Deep Sky Object Visibility
This chart graphically indicates the conditions of visibility for extended (not point-like) deep sky objects as a function of object and [object + sky] surface brightness (SB). For selected values of sky SB and object SB, the computed vales of SB for [object + sky] are drawn as curves. This permits to see at a glance when [object + sky] vs sky SB is sufficient to provide the required contrast for detection.
The human visual system is first and foremost a contrast detector. But in the low brightness regime of DSO observing visual system noise is a significant factor, worsening with decreasing scene brightness. This imposes a limit of roughly 27 magnitudes per square arcsecond (MPSAS), beyond which visual noise dominates any signal. Hence the lower SB limit of 28 MPSAS plotted here.
As the image dims, visual resolution decreases, and the required angular size for detection increases. At the poorest contrast from which a signal difference can be detected, an object must subtend an apparent angle of several degrees, with some small gain accruing perhaps up to a couple tens of degrees. This chart plots curves for angular size detection thresholds of 0.5, 1, 2 and 6 degrees. Even for a 20 degree limit, the curve plotted for such might lie but a couple tenths of a magnitude to the left of that for 6 degrees.
Data from the RASC Observer's Handbook is incoprorated, some of which in turn comes from Blackwell (1946).
CHART LAYOUT
Object SB is plotted on the X (horizontal) axis, increasing toward the right, and ranging between 28 and 13 MPSAS. For perspective, the brightest nebulae have SB ~14 MPSAS. Galaxy and globular cluster cores can peak at 15-16 MPSAS. Color is seen to about as faint 18-19 MPSAS. A dim nebula such as the North America has SB about 24 MPSAS.
[Object + sky] SB is plotted on the Y (vertical) axis, increasing upward, and ranging between 28 and 13 MPSAS. This scale represents also the sky SB, in the absence of any superimposed source. For perspective, a Full Moon or heavily light polluted urban sky is about 17 MPSAS. A surburban sky might be 19.5 to nearly 21 MPSAS. A pristine site's sky gets to as dark as 22 MPSAS.
Immediately apparent is the empty triangular region in the lower right. No combination of object or sky brightness can be plotted here. The straight-line limit running diagonally, and connecting equal values for object and [object + sky] SB, results from the simple fact that [object + sky] SB essentially equals object SB when the latter is at least several magnitudes brighter than the sky SB. The region inside which data can be plotted is colored. Fainter than the color detection threshold of 18-19 MPSAS for [object + sky], this region is tinted a cool blue-grey. Brighter than this, where the eye can perceive color, the tint is a warm peach. Where the object SB has its contribution greater than that of the sky, and hence dominating, a green wash of color is applied.
For each full magnitude interval in sky SB, from 17 to 27 MPSAS, a mostly horizontal curve is plotted for the combined SB of [object + sky]. That curve for a pristine sky of 22 MPSAS is highlighted in purple (but just to object SB = 19 MPSAS; no brighter, so as to not further crowd the converging curves.) Where object SB is much less than Sky SB, [object + sky] SB essentially equals that of sky SB. This is why the plotted curves are flat well to the left of (much fainter than) the rightmost limit. Once Object SB gets to within about 4 magnitudes of sky SB, the combined light begins to result in an upward deflection. All curves are precisely identical, the next one lower being simply displaced down and to the left by 1 magnitude in each axis.
A key value of [object + sky] SB has it 2X brighter than sky SB. When object SB and sky SB are equal, the combined light is twice as bright, which is a difference of 0.75 magnitudes. On every curve for sky SB this datum is plotted as a larger blue dot, and connected by a blue line. For example, on the curve for Sky MPSAS = 20, this point on that curve occurs where object SB is also 20, and the [object + sky] SB = 20 - 0.75 = 19.25 MPSAS.
Another key value of [object + sky] SB has the object SB 3 magnitudes fainter than sky SB. These data are plotted as red dots connected by a red line. Here [object + sky] SB is 0.06 magnitude, or about 6% brighter than sky SB. For most observers, their mediocre sky will permit to see objects at about this level of contrast when the exit pupil is moderate to large. (More specifics later...)
If the human visual system had almost infinite dynamic range, or at least a *very* much lower limit for scene brightness before visual noise becomes of any consequence, the same limits of contrast and object apparent angle would apply for the whole region bounded by this chart. But that is not so.
As scene brightness decreases, visual system noise worsens and ever more greatly impairs the detection of contrast. This requires that an object subtend a larger apparent angle in order to be detected. Or conversely, at given angular size the object becomes progressively harder to detect. To visualize the trends that result, four black curves are drawn for size detction limits of 1/2, 1, 2 and 6 degrees. These limits have the required minimum size increasing to the left, toward fainter object SB. Note how the limit values would become exponentially more crowded together toward the left if the interval was linear. That these curves plunge more vertically than the aforementioned limits very directly reveals the impact of the worsening visual noise. And the steepening toward fainter [object + sky] SB shows how the *rate* of increase in visual noise increases. The 6 degree limit curve both intercepts the limit for plotted data *and* becomes nearly vertical at 27 MPSAS; a consequence of the limit where visual noise dominates any signal at about 27 MPSAS. (As noted earlier, extending to a size limit for detection of a couple tens of degrees could gain a couple tenths of a magnitude.)
