5 mm exit pupil) left, and 10x (5 mm exit pupil) right in the same telescope.
If more light is entering your eye the view will be brighter. This means that larger exit pupils provide brighter views of objects. Bigger telescopes produce better views by producing a brighter and sharper image at the same magnification as smaller telescopes.
Great! If a larger exit pupil means a brighter image then why is a 2 - 3 mm exit pupil usually recommended over 6 - 7 mm? A larger exit pupil not only means the object is brighter but that the sky is as well. The goal is visual astronomy is not a bright image but one that provides the highest contrast. Using a smaller exit pupil will dim the background sky and the object but the contrast will likely improve. For a single telescope, a smaller exit pupil is synonymous with higher magnification. The view in two different telescopes with the same exit pupil has the same sky and object surface brightness regardless of any other telescope properties.
In most cases, the large objects are best at large exit pupils (5 mm +) and for star-like objects exit pupils around 2 mm will be best. If filters are being used the more restrictive the filter the larger the exit pupil should be to compensate for the reduction in light. A UHC filter will perform well at 3-4 mm while an O-III is better suited for 5-6 mm.
Large galaxies and diffuse nebulae are best at large exit pupils. For galaxies that are small or have bright central regions such as Seyfert galaxies, smaller exit pupils will work to improve contrast.
Globular clusters and planetary nebula are best at medium and small exit pupils 2 - 3mm. Globular clusters will partially resolve into stars that will benefit from the sky-darkening at small exit pupils but to see the diffuse glow of the cluster larger exit pupils are needed. Planetary nebulae are small and bright and often look like stars until seen with high magnification. The ideal exit pupil varies on the object and is best found through trial and error by looking into an eyepiece.