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ODM Matrixes
Mel Bartels created ODM (Optimum Detection Magnification) as a tool to explore how aperture
and magnification impact the visibility of faint extended objects in telescopes. I discovered ODM a few
years ago and have been using this nifty little application to explore my own questions about visual observing
ever since. For instance, how is it that we're able to more faint galaxies as aperture increases? The answer
is not as obvious as it may seem.
For example, a common misconception among amateur astronomers is that objects become
brighter as aperture increases. Afterall, objects look brighter in larger telescopes. So, they must be brighter.
However, a telescope also magnifies the object at which it is pointed. And in doing so, the light collected
is spread over a larger area. As a result, the object's surface brightness will, at best, equal that of the
naked eye view or, in most instances, actually be reduced. Galaxies and nebulae do not become brighter
as aperture increases.
Still, a telescope can show us more galaxies than the naked eye. And as we use larger and
larger aperture, we are able to see galaxies and other extended objects of lower surface brightness.
How does this work? This is the question that has consumed me these past few years. Mel Bartels'
ODM has been a useful tool in exploring that question.
Here's how ODM works. The software leads you through setting several basic parameters:
sky surface brightness in magnitude per square arcsecond; object magnitude; and object size. Then, ODM
uses that data to calculate the object surface brightness to the eye; the surface brightness of the object
in the telescope; the surface brightness of the background sky in the telescope; the threshold contrast
at which objects are seen; and the "log object contrast," which is essentially a measure of the contrast
ratio of the object against the background sky. Then, using these data, ODM predicts whether or not the
object will be visible. And if the object is deemed within reach, ODM suggests an optimum magnification
at which detection will occur.
My experience suggests ODM does a pretty good job, at least as far as my 10-inch Newtonian
under a dark sky is concerned. But don't just take my word for it. See for yourself. ODM is free software
and is available to download at Bartels' web site: http://members.efn.org/~mbartels/aa/visual.html
At Bartels' suggestion, I have built the following matrixes to aid in isolating the effect of the
different variables involved in ODM: sky brightness, aperture, object brightness and object size. Each
of the eight matrixes concerns a different pair of variables. After each, I've summarized the results. At
the bottom of this page in the "Discussion" section, is my attempt at interpreting those data. And I invite
you to do the same. It is my hope that making these data publicly available will encourage other to
study ODM's predictions and offer their own interpretations. For it is only through the open exchange
of ideas that we will come to understand the nature of deep-sky observing. And in doing so, we can
help current and future generations of astronomy enthusiasts to get the fullest possible enjoyment
from their telescopes.
Bill Ferris, June 2004
ODM Matrixes: Key
Sky SB: Sky surface brightness (mag./arcsec.^2) SB Reduc. At ODM: Surface brightness reduction at optimum magnification (mag./arcsec.^2))
Aperture: Telescope aperture (inches) Obj. SB at ODM: Object surface brightness at optimum magnification
Obj. Mag.: Object magnitude Obj. + Sky SB at ODM: Combined surface brightness of object and sky at optimum magnification
Maj. Axis: Object greatest dimension (arcminutes) Sky SB at ODM: Sky surface brightness at optimum magnification
Min. Axis: Object smallest dimension (arcminutes) Log Obj. Cont.: The log (base 10) of the contrast ratio between the object and the sky
Obj. SB: Object surface brightness (mag./arcsec.^2) Log Thresh. Cont.: The log of the threshold contrast at which objects become visible
Min. Mag.: Minimum usable magnification in that aperture Log Cont. Diff.: The log of the difference between threshold and object contrast
St. Lim. Mag.: Telescope limiting magnitude ODM: Optimum magnification for detection
Matrix #1: Sky Surface Brightness and Aperture Remain Constant
Sky SB Aperture Obj. Mag. Maj. Axis Min. Axis Obj. SB Min Mag. St. Lim. Mag. SB Reduc. at ODM Obj. SB at ODM Obj. + Sky SB at ODM Sky SB at ODM Log Obj. Cont. Log Thresh. Cont. Log Cont. Diff. ODM
22.00 5.00 13.35 1.00 1.00 21.98 17 14.38 2.95 24.92 24.18 24.95 0.01 0.01 0.00 55
22.00 5.00 13.75 0.50 0.50 20.87 17 14.38 4.02 24.89 24.56 26.02 0.45 0.45 0.00 90
22.00 5.00 13.96 0.25 0.25 19.58 17 14.38 3.56 23.14 23.03 25.56 0.97 0.97 0.00 73
22.00 5.00 14.21 0.13 0.13 18.41 17 14.38 9.09 27.50 27.46 31.09 1.44 1.44 0.00 931
The first four matrixes pair sky surface brightness (SB) with each of the other three variables.