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Useful formulae and data are included in the othewrwise unused region of the chart.
The professional community expresses SB as magnitudes per square arcsecond (MPSAS), and I encourage the amateur to do so, too. Magnitudes per square arcminute has the advantage of resulting in a number nearer to the integrated magnitude, which does correspond exactly when the object has an area of 1 arcminute^2, a size corresponding to numerous planetary nebulae and distant galaxies. The conversion factor from MPSAS to MPSAM is included, for convenience. But it is better to become familiar with and use exclusively MPSAS, in my opinion.
To assess the telescopic visibility when the exit pupil is smaller than the observer's iris, it's necessary to calculate the amount of dimming compared to the maximal, iris-equaling exit pupil diameter. One's own fully dilated iris is the benchmark for the full-brightness image. It suffices to relate the exit pupil to *your own dark adapted iris*, whether larger or smaller than the average; there is no pressing need to establish an absolute fiducial corresponding to a particular iris diameter. A more extreme example, to show how significant dimming via a small exit pupil can be: Compared to a 7mm iris, an exit pupil of 0.7mm dims the image by a factor of (7 / 0.7)^2 = 100, or LOG(100) * 2.5 = 5 magnitudes. If the sky is 22 MPSAS, the telescopic sky image will be 22 + 5 = 27 MPSAS.
Filters are used to enhance object contrast by transmitting as much light from the object as possible while rejecting the maximum of unwanted light. The filters for which the effect is most consistent are those intended for emission nebulae. Typically 90+ per cent of the visible nebular emission is transmitted, and a good fraction of sky glow is blocked. Two well known Lumicon filters are represented; their equivalents by other makers should have fairly similar efficacy. You can safely enough treat your filter as though transmitting all nebula light, or you can reduce the object SB by 0.1 or 0.2 MPSAS if filter transmission is 90% or 80%, respectively. (Unless you wish to be rigorous and calculate the loss more formally.)
HOW TO USE THE CHART
While intended primarily to reveal trends, and thereby aid in developing an appreciation for the dimensions and magnitude of impact for the relevant variables, this chart can serve as something of a predictor of object visibility. The two variables you *must* have are sky SB and Object SB. These will at least indicate the contrast, and hence if the object could ever be seen given sufficient aperture. When you know the object size, you can see if it meets the detection threshold for size at given magnification and exit pupil dimming (for a particular objective aperture.)
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First we will assume a telescope used at lowest magnification, without filters. The exit pupil equals the eye's iris, and the scene surface brightness is maximal, just about equaling the unaided eye brightness. There is no exit pupil dimming to account for.
Case #1
Sky SB = 21 MPSAS
Object SB = 22 MPSAS (1 mag. dimmer than the sky)
Object size = 10 arcminutes (1/6 degree)
Telescope aperture = 200 mm
Exit pupil = 6mm (same as eye's iris)
Magnification = 200 / 6 = 33.3X
Given the above, we calculate the object's apparent angular size as 33.3X * 10' = 333 arcminutes, or 333 / 60 = 5.6 degrees.
On the graph, locate the curve for Sky MPSAS = 21 (it's the next one above the purple curve for a sky of 22 MPSAS.) Follow it to the point where it intercepts Object MPSAS = 22, on the X-axis. Estimate the [object + sky] SB from the Y-axis scale. You should obtain ~20.6 MPSAS. This is ~0.4 magnitude, or roughly 40% brighter than the surrounding sky, and represents not particularly poor contrast.
Compare this point with the curves for the detection limit for size. We see it lies between the curves for 1/2 and 1 degree, nearer to the former, and can say the object should subtend ~35 arcminutes in order to be seen. This is well below the observed extent of 5.6 degrees, and so the object can be expected to be fairly well seen.
We might ask, "What would be the minimum aperture required to see this object, if the exit pupil is also 6mm?"
We know the minimum size for detection to be ~35'. A 10' object magnified to 35' requires 35 / 10 = 3.5X. At a 6mm exit pupil, the objective apreture must be 3.5 * 6 = 21mm. A 4X20 bino, or a 6X30 finder should serve to detect this object with reasonable surety.
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Now we consider an object observed using a smaller exit pupil, which dims both the object and the sky equally.
Case #2
Sky SB = 21 MPSAS
Object SB = 22 MPSAS (1 mag. dimmer than the sky)
Object size = 2 arcminutes (1/30 degree)
Telescope aperture = 200 mm
Exit pupil = 2mm
Eye's iris = 6mm
Magnification = 200 / 2 = 100X
We calculate the object's apparent angular size as 100X * 2' = 200 arcminutes, or 200 / 60 = 3.3 degrees.
