Double Star Astronomy

Part 2: Training the Binary Eye


1. Recognize the Diffraction Artifact
Visualize the Airy Disk
Your Resolution Exit Pupil

2. Identify Your Detection Limit
Visualize the Diffraction Limit
Your Detection Exit Pupil

3. Look Behind the Turbulence

4. Visualize Angular Separation
The Standard Eyepiece
Calculating Your Resolution Limit
How Big Is Very Small?
Training Your Eye to Estimate Separation
Describing Close Separations

5. Visualize Position Angle
Measures of Angular Direction
Field Presentation
Pointing Rotation
Observer Rotation
Estimating Position Angle

6. Use Active Magnification
A Minimal Eyepiece Selection
The Maximum Eyepiece Usage

7. Choose the Right Telescope
Equatorial Mount & GoTo Pointing
How Much Aperture?
What Type of Telescope?

8. Learn the Factors that Affect Resolution
The Seven Factor Groups
The Most Important Factors
Resolution Ratio & Magnitude Difference
A Troubleshooting 'Rule of Thumb'

9. Develop Your Observing Technique
System Brightness and Visual Response
Other Observational Factors
Enjoy the Journey


Observing double stars is one of the great pleasures of visual astronomy. These gravitationally bound stellar systems combine all types of stars in an enormous variety of dynamic configurations, from matched binary suns to miniature star clusters. As observational targets they are robust against light pollution and poor seeing, within reach of telescopes of any aperture, and visible at all times of the year.

Double stars present unique challenges to the visual astronomer that require training the eye through knowledge and motivated practice. This page describes nine steps to develop your visual abilities, with information that can anchor your understanding of the visual challenges in double star astronomy. (See the page A Double Star Primer for an overview of double star astronomy.)

1. Recognize the Diffraction Artifact

Despite the preconception that stars appear to the observer as "point" light sources, stars never appear as points in an optical system. Instead, due to the wave nature of light, they form a specific type of diffraction artifact: a circular, evenly bright Airy disk that (in bright stars) is encircled by one or more concentric diffraction rings.

This diffraction pattern is so tiny that the eye cannot distinguish it from a point when low magnification is used. But with the high magnification necessary to separate very close double stars, it defines the star image.

Visualize the Airy Disk. The first step in double star astronomy is to learn what the Airy disk and diffraction rings look like in their "pure" form, as they will appear under good seeing with sufficient magnification.

This is easy to do by means of an aperture mask (diagram, below) that reduces the entrance pupil of a telescope and thereby enlarges the diffraction artifact for easier viewing.

To make the mask:

(1) Choose a suitably large piece of cardboard, matt board or thin foam core board, and cut it to a circular shape that will closely but not snugly fit inside your telescope tube or dew shield.

(2) Cut a single circular hole in the mask that is about 1/3 your aperture diameter or less, and no more than 80 mm in apertures larger than 200 mm. This hole can be in the center of the circle for a refractor, but must be to one side in a reflector to avoid the obstruction of the secondary mirror mounting.

For a more durable mask, an excellent material is the stiff plastic used to make real estate and business signs (sold in most hardware stores), but to cut this material you will need a utility knife or heavy shears — and a strong hand.

If necessary to stabilize the mask, cut a large central hole (shown in the diagram) to fit the mask over the secondary mirror mounting of a Schmidt Cassegrain telescope, or a smaller hole to pass the secondary adjustment knob on a Newtonian reflector. This allows the mask to rest against the corrector plate rim mounting or the vanes of the secondary spider. Refracting telescopes won't need this hole.

To use the mask, first center and focus the telescope on a bright (magnitude 2 or brighter) star with a your medium magnification eyepiece. Then place the mask over the front end of your telescope, resting it against the refractor lens, corrector plate or reflector secondary mirror support.

The star should now appear as a distinct, round disk, distinctly edged and evenly bright (not fuzzy as in theoretical diffraction), and surrounded by one or more concentric, evenly spaced rings (diagram, below). This is the star diffraction artifact in a circular aperture — the best possible image of a single star through an astronomical telescope.

The Airy disk viewed through the stopped down mask is much larger than it appears in your telescope at its full aperture; if the aperture mask is 1/3 your full aperture, the Airy disk will appear 3 times larger. But that's the goal: you want to learn what the star diffraction image actually looks like, magnified and clarified, so that you are able to recognize it in your observations.

Your Resolution Exit Pupil. Take your time, and try the full range of focal lengths in your eyepiece collection (or a zoom eyepiece with explicitly marked focal lengths), including the eyepieces with a Barlow lens if you have one. If you do not have a sufficient range of eyepieces, you can create several masks with holes of different sizes.

After you have carefully studied the effect of different eyepiece magnifications on the diffraction image, choose the lowest magnification (or largest mask hole diameter) that produces a comfortably fat, clear and easily examined Airy disk. Do not make the disk larger than necessary to clearly see (1) the breadth of the Airy disk and (2) the dark interval between the disk and the first diffraction ring. (If you do not see a first diffraction ring, observe a brighter star.)

Now calculate the exit pupil at which you are viewing the Airy disk. The exit pupil is inversely proportional to your diffraction magnification: the smaller your exit pupil, the larger the Airy disk will appear. Use this formula:

de = ƒe x A/ƒo

where ƒe is the eyepiece focal length, ƒo is your telescope focal length, and A is the diameter of the circular hole in your aperture mask. (All measurements in millimeters: one inch equals 25.4 millimeters.)

In my 305 mm ƒ/10.4 SCT with a 76 mm aperture mask, I find an eyepiece with a 7 mm focal length produces an ample, clear Airy disk, so:

de = 7 x 76/3170 = 0.17 mm

so my "comfortable" diffraction magnification is at an exit pupil de = ~0.2mm.

The exit pupil you calculate in this way is your resolution exit pupil. For most double star astronomers, the star diffraction artifact is "adequately magnified" only when the exit pupil is equal to or less than ~0.5 mm. The great double star astronomer Sherburne Burnham recommended that:

"With a six-inch aperture, the best general working eyepiece will be a power of about 200. That will show most of the minute stars as well as any other power, and with it the practiced eye will rarely pass any pair down to 0.4" in distance, without suspecting something, and trying a higher power."

Calculation shows he is recommending an exit pupil of ~0.75 as the routine working magnification; note that a pair 0.4" apart is only half the minimum separation of the Abbe or Dawes limit for a six inch aperture!

Now, if you have the eyepieces available to do this, remove the mask and insert an eyepiece that gives you an equivalent resolution magnification with your full aperture. The focal length of this eyepiece will be equal to:

ƒe = de x N

where de is your resolution exit pupil and N is the relative aperture (focal ratio) of your telescope primary mirror or lens. You may need to use a barlow lens with your smallest focal length eyepiece in order to achieve this high diffraction magnification.

You may notice that the Airy disk produced by the mask is more robust against poor seeing — it appears clearly even when the atmosphere is turbulent or focus is not quite accurate. This is because the mask has drastically reduced the effective aperture of your telescope, and smaller aperture instruments produce better views when seeing is poor. But it is also a basic property of the Airy disk itself.

2. Identify Your Detection Limit

The second step in training your binary eye is to identify the minimum angular separation that you are able to detect with your telescope and your eyes. Here we are concerned with a different visual test: whether you can notice "something unusual" about a star that might indicate that it is a double star.

Visualize the Diffraction Limit. Now, instead of reducing the aperture in order to magnify the diffraction artifact (as we did in Task 1), we want to view at its actual size the angular width of your telescope's full aperture resolution. That is also accomplished by a special mask, described by Christopher Taylor in Bob Argyle's Observing and Measuring Visual Double Stars (2nd edition) (Springer 2012, p.131).

The logic of this mask is simple: we want to see the smallest interval that can be focused on the image plane of your telescope. This is a function of the telescope's full aperture — the diameter of the primary mirror or objective lens. So the mask used in this case transforms the telescope aperture into a double slit interferometer whose resolution is determined by the distance between the centers of two slits.

To make the mask:

(1) On a piece of cardboard, matt board or foam core board, draw a line 3 times your aperture diameter, and mark the center point of the line.

(2) Set a compass equal to the interior radius of your telescope tube (IR in the diagram), and use this radius to draw a circle around the center point.

(3) Measure, from the center point along the line, a distance equal to 0.82 times your aperture diameter or 1.64 times your aperture radius (r in the diagram) — for example, this is 24.6 cm for a 300 mm aperture (9-13/16 inches for a 12 inch aperture), and mark this point on the line. Set the compass equal to the aperture radius and draw an arc around this point and through the circle (see diagram, above).

(4) Do the same in the opposite direction on the line.

(5) Cut out the large circle, then cut away the two sections where the circle was scribed by the two arcs. (As before, for a Cassegrain, SCT or Newtonian, you may want to cut out a central circular area to fit over the secondary mounting and make the mask easier to rest against your corrector plate or secondary spider vanes.)

Center and focus your telescope on a convenient bright star (visual magnitude 3 or above), then place the mask inside the tube. The star will now appear as a tiny oblong smear. At low diffraction magnification (an exit pupil de = ~3) the smear will appear continuous; at medium diffraction magnification (de = ~2) the smear will seem vaguely to shimmer or iridesce. Finally, at an exit pupil of around de = 1.0, the smear will appear clearly divided by a series of ten or more very narrow, parallel and equally spaced dark bands (diagram, below).

This is an interference pattern of reinforced and canceled light waves that displays the Rayleigh resolution criterion for your aperture, the minimum separation between two Airy disks that produces a clearly perceived dark gap between them.

(The spacing of light and dark interference bands, such as the diffraction rings around the Airy disk, is defined as λ/D, the Abbe resolution limit, but the aperture slits provide a resolution of λ/0.82D or 1.22λ/D, the formula for the Rayleigh criterion. Alternately, the mask simulates the diffraction spacing of the Abbe resolution limit in an aperture that is 0.82 times as large as the aperture you are using.)

Your Detection Exit Pupil. What is the lowest magnification that allows you to see these small intervals clearly? Try observing the interference pattern with all your eyepieces, or with a zoom eyepiece if you have one, this time looking for the longest focal length eyepiece that still resolves the bands clearly — for example, as the diagram above appears when viewed about 12 feet from your computer screen. Because we're still examining a diffraction artifact, we need to state this as a diffraction magnification. So calculate your exit pupil a second time, as:

de = ƒe/N

where N is the focal ratio or relative aperture of your telescope. This is your detection exit pupil. It will usually be around 1.0 mm or a bit more, and much larger than the resolution exit pupil you calculated in the previous task.

Your detection limit is actually the combination of two optical systems — the telescope and your eyes. The values normally cited for the human visual resolution vary widely, from under 100 to more than 200 arcseconds. Rather than rely on these conflicting pronouncements, the minimum magnification you need to display these interference bands clearly can be used to calculate your visual resolution limit. This personal constant is given (in arcseconds) as:

Rv = M x 138/Do (aperture measured in millimeters)

For example, if a 12.5 mm or shorter focal length eyepiece is necessary to show the diffraction bands distinctly in a 300 mm ƒ/10.4 SCT, then the magnification is about 250 and the resolution exit pupil is 1.20. This equates to a personal dark adapted resolution limit of Rv = 250 x 138/300 = 115 arcseconds (1.9 arcminutes).

Unless you require a low power "finder" eyepiece to aid in locating double stars by star hopping — or you enjoy "spacewalk" views through a low power, wide angle eyepiece — your detection exit pupil suggests the lowest power (longest focal length) eyepiece that will be routinely useful when observing double stars.

Your experience may differ somewhat, but most visual double star astronomy is conducted at what lunar and planetary astronomers would consider high magnification — using exit pupils from 1.5 to 0.3 or less.

3. Look Behind the Turbulence

Seeing — the effect of atmospheric turbulence on a telescope image — is the single greatest observing difficulty in terrestrial astronomy. All telescopes with apertures greater than about 150 mm are, on most nights of the year, "seeing limited" rather than diffraction limited.