In this first matrix, sky SB and aperture are kept constant. I envision this as an observing session by
a lone amateur astronomer with a 5-inch aperture telescope under a pristine dark sky. As you scan
across the matrix, find the column labeled "Log. Cont. Diff." or the log of the difference between the
threshold contrast and the object contrast. All five objects have values of 0.00 in this column. In other
words, the contrast ratio between these objects and the sky background is right at the predicted
 threshold for this aperture under this sky.
Matrix #2: Sky Surface Brightness and Object Magnitude Remain Constant
Sky SB Aperture Obj. Mag. Maj. Axis Min. Axis Obj. SB Min Mag. St. Lim. Mag. SB Reduc. at ODM Obj. SB at ODM Obj. + Sky SB at ODM Sky SB at ODM Log Obj. Cont. Log Thresh. Cont. Log Cont. Diff. ODM
22.00 5.00 13.35 1.00 1.00 21.98 17 14.39 2.95 24.92 24.18 24.95 0.01 0.01 0.00 55
22.00 7.90 13.35 1.60 1.60 23.00 27 15.39 1.99 24.99 23.63 23.99 -0.40 -0.40 0.00 56
22.00 12.50 13.35 2.34 2.34 23.82 43 16.38 0.83 24.66 22.65 22.83 -0.73 -0.73 0.00 52
22.00 20.00 13.35 2.89 2.89 24.28 68 17.40 0.40 24.68 22.27 22.40 -0.91 -0.91 0.00 68
In Matrix #2, sky SB and object magnitude remain constant. Our lone observer has been
joined by three deep-sky enthusiasts. The four telescopes are of different apertures, each size
difference corresponding to a whole magnitude interval in light-gathering power. Threshold object size
increases and surface brightness decreases as aperture increases.
Matrix #3: Sky Surface Brightness and Object Size Remain Constant
Sky SB Aperture Obj. Mag. Maj. Axis Min. Axis Obj. SB Min Mag. St. Lim. Mag. SB Reduc. at ODM Obj. SB at ODM Obj. + Sky SB at ODM Sky SB at ODM Log Obj. Cont. Log Thresh. Cont. Log Cont. Diff. ODM
22.00 5.00 13.35 1.00 1.00 21.98 17 14.39 2.95 24.92 24.18 24.95 0.01 0.01 0.00 55
22.00 7.90 13.89 1.00 1.00 22.52 27 15.39 1.99 24.51 23.47 23.99 -0.21 -0.21 0.00 56
22.00 12.50 14.36 1.00 1.00 22.99 43 16.38 2.00 24.99 23.63 24.00 -0.40 -0.40 0.00 89
22.00 20.00 14.84 1.00 1.00 23.47 68 17.40 1.65 25.12 23.40 23.65 -0.59 -0.59 0.00 121
Sky SB and object size are held constant in the third matrix. We're working with the same
group of observers and, in this scenario, they're going after the faintest 1' diameter galaxy that can
be seen in their respective telescopes. As we've seen in the first two matrixes, ODM predicts that
threshold object brightness decreases as aperture increases. This is true, both in terms of object
magnitude and surface brightness.
