Telescope Equations

How to Size Up & Set Up a Scope

Theory

The telescope equations as presented here are quite simple, so it's entirely possible to analyze a scope, at least to reasonable estimates, in your head -- while at the scope. It's useful to follow a consistent process, and as described here the process goes in two steps, looking first at intrinsic properties of the scope, then setting its operating points.

Note: for explanations of the equations below go to the Equations Summary and click on the "Explain This" button next to the equation.

Telescope Properties

Properties that are intrinsic to the scope are those that depend only on the objective lens or mirror.

1. Find the Specifications on the Objective
   Notice that there are only two values we need to completely assess the scope and determine its operating points:
• Diameter of the objective, D_O
• f-Ratio of the objective, f_R


2. Calculate Intrinsic Resolution & Brightness
   • Resolving Power, P_R = 120/D_O
   • Magnitude Limit, L_mag = 2 + 5 log(D_O)
     This can be estimated in your head. To see how, check out the Logarithms Page.

Operating Points

The operating points depend on the eyepieces, and it's useful to bracket a range of eyepieces, looking for the shortest and longest focal lengths that will be used. We will look for the high resolution end of the scale, followed by the high brightness end of the scale.

1. Highest Resolution
1. Maximum Magnification, M_max = D_O
   start with this because you can read it right off the scope (or its specs) and because you may find it goes beyond the limitations of the atmosphere, ~200. If so, you may choose to limit it to around 200, and that in turn will influence the choice of eyepiece.
2. Exit Pupil, D_ep = D_O/M,
   where M will depend on your decision above. If you can make use of the intrinsic max magnification of the scope (D_O), then the exit pupil will be 1mm.
3. Focal Length of the Eyepiece, f_e = D_ep * f_R.
4. Resulting Surface Brightness, SB = 2*D_ep².
2. Highest Brightness
1. Focal Length of the Eyepiece, f_e = 7 * f_R
   start with this because you may find that trying to reach 100% surface brightness leads to an eyepiece focal length you can't get. If that's the case, you choose the longest focal length you can get, and proceed with the analysis based on that.
2. Exit Pupil, D_ep = f_e/f_R, where f_e depends on the result of your assessment above. If you can get to the maximum required focal length then the exit pupil will be 7mm.
3. Minimum Magnification, M_min = D_O/D_ep
4. Resulting Surface Brightness, SB = 2*D_ep², which will be 100% if you were able to get the maximum focal length eyepiece (i.e. D_ep = 7mm).
3. Intermediate Points
The end points of the operating range, highest resolution and highest brightness, serve as reference points for you. In all likelihood you will want other eyepieces as well. These are some additional operating points to consider:
1. Half-Maximum Magnification: or twice the minimum eyepiece focal length, which works out to twice the f-ratio and an exit pupil of 2mm. This hits the optimum resolution spot where your eye and the scope work best together.
2. Half-Maximum Brightness: you get to this point by setting a 5mm exit pupil, which works out to an eyepiece focal length of 5*f_R , and sets the scope to 50% surface brightness. This is often a good compromise of magnification and brightness for deep-sky imaging.
3. Extra-High Magnification: since M_max gets you to where your eye can just see the smallest detail the scope can show, you might want to go a little bigger to make that detail easier to see. A good practice is to keep it to no more than 50% bigger (1.5*M_max) which means an eyepiece focal length that is two-thirds of f_e-min , which works out to be 0.67*f_R. With an exit pupil of 0.67mm and a surface brightness of 1%, you will want to have brightness to spare to see much detail... for example this might work when looking at planets.

As explained on the Surface Brightness page the range of exit pupils from 1mm to 7mm, with an extension to a D_ep of 0.67mm for pushing our luck on the high-power end, defines the performance range for the scope as diagrammed below.

A Note on the Exit Pupil

The analysis of the operating points of the scope is conducted here entirely using the exit pupil formulas. There are many ways to tackle the problem and I find this to be the most straightforward, providing the most information the fastest and with the least computation, since the exit pupil is such a good index to performance. This is important if you want to be able to size the situation up in your head.