In training task 2, you probably noticed that the diffraction bands appear to shift and shimmer due to atmospheric turbulence. This is because the arriving light is reduced to two narrow beams. Where wavefronts meet the bands appear; disruption of the separate beams by turbulence shifts this meeting point to one side or another.

However, if you switch from the resolution mask used in task 2 to the detection mask used in task 1 at the same exit pupil (which means you must switch eyepieces, because mask 1 has a larger relative aperture and requires a lower power eyepiece), you will observe that the Airy disk is less disrupted than the diffraction bands by the prevailing atmospheric turbulence.

This illustrates a fundamental feature of the diffraction artifact: under mediocre to poor seeing, the Airy disk is more robust than the diffraction rings. It is common to see the Airy disk clearly when the rings are disrupted into randomly flashing short arcs or "speckles" of light, and this is one of the appearance features of a bright star under mediocre seeing (Standard Scale 4-6).

So the next step in your training is to look at a single moderately bright (around visual magnitude 2 to 3) star, in perfect focus, under mediocre seeing with an eyepiece that produces your resolution exit pupil — at or below 0.5 mm.

The star will appear to shimmer, slither, waffle, undulate, scintillate and tremble. Novice double star observers may assume that this is what one is supposed to look at: in fact it's what you are supposed to ignore. The pure Airy disk is still there, hidden underneath the atmospheric turbulence, in the same way that a bull's eye pattern painted on the bottom of a swimming pool is still visible underneath a rippling or sloshing surface. Unless the seeing is very bad, or the star is too faint, you will soon recognize the Airy disk: it will appear distinctly inside or behind the fragmented diffraction rings or "speckles" that surround and sometimes obscure it. The effect is striking, rather like car headlights shining through heavy rain (see diagram C, below).

If you have a Barlow lens, see what happens when you double the magnification of your resolution limit, and try to focus on the Airy disk. This will be at an exit pupil no lunar or planetary astronomer would ever dream of using, but you may find that the Airy disk is actually easier to see!

There are limits, of course. You have exceeded the useful magnification when (1) the star image is too faint to recognize with direct vision; (2) magnification has made the turbulent motions so large that the jumping movements confuse your eye; or (3) the faintness of the image makes it impossible to find the best focus, even by randomly adjusting the focus and then waiting to see if this allows glimpses of the Airy disk.

Under the worst seeing conditions the Airy disk of a bright star will be completely dissolved into a bloated, swarming mass of "speckle" interference and the star diffraction artifact will be inflated to two, three, six or more times its normal size. No magnification can pierce this chaotic tumult, and only a much lower power — usually two or more times your detection exit pupil — will provide acceptable views of wide double stars and rich star fields. Note however that the faintest stars will be so disrupted by the turbulence that they become invisible, as if you were observing under a full Moon or with a smaller aperture. In terms of the telescopic limit magnitude, your telescope will still be "seeing limited": there is always a third optical component — the refracting effects of the atmosphere — in the image delivered by your telescope to your eyes.

You develop two skills with this task. The first is the ability to sit, watch and wait for the moments of relative calm that yield glimpses of the diffraction pattern you are already familiar with. The second is the ability to mentally piece together these mere glimpses or "revelation peeps" into a stable visual impression of the primary star and its close companion. These two skills, patience and constructive observation, are the keystones of visual double star astronomy.

You also develop experience with the effects of seeing that will allow you to diagnose the quality of seeing through the appearance of a highly magnified, medium magnitude star. This is the basis of the original Standard Scale developed by W.H. Pickering and A.E. Douglass, described on another page as my Compact Seeing Scale.

4. Visualize Angular Separation

The previous task has trained your eye by observing a single star image. Now it's time to become familiar with the visual appearance of matched (equal magnitude) double stars stars through a given eyepiece on your telescope. This develops your ability to recognize angular separation as a fundamental parameter of double stars.

This is a very difficult observing skill to learn. The reason? Our vision is adapted to judge the size of objects in a room or a landscape of three dimensional space viewed with the fixed magnification of our naked eye vision. In double star astronomy, we must estimate the angular width between two points of light viewed in depthless space with a variable or uncertain magnification and apparent field of view.

The Standard Eyepiece. The essential piece of equipment to overcome these difficulties is a standard eyepiece that you use routinely to observe double stars. The eyepiece acts as a constant frame or window at a constant apparent "distance" or magnification. This constancy can anchor your estimates of visual separation. The standard eyepiece should have:

• an exit pupil (de = ƒe/N) that is approximately equal to your detection exit pupil, so that you can reliably detect even a close double star within a rich field of stars;

• a moderately wide apparent field of view, large enough to include a generous area around the double star but small enough that the circumference of the field stop is visible in your peripheral vision when you focus your gaze at the center. For most people this is an apparent field of view of around one radian (60°) — for example, a width of one meter viewed from a distance of one meter or more. You should specifically avoid "space walk" or super wide angled eyepieces.

A well chosen standard eyepiece will create a circular "frame" around the observing target at an exit pupil that makes the smallest details of the image barely available to your eye. If you do not already have an eyepiece that fits this description, then it is worthwhile to acquire one.

Calculate Your Resolution Limit. Next, you must calculate the resolution limit for your telescope. There are three resolution criteria commonly used to calculate the minimum separation between components of equal magnitude that can be resolved by a specific telescope aperture. These multiply the fundamental diffraction limit of the aperture, the λ/D or Abbe resolution limit, by a scaling factor k. (For the λ/D resolution limit, k = 1.00.) The three common criteria are:

(1) the Rayleigh resolution limit (k = 1.22), calculated as 138/Dmm or 5.45/Dinches. This is a theoretically justified and rather generous criterion; at this separation the stars are clearly separated by a dark diffraction band (see illustration, below).

(2) the Dawes resolution limit (k = 1.02), calculated as 116/Dmm or 4.56/Dinches. This is an observationally derived criterion that is essentially identical to the Abbe limit derived from diffraction theory. For that reason, I prefer the λ/D formula (113/Dmm or 4.5/Dinches) instead. At this separation the stars are divided by a barely discernible faint band, or appear as two circular disks touching at their edges.

(3) the Sparrow resolution limit (k = 0.95), calculated (usually) as 108/Dmm or 4.24/Dinches. This is really a detection limit because the Airy disks overlap by roughly 1/3 of their theoretical diameters and the separation is actually below the diffraction limit of the aperture, at k = 0.95. At this separation the stars snugly touch or slightly overlap at the edge and there is no dark gap between them.

These are only guidelines to a telescope's potential performance. Most optical textbooks rely on the Rayleigh criterion, while most visual double star astronomers prefer the Dawes criterion. The practical differences among them are inconsequential. For example, I have a 305 mm (12 inch) SCT telescope, so my resolution criteria can be calculated as 0.45" (Rayleigh), 0.38" (Dawes), 0.37" (lambda/D) or 0.35" (Sparrow). Because nearly all double star catalogs quote separations only in tenths of an arcsecond and therefore have a ±0.05" margin of error to begin with, 0.4 arcseconds is close enough for most purposes.

How Big Is Very Small? Now let's simulate on your computer screen, as far as practicable, a visual example of the resolution interval as it would appear through a telescope using the standard eyepiece described above.

The diagram below (left) assumes a 305 mm ƒ/10.4 SCT used with a 10 mm eyepiece for an exit pupil of 0.96 and an object magnification of 317x. When viewed from a distance of 24 inches or 60 centimeters (and provided the "2 inch/5 centimeter" interval is displayed at that width), the yellow dots in the diagram show ten times the 0.4 arcsecond resolution limit. The white dots just above them show a separation corresponding to 0.7" — the smallest interval I can represent in a pixel display. Below these, the magenta dots represent a 60 arcsecond (1 arcminute) interval.

For comparison, the diagram (right) shows the equivalent intervals in a 140mm ƒ/7 refractor with a 6.7mm eyepiece, which delivers a 0.96 exit pupil at an object magnification of 146x. Now the two white dots represent a 1.6" interval, and the yellow dots a 16" interval; at this lower object magnification, the 1 arcminute interval between the magenta dots appears much smaller.

These examples demonstrate two important optical principles:

• Diffraction magnification is proportional to the exit pupil only. The yellow dots, which represent the resolution limit in two different optical systems at two different magnifications, are exactly the same visual distance apart. This is because the resolution limit is a diffraction limit, and the diffraction magnification (the exit pupils) are identical.

• Diffraction magnification is not object magnification. The distance between the magenta dots representing one arcminute on the celestial sphere varies with object magnification while the exit pupil remains the same. The white dots illustrate that the combination of greater resolution at higher magnification allows smaller intervals to be visualized at the diffraction limit.

The framing effect of the eyepiece apparent field of view has a very helpful effect on the appearance of these visual intervals in a field of stars. This is the first of two reasons why you want a standard eyepiece where the field stop is just visible in the periphery of your visual field: it anchors the sense of proportion necessary to estimate small angular widths.

Training Your Eye to Estimate Separation. These intervals can only be learned at the telescope. Using a reliable source of double star information (see the page on Double Star Datasets), select up to a dozen double stars that are between your resolution limit and 10 times your limit (from about 0.4" to 4" with a 300 mm SCT, or from 0.8" to 8" with a 140 mm refractor) and that will be visible at your location in a single night. Limit your selection to double stars where both stars are between magnitudes 3 and 7, and are of roughly equal magnitude (magnitude difference no greater than ~0.7), so that the principal contrast is in the visual separation between the pairs. (This page provides an illustrative list for fall and spring.)

The critical point is to observe these targets using the standard eyepiece (the one you will use most frequently) to familiarize yourself with small visual intervals as seen through your equipment. You should be able to get down to about twice the theoretical limit without much trouble and using your detection exit pupil — again, on moderately bright, equal magnitude pairs in fair to good seeing. In other situations (fainter pairs, unequal magnitudes, mediocre or poor seeing) observing skill plays a greater role, although practice will improve your abilities quickly, and the resolution exit pupil will be necessary.

Once you have established what the different intervals look like in your standard eyepiece, try estimating larger angular widths. To do this, calculate your standard eyepiece true field of view (TFOV = AFOV/M). For example, a 7 mm Pentax XW eyepiece with a TEC 140mm ƒ/7 refractor has an apparent field of view of 70° and yields magnification of 140x at an exit pupil of 1.0, so its true field of view is 70°/140 = 0.50° or 30 arcminutes — the same diameter as a full Moon, and an ample area of sky. It is therefore 15 arcminutes from the center of the field to the edge of the field.

Use this fact to estimate the angular separations in very wide double stars — wider than 60 arcseconds or 1 arcminute. In the Pentax example, half the distance from center to edge is 7.5 arcminutes, 1/3 the distance is 5 arcminutes, and so on. This is the second reason you want to clearly see the eyepiece field stop: it allows you to estimate large separations without using a micrometer eyepiece or drift timing.

The goal of this task is to learn how to estimate the separation between two stars when they are observed with your standard eyepiece and your telescope. This skill will help you look for faint companions, to know (by comparison with catalog data) whether you are looking at the right double system, and to identify an unknown system in a double star catalog simply by visual estimates of its celestial coordinates, magnitude and separation.

Describing Close Separations. In the late 18th century, double star astronomers used the diameter of the Airy disk as a unit of angular width to describe the separation between double stars. Unfortunately, as you already discovered in task 1, the Airy disk angular width changes depending on the aperture you use and it also changes with magnitude — the disk becomes smaller in fainter stars.

Despite that, some double star astronomers deploy a complex vocabulary of discriminations in order to describe the appearance of very small angular separations. Sissy Haas, for example, suggests using rod shaped, figure eight, touching or kissing, split by a hair, a tiny gap apart, fairly close, wide, very wide and super-wide. Others use rod shaped, oblong, egg shaped, elliptical, notched and (of course) barely notched.