Now the exit pupil dimming must be accounted for. The brightness ratio = (6 / 2)^2, which is a factor of 9; compared to a 6mm exit pupil, a 2mm exit pupil dims *both* the object and sky to 1/9 their undimmed brightness. In magnitudes this LOG(9) * 2.5 = 2.4. And so through that 2mm exit pupil the Sky SB = 21 + 2.4 = 23.4 MPSAS, and the object SB = 22 + 2.4 = 24.4 MPSAS.
On the graph, interpolate for the curve for Sky MPSAS = 23.4, and Follow it to the point where it intercepts Object MPSAS = 24.4, on the X-axis. Estimate the [object + sky] SB from the Y-axis scale. You should obtain ~23 MPSAS. Because the object and sky SB differ by the same 1 magnitude as for Case #1 above, the same ~0.4 magnitude, or roughly 40% brighter than the surrounding sky, obtains for the object relative to the surrounding sky. We have the same intrinsic scene contrast, as expected.
Compare this point with the curves for the detection limit for size. We see it lies barely to the right of the curve for 2 degrees, and can say the object should subtend probably something like 1.8 degrees in order to be seen. This is safely enough smaller than the observed extent of 3.3 degrees, and so the object can be expected to be seen without too much trouble.
The important lesson here is that for the same object and surface brightness, a dimming by 2.4 magnitudes via a smaller exit pupil demands that for its detection an object must increase in minimum apparent angle by a factor of 3!
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Now we introduce a nebula filter, which passes (nominally) all visible nebular light of consequence but darkens the sky. We will use all conditions for Case #2, the only differences being magnification/exit pupil and the addition of a line filter which we will assume to not dim the nebula at all.
Case #3
Sky SB = 21 MPSAS
Object SB = 22 MPSAS (1 mag. dimmer than the sky)
Object size = 2 arcminutes (1/30 degree)
Telescope aperture = 200 mm
Exit pupil = 6mm
Eye's iris = 6mm
Magnification = 200 / 6 = 33.3X
Line filter (e.g., O-III or H-beta), which dims the sky by 2.6 mag.
We calculate the object's apparent angular size as 33.3 * 2 = 66.6 arcminutes, or 66.6 / 60 = 1.1 degrees.
Now the dimming of the sky by the filter must be accounted for. The sky SB = 21 + 2.6 = 23.6 MPSAS.
The object is not dimmed due to the iris-equalling exit pupil of 6mm and the filter's transmission of all visible nebular emission, and so still has SB = 22 MPSAS, which is now 23.6 - 22 = 1.6 magnitudes *brighter* than the surrounding sky. Contrast has been greatly boosted!
On the graph, interpolate for the curve for Sky MPSAS = 23.6, and Follow it to the point where it intercepts Object MPSAS = 22, on the X-axis. Estimate the [object + sky] SB from the Y-axis scale. You should obtain ~21.75 MPSAS. And so the combined light of object and sky make the object appear to be brighter than the sky by 23.6 - 21.75 = 1.85 magnitudes. (versus 0.4 magnitudes when not filtered.)
Compare this point with the curves for the detection limit for size. We see it lies well to the right of the curve for 1/2 degree, meaning it should be detected when larger than perhaps 20 arcminutes. This is comfortably smaller than the subtended size of 1.1 degrees, and so can be expected to be reasonably well seen.
FINAL NOTES
The threshold for color detection must surely vary depending on the degree of color purity and difference. For instance, if the surface brightness for object and sky are the same, the blue-green for an emission nebula seen against a reddish sky glow could well have its hue be more pronounced than for a less spectrally pure object seen against a sky glow of not so different color to that of the object. And so the threshold of 18-19 MPSAS, as well as the demarcation separating the regions where object and sky are dominant, are probably somewhat indistinct. One of the glows could be a bit dimmer, yet nonetheless exhibit its color more strongly.
The series of curves for various sky SB, as noted, all have the same form as defined by the brightness *ratios.* They superficially resemble hyperbolas, where the extensions away from the region of greatest inflection converge asymptotically to the left and upper right, as set by the limits of sky SB and object SB, respectively. Outside of 4 magnitudes removed from the intercept where object and sky SB are equal, the curves are essentially flat. This is because by the time one or the other of sky and object SB exceed the other by 4 magnitudes, the dimmer of the two adds practically no light of consequence, and the curve thereafter assumes an essentially straight line.
Hopefully the examples provided will sufficiently instruct on the chart useage. At the very least study and appreciate the implication of the impact induced by the differing and changing slope of the curves for minimum size for detection. This reveals much behind the deficiencies of the visual system at such low brightness levels.
Perhaps the most useful rule of thumb to derive: In the brightest sky under which one might do any kind of DSO observing, the object can be detected when its SB is as much as 4 magnitudes fainter than the sky SB. But near the limit of sensible sky brightness as seen through a small exit pupil, the object must have SB about equal to sky SB.
And to remind: Between the brightest sky just about conceivable for DSO work (17 MPSAS) and the dimmest the visual system can work with (27 MPSAS), we have 10 magnitudes, or a brightness range of factor 10,000.
Edited by GlennLeDrew, 30 September 2016 - 05:18 AM.
