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Introduction |
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The Process |
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Terms & Symbols |
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Scope Equations |
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Special Cases |
Stars are so unimaginably far away that the light we receive from them arrives in rays that are perfectly parallel. Your eye is designed to focus these parallel rays to a point, allowing you to identify where the light is coming from.
A telescope, in its original configuration (refractor), consists of two lenses. The first one, the objective lens, collects light and focuses it to a point. (Note that the objective mirror in a reflecting telescope does exactly the same thing.) The second lens, the eyepiece, catches the light as it diverges away from the focal point and bends it back to parallel rays, so your eye can re-focus it to a point.
Notice how the telescope has taken all the light passing through the objective lens and compressed it down to a column of light that will pass through the pupil of the eye. This is one of the three major tasks of the telescope, the full list being:
The equations on this page permit you to find just exactly how well the telescope will perform these tasks, and along the way I also show how the tasks are accomplished, by explaining both the theory and the practice.
CAUTION - telescope manufacturers will often advertise the magnification of the scope, and give really big, impressive numbers. The problem is that the number is essentially meaningless.
The magnification of a telescope is a combined function of the scope and the eyepiece that is used, so the user can set the magnification to almost any arbitrary value by selecting a suitable eyepiece. Whether the resulting image is clear, or barely visible, depends on other properties of the telescope. Therefore the magnification is not the most important measure of a telescope.
What actually is the most important measure is the diameter of the objective, or more simply the scope diameter, because that determines both the resolving power (the smallest detail you can see) and the light-gathering power (the faintest objects you can see). How the scope diameter determines the performance of your telescope is explained through the equations below.
Of course you can use the equations below however you like. Just in case it seems like a lot of equations to you, or maybe looks a bit overwhelming, here's a procedure I can suggest for you. It's in two parts, three steps each (this is the process I use).
Determine the Deep Sky Performance
This is to figure out how well the scope will show you galaxies and nebulae... something that's important to me, so I figure this out first. If you're not interested in deep sky you can skip to the (very easy) next part on star & planet performance.
Determine the minimum magnification for the scope, and the eyepiece focal length required to get it. Then identify whether an eyepiece at that focal length is available to you. Note that these are very simple calculations.
If you can get the eyepiece that gets minimum magnification, you can achieve 100% surface brightness with the scope -- it's a good one for deep sky imaging. Otherwise for the eyepiece with the longest available focal length, calculate the scope brightness. The closer to 100%, the better for deep sky imaging.
Note that all scopes will give the same non-star image brightness at 100% scope brightness (assuming the same quality of optical glass & coatings). So this scale of 0 - 100% is a "universal" brightness scale.
With the surface brightness determined, check the magnification for this eyepiece -- anywhere from around 30-60 or so gives you a good (wide) view of the sky. You can use the Scope Field of View equation to determine how much of the the sky you will actually see with this scope & eyepiece.
Determine the Star (& Planet) Performance
This is to figure out how well this scope will capture and resolve individual stars, including its ability to resolve clusters.
A very simple calculation and very important for determining whether you can split double stars that are close together.
Some books will have you calculate the "light grasp", but that tells you nothing by itself. Finding Gmag will tell you which stars you will be able to see in your scope.
This determines how high your scope is capable of going in magnification before stars start to turn fuzzy and fade away. This is also important for planetary viewing.
Note that the maximum (useful) magnification you can achieve, and the eyepiece that gets it, are incredibly easy to find. The maximum magnification is the scope diameter in mm, and the eyepiece to get it, in mm, is the f-ratio. This is a rather remarkable, and extremely useful, result of these equations.
| Symbol | Meaning |
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| Dep | Diameter of the exit pupil. The exit pupil is where the light leaving the eyepiece converges to its smallest circle -- you find the exit pupil when you bring your eye up to the eyepiece until you can see the whole image. |
| DO | Diameter of the objective. The "objective" can be either the large lens at the front of the telescope (in a refractor) or the large mirror at the back of the telescope (in a reflector). |
| fe | Focal length of the eyepiece. The distance from the center of the eyepiece lens to the point at which light passing through the lens is brought to a focus. |
| fO | Focal length of the objective. The distance from the center of the objective lens (or mirror) to the point at which incoming light is brought to a focus. |
| fR | f-Ratio. Simply the ratio of the focal length to the diameter of the objective, or fO/DO. This is written "f/" and then the value. An example would be an "f/10" telescope, meaning the focal length is 10 times the diameter of the objective. This is commonly given along with the diameter of the objective to describe a scope, and is a surprisingly useful parameter for characterizing its performance, as seen below. |
| FOVe | Field of view of the eyepiece. A measure of the area you can see when looking through the eyepiece alone. This is expressed as the angle from one side of the area to the other (with you at the vertex). The two parameters fe and FOVe are the two primary specifications for the eyepiece. |
| FOVscope | Field of view of the scope. Tells you how much of the sky you see in the image in the telescope. This is the distance from one side of the eyepiece image to the other, expressed in degrees or minutes of arc across the sky. |
| Gmag | Gain in visible star magnitudes. The increase in star magnitudes that you can see by looking through the scope (compared to looking by eye). So for example if the faintest star you can see by eye is magnitude 5, a gain of 7.3 would mean you could see stars of magnitude 5+7.3 = 12.3 in the scope. |
| M | Magnification. The apparent increase in size of an object when looking through the telescope, compared with viewing it directly. |
| PR | Resolving Power. The smallest separation between two stars that can possibly be distinguished with the scope. This is an indication of the finest detail the scope is capable of seeing -- regardless of the magnifying power. |
Note 1: You might notice the number 7 shows up a lot in these equations. This is because I am taking the diameter of the pupil of a dark-adapted eye to be 7 mm, as is customary for these calculations, and matching the diameter of the exit pupil, Dep, to that. You could reasonably replace all the 7's with "Deye", but I find simply calling it out as 7 is more practical.
Note 2: I've made these equations as simple and easy to use as possible, so you can do most, if not all, of these calculations in the field in your head. An additional pointer: always work in millimeters. The fact that values are sometimes given in inches and sometimes in centimeters can make things more complicated than they need to be. Millimeters work out really well, so always convert. Multiply inches by 25 and centimeters by 10 to convert to millimeters.
| Term Computed | Equation | Theory & Practice |
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| Resolving Power (in arcseconds) |  |
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| Magnification |  |
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| Scope Field of View |  |
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| Diameter of the Exit Pupil |  |
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| Focal Length of the Eyepiece |  |
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| Gain in Visible Star Magnitudes |  |
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| Surface Brightness at Magnification M using Eyepiece Focal Length fe |
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| Term | Exit Pupil | M | fe | SB | Theory & Practice |
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| Minimum Magnification | 7 mm | DO ÷ 7 | 7 × fR | 100% |
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| Optimum Magnification | 2 mm | DO ÷ 2 | 2 × fR | 8% |
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| Maximum Magnification | 1 mm | DO | fR | 2% |
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Your questions and comments regarding this page are welcome.
You can e-mail Randy Culp for inquiries,
suggestions, new ideas or just to chat.
Updated 05 July 2010
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