I think this is entirely too much frosting on the cake. The two benchmark criteria are detection, which occurs when a double star appears as a visually peculiar or misshapen single star, and resolution, when the double star appears as two distinct and well formed Airy disks, which to purists must be separated by a dark interval however thin it may be. Everything on the side of resolution can be accurately and consistently described by the arcsecond or arcminute angular separation, and verbal descriptions are superfluous.

Detection is a different matter. Several experienced observers report that reliable detection of an equal magnitude binary, across a wide range of apertures and under good to excellent seeing, is possible down to separations as small as k = 0.50 (see diagram, above) with an appearance described as elongated. As the system widens, a distinct notched appearance seems to occur relatively close to resolution, at around k = 0.80 to 0.90, and this persists up to a resolution threshold that is closer to the Rayleigh limit as aperture gets smaller. Oddly, a study by Bob Argyle suggests that detection below the resolution limit requires larger separations (that is, larger values of k that are still less than 1.0) in larger apertures, which seems likely to be due to the increased effects of seeing. As a result, the transition from elongated to resolved may be spread over a wider angular interval in smaller telescopes.

Note that these considerations apply to relatively bright and matched or nearly matched double stars, typically 8th magnitude or brighter. Fainter or unequal double stars are less easily described, and the fainter component may not display a clear Airy disk at all. For these reasons, Paul Couteau suggests that 10th magnitude stars are the lower limit for visual double star astronomy, regardless of aperture.

5. Visualize Position Angle

At some point the elongated appearance of close double star components allows the observer to estimate visually the position angle or, in a matched binary, a pair of position angles 180° apart. If the appearance of the system allows you to estimate the position angle to an accuracy within ±20°, prior to checking this value in a double star catalog, then an unambiguous detection or resolution has been made. To do that, you must be able to estimate position angle — and this means you always know which way is "up" in your field of view.

This is a major point of difference between double star astronomers and deep sky or planetary/lunar astronomers. Those folks rarely need positional information because their target star cluster or nebula forms a recognizable gestalt, or displays orientation directly — as the alignment of Jupiter's belts, Saturn's rings, or the terminator on the Moon.

Unfortunately, visually estimating position angle can be even more difficult than estimating separation. When I first began using an altazimuth mounted telescope with a 2" mirror diagonal, I was completely befuddled by this task.

The solution to the difficulty is to break the challenge down into its three component problems and tackle them one at a time, from easiest to most confusing. To assist your learning, direct observation of the half Moon with its linear shadow edge or an artificial target can greatly help you understand what is going on.

Measures of Angular Direction. First, a review of how position measurement works. As the celestial sphere is viewed by the naked eye, or represented in a star chart or planetarium software, position angle is measured from the line to celestial north, and increases through east in a counterclockwise direction. If you are facing due south, then north is the meridian at 0° position angle, east is on the left at 90° position angle, and west is on the right at 270° position angle, as shown in the "naked eye (star chart)" diagram (below left).

In the double star literature separation is sometimes notated as rho (ρ) and position angle as theta (θ), or identified by the symbols for their units of measurement, " (arcseconds) and ° (degrees).

A secondary but very useful system of directional notation describes the quadrant position of an object, as north, south, preceding (west) or following (east) of the target object (primary star). Instead of discerning one unit out of 360°, you choose one of only four possibilities, which greatly simplifies record keeping and also signals that your directional information is only approximate. For example, the label "south" may point to an object with a position angle anywhere between 135° to 225°.

A less ambiguous method, especially when the object is located between two cardinal points, is to combine the two relevant labels to create eight octants of 45°. In this convention north or south always given first when combined with preceding or following. Using this octant system, north following would indicate an object between ~20° to ~70°, following an object between ~70° to ~110°, and south following an object between ~110° to ~160° (diagram, above left).

Field Presentation. The first challenge is to determine the presentation of the celestial sphere in your eyepiece field of view. Presentation involves (1) the location of north (or some other cardinal point), and (2) the direction in which position angle is measured.

Without a mirror diagonal, a refractor, finder scope or Newtonian reflector will both invert and revert the naked eye orientation, effectively presenting to your eye an image rotated by 180°. In this condition the position angle still increases in a counterclockwise direction in the field of view (diagram above, "in a telescope"). If you are observing a southern object on the meridian and you are facing south when observing (with a refractor), then north is at the bottom of the field of view and west or preceding is on the left. If your sidereal drive is turned off, stars will appear to drift from right to left.

If you are using a mirror diagonal (for example, in an Schmidt Cassegrain telescope), then the field is flipped left to right. Now, if you are observing a southern object on the meridian, the eyepiece in the mirror diagonal is vertical, and you are facing south when observing, north will be at the top of the field of view but west will still be on the left, and the position angle increases in a clockwise direction (diagram above, "with a mirror diagonal"). If your sidereal drive is turned off, stars will still appear to drift from right to left.

Note that no matter what part of the sky you observe, position angle is always measured in the same direction: counterclockwise in a "direct view" inverting telescope, and clockwise with a mirror diagonal. When locating visual compass points, it is less confusing always to start from north and move in the same direction as position angle increases. In a refractor used without a diagonal, this is in the counterclockwise direction: east is 90° counterclockwise from north (bottom), and west is 90° counterclockwise from south (top). In a telescope used with mirror diagonal, it is in the clockwise direction: east is 90° clockwise from north (top), and west is 90° clockwise from south (bottom).

Most observers have a standard setup determined by their choice of telescope and mount — owners of Newtonian and Dobsonian reflectors routinely do not use mirror diagonals, owners of refractors and Cassegrains (for example, a Schmidt Cassegrain, Maksutov Cassegrain or Dall Kirkham) routinely use mirror diagonals. So identifying the presentation of your field of view is a problem you only have to solve once.

Pointing Rotation. The second challenge is to locate the direction of north within the field of view in relation to the telescope itself. This depends on which part of the sky you are observing — the direction your telescope is pointed in relation to celestial north pole — and on the type of mount you are using.

In telescopes on a fork declination with a polar wedge, there is no pointing rotation anywhere in the sky. The north/south coordinates are always in the same location.

With a German equatorial mount with rotation rings, the tube rotation is a form of observer rotation (see below), so the appropriate method is to start with the tube rotated so that the axis of the eyepiece (focuser tube) is perpendicular to the axis of the declination axis to establish the field orientation, then use a configuration of field stars to transfer this orientation into the field itself. Then the tube can be rotated to a more comfortable position and the orientation recognized from the field.

With a German equatorial mount without rotation rings, the telescope tube is in a fixed position on the declination axis and both the declination and sidereal axes of rotation are aligned to the celestial coordinates. The field orientation is constant across the entire area of sky on either side of the meridian. Pointing rotation occurs when the telescope is turned to the opposite side of the meridian, passing through celestial north (diagram, left). This "northern passage", necessary for a meridian flip, rotates the field by 180°, so the observer basically has only two position orientations to cope with.

With a German equatorial mount or a right ascension fork mounted on a polar wedge, locating the east/west dimension is quick and convenient: slew slightly a small distance east in right ascension, and a star centered in the eyepiece field will appear to move due west. North is then 90° counterclockwise (no mirror diagonal) or clockwise (with a mirror diagonal) from west in the eyepiece field (refer to the previous diagram).

Why not use the north-south declination slew instead? Because most commercial mounts are now servo driven with some form of hand paddle or handset control. Two opposing buttons or joystick motions control declination slews north or south, but these typically reverse function when the telescope performs a meridian flip (image, right). This occurs because a clockwise rotation in declination when the telescope is on the west side of the mount will point the telescope farther south, but a clockwise rotation when it is on the east side of the mount will point it farther north. The right ascension controls never change their directional effect, and provide unambiguous cues to celestial east and west. (If you manually adjust your telescope, there is less possibility of confusion. Adjusting the tube toward south declination will cause stars in the field to move north.)

With an altazimuth mounted telescope, pointing rotation occurs every time you change the pointing direction and as an object is tracked in its sidereal movement. The diagram (below) shows the main variations for a northern hemisphere observer using a "rear view" refractor or Cassegrain format telescope, with a mirror diagonal on an altazimuth mount.

These variations occur because an azimuthal slew is not aligned to polar north but to the zenith.

The most reliable method to identify field direction is to center the target object, insert a high power and roughly parfocal eyepiece (so that it is not necessary to adjust the focus), then turn off the sidereal drive and allow the star to drift across the field of view, which establishes the east to west (following to preceding) directions. Then north will be perpendicular to this line, on one side or the other depending on the direction in which position angle increases. But this eventually becomes impractical for objects near the celestial pole because they drift too slowly.

With my LX200 SCT, a vertically positioned mirror diagonal and an observing position facing the visual back, I developed a method of modeling the field of view position angle with my hands:

1. Face the point on the horizon that is directly below the object you want to observe, extend your arm, and hold the edge of your flattened hand and fingers, or a single finger, in a vertical position (pointing to the zenith).

2. Raise this vertical hand or finger to the altitude of the object you want to observe, and with the other hand or a second finger, point as accurately as possible from this position to the north or south celestial pole.

3. Flip the angle you have formed left to right or right to left around the vertical direction. (Accuracy is often improved if you state out loud the angle this forms (e.g., "45° from north"). This is the angle from the top of your field of view to celestial north as it will appear in your telescope.

After some practice, I am able to estimate position angles to within ±20° of catalog values using this rough method.

Observer Rotation. The third challenge is to determine the field rotation caused by changes the physical orientation of your head to the telescope — the observer orientation.

The simplest way to visualize this: starting at your habitual viewing position and an eyepiece clamped (with the compression ring) in the focuser, place two small strips of tape on opposite sides of the eyepiece mount or eye guard, so that a line between them corresponds to the line connecting the pupils of your eyes. Then any observing position you adopt where the tape marks are not in the same alignment with your eyes means that an observer rotation has occurred.

Observer rotation occurs when you (1) reverse your observing position from one side of the telescope tube to the other, for example because you have flipped an equatorial telescope from one side of the meridian to the other; (2) rotate the telescope tube in rotation rings; (3) rotate the mirror diagonal in the focuser; or (4) change your orientation to the telescope tube due to a change in telescope pointing, for example from vertical to horizontal pointing in a Dobsonian telescope.

The only situation in which observer rotation does not occur is with an altazimuth mount, a "rear view" refractor or Cassegrain format, a fixed mirror diagonal with the eyepiece aligned vertically when the telescope is pointed horizontally, and the observer standing or sitting in a vertical position directly behind the optical tube assembly and facing in the same direction as the optical axis. This astronomer never experiences observer rotation and only has to deal with pointing rotation (as described above). In contrast, any telescope on a German equatorial mount or fork mount with polar wedge will necessarily induce one or more forms of observer rotation, in addition to the pointing rotation caused by a polar flip.

An observer working with an altazimuth mounted Newtonian (Dobsonian) experiences both pointing and observer rotation, and has no slew aligned to the celestial sphere, and therefore suffers continual difficulty in establishing the field orientation. Star drift is the only reliable way to disambiguate the situation, which is convenient because these telescopes often do not have a sidereal drive.

However, with a refractor or Cassegrain scope on a German equatorial mount, eliminating observer rotation is easy. Position and lock the diagonal in the focuser mechanism so that the eyepiece is exactly parallel to the declination axis: then the field presentation will always be rotated so that the east/west axis is parallel with the optical axis and the optical tube assembly. North will be on the left if you are observing west of the meridian, and on the right if pointing east of the meridian. Now the sides of the mirror diagonal, the extended Crayford focuser or sides of the telescope tube itself provide edge cues for the east/west (following/preceding axis), and this is more than adequate to judge quadrant or octant locations and to eliminate the uncertainties caused by changes in your physical position.

Estimating Position Angle. Once you have made the equipment adjustments and mastered the conceptual basics of necessary to identify north, east/west and the direction of position angle measurement in your eyepiece field of view, you are ready to visually estimate position angle.