Practice

Various Random Interesting Examples

I literally will just wander around the web here and provide an analysis of whatever scopes catch my attention, to give you a sense of how the equations can work for you.

Meade ETX-80BB Backpack Observatory Telescope

Portability of a scope can be critical, because if you can't take the scope where you're going then it's no scope at all. This one is an f/5 with an 80mm aperture, comes with an extra-light tripod and its own backpack.

Telescope Properties

• D_O = 80mm
• f_R = 5
• P_R = 120/D_O = 120/80 = 1.5 arc-seconds
• L_mag = 2 + 5 log(D_O) = 2 + 5 log(80) = 11.5

Operating Points

1. Highest Resolution
1. M_max = D_O = 80 is within atmosphere limits
2. D_ep = D_O/M = 80/80 = 1mm (because we are at max magnification, not limited by the atmosphere)
3. f_e = D_ep * f_R = 1*5 = 5mm
4. SB = 2*D_ep² = 2*1² = 2%
   Meade claims the highest useful magnification of this scope is 160. This analysis shows that is past the point of matching your eye's resolving power by a factor of 2, so the resulting image would be blurry. Also, our analysis shows you would need a Barlow doubler with a 5mm eyepiece or else a 2.5mm eyepiece (tough to find) to double the M_max of 80 and get to 160x.


2. Highest Brightness
1. f_e = 7 * f_R = 7*5 = 35mm (certainly available)
2. D_ep = f_e/f_R = 35/5 = 7mm (because you can reach maximum brightness)
3. M_min = D_O/D_ep = 80/7 = 11.4 (hmm... let's discuss this, below)
4. SB = 100% since we are at minimum magnification.
   The brightness number comes out looking good, but the small diameter means you get a bright image with pretty lame magnification. At a magnification of 11.4, a good pair of 10x50 binoculars would give this scope, using a 35mm eyepiece, some stiff competition. That means it's not likely you want to operate there. The scope actually comes with two eyepieces, a 9.7mm and a 26mm. Let's see what the operating point is for each.


3. The 26mm Eyepiece
   In order for you to get a sense for how I really do this, I will round the focal length off to 25mm as soon as I see it's easier to divide by an f-ratio of 5.
1. f_e = 26mm, call it 25mm to calculate faster with only a 4% error.
2. D_ep = f_e/f_R = 25/5 = 5mm
3. M = D_O/D_ep = 80/5 = 16
4. SB = 2*D_ep² = 2*5² = 50% (they picked the half-max brightness point)
This delivers a more decent (not awesome) magnification than the 35mm eyepiece at a reasonable brightness. Could be useful to survey the sky, and to show your friends a good wide-field view of things like the moon or the Pleiades. See how rounding made it easy to do these in your head?


4. The 9.7mm Eyepiece
1. f_e = 9.7mm, call it 10mm since that's clearly easier to calculate with only a 3% error.
2. D_ep = f_e/f_R = 10/5 = 2mm
3. M = D_O/D_ep = 80/2 = 40 (they picked the half-max magnification point)
4. SB = 2*D_ep² = 2*2² = 8%
   As mentioned elsewhere, a 2mm exit pupil provides the optimum for the resolving power of the eye, which makes this an interesting choice on the part of Meade. This is probably the eyepiece you would like with this scope.

Celestron AstroMaster 114 EQ Reflector Telescope

A bigger scope with a longer f-ratio, on a German equatorial mount (my favorite) and excellent reviews, so let's have a look at this one.