Matrix #4: Sky Surface Brightness and Object Surface Brightness Remain Constant
Sky SB Aperture Obj. Mag. Maj. Axis Min. Axis Obj. SB Min Mag. St. Lim. Mag. SB Reduc. at ODM Obj. SB at ODM Obj. + Sky SB at ODM Sky SB at ODM Log Obj. Cont. Log Thresh. Cont. Log Cont. Diff. ODM
22.00 5.00 13.35 1.00 1.00 21.98 17 14.39 2.95 24.92 24.18 24.95 0.01 0.01 0.00 55
22.00 7.90 14.35 0.63 0.63 21.98 27 15.39 2.95 24.92 24.18 24.95 0.01 0.01 0.00 87
22.00 12.50 15.34 0.40 0.40 21.98 43 16.38 2.95 24.93 24.19 24.95 0.01 0.01 0.00 138
22.00 20.00 16.36 0.25 0.25 21.98 68 17.40 2.96 24.93 24.19 24.96 0.01 0.01 0.00 221
In the final pairing of sky surface brightness with the other variables, object surface
brightness is held constant. Our intrepid observers now seek out threshold objects having the same
surface brightness. As aperture increases, the size and integrated magnitude of threshold objects
decrease. Check out the "Log Obj. Cont." column. All four objects have the same 0.01 value.
Matrix #5: Aperture and Object Magnitude Remain Constant
Sky SB Aperture Obj. Mag. Maj. Axis Min. Axis Obj. SB Min Mag. St. Lim. Mag. SB Reduc. at ODM Obj. SB at ODM Obj. + Sky SB at ODM Sky SB at ODM Log Obj. Cont. Log Thresh. Cont. Log Cont. Diff. ODM
22.00 5.00 13.35 1.00 1.00 21.98 17 14.39 2.95 24.92 24.18 24.95 0.01 0.01 0.00 55
21.00 5.00 13.35 0.63 0.63 20.98 17 14.39 3.97 24.94 24.20 24.97 0.01 0.01 0.00 88
20.00 5.00 13.35 0.39 0.39 19.93 17 14.39 5.01 24.94 24.22 25.01 0.03 0.02 0.01 142
19.00 5.00 13.35 0.25 0.25 18.97 17 14.39 5.97 24.93 24.20 24.97 0.01 0.01 0.00 221
The next three matrixes pair aperture with the other ODM variables. We've already looked at
aperture and sky SB in the first matrix so, to begin this sequence, we'll pair aperture with object
magnitude. In matrix 5, we explore the performance of a 5-inch telescope under increasingly bright
skies. Notice that threshold object surface brightness increases as the background sky brightens. Also
notice, with each one-magnitude change in sky brightness, object surface brightness also changes by
about one magnitude. And as in the fourth matrix, all four threshold objects have essentially the same
contrast value.
Matrix #6: Aperture and Object Size Remain Constant
Sky SB Aperture Obj. Mag. Maj. Axis Min. Axis Obj. SB Min Mag. St. Lim. Mag. SB Reduc. at ODM Obj. SB at ODM Obj. + Sky SB at ODM Sky SB at ODM Log Obj. Cont. Log Thresh. Cont. Log Cont. Diff. ODM
22.00 5.00 13.35 1.00 1.00 21.98 17 14.39 2.95 24.92 24.18 24.95 0.01 0.01 0.00 55
21.00 5.00 12.90 1.00 1.00 21.53 17 14.39 2.98 24.51 23.46 23.98 -0.21 -0.21 0.00 56
20.00 5.00 12.37 1.00 1.00 21.00 17 14.39 3.99 24.99 23.63 23.99 -0.40 -0.40 0.00 89
19.00 5.00 11.83 1.00 1.00 20.46 17 14.39 4.66 25.12 23.41 23.66 -0.58 -0.59 0.01 121
Keeping aperture and object size constant, we again see that surface brightness increases
for threshold objects viewed against increasingly bright skies. Also notice that the threshold contrast
value reduces as sky brightness increases. Another interesting result is that a one-magnitude change
in sky brightness does not produce an equivalent change in threshold object brightness. The surface
brightness and magnitude of a threshold object changes by about half a magnitude with each whole-
magnitude change in sky brightness.
Matrix #7: Aperture and Object Surface Brightness Remain Constant
Sky SB Aperture Obj. Mag. Maj. Axis Min. Axis Obj. SB Min Mag. St. Lim. Mag. SB Reduc. at ODM Obj. SB at ODM Obj. + Sky SB at ODM Sky SB at ODM Log Obj. Cont. Log Thresh. Cont. Log Cont. Diff. ODM
22.00 5.00 13.35 1.00 1.00 21.98 17 14.39 2.95 24.92 24.18 24.95 0.01 0.01 0.00 55
21.00 5.00 12.39 1.56 1.56 21.98 17 14.39 2.98 24.97 23.62 23.98 -0.39 -0.39 0.00 56
20.00 5.00 11.12 2.79 2.79 21.98 17 14.39 2.41 24.39 22.25 22.41 -0.79 -0.79 0.00 43
19.00 5.00 9.35 6.30 6.30 21.98 17 14.39 0.40 22.37 19.33 19.40 -1.19 -1.19 0.00 17
In this matrix, aperture and object surface brightness are held constant while our lone
observer uses his telescope under brighter and brighter skies.