To do this, train your binary eye by observing a series of wide (ρ > 20"), bright double stars. In the first training step, target the star with your finder scope or computer GoTo controls, then before you look through the eyepiece, read the catalog data for position angle and separation, and using your experience with separation and field rotation, try to imagine the appearance of the star in the field of view. Then look at the star, and use the visual fact to correct your estimates. Repeat this until you feel confident in your ability to imagine the visual appearance from the data alone.

Then reverse the procedure. First look at a widely separated star without consulting the catalog data, estimate the separation and position angle visually, and finally compare your visual estimates to the catalog values.

You will find that you can quickly estimate position with remarkable accuracy. I consider a position angle estimate within ±20° of the catalog value as a successful detection of a faint or very close companion. In fact, with a properly aligned and clamped star diagonal on an equatorial mount, I routinely achieve an accuracy that is within ±5° of the catalog values.

Two complementary observing habits are involved here. One is discovery: examine a double star at the eyepiece before consulting the catalog data: this enhances the surprise (or disappointment) at what you might see, and develops the habit of active looking and problem solving that can reveal interesting specifics in the field of view (faint nearby doubles, unusual asterisms, additional components in multiple stars catalogued as binary). The other is confirmation: check the catalog data after you feel confident that you have detected a difficult companion, rather than rely on the dubious method of letting the data seduce you into "thinking maybe" you see the companion where the catalog says it should be. As added benefits, these habits help you to identify cases where the catalog data are wrong, and teach you to visualize the appearance of a system in your standard eyepiece based on the catalog data alone.

6. Use Active Magnification

I've emphasized the benefits of habitual viewing with a standard eyepiece, but you will need to expand your range of eyepieces, and learn the importance of active magnification, to derive the most pleasure from double star astronomy.

A Minimal Eyepiece Selection. A minimal (necessary and sufficient) eyepiece selection should include these three, chosen to work together:

(1) The standard eyepiece — the eyepiece that provides your detection exit pupil, the standard visual scale (object magnification) for judgments of separation, position angle, magnitude and distances to noteworthy field features, and a clearly visible field stop (an apparent field of view of 70° or less) used to estimate large field distances. This is the benchmark of your selection.

(2) A resolution eyepiece — the "nutcracker" that provides a diffraction magnification close to your resolution exit pupil and is used to resolve the closest and most difficult double stars. A high quality zoom eyepiece that includes the desired focal length is better than a fixed eyepiece, because this allows you precise adjustment in the magnification that affects the limit magnitude determined by the background sky brightness, the contrast between a faint companion and the glare or diffraction rings of the primary star, and the apparent range of motion in the "dancing" caused by atmospheric turbulence. Note that the field of view is irrelevant, and in terms of increasing light transmission and reducing glare and internal reflections (ghosting), a traditional AFOV (≤ 50°) eyepiece with only 3 or 4 lenses (such as a genuine Orthoscopic, genuine Plössl or RKE) may be optimal.

(3) A finder eyepiece — a wide true field eyepiece is often necessary to find target stars when star hopping with a star atlas, but it's also useful as a "landscape" eyepiece that can frame a double star within a large star cluster, a rich field of stars, a dark field (spectacular in its own way, especially with a wide or "fragile" binary), or just for recreational viewing when the seeing is bad. Here you will likely prefer something around twice your detection exit pupil (de ≥ 2), so that stellar Airy disks disappear, with an apparent field of view ≥ 70°.

Although infrequently called out as a specification, your eyepieces should be comfortable to use, with adequate eye relief and a comfortable eye rest. In combination with your choice of telescope, they should produce a flat field and be free of field height aberrations such as astigmatism. Finally, it is helpful to have parfocal eyepieces that do not require refocusing when one is replaced by another — this makes the practice of active magnification more convenient.

The Maximum Eyepiece Usage. There are four competing considerations that influence the choice of magnification in specific situations. These clarify the reasons why you have chosen a set of three different oculars, and when you should use one instead of another:

• Change eyepieces often. This advice from Paul Couteau is probably the best single rule: he compares it to shifting gears while driving a sports car. Passive magnification or viewing everything with a single eyepiece has serious drawbacks. By systematically hunting with a high power eyepiece, you will miss some of the beauties of the field, including doubles near to the target system. If you are cruising with a low power eyepiece, you will fail to detect close pairs. The recommended selection of three eyepieces is not burdensome. Take the time to use them all.

• When in doubt, reduce your exit pupil. The common rule for planetary and lunar observing — use the magnification that is not affected by atmospheric turbulence — is often wrong for double star observing. It can be helpful to increase diffraction magnification as the seeing worsens, especially when double star components are of similar magnitude. I regularly use exit pupils of 0.5 or higher specifically when seeing is bad. Only the worst, "boiling" air turbulence will obliterate the star Airy disks; until that happens, increasing magnification can make the Airy disks easier to see through the turbulence.

For double star astronomy, the upper limit of magnification has been passed only when (1) the image is too faint to be seen clearly in the fovea or to allow the optimal focus to be found; or (2) the side to side motions of the turbulence become so wide or rapid that they are confusing to the eye. In either case, inspection of the image for up to half a minute is advisable before making a judgment, as a seemingly hopeless image can briefly display the stellar disks to patient observation. It is much better to pass these visual limits, then retreat to the highest useful magnification, then to timidly work within your preconception of "what the atmosphere permits".

• Find the optimum for pairs of contrasting magnitude. An important exception to the previous rule is the case of a star system with a close separation and a significant difference in magnitude between the primary and its component(s). Sirius (alp CMA) is the classic example, but there are many others. In these instances you will miss viewing the faint secondary if you use either too low or too high a power. Too low, and the secondary will be indistinguishable from the glare; too high, and the secondary will be smeared out so much by the magnification that it becomes dimmer and even nebulous. In effect, the planetary and lunar advice applies: you need to find the magnification that increases the visibility of the separation without losing the secondary in the seeing.

• Stop and enjoy the view. Despite the emphasis on high magnification that comes with splitting close double stars, all stars are located in a star field and a low power and/or wide angle eyepiece is necessary to place the star in its proper setting. On the one hand, low power views are especially rewarding with stars located in or near the Milky Way (e.g., Cygnus, Lacerta, Cassiopeia or Orion), stars within open star clusters or nebulae (STF 1121 in M47, the ORI in M42), "double double" stars that can be viewed side by side (epsilon 1 and 2, STF 2470 and 2474 in Lyra), stars of striking brightness or color contrasts (chi CYG, bet CYG). On the other hand, the density of field stars at a specific location is the benchmark to evaluate whether a group is actually an optical double star — a wide pair or a faint companion observed in an otherwise dark field, almost empty of other stars, is much more likely to be a physical system. And these systems are a stark image of the isolation and extreme distances of space.

You can explore and identify your own capabilities with magnification, using star images modified by aperture masks. It is not necessary to accept "common wisdom" in its place. Exit pupils less than 1.0 do not furnish "wasted" magnification as sometimes claimed: they make the diffraction artifact larger and easier to analyze, especially when atmospheric turbulence puts it in motion.

As another example of a misguided rule, Alan Alder in Sky & Telescope suggests using an object magnification equal to 750 divided by the arcsecond separation of the binary. Although the rule does encourage you to use high magnification with close double stars, it becomes impractical with wider pairs. A double star separated by 17" — the average separation of pairs listed in WDS — yields a magnification of 44x by Adler's rule, which is a 50 mm eyepiece in a 8.5" ƒ/10 Schmidt Cassegrain. Worse, by continually adapting magnification to separation, you lose the standard frame of reference that allows you to judge separation and magnitude visually.

It also weakens the visual appreciation of a system's spatial context. It is more dramatic and informative to see the same close or wide double stars in the same visual presentation, in other words at the same magnification in the same apparent field of view. This is the merit of having a single wide field or super wide field eyepiece that can you use to place the system in context, then using higher magnifications as needed to zero in on system details or resolution.

Note that most "astrometric" or measurement eyepieces come in only one focal length (usually around 12mm) ... so you don't have much discretion there. You can apply a 2x, 3x or 5x barlow to increase the magnification into the resolution range of exit pupil, but this will also increase the motion of the system caused by atmospheric turbulence.

7. Choose the Right Telescope

A longer term part of training your binary eye is an "eyes on" sense for what different telescope systems can accomplish in terms of light grasp, resolution, transmission and sensitivity to the atmospheric turbulence and light pollution typical of your observing site, and a "hands on" familiarity with the different usage requirements they impose in terms of observing comfort, collimation, cool down, pointing accuracy and portability.

The criterion here is your strong preference: you would rather use your chosen double star telescope than any other. The only way you can develop this preference is to use many different telescopes. Choosing the right telescope is therefore a sociable task. You should attend star parties, visit the observatories and examine the equipment of other astronomers in your area, and talk to vendors and manufacturers about the suitability of their products for your interests.

The Equatorial Mount & GoTo Pointing. Perhaps the most important piece of equipment for the double star astronomer is not the telescope or the eyepiece, but the mount. Here the essential component is a computer controlled servo mount that can point on command and track reliably. The computer controlled pointing should allow the manual input of celestial coordinates, or should contain from the manufacturer a large double star database that is easy to access from the handset.

The discussion about field orientation (above) illustrates the desirability of a German equatorial mount as an aid to the identification of celestial coordinate directions in the field of view. But an altazimuth mount is very comfortable to use: the physical location of the eyepiece never varies in a vertical direction by more than the distance from the fork (altitude) axis to the eyepiece. And I found that the pointing rotation that occurs with an altazimuth mount could be mastered with a bit of practice. Physical comfort dominates as the primary requirement for long nights of enjoyable viewing.

Most commercial telescopes with GoTo mounts will perform reliably if first aligned to two or three known stars. So a secondary consideration is whether you intend to use the telescope as a portable instrument, and how easily it can be packed up, physically moved, powered and operated in the field. If you are lucky enough to have a home or remote observatory, these considerations take second place to the GoTo accuracy of the mount after it is properly aligned on a fixed pier.

The quality of double star data as currently (2016) packaged with commercial mounts is, on the whole, poor. The selection, if there is one, is likely to be superficial, accessing the data through a handset is cumbersome, and you may be required to identify a multiple star through the catalog label for a single pair of components. In this situation it is very helpful to manually enter double star coordinates from a database such as WDS or a double star observing guide.

Properly aligned, the pointing accuracy of most commercial mounts is almost spookily good. Even so, confirm that your preferred mount can consistently put your target double star system within the field of view of your standard eyepiece, regardless of where it is in the sky. Test this by pointing to bright, named stars on both sides of the meridian and in the north and south directions of the celestial hemisphere.

How Much Aperture? The direct benefits of increased aperture are a smaller resolution limit (as 1/Do), greater light grasp (as Do2), a deeper limit magnitude (as log[D]) and, at a constant exit pupil (diffraction magnification), greater object magnification (as Do). This might suggest that a double star astronomer should acquire the largest practicable aperture, but we can see from their different mathematical definitions that the factors will not change in equal proportion as aperture increases.

The discussion on another page evaluates this issue as the change in the aperture factors across the range of apertures that might be used for astronomical observing — from a 40 mm binocular ("min" aperture) up to the largest telescope most astronomers could afford or find practical to use (assumed to be 20 inches or 500 mm, the "max" aperture).

The chart (right) illustrates the relative decrease in the resolution limit (1/Do) and the increase in the light grasp (D2) as aperture gets larger. The resolution limit falls dramatically with increasing aperture and the gains bottom out quickly, so that an aperture increase that is just 20% of the maximum affordable or practicable aperture (~140 mm) delivers 75% of the potential resolution gain; half the maximum aperture (250 mm) achieves 93% of the largest aperture resolution. (Magnification increases linearly with aperture and follows the dotted diagonal line.)