Telescope Properties

• D_O = 114mm
• f_R = 9
• P_R = 120/D_O = 120/114 = 1.1 arc-seconds
• L_mag = 2 + 5 log(D_O) = 2 + 5 log(114) = 12.3

Operating Points

1. Highest Resolution
1. M_max = D_O = 114 is within atmosphere limits
2. D_ep = D_O/M = 114/114 = 1mm (because we are at max magnification, not limited by the atmosphere)
3. f_e = D_ep * f_R = 1*9 = 9mm
4. SB = 2*D_ep² = 2*1² = 2%
   Celestron claims the highest useful magnification to be 269 -- they are using the formula 60× scope diameter in inches, which you will see in some references and some sites on the internet. Since it is more than double the point where your eye sees all of the scope's detail, the images will be very blurry, and dim with an exit pupil of 114/269 = 0.42mm and SB = 2*0.42² = 0.4%.
   The operating point we find here with a 9mm eyepiece is quite good, though, and matches up well with the 10mm eyepiece they provide with the scope. Along with the 10mm you also get a 20mm eyepiece which puts you at the half-max magnification point, close to the optimum resolution. A slightly shorter focal length of 6 or 7mm might also be handy to have.


2. Highest Brightness
1. f_e = 7 * f_R = 7*9 = 63mm -- out of practical range, so we will have to go with something more obtainable, call it 40mm
2. D_ep = f_e/f_R = 40/9 = 4.4mm
3. M_min = D_O/D_ep = 114/4.4 ≈ 26
4. SB = 2*D_ep² = 2*4.4² ≈ 40%
   The higher f-ratio limits the surface brightness you can achieve with this scope, but gets you to maximum magnification with a reasonable eyepiece focal length.
   Seeing faint nebulas or galaxies with this scope will be a challenge. A 40mm eyepiece gets to 40% surface brightness and yields a 26x magnification, which is a bit low, and the 20mm eyepiece, the longest focal length provided with the scope, has a D_ep = 20/9 = 2.2mm, a magnification of M = D_O/D_ep = 114/2.2 = 51, pretty decent, and SB = 2*2.2² = 10% -- not horrible but not particularly good for deep sky observing.

Special Note: The deep-sky (brightness) vs. magnification limitations you see with this scope and the previous one are not a reflection against the scopes -- they are both great for what they do and have gotten great reviews. What this is showing is the motivation that astronomy enthusiasts have for going with larger diameter, also heavier and costlier, telescopes.

Orion SkyQuest XT8 Classic Dobsonian Telescope

A really popular 8-inch scope on a Dobsonian mount.

Telescope Properties

• D_O = 203mm
• f_R = 6
• P_R = 120/D_O = 120/203 = 0.6 arc-seconds
• L_mag = 2 + 5 log(D_O) = 2 + 5 log(203) = 13.5

Operating Points

1. Highest Resolution
1. M_max = D_O = 203 is within atmosphere limits (just barely)
2. D_ep = D_O/M = 203/203 = 1mm (because we are at max magnification, not limited by the atmosphere)
3. f_e = D_ep * f_R = 1*6 = 6mm
4. SB = 2*D_ep² = 2*1² = 2%
   Orion claims the highest useful magnification to be 300 -- and this one can be defended, as it aligns with the recommendation made in the "Theory" section above.
   On the other hand, the atmosphere will not allow you more than a magnification of 200, and usually much less. So getting to a magnification of 200 with a 6mm eyepiece is likely to be all you would ever need with this scope. A 4mm eyepiece (or 8mm with a 2x Barlow) might be interesting to use for testing purposes, and would be less useful for observing.


2. Highest Brightness
1. f_e = 7 * f_R = 7*6 = 42mm. Period. Get one of those for this scope.
2. D_ep = f_e/f_R = 42/6 = 7mm
by definition since we are at minimum magnification.
3. M_min = D_O/D_ep = 203/7 = 29
4. SB = 100% since we are at minimum magnification.
   Check it out -- maximum telescope brightness reached at a magnification of 29. Very decent. A 10" scope hits this point with a magnification of 36, provided it has a low enough f-ratio (6 or less).


3. Intermediate Points
• Half-Max Magnification: D_ep = 2 means an eyepiece focal length of f_e = D_ep*f_R = 2*6 = 12mm.
• Half-Max Brightness: D_ep = 5 means an eyepiece focal length of f_e = D_ep*f_R = 5*6 = 30mm. Orion gives you a 25mm eyepiece with the scope, you could count that as close enough.
• So with an eyepiece set of 6mm, 12mm, 30mm, and 42mm, you have the full range nicely covered, all the way from the maximum magnification allowed by the atmosphere to an image that is 100% brightness. Pretty sweet.

