Understand the relationship
The reasoning behind the result
Focal lengths determine angular magnification
M=Feffective/feyepiece; Feffective=Fnative B
The positive magnification M compares apparent angular size through the telescope with the unaided angular size in an ideal afocal arrangement. Fnative is the native telescope focal length, B is an entered effective focal-length multiplier, and feyepiece is the eyepiece focal length. Both focal lengths must use the same length unit before division.
A shorter eyepiece focal length increases magnification for the same effective telescope focal length. A known amplification or reduction factor changes the effective focal length and f-number while the physical aperture stays the same. The factor is an operating-configuration input, not a universal property inferred from an accessory label.
Exit pupil decreases as magnification increases
pexit=D/M; effective f-number=Feffective/D
The exit pupil is the image of the entrance aperture formed by the eyepiece. For the ideal unobstructed system here, its diameter is aperture D divided by angular magnification. All pupil outputs are in millimetres. Equivalently, eyepiece focal length divided by the effective f-number gives the same pupil diameter.
The live curve holds the entered aperture fixed and plots this reciprocal relationship. Each named setup is marked on the curve; two different eyepiece/accessory combinations can land on the same point because they have the same magnification and pupil. Equal points do not establish equal field of view, eye relief, distortion or image quality.
A target can be solved without repeated trial entries
ftarget=Feffective/Mtarget; Mtarget=D/ptarget
A chosen magnification directly determines the eyepiece focal length under the entered effective focal length. A chosen exit pupil first determines magnification, then the required eyepiece focal length. Named targets allow the visitor to compare several alternatives without retyping the telescope dimensions.
These inverse results are continuous arithmetic values. The calculator does not assert that a matching eyepiece exists, can reach focus or is optically appropriate. Known-magnification mode needs only aperture and power; without a focal length, it deliberately leaves eyepiece and f-number outputs undetermined.
An entered observer pupil is a separate geometric stop
Daccepted=min(D, M·peye); admitted area fraction=min(1, peye/pexit)²
If an observer pupil is supplied, the optional comparison assumes centered, concentric, unobstructed circular pupils. When the exit pupil is larger, the smaller observer pupil admits only part of that ideal beam. The effective accepted entrance diameter and admitted area fraction describe this geometry.
This is not a prediction of retinal brightness, contrast, faint-object visibility or visual acuity. Central obstructions, decentering, vignetting, eye relief, transmission, aberrations and changing pupil size are outside this simple overlap model. No pupil size is supplied as a biological default.
Follow the numbers
Two ways to produce 100x
- A 200 mm aperture telescope with 1,000 mm native focal length and a 10 mm eyepiece gives 1,000/10=100x. Its exit pupil is 200/100=2 mm.
- Using a 20 mm eyepiece with a known 2x effective focal-length multiplier instead gives 2,000/20=100x and the same 2 mm exit pupil. The effective f-number changes from f/5 to f/10.
- To obtain a 4 mm exit pupil natively, the target magnification is 200/4=50x and the required eyepiece focal length is 1,000/50=20 mm.
Magnification and exit pupil reconcile, while field of view and accessory performance still require their own information.
Quick guide
How to use this calculator
- Choose known equipment, a target magnification, a target exit pupil or already-known magnifications.
- Enter telescope dimensions in their selected unit. Eyepiece and pupil diameters always use millimetres.
- Add named setups. Accessory factors must be known for the actual configuration; the tool does not infer them from a product name or spacing.
- Compare the full ledger. If an observer pupil is independently known, enable the separate centered-beam comparison; it does not estimate perceived brightness.
Calculation method
Calculation and interpretation
Keep magnification, effective focal length and exit-pupil diameter connected in one equipment comparison.
Feffective=Fnative·B; M=Feffective/feyepiece; pexit=D/M=feyepiece/(Feffective/D); ftarget=Feffective/Mtarget=Feffective·ptarget/D.
Worked example
Two ways to produce 100x
Magnification and exit pupil reconcile, while field of view and accessory performance still require their own information.
Feffective=Fnative·B; M=Feffective/feyepiece; pexit=D/M=feyepiece/(Feffective/D); ftarget=Feffective/Mtarget=Feffective·ptarget/D.
Supported inputs
Precision and limits
Ideal afocal and unobstructed geometry
This model does not trace rays, account for central obstructions or evaluate focus travel, collimation, eye relief, vignetting, optical quality or accessory compatibility.
No useful-power or visibility guarantee
A computed magnification does not establish additional resolved detail, a useful observing limit or safe viewing. Atmospheric conditions, target contrast and equipment quality are not modeled.
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