In contrast, the chart shows that light grasp accumulates rapidly only in the largest apertures — half the maximum aperture yields less than 30% of the total possible gain in light grasp, and 75% of the gain in light grasp is obtained only with 85% of the maximum feasible aperture (425 mm or about 17 inches). Although not shown in the diagram, limit magnitude changes in the same way as resolution. A 140 mm aperture achieves roughly 50% of the total possible gain in limit magnitude, and a 250 mm aperture achieves 75% of the practicable gain.

These contrasts divide the range of practical or affordable apertures that an astronomer might use into two regimes. Across the smaller half of the affordable or practical aperture range, increasing aperture primarily benefits resolution and limit magnitude; across the larger half of the range, increasing aperture primarily benefits the light grasp useful to see extended targets with low surface brightness. This is the origin of the "aperture fever" that affects deep sky astronomers and their preference for apertures in the "light grasp" half of the aperture range. For the double star astronomer, however, resolution and limit magnitude are the most important image quality criteria, which shifts the aperture preference to the smaller or "resolution" half of the practicable aperture range.

An alternative evaluation can ask, how many new catalogued double stars does an increase in aperture provide? As aperture increases the total number of double stars must also increase as well — but there a point of diminishing returns. For the total count of optical pairs or "high probability" physical double stars listed in the WDS, the graph (left) indicates that an aperture of around 225 mm (9 inches) is where the yield in optical pairs begins to decline, and for the "high probability" double stars the tally becomes essentially flat. We don't add more double stars beyond this implicit 11th magnitude limit because in most cases we lack sufficient proper motion, radial velocity, relative positional change or spatial distance data to identify double stars reliably.

An important additional consideration is the sensitivity of larger apertures to thermal turbulence or seeing, which becomes more noticeable at apertures above ~150 mm (depending on local observing conditions). Larger apertures deliver very high resolution in theory, but their sensitivity to atmospheric turbulence makes it difficult to exploit this resolution advantage in full, and in larger reflectors the mirror mass can prolong cool down and can create thermal currents across the mirror (aperture seeing).

For the visual or CCD double star astronomer, there are excellent reasons to use apertures no greater than 250 mm in a reflector or 200 mm in a refractor. These yield λ/Do resolution limits of 0.45" and 0.57" and averted vision, dark sky limit magnitudes of 14.6 and 14.1, respectively. Doubling the aperture to 500 mm does not really increase the variety of interesting targets that can be observed. Nearly all "showpiece" doubles are naked eye magnitude pairs; as increased aperture resolves the closest pairs, it discovers new barely resolvable pairs; as it improves the visibility of very faint double stars and faint companions in bright double stars, it increases the number of targets that can only be seen with averted vision. In the 2016.01 Washington Double Star Catalog there are almost 13,700 "high probability" physical double star systems — 12,100 binaries, 1580 multiple stars — with primary magnitude down to 10.5, secondary magnitude down to 13.5 and a separation of 0.5" or wider. Owners of moderate aperture telescopes will have plenty to occupy their interest, and their smaller apertures will be noticeably less sensitive to the optical defects of a turbulent night sky.

What Type of Telescope? The third critical aspect of telescope preference is the optical design. Among commercially available telescopes, designs divide into the main families of refractors (dioptric), reflectors (catoptric) or the modern hybrid catadioptrics. Reflectors include the peerless Newtonian and the many Cassegrain variations, including the classic Cassegrain, Dall Kirkham (DK), and Ritchey Chrétien (RC). Catadioptrics include the Schmidt Cassegrain (SCT), Maksutov Newtonian (MN) and Maksutov Cassegrain (MC). The following comments can guide your expectations with each format.

• Consider the focal ratio. The relative aperture of the instrument determines the range of diffraction magnification that you can achieve with a commercial range of eyepieces without added optics such as a Barlow lens or Powermate. A large relative aperture (above ƒ/10, optimally ƒ/20 or more) will enhance your ability to exploit the aperture resolution at exit pupils down to at least 0.2 mm.

• Consider your viewing comfort. The physical and mass characteristics of the instrument are limiting factors. Focal length determines tube length, which makes large aperture, large focal ratio refractors or Newtonians both massive and unwieldy. The popularity of the commercial Schmidt Cassegrain and Dall Kirkham designs is due in part to their compact primary mirror focal length and magnifying secondary mirror, which yields an effective focal length that is four or five times the physical tube length. Weight is also a consideration, and usually requires fixed pier mounts in SCTs larger than 12" aperture and refractors larger than 6". A long focal length refractor or Newtonian reflector is the most awkward to use, as the instrument will require you either to sit almost on the ground or use a ladder to reach the eyepiece; the optical tube assembly will be heavy and may require two people to mount and dismount it safely.

• Refractor images are not reflector images. There are several differences between the diffraction artifacts produced by catoptric or catadioptric versus dioptric telescopes, due to the presence of a central obstruction. Reflector diffraction rings are brighter and the Airy disks slightly smaller than in the same aperture and focal ratio refractor, and catoptric designs create diffraction spikes from the star image caused by the secondary mirror supports. This is a tradeoff rather than a clear advantage: the brighter rings make close, faint companions more difficult to detect in a reflector, but the smaller Airy disk can make close, matching double stars easier to resolve in a reflector. The actual advantage is esthetic: star images are prettier in a refractor.

• Catoptric systems yield more accurate star color. There is no transmission through glass in a reflecting telescope, so the color rendering in a reflector can be slightly more accurate (excluding the effect of the eyepiece glass and the glare of bright stars in large apertures).

• Across systems, aperture is apples to oranges. The specific design details of a telescope are as important as raw aperture. "Corrected" Dall Kirkham telescopes, or small focal ratio Newtonians used with a coma corrector, are actually catadioptric instruments. Antireflection coatings and baffling in a refractor tend to produce a darker background sky, and the contrast in delicate planetary detail tends to be more pronounced, than in a reflector of the same aperture. Well built focusing mechanisms can made observing a precision activity, poor ones a frustrating struggle.

Telescopes are physical objects and engineered mechanisms, and these affect what the telescope can do, how it is used, and how it should be maintained. These are considerations left to the individual observer. In general, for double star astronomy, any high quality telescope with an aperture up to about 250 mm will provide the necessary optical resources for many years of visual double star astronomy.

8. Learn the Factors that Affect Resolution

If you have engaged the tasks described on this page, then you have come very far as a double star observer. You understand the structure of the diffraction artifact, its visual dimensions, how diffraction magnification is distinct from object magnification. You can recognize and estimate both separation and position angle at the eyepiece, and visualize a system configuration from catalog data. You understand how to use magnification to grapple with poor seeing, and have chosen an appropriate telescope and eyepieces for your pursuit. All that remains is to continue your observing to gain experience and refine your observing techniques.

The pleasures and rewards of visual double star astronomy are greatly enhanced by knowledge of the origin and variety of these systems and their unique place in Galactic star formation and stellar evolution. Those topics are explored on another page. I urge you to learn what you can about the natural history of these remarkable systems.

In this final section I outline what I have learned over the past few years about the visual pursuit of double stars, using instruments from 80 mm and 140 mm achromatic and apochromatic refractors to a 180 mm Maksutov Cassegrain, a 250 mm Dall Kirkham and a 305 mm Schmidt Cassegrain.

The Factors that Affect Resolution or Detection. What are the factors that affect your ability to observe, detect and resolve a double star system? The list of potential factors turns out to be a long one. I divide them into seven groups: each group contains variables that either depend on or influence the other factors in the group and, as a group, can determine the importance of other factors.

• (A) The system brightness factors are (1) the magnitude of the brighter (primary) star (m); (2) the magnitude of the fainter (secondary) component (m'); (3) the magnitude difference m'–m between the primary and secondary components (delta–m or Δm); and (4) the primary and secondary spectral types.

• (B) The system configuration factors are (5) the component angular separation (ρ); (6) the component position angle (θ); and (7) the system annual change in separation (Δρ) or in position angle (Δθ).

• (C) The aperture factors are (8) the aperture (entrance pupil) diameter Do; (9) the aperture resolution limit Ro (proportional to 1/Do); (10) the aperture limit magnitude LM (proportional to log[Do]); and (11) the relative aperture or objective focal ratio (No).

• (D) The enlargement factors are (12) the objective focal length (ƒo); (13) the eyepiece focal length (ƒe); (14) the object magnification (Mt) calculated as the ratio of [12]/[13] or ƒo/ƒe; and (15) the diffraction magnification or exit pupil (de), calculated as the ratio of [13]/[11] or ƒe/No.

• (E) The instrumental factors include (16) the collimation of all optical components, including the eye, to the optical axis of the objective; (17) the accuracy of focus; (18) net light transmission, after absorbance and scattering by all optical materials and surfaces; (19) the optical quality of the image, or the sum of all aberrations caused by the optical surfaces and any glare or ghosting in the image; (20) the thermal currents within the instrument itself, or instrument seeing; and (21) in catoptric or catadioptric telescopes, the telescope central obstruction ratio (CO).

• (F) The environmental factors are (22) the atmospheric seeing (thermal turbulence); (23) the atmospheric diffusion (transparency); (24) the amount of dewing or condensation on optical surfaces; (25) the sky brightness or naked eye limit magnitude (NELM); and (26) the altitude and azimuth of the target.

• (G) The observer factors include the observer's (27) visual acuity; (28) dark adaptation (as this affects the visual thresholds LA, LF, and LD); (29) criteria for "resolution" and for the maximum separation accepted as a double star; (30) familiarity with the instrument and site; and (31) "personal equation" — including motivation, patience (duration of observation), alertness, observing skill, and the expectation created by knowledge of the system configuration and visual appearance.

• (H) The last factor is often unsuspected: (32) the quality or accuracy of the catalog data, verbal descriptions or diagrams that the observer relies on to identify and evaluate the system.

I discuss the effect of these factors in detail, below. Keep in mind that in many situations the observer factors (D) are comparable in importance to the instrumental and environmental factors combined. In particular, the element of observer skill means that you treat the detection or resolution of close companion stars as a problem to be solved. The observational challenge is not whether you can resolve a close pair on first examination, but how well you can identify the reasons for a failure to detect the pair, and adapt to improve your chances of successful detection or resolution.

The Most Important Factors. Several observational studies or theoretical arguments have attempted to estimate the detection of a faint companion or the resolution of a close double star using a limited selection from the many factors [1-32] listed above.

This approach cannot (and does not) reliably predict individual performance, due to the wide variation in individual capabilities and observing skill, the complexity of the visual stimulus, inaccuracies in catalog data, and variation in the quality and adjustment of astronomical telescopes. It will be useful, however, to discover that this approach has focused on a limited number of factors.

By their selection from the 32 factors, all the predictive models assert that the selected factors are sufficient for prediction; the actual formula then describes how they are significant. Excluded factors are assumed to be either optimal or "average" and, if that assumption is correct, they can be ignored.

The Estonian astronomer Ernst Öpik (1924) examined the parameters of double star catalogs and on statistical grounds proposed a double star "subjective measure of difficulty" c that combines the effects of "instrument, observer and conditions of seeing" (and implicitly also the magnification) from the aperture [8] alone as:

c = Δm – 4·log(ρ) – 4·log(D'/D)

where Δm – 4·log(ρ) represents an "objective measure of difficulty" of the double star as a visual stimulus, or "the combined action of diffraction, chromatic aberration of the objective, atmospherical dispersion of light and the physiological effect of contrast in the human eye upon the possibility of seeing a companion at a given angular distance from a brighter star." (Öpik adds that the major part of the effect of contrast in the objective measure of difficulty "may be eliminated by the use of the greatest possible magnifying power.")

The term 4·log(D'/D) represents a "subjective measure of difficulty" that will depend on "the instrument, observer and conditions of seeing." Despite this combination of three very different factors, the subjective measure is based only on the aperture in use (D') as a ratio to a baseline aperture (D), and assumes that the magnitude of the fainter component "is not too near the limiting magnitude of the telescope."