Obsession Telescopes 18-inch Dobsonian

One last example, we might as well take a look at a monster.

Telescope Properties

• D_O = 18" * 25.4 = 457mm
• f_R = 4.5
• P_R = 120/D_O = 120/457 = 0.3 arc-seconds
• L_mag = 2 + 5 log(D_O) = 2 + 5 log(457) = 15.3
  Wow. Pulling in the eighth brightest moon of Saturn (Hyperion).

Operating Points

1. Highest Resolution
1. M_max = D_O = 457
   Oh, ah... No.
   No, no, no. Not on this planet, as long as we continue to enjoy a breathable atmosphere. In this case we are going to define M_max to be what we think is achievable, and I would call that 200. It's a judgement call, and you might go with a different value, the arithmetic follows the same process as here:
2. D_ep = D_O/M = 457/200 = 2.3mm
3. f_e = D_ep * f_R = 2.3*4.5 = 10mm
4. SB = 2*D_ep² = 2*2.3² = 11%
   Going beyond this point in magnification will simply provide a really close, detailed view of the blurring and distortion being produced by atmospheric conditions. Look at the surface brightness, though, at 200x: 11%, more than 5 times what the 8" gives us at that magnification. This is the point of the bigger scope.


2. Highest Brightness
1. f_e = 7 * f_R = 7*4.5 = 32mm. Easy to get, this would be highly recommended... interesting to note this is the longest focal length eyepiece you should be using with this scope.
2. D_ep = f_e/f_R = 32/4.5 = 7mm by definition since we are at minimum magnification.
3. M_min = D_O/D_ep = 457/7 = 65.3
   Wow. This is the minimum magnification for this scope.
4. SB = 100% since we are at minimum magnification.
   Since the magnification is so high at maximum brightness, the deep sky images are to die for. The Ring Nebula jumps out at you even under a full moon. A scope like this is a deep-sky explorer's dream.


3. Intermediate Points
• Half-Max Magnification: that's pretty much what we ended up picking when we realized that max magnification would be obliterated by all the air up there.
• Half-Max Brightness: D_ep = 5 means an eyepiece focal length of f_e = D_ep*f_R = 5*4.5 = 22.5mm. Televue makes a nice 22mm (I didn't say cheap) and I think Celestron's got one as well, both with 2" barrel which would fit this scope. 20mm is a little easier to find... in any event this operating point is reasonably achievable.
• We're not even going to discuss extra-high magnification with this scope. So then you're really pretty much set with an eyepiece that's got about a 10mm focal length, one that's about 20mm, and one that's about 30mm. More than just these three does not make a lot of sense.

In Summary...

You can see how the performance scale provides a framework that helps with sizing up and setting up a scope.

• The exit pupil serves as a central reference point for calculations. Always going first to the exit pupil does three things for you:
  1. Tells you where you are on the performance scale.
  2. Keeps it simple: you always know what to do first
  3. Positions you to quickly calculate magnification, eyepiece, and surface brightness.
• We use the end points of the scale to bracket the operating range: highest magnification, then highest brightness.
• We included practical considerations in the analysis: whether the highest achievable magnification is limited by the atmosphere, and whether the longest focal length eyepiece is decided by what's available.
• The two intermediate operating points — half the scope's maximum magnification and half the maximum brightness — serve as additional guideposts which help us to understand our choices of operating points for the scope.

We didn't try to nail down an answer to five-decimal-place accuracy. We found what eyepieces to use with the scope and how to use them. There are only so many choices for eyepieces that are available on the market, and ultimately performance is subjective. It will depend on the target being observed, the observing conditions and YOU, the observer. The goal is finding where you want to be, then getting as close as reasonable. Like a game of horseshoes, close counts.

Back to Telescope Equations Home Page

Questions

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 17 May 2019