The Majesty Factor
The Nexus of Contrast, Power, Field — essay by Al Nagler
"Majesty Factor" Summary

To illustrate the dramatic effect of combining a larger apparent field (yielding greater deep sky details) with smaller exit pupils (yielding fainter stars with darker sky background) we propose the "Majesty Factor". We define it simply as the cube of the ratio of any two different apparent field eyepieces having the same field stop diameters (same true field). Examples:

(100°/70°)3= 2.92 "M.F."     or     (70°/50°)3= 2.74 "M.F."    or    (100°/50°)3= 8 "M.F."

After showing the first Ethos 13mm at a number of events since April 2007, we can safely conclude it brings the observing experience to a new level. This is based solely on user's reactions to views of familiar objects, not on any prejudgments, publicity or hype on our part. While we were quite confident of success, we wanted, and still want, to explore all the ramifications of what a sharp 100° field really represents.

Right after NEAF in April, Rodger Gordon, the acknowledged "eyepiece junkie" of all time, wrote me "Definitely the finest wide-angle eyepiece I've ever seen. If God is an astronomer, this is the wide-angle eyepiece he'd choose. You can quote me." Thanks, Rodger. I waited until now to avoid "priming the pump", so to speak before quoting your unbridled enthusiasm publicly.

For some time, I've been pondering just why the response has been so overwhelmingly positive. And if I really understand why, is it possible to quantify? My views of the Double Cluster at Stellafane pointed the way.

The 1991 article I wrote for Sky and Telescope on magnification provides the key. A major conclusion for low power states: "The best view occurs with the highest power that comfortably includes the target object. Higher powers darken the background sky, reveal fainter stars and show more detail. The resulting smaller exit pupil also minimizes the effects of eyesight defects."

Considering the potential of Ethos, let me posit a more general conclusion.

For deep sky viewing of star fields, open and globular clusters, nebulae and galaxies, choose the highest power that frames the subject, so long as the sky background does not reach black, and the atmosphere does not degrade the resolution. The smaller exit pupils permit a darker sky background which achieves greater contrast against the fixed brightness of stars, while the greater magnification reveals more structural details on extended objects. Using eyepieces with larger apparent fields maximizes the viewing experience.

The result is an increase in what I would call the Majesty Factor, the nexus of contrast, power and field.

It's clear that the largest possible apparent field for a given true field yields the most magnification for greater resolution, with a darker sky background for more contrast as a result of the smaller exit pupil. I believe this combination of contrast, power and field causes the typical "wow" reaction — the Majesty Factor. I think Tom Trusock said it most succinctly in his Starfest (Canada) report: "The same true field at higher magnification means that you'll see blacker skies and more detail." Dennis di Cicco in his 5-star review of Ethos in his October 2007 Sky & Telescope review noted something similar: "Observing with the 12-inch scope, I typically bounce between a wide-field eyepiece for star-hopping and a high-power one for detailed views. But the Ethos gave me both. The field was large enough to star-hop, and the magnification was high enough to bring out faint stars and resolve details in galaxies and star clusters." (He coincidently also illustrated field sizes using the Double Cluster.)

Basic Telescope and Eyepiece Concepts
• apparent field: perceived span of sky seen through eyepiece (without telescope). Not used in true field (see) calculation.
• exit pupil: image of objective formed by eyepiece. Location where full apparent field is seen.
• f/#: a ratio that describes the relation between the aperture and focal length of the telescope -- important for photography
• field stop: ring inside the eyepiece barrel that limits true and apparent field size
• focal length: effective distance from entrance of an optical system to focal point
• magnification: relative change in angular size of object
• true field: span of sky seen through telescope/eyepiece combination
Basic Telescope and Eyepiece Formulas

Let's try to quantify the so-called Majesty Factor. While we cannot quantify the majesty of a great symphony, work of art or edifice, I think a meaningful Majesty Factor is quantifiable for those great deep sky views. Here's how:

Let's consider a range of possible eyepieces with apparent fields of 50°, 60°, 68°, 82° and 100°. Now let's pick an object, (like the Double Cluster) and let's say it's properly framed in the field of a 50° Plössl with a 26-mm focal length in an f/4 telescope so the exit pupil = 6.5-mm. Let's arbitrarily assign a factor of 1 to the power (magnification) of this telescope and a factor of 1 to represent the contrast for the 6.5-mm exit pupil. Therefore, for the given true field:

Majesty Factor = 1 (power factor) x 1 (contrast factor) = 1

Now let's replace the Plössl with a 100° (apparent field) Ethos with a 13-mm focal length. This yields the same true field of view at twice the power with twice the apparent field and half the exit pupil. The 3.2-mm exit pupil is only ¼ the area of 6.5-mm, so the sky background darkens by a factor of 4 (contrast factor). The magnification power factor yields twice the detail or resolution. Therefore:

2 (power factor) x 4 (contrast factor)  = 8x Majesty Factor

Working out the math for all the apparent fields listed above, we have:

Majesty Factor for Various Apparent Fields for Eyepieces Yielding Same True Field
Apparent Field (°) Power Factor   Contrast Factor   Majesty Factor
Plössl 50 1.00 x 1.00 = 1.00
Panoptic 68 1.36 x 1.85 = 2.52
Delos 72 1.44 x 2.07 = 3.00
Nagler 82 1.64 x 2.69 = 4.41
Ethos 100 2.00 x 4.00 = 8.00

A simple rule of thumb is that for any two eyepieces having the same true field of view, the Majesty Factor equals the cube of their apparent field ratios. Example is (100°/70°)3=2.92.

Majesty Factor Visualization

13-mm Ethos (left) and 26-mm Plössl (right) have same true field of the Double Cluster. But Ethos has twice the power factor and four times the contrast factor. Ethos therefore has a Majesty Factor of eight relative to the Plössl's Majesty Factor of one. For reference, the 0.9° true field shown could be produced by a 14" f/4 telescope using the 100° Ethos at 109x with a 3.2mm exit pupil or the 26-mm 50° Plössl at 55x with a 6.5mm exit pupil. (TV-60is image of Double Cluster by Robert Reeves.)