However, log(D'/D) is linear on log(D'/λ), the inverse of the calculated angular resolution criterion (R) of the aperture in use, and the difference of logs is the log of a ratio; so the Öpik formula can be rearranged and standardized on units of the resolution of the aperture as:

Δm = c + k·log(ρ/Ro)

where Δm is the largest magnitude difference that permits detection at the given separation ρ, and ρ/Ro is the resolution ratio or the pair separation measured in units of the aperture resolution limit. (Throughout, I use the Abbe resolution limit or λ/D, which is equivalent to the Dawes resolution criterion, as the unit of choice.) In this standardized form, the shape of the resulting curve can be adjusted to fit observational evidence or optical theory by the choice of c (the maximum allowed Δm at the resolution limit) and k (the rate of increase in Δm as ρ/R increases).

Summarizing all of Öpik's comments, we learn that magnitude difference [3], separation [5], aperture resolution limit [9], aperture limit magnitude [10], magnification [14 or 15], seeing [22], dispersion [23], image quality [19] and overall observer factors (G) were identified as critical to double star detection.

In a frequently cited study, amateur astronomer H. Peterson (1954) developed a formula from his own observations to predict the faintest double star companion m'max that could be resolved or detected with a given telescope as:

m'max = LM – 2.4 + 1.6·log(ρ/Ro), ρ/Ro ≥ 1.0

which claims that the limit magnitude of component detectability depends only the telescope limit magnitude LM [10], and the resolution ratio of the pair viewed through a given aperture. (Peterson adds that magnification [14 or 15] sufficient to see the resolution limit is assumed.)

A recent, more sophisticated model developed by T. Napier-Munn (2008) claims to estimate with a high (~86%) cross validated accuracy the probability p(Y/N) that a double star can be resolved by a given aperture:

x = 1.6225 – 1.2026(m'–m)/ρ – 0.5765m'/ρ + 1.9348D2Z/105;
p(Y/N) = ex / (1 + ex)

where e = 2.71828 and Z is a 1 to 5 "Danjon" rating of the seeing (5 being best) which, as a visual rating, will combine atmospheric and instrumental sources of turbulence.

The resulting function does not have a perceptual interpretation, as it is based on a linear rather than log relationship between separation and magnitude difference. The probability forms an ogive curve that (given a 250 mm aperture, a 6th magnitude primary star and a Δm = 1) predicts a 75% (3 in 4) probability of detection at values of ρ/R between 6.8 (in very poor seeing) to 1.75 (in excellent seeing).

Napier-Munn's choice of factors indicates that secondary magnitude [2], magnitude difference [3], component separation [5], aperture [8] and atmospheric plus instrumental seeing [18 and 22] are the important variables. (Adequately high magnification [14] was also found to be important, but was excluded from the model "for practical reasons.")

Finally, amateur Christopher Lord has proposed a model based on a detailed review of the diffraction artifact and historical double star discoveries:

ρ/Ro = 1.033·10((Δm–δm)/n)

where δm is the minimum magnitude difference of visual significance to resolution (which Lord gives as 0.1), and n is "an index of telescope performance" that is the sum of tabulated values for the system resolution ratio [5 and 9], the central obstruction ratio [21] and the rated atmospheric seeing [22], yielding a range in the values of n from 1.25 to 12.0 with a median of 7.

However, the Lord equation is functionally identical to and graphically indistinguishable from the standardized Öpik function when c = 0 and k = n, so Lord's innovation is entirely in a table of lookup values for the aperture [8], the central obstruction ratio [21] and the rated seeing [22], which are summed to determine the value of n in the calculation. We have come full circle in the theoretical analysis of double star resolution.

These theoretical approaches gloss over three difficulties. The first is a focus on the effect rather than the cause: some factors are determined by others, and it's really the determinative factors that we want to identify and control. Peterson's rule actually says that aperture [8] and sky brightness [25] alone are sufficient predictors of detection, as these determine both the resolution ratio and telescope limit magnitude.

The second difficulty is the assumption of ideal conditions in the excluded factors — for example, that sufficient magnification is used. Larger apertures [8] are more sensitive to atmospheric turbulence or seeing [22] and more likely to generate instrument thermals [20] that degrade resolution. Making a direct connection between aperture and resolution requires complete cool down (thermal equilibrium) and a perfectly tranquil atmosphere so that any increase in resolution is not masked by turbulence.

The last difficulty is that the cause and effect relationships change depending on the specific situation — if one of the excluded factors is less than ideal, the resolution limits can change. The target, instrument, site conditions and observer skill and acuity can make any single factor sometimes irrelevant and sometimes important. Experience is necessary to judge how the factors affect each other in a specific situation and how to adapt as necessary when the situation changes.

Resolution Ratio & Magnitude Difference. Let's turn from theory to an assessment of actual observer performance. Patrick Treanor (1946) adopted a method of graphical analysis, first used by Ernst Öpik (1924), to study the detectability of visual double stars by over 40 astronomers. This Treanor plot uses the bare minimum of analysis variables: the resolution ratio (ρ/Ro, [5] and [9]) and the magnitude difference between double star components [3].

I've recreated his plot with 2300 double stars discovered by means of visual detection (diagram, below). The key (lower left) identifies the eight 19th and 20th century astronomers, with the aperture (Dmm) attributed to each astronomer at the time of their discoveries — refractors in all cases excepting John Herschel. The angular separation was recorded within a few years of the pair discovery and this separation is divided by the assigned aperture resolution limit to calculate the resolution ratio.

Following Treanor, I've identified with small crosses the Δm (from the Airy disk peak luminance) and standardized radius of three bright features of the diffraction artifact: the Airy disk full width half maximum (FWHM) and the first two diffraction rings. Dashed lines indicate the standardized radius of the three dark intervals enclosing these features, including the first dark interval that defines the Rayleigh resolution criterion. Treanor connected the dark intervals at points equal to the Δm of the adjacent inner bright feature to define a "rather optimistic" limit to visual detection "under astronomical observing conditions" (gray curve in diagram).

The uneven distribution of discoveries by different astronomers illustrates that different working methods and telescopic equipment shape the detection limits of different astronomers. The discoveries by Wilhelm Struve or John Herschel, who surveyed the available sky at a brisk pace, are most often double stars of similar magnitude and easily resolved separation. Otto Struve, Burnham, Muller and Innes — who seem to have had unusual visual acuity and were also searching after the "easy pairs" had already been identified — culled a number of double stars below the Rayleigh resolution limit (1.22Ro).

Öpik first used a similar plot to estimate the proportion of potential binary pairs that were beyond visual detection and would remain undiscovered by visual astronomers. This was an important issue in early 20th century astronomy because it indicated the dynamic characteristics of double stars (mass ratios and orbital distances) that would be missing from supposedly exhaustive double star catalogs. This goal of correcting for physical systems omitted from the corpus of double stars already discovered has been superseded in recent amateur astronomy by an interest to "predict" in advance whether a double star can be split by an average observer using a given aperture — a quite different problem.

To delve this issue, I've included in the diagram, in addition to the Treanor limit: (1) my standardized version of the "subjective" detection boundary proposed by Ernst Öpik and calculated as Δm = 1.0 + 4*log(ρ/Ro) (yellow curve); (2) the logarithmic equivalent of a detection boundary defined by Gerard Kuiper and calculated as Δm = 0.75 + 6*log(ρ/Ro) (white curve); and (3) the resolution equation developed by Chris Lord (green curve), shown here in the form Δm = 0.0 + 10.0*log(ρ/Ro), which is identical to the curve where n = 10. The Öpik curve is the general boundary for component detection by professional astronomers; the Lord curve describes (for example) the resolution capability of a 250 mm refractor under excellent seeing. In each case, the boundary defines an area to the right and above the curve where detection or resolution is expected to occur.

Note that all the "predicted resolution limits" (with the exception of Treanor's) can be reproduced as the original Öpik function with different values of c and k.

All the resolution limits fail to include a number of pairs with separations between 0.5 < ρ/Ro < 2 and Δm < 3, in the area enclosed by the ellipse. They are likely not errors in the assigned apertures, as the same exceptions appear in Treanor's original plot. In some cases these may be incorrect placements due to inaccurate measurement of the component magnitudes or separation when the secondary is on or inside the second diffraction ring. Most of them probably indicate the exceptional visual acuity of certain astronomers. The fact that all the resolution limits seem incorrectly to exclude these close unequal pairs from detection indicates that the resolution functions lose validity in the very pairs where an accurate definition of visual limits would be most useful.

Telescope design and observing conditions can have a large impact. The diagram (right) illustrates the family of resolution limits produced by the Öpik equation when c = 0. Within the aperture standardized Treanor plot, the function changes only with the value assigned to k, identical to n in the Lord equation, as it varies from 1.25 (representing a small aperture reflector with a large central obstruction used in poor seeing) to 12 (a large aperture refractor in excellent seeing). The effect of increasing the aperture, minimizing the central obstruction and observing in calm skies is to straighten the curve vertically near the theoretical resolution limit. (The dotted line "rule of thumb" limit is approximated by the median value of n = 7.) Interpreting the differences in more detail is complicated by the many omitted variables, such as magnification (a problem mentioned by Treanor).

The observational data plotted in the diagram (above) suggest three conclusions:

(1) The number of matched or nearly matched pairs at ρ/Ro < 1.0 indicates that detections at separations less than the Abbe resolution limit are possible. Similar exceptional detections are found in Treanor's original plot and attributed to pairs in which "both components are bright" (far above the telescopic limit magnitude).

(2) Within the radius of the second dark interval (ρ/Ro  2.2) there are few detections below delta–m > 2.0, which suggests the difficulty of detecting unequal pairs that are obscured by the first diffraction ring. (There also a hint that components located on the first or second diffraction dark interval are easier to detect than those located on the first bright ring.)

(3) Detection or resolution boundaries with a flattened slope take into account the extended effect of obscuring glare on the visibility of clearly separated but faint components — and if the primary star is bright enough or atmospheric diffusion is extensive enough, glare can obscure even those components that are "not too near the limiting magnitude of the telescope."

Note that the "prediction" curves are all based on a sample of double stars observed and detected only, omitting those stars that were observed and not detected. Unless we can compare the characteristics of both the successful and failed detections, we cannot statistically define a boundary between them that represents the observational probability of detection or resolution.

A Troubleshooting "Rule of Thumb". However, a "predicted resolution limit" can be useful to gauge the relative difficulty of visual detection or resolution. Here the aim is to decide whether something is amiss — there's an observational problem to be solved — when detection or resolution fails.

All the resolution prediction methods described above require a calculator or spreadsheet to implement. However the three conclusions from the preceding section suggest a simple "rule of thumb" (RoT) that can be easily calculated mentally, at the eyepiece. This is stated in units of your telescope's λ/D or Abbe resolution limit (R), which you should calculate in advance. Then for any pair you estimate the resolution ratio ρ/Ro and the magnitude difference between components (Δm).

The RoT consists of two parts:

• Δm less than 1: If the magnitude difference is less than or equal to 1.0 then resolution is possible provided the separation is equal to or greater than the telescope's resolution limit and the matched pair is brighter than the foveal magnitude limit (usually about 5 magnitudes brighter than the aperture limit magnitude [see below]; ρ ≥ Ro, m' ≤ LM–5).

• Δm greater than 1: If the magnitude difference is greater than 1.0, then resolution or detection is possible provided the resolution ratio is larger than the magnitude difference and the secondary is somewhat (at least 2.5 magnitudes) brighter than the aperture limit magnitude (Δm ≤ ρ/Ro, m' ≤ LM–2.5).

This RoT provides a useful approximation to Öpik's C (when c = 0 and k = 7) at small separations (ρ/R < 5) under "normal" observing conditions. Note that, for apertures in the range 100 to 130 mm, Ro is close enough to 1.0 that the magnitude difference can be compared directly to the separation. These are also apertures of a size that is minimally affected by the seeing, which eliminates a significant source of uncertainty.

This "rule of thumb" is plotted in the diagram (above) as a dotted line. It correctly identifies 96% of the double stars in the diagram as "detectable". Failure to detect or resolve a double star inside the "rule of thumb" limit (above and to the right of the dotted line) implies something is wrong — usually, a problem with the instrument (objective or eyepiece dewing, insufficient magnification), with the conditions (diffusion, seeing, light pollution), with the data (outdated or misprinted values for magnitude difference and separation), or with the observer (looking at the wrong star, not sufficiently dark adapted, impatient). Regardless of the reason for the failure, the problem becomes an observing challenge for skill and perseverence to overcome.

For example: the magnitude difference between Sirius (v.mag. –1.47) and Sirius B (v.mag. 8.3) is about 9.8, and their catalog separation is ρ = 9.7″. Working in round numbers, the resolution limit is Ro = 0.5″ in a 250 mm aperture, the resolution ratio ρ/Ro = ~10/0.5″ = ~20, the magnitude difference Δm = ~10 and and both stars are brighter than the foveal magnitude limit of ~10.0. Therefore, with Δm < ρ/Ro (10 < 20) Sirius B should be easily detected. In the attempt, I was unable to detect Sirius B despite careful observation, so I began to consider factors that might cause this failure. I settled on glare [19] arising from the magnitude of the primary star [1] as the likely cause. I explored ways to make Sirius B visible that involved reducing the light from Sirius A (observing in twilight, obscuring the primary with the edge of the eyepiece field stop, observing when humidity was low, tapping the telescope slightly to separate Sirius B from scattered light, etc.). With these troubleshooting methods I was successful in detecting the companion.

This is a general illustration of how a "predicted resolution limit" can be used to identify an observing problem and guide the observer to identify the source of the problem and the methods most likely to resolve it.

9. Develop Your Observing Technique

By looking for mathematical prediction, we've made the transition from a large number of possible observational factors in visual double star astronomy to a core set of system features and observational conditions — magnitude difference, resolution ratio, magnification, central obstruction and seeing — that most affect detection or resolution.

Now it's appropriate to reconsider the entire list of factors (1-32) that affect the visual detection or resolution of close double stars and ask how knowledge of their potential effects can help an experienced observer use active problem solving to overcome obstacles to observation.

The factor relationships in the first group, (A) system brightness, illustrate how the factors can influence each other depending on the circumstances. The remaining groups will be treated in outline: you can explore these through your own observing experience.

System Brightness and Visual Response. The main effects of system brightness arise through the primary star brightness [1], and the effects of primary star brightness include the spreading out of the light by glare [8], atmospheric diffusion [23], light scatter or reflections within the instrument [19], an increased number and brightness of diffraction rings, especially as amplified by a central obstruction [21], and — for bright primary stars viewed in large aperture — a change in the dark adaptation (resolution and contrast sensitivity) of the eye [28].

Let's start with the relationship between the telescopic limit magnitude LM [10] and the double star's delta-m [3], as this is perhaps the most important stimulus relationship that is not represented in a Treanor plot.

If the primary star is faint enough, then a large delta-m will place the magnitude of the secondary near the telescopic limit magnitude. This limit may be too low if the system is observed through the effects of dispersion, which will appear as a nimbus or glowing area around the star (diagram A, below). This brightening reduces the luminance contrast between a faint companion and the background sky brightness, making the companion more difficult to detect.

The nominal limit magnitude of the aperture [10] is calculated on the basis of an average observer's dark adapted, naked eye limit magnitude using averted vision [28]. This is calculated with a standard formula that assumes an optimal value of NELM, although the actual NELM will vary according to the sky brightness [25], the seeing [22], the atmospheric diffusion [23], dewing on the optical surfaces [24] and the observer's dark adaptation [28]. The telescopic LM is also affected by the magnification [14] — up to a point, higher magnification reduces sky brightness in the field of view and makes fainter stars visible.

The diagram (right) plots a sample of 5000 double star detections by our eight famous astronomers, out to ρ/Ro = 100 on a log scale. The ordinate shows the difference betweeen the nominal telescopic limit magnitude and the magnitude of the fainter component [LM – m']. The average primary magnitude was telescopic — greater than 6.5 — and the average telescope limit magnitude was around 15.4.

The white dots show the average of the reported detections within whole number intervals of the log resolution ratio scale. These are fit with a piecewise slope (the white line), from an average ΔLM of 6 magnitudes at ρ/Ro = 1 to to about 4.5 magnitudes at ρ/Ro > 10, and a constant (flat) value of 4.5 at greater separations. Keep in mind that we're considering overall the performance of large aperture refractors; smaller apertures and reflectors will behave somewhat differently.

The detections produce a very large scatter — which we expect in a complex perceptual task measured in a natural environment across different observers — but they suggest that detection is limited by primary star brightness out to a radius of about 10ρ/Ro, and is constant after that. At the larger separations sky brightness, and what is most likely the observer's concept of a "field star" or the widest separation accepted as a double star, will limit the identification of a double star component. (As Öpik put it, "the choice of the observers tended to reject many wide pairs" that could be detected.)

The frequently cited Peterson threshold (given above) is inserted for comparison, and it is clearly too optimistic as a predictor of detection. My suggested criterion is shown as the orange line, sloping from about 4 magnitudes above the limit magnitude at the resolution limit, to a constant 2.5 magnitudes at separations greater than ~10 ρ/Ro. This criterion identifies detectable or recognizable components with a >95% probability.

To put this information in the context of human visual acuity, we must calculate the actual illuminance values of the limit magnitude and secondary magnitude. The graph (left) shows the illuminance (in lux) delivered into the retinal spot area by telescopes of 100 to 400 mm aperture operating at unit exit pupil (and assuming Vega has an illuminance of about 2.46x10–6 lux).

Even in the 400 mm aperture, only stellar magnitudes above ~5.0 are telescopically photopic at unit exit pupil, and only Venus is photopic to the naked eye. Even in the largest aperture (400 mm) at a 3.5 mm exit pupil when the retinal spot is smallest, stars below ~7.0 are in the mesopic zone.

Stellar magnitudes equivalent to planets are indicated by vertical lines; Mars and the orange stars Antares and Etamin illustrate the naked eye illuminance range of color perception. Overall, it is clear that, regardless of aperture, most double stars are observed at mesopic levels of illuminance at the unit exit pupils necessary to resolve the closest matched pairs.

These variations in the retinal flux have a profound effect on visual acuity across the telescopic magnitude range, and the table (below) summarizes the approximate magnitude location of different star appearance thresholds as increments above the telescope limit magnitude based on a "dark site" naked eye limit magnitude (NELM = 6.5) and a telescope exit pupil of 1.0. Note that the thresholds have a constant visual (lumens) value regardless of aperture, but do change with the exit pupil, as shown by the dotted line for de = 6 mm in a 400 mm aperture.


aperture (mm)80140200250300400

Ro (arcsec)1.420.810.570.450.380.28
limit magnitude (lux) (2.3x10–4)12.013.214.014.514.915.5

threshold (lux)magnitude of thresholddescription
LM–13.0 (3.7x101)–1.00.21.01.51.92.5photopic/mesopic threshold (glare threshold)
LM–7.0 (1.5x10–1)5.06.27.07.57.98.5foveal resolution degrades (color threshold)
LM–5.0 (2.3x10–2)7.08.29.09.59.910.5mesopic/scotopic threshold (foveal threshold)
LM–2.5 (2.3x10–3)9.510.711.512.012.413.0detection at ρ/Ro >10
(averted vision)

This table is only suggestive of the changes in an observer's visual capabilities in relation to different target attributes (color, separation, magnitude difference, image quality, etc.) and will certainly vary with the capabilities or "personal equation" of different observers.

These perceptual effects cause changes in the appearance of the Airy disk, due to the decreasing foveal resolution capability with decreasing brightness, the amount of luminance contrast against the background sky brightness. As stars become fainter the Airy disk becomes smaller in diameter (an effect recognized and described by both William and John Herschel), down to a limit where the Airy disk diameter is perhaps one third the diameter of the disk that is produced by a very bright star (diagram, below left).

In apertures I have used, this Airy disk threshold (LD) seems to be around 7 magnitudes brighter than the limit magnitude of the aperture (LM), assuming a dark adapted eye with a naked eye limit magnitude (NELM) of about 6.5 in the limit magnitude calculation.

Within about 2 magnitudes below this threshold, the star can still be visualized by looking at it directly although it only appears as an indistinct point lacking both disk and rings. Below the foveal threshold (LF), which seems to lie at about 5 magnitudes above the aperture limit magnitude, the star will completely disappear if looked at directly, but will be clearly visible with averted vision (diagram, below center).

Presumably, at the aperture limit magnitude (LM) all faint components and field stars become undetectable even with averted vision. However, the chart (above) illustrates that this peripheral detection limit (LA) is actually around 2.5 magnitudes above the theoretical aperture limit magnitude.

The secondary magnitudes in the diagram (above right) suggest there are very few detections at ρ/R ≤ 5 at magnitudes less than ~3 above the limit magnitude, demonstrating that close components can still be detected just below the Airy disk threshold (LD). This is a brightness ratio of about 16 times, far less than the roughly 50 times greater luminance sensitivity of peripheral to foveal vision (at 550 nm), which implies that glare from the primary and the diffraction rings had added about 1 magnitude on average to the background brightness.

Finally, stellar spectral type can affect the apparent magnitude differences. Red stars (spectral types K,M,N) appear visually brighter than their listed magnitude might suggest (especially the luminous giant red stars), and blue stars (spectral types O,B) appear fainter. This makes a double star catalog that actually lists spectral info rather handy — the Washington Double Star Catalog includes spectral information when available, but most other catalogs omit it.

The effects of stellar brightness depend on many factors, but choosing the best magnification is frequently important. Magnification can, by decreasing the background sky brightness, increase the limit magnitude; it also enlarges the angular separation and can punch through mediocre or poor seeing to display the relatively robust Airy disks of bright double stars. If glare is the problem, then observing during the hour or so of twilight, when sky brightness is higher than it will be at night, can reduce the primary's apparent magnitude and reveal a close and much fainter companion.

Other Observational Factors. The factors within the system brightness group have illustrated how perceptual attributes are mutually dependent. The remaining factor groups will be summarized briefly to highlight their major effects.

(B) System Configuration Factors. Separation [5] is the critical system attribute. A close, fainter companion, even one above the Airy disk threshold, is easily masked by speckles or diffraction rings and takes on an appearance deceptively unlike the usual Airy disk. It will appear as a thickening, brightening or irregularity in the diffraction ring around the brighter star (diagram above, B) and familiarity with the effect is required for detection. Matched or equal magnitude double stars allow the greatest possible diffraction magnification [15], and in mediocre or poor seeing conditions [22] the use of exit pupils below 0.5 mm often produces an appearance similar to car headlights seen through heavy rain (diagram above, C).

Position angle [6] can be significant if it coincides with a diffraction spike caused by the secondary mirror supports. It is also sensitive to seeing cross currents. During an extended observation of the "double double" eps 4,5 LYR, whose position angles are roughly 90° apart, I found that one pair could be resolved at the same time the other was obscured by seeing, due to a directional difference in the elongation or motion of thermal cells. Thermal currents will rise vertically off a cooling mirror, and obscure close separations that lie in the same direction (e.g., PA = 0° when observing due south with an altazimuth mount).

The annual change in system parameters [7] becomes important when it can affect the accuracy of double star data. Most amateur and software datasets do not provide the epoch of last measurement for double star parameters, and several close systems (alp CMA, gam VIR) will show visible change within a single year. It is always advisable to refer to the latest edition of the Washington Double Star Catalog online, which is updated frequently.

(C) Aperture Factors. Aperture diameter [8] determines resolution [9], limit magnitude [10] and contributes to image brightness [11]. On the whole, a larger aperture is always better, and per inch of aperture a refractor will usually produce better resolution and fainter diffraction rings than a reflector. However, as explained above, the images in apertures larger than about 200 to 250 mm are routinely degraded by atmospheric turbulence [22], which means any theoretical increase in resolution cannot be consistently utilized.

It is extremely helpful to become familiar with a standard visual rating scale to guide your judgment of the quality of the in focus Airy disk image — the best single test of telescopic image quality — and use it to routinely evaluate the quality of seeing throughout the observing session.

Aperture limit magnitude is calculated as LM = mne – 5·log(δ) + 5·log(Do), where mne is the faintest star that can be seen with naked eye averted vision and δ is the eye pupil diameter. It is actually a nominal measure of aperture light grasp standardized on receptor (observer) detection threshold, ignoring the image effects of glare [19] and light loss [17]. Both atmospheric diffusion [23] and objective dewing [24] have a very large impact on LM, and both are associated with atmospheric humidity.

This nominal aperture limit magnitude determines the visual magnitude thresholds for (a) averted vision LA, (b) foveal vision LF, and (c) the appearance of the Airy disk LD (diagram, above). The observer should test the magnitude value of these thresholds using double stars stars with faint companions of known magnitude, aware that the absolute values will change if the sky brightness or aperture are changed.

Focal ratio or relative aperture [11], when it is small (less than ƒ/6 in Newtonian reflectors, less than ƒ/10 in achromat refractors) can affect both the image quality toward the edges of the field of view and (in reflectors) the difficulty of achieving and maintaining optimal collimation. It is always important to observe double stars centered in the field of view, where all optical aberrations will be minimized.

(D) Magnification Factors. These are discussed above and in the sections on magnification and exit pupil in another page. The observer should take the trouble to calculate his or her detection exit pupil, as this is the magnification at which both the diffraction artifact and the resolution limit of the aperture are visible to the eye.

Most efforts to create a prediction formula for double star detection or resolution inexcusably omit magnification as a significant factor. Magnification is perhaps the single most important tool that the observer can use to suppress glare, raise the visibility of faint stars, suppress the effects of mediocre seeing (by choosing an optimal magnification), and enhance the visibility of the Airy disk behind poor seeing (by choosing a very high magnification). Choice of magnification can be compared with the choice of clubs in golf, and magnification should be chosen in a similar spirit of the right tool for the task at hand.

(E) Instrumental Factors. In a Newtonian or Cassegrain type reflector, a large central obstruction ratio [21], Ds/Do > 0.30) will increase the brightness and number of the diffraction rings around the Airy disk. In mediocre or poor seeing (especially in apertures above 200 mm), the diffraction ring(s) will be broken up into mobile arcs or speckles, and the secondary star will appear as an arc or speckle that moves around less than its surroundings, or reappears in the same location. Poor baffling, optical surface roughness and inadequate lens coatings within the optical system can produce glare [18] that increases the background brightness around the primary star or spreads light from other bright objects in the field of view.

Aperture diameter [8] determines the mass of a primary mirror, which affects cool down time and sensitivity of the optics to temperature changes during the night. An enclosed tube (Maksutov Cassegrain, Schmidt Cassegrain) retards mirror cool down, but does not appreciably affect refractor cool down because the objective is exposed.

Focus [18] is not fixed optical condition but a central or average image quality as this is varied by thermal currents. Patient viewing is usually more productive than continual attempts at refocusing. Focus can be refined by using averted vision to watch for the sudden appearance of very faint stars which defocus will render invisible. Focus changes with zenith angle. In situations where poor seeing arises low to the ground, it effect can be weakened be defocusing in the intrafocal direction (toward the objective). Since the star is at a much greater distance than the atmosphere, its focal point is in front of the atmospheric focal point, and there is often a point where the star image is defocused much less than the turbulence.

After seeing, accurate collimation has the largest effect on image quality, and the astronomer should make every effort to obtain the best collimation possible with the equipment. Check collimation with slightly defocused star image at the start of every observing session, and to look for mirror thermal currents; then use the focused image to assess seeing.

(F) Environmental Factors. Seeing [22] is treated on another page. In addition to the obscuring effects of the diffraction rings and poor seeing, light pollution and diffusion can spread glare that increases the brightness of the sky. This brightened sky reduces contrast with a faint companion, so that it can be invisible well beyond the last visible diffraction ring and well above your telescope's magnitude limit. Diffusion [23] can be diagnosed by observing a 6th to 8th magnitude star with the standard eyepiece, using averted vision to view the diameter of the visible nimbus. Dewing [24] is more often recognizable by a reduction in the limit magnitude, which may go undetected because it is gradual; eyepiece eye lens condensation will make the exit pupil visible as an illuminated disk of roughly even magnitude around a centered bright star or planet. Both seeing and sky brightness vary significantly with the direction of view [26]; light domes are usually have a fixed location on the horizon, but the direction of most favorable seeing (if there is one) will change with surface air currents, the prevailing breeze or cloud motion, and high altitude jet stream.

(G) Observer Factors. E.A. Douglass remarked that "The result of our own experience in studying planetary detail has been to regard the atmosphere as of the first importance [to visual detection], the energy and the intelligence of the observer as of the second and to put last of all, the instrument." Energy and intelligence could be specified more concretely, but there is no question that observer attributes and "personal equation" are of capital importance in double star astronomy.

Of the many possible observer advantages, rest, enthusiasm, healthy vision, experience with the telescope, skillful looking and persistence are the main ones, and fatigue, frustration, boredom, inexperience and discomfort (posture strain, cold, hunger) the major impediments. Note that none of these are fixed attributes: the observer is continually changing during the evening. It is useful to be self aware, monitor one's own acuity, interest, comfort and capability, and know when to adapt ... or quit for the night.

Experience with the same equipment at the same site [31] provides an invaluable basis for observations, clearing away minor impediments or frustrations and making work easier and observing time more productive. The observer should look for ways to make procedures standardized so that they can become habitual, and find ways to eliminate or streamline any awkward or distracting tasks. Rather than blundering past minor frustrations — dropping an eyepiece, tripping over a cable or cord, straining to read a chart, stooping to look into the eyepiece — stop, take a breath, and think for a moment how you can rearrange your set up so that it doesn't happen again.

Although observer dark adaptation [28] is important, amateur astronomers often obsess unnecessarily about it. On many nights the sky brightness will exceed the observer's potential maximum dark adaptation, so complete dark adaptation will never be reached. The change in threshold produced by transient exposure to even a moderately bright light normally dissipates within at most a dozen minutes. Purchase a red light flashlight with adjustable brightness, and don't hesitate to use it when you need to read a chart or adjust equipment. Regardless of the color of the light, always try to use the minimum amount of light necessary for a specific task (changing eyepieces, reading a star catalog).

With experience, a variety of methods for tactical looking or visual problem solving become part of the double star observer's arsenal. There is not a widely adopted lore about these things, you discover them for yourself by paying attention to how you are looking at a difficult system and how that affects what you see. For example, if you must use averted vision to locate a faint companion, direct your gaze to different parts of the field in order to use different areas of your peripheral vision, at different distances from the primary star: you will discover the angle that produces the maximum sensitivity, and the direction in which the sensitivity seems most acute.

Some looking techniques can be subtle. An illustration is the method of foveal coaxing. In many targets where a bright primary is accompanied by a faint secondary near the foveal threshold, the first view will only reveal the secondary with averted vision (diagram panel 1, below): if the foveal gaze is directed back at the system, the secondary disappears entirely (panel 2, below). However, by incrementally moving vision back and forth between averted and direct gaze, each time reducing the amount of displacement, the secondary can eventually be held in view with direct vision (panels 3-6).

This illustrates that skillful looking can overcome situations in which the stimulus seems at first to be below a detection threshold. It is unlikely that the foveal sensitivity has changed during this procedure, since the eye is already dark adapted. Instead, the cognitive component of detection has been guided to identify a very faint retinal signal as a reliably stable stimulus in visual space. This brings out the important point that perception is not simply a passive receptor process equivalent to using a camera CCD: it is supported and guided by cognitive processes that respond to attentive looking and training.

A related process of pattern recognition occurs when attempting to detect or resolve a faint companion under conditions of mediocre or poor seeing. In this case the eye is presented with rapidly changing images of the double star produced by the random optical deflections and magnifications caused by thermal eddies in the atmosphere crossing the optical path. The visual cortex can extract a consistent signal from these time variable changes in the raw visual stimulus, and once it has recognized a consistent signal within the random noise it is able to lock onto the signal as a visual fact. This might be called the Waldo effect, because the Waldo component is at first hidden within the teeming crowd of speckles, but once located is difficult to ignore. Again, this is a cognitive rather than sensory attribute of vision: the retinal stimulus was there all along, but is only accessible to consciousness through a process of recognition.

(H) Accuracy of Catalog Data. A common source of failure to detect a close companion is inaccuracy in the double star data [32]. This applies to the position angle and separation of pairs that have not been measured in several decades, and to the magnitudes and magnitude differences in close pairs. Misprints and incorrectly entered data also occur, including celestial coordinates in which the sign of the declination has been reversed. Weirdness does happen!

The simplest way to determine the accuracy of the separation estimate is to be able to estimate separation visually. A more robust approach is to search WDS for a double star with similar primary magnitude and separation but a somewhat brighter secondary; this will show you the separation and approximate magnitude contrast you should find in the challenge pair. For fast changing (close orbiting and nearby) systems, such as Porrima or eta Coronae Borealis, you may need to refer to an ephemeride that will give the separation and position angle for each year. With experience, and by using the standard eyepiece to evaluate separation, you will anticipate the appearance of any separation interval: if a disconcerting discrepancy is observed, check the catalog data as the first source of the problem.

Enjoy the Journey. If this topic seems complicated, it is. But it appears more daunting than it actually is because I have summarized the many possible issues in one place. In fact, the difficulty is part of the challenge and the reward — the enjoyment — of visual astronomy.

Patience and persistence count for a lot: astronomers typically make several attempts to detect Sirius B and similarly difficult double stars before they succeed. The success is a learning experience. Double stars of unequal magnitude are a dimension of double star astronomy where skill and opportunity have the greatest effect. Every challenge is an opportunity to improve your observing skill.

As you apply the suggestions I've offered you will observe dozens and then hundreds of different double stars. What I found remarkable in my own experience was that through this training process I acquired a really acute sense of my instrument and my vision (both capabilities and limitations), the observing conditions, and the variety of double star star configurations; I developed the kind of integrated perception that atheletes have for their sport — pushing the limits, using proven techniques, waiting for opportunities, and problem solving the challenge.

It's difficult to express ... but I would come across star pairs where the relative magnitudes, the colors, the separations, the surrounding field, the magnification, the shimmering of the air, all created an inexpressible gestalt of contemplation and awe. When it is a difficult star pair, and I persist in working on it, until — suddenly! — the separation appears, or the companion emerges from the glare. It's a unique thrill.

Further Reading

The most comprehensive and up to date source of data on double stars is the Washington Double Star Catalog (WDS), available as four very large text files from the USNO web site, or as a 40 megabyte Excel spreadsheet (.xlsx) edited by me.

Observing and Measuring Visual Double Stars (2nd edition) (2012) by Bob Argyle - "The definitive book for those who are serious about this fascinating aspect of astronomy" ... really!

Double Stars (1978) by Wulff Heintz – A classic in the field, compact and informative, although dense and in many places technical.

Observing Visual Double Stars (1981) by Paul Couteau – Reader friendlier than Heintz, more focused on observing methods, and with many topics of interest to the double star observer.

On the telescopic resolution of unequal binaries (1946) by P.J. Treanor - An early attempt to derive a "predictive" double star detection limit.

Limits for splitting double stars by K.A. Fisher - Evaluation of various methods to predict the resolution of close double stars of unequal magnitude.

A Mathematical Model to Predict the Resolution of Double Stars by Amateurs and Their Telescopes (2008) by Tim Napier-Munn - A recent attempt to synthesize prediction models for close double stars of unequal magnitude.

 

Last revised 3/4/16 • ©2016 Bruce MacEvoy