Matrix #8: Object Magnitude and Object Size (and Object Surface Brightness) Remain Constant
Sky SB Aperture Obj. Mag. Maj. Axis Min. Axis Obj. SB Min Mag. St. Lim. Mag. SB Reduc. at ODM Obj. SB at ODM Obj. + Sky SB at ODM Sky SB at ODM Log Obj. Cont. Log Thresh. Cont. Log Cont. Diff. ODM
22.00 5.00 13.35 1.00 1.00 21.98 17 14.39 2.95 24.92 24.18 24.95 0.01 0.01 0.00 55
21.00 7.74 13.35 1.00 1.00 21.98 27 15.34 2.99 24.97 23.62 23.99 -0.39 -0.39 0.00 87
20.00 13.90 13.35 1.00 1.00 21.98 47 16.61 2.44 24.42 22.27 22.44 -0.79 -0.79 0.00 121
19.00 31.50 13.35 1.00 1.00 21.98 107 18.39 0.37 22.35 19.31 19.37 -1.19 -1.19 0.00 106
In this final matrix, object magnitude and size are kept constant. By extension object surface
brightness is also kept constant. The is a test of the minimum aperture required to detect the same
1' diameter galaxy under a variety of sky conditions. The same galaxy that is seen in a 5-inch aperture
under a pristine sky requires nearly 32-inches of light-gathering oomph to be detected in urban skies.
ODM Matrixes: Discussion of Results
What does all this mean? I believe the results shed light on the role aperture plays in the
detection of deep-sky objects. Also, the matrixes speak to the relative impact of aperture and sky
darkness in deep-sky observing.
We might imagine that Matrix 1 concerns the same galaxy. Miraculously, we're able to move
the galaxy away from Earth and adjust its brightness. For example the first object in the matrix is
13.35 magnitude and 1' in size. Suppose we doubled the distance to this galaxy. It would diminish in
size by one-half, which is the value we see in the second line of the matrix. Being twice as far, it would
appear one-fourth as bright. That brightness ratio corresponds to a change in magnitude of 1.51. Its
integrated magnitude would drop to 14.86.
However, the surface brightness would remain unchanged. Since that light is packed into a
smaller surface area, the SB would still be 21.98 MPSAS. Yet, ODM predicts this object would lie
below the threshold of visibility because it would be too small.
How much brighter does this galaxy need to be before it again is visible to the eye? ODM
predicts the surface brightness needs to increase by 1.1 magnitude. This corresponds to a 2.8-times
change in brightness. Since the sky background remains constant, the contrast ratio between the
sky background and the object improves and the object is seen.
In the first matrix, ODM predicts the surface brightness of a threshold object must increase
as it gets smaller, if that object is to remain visible in a fixed aperture. This illustrates a defining
characteristic of human vision. Under low light conditions, the eye is primarily a contrast detector.
People aren't good at discerning color or resolving small angular features in the dark. Rather, we detect
brightness variations; the larger the better.
As a result, we're better at detecting large, faint objects than small, faint objects. In the first
matrix, the integrated magnitudes of threshold objects grow progressively fainter as object size gets
smaller. However, their surface brightnesses increase. The objects get brighter with respect to the
background sky, contrast improves and the objects are still visible.
One question I've been exploring the last few years is, what is the relationship between
aperture and threshold contrast? As we scan down the "Obj. SB" column in Matrix 2, we see that
going from a 5-inch aperture to a 7.9-inch aperture produces an object SB reduction of about one
full magnitude per square arcsecond. The next jump in aperture represents another one-magnitude
gain in light-gathering, but threshold object surface brightness drops by just 0.8 magnitude. The final
increase in aperture--another whole magnitude gain in light-gathering--produces a 0.5-magnitude drop
in threshold object surface brightness.
It would appear that performance changes non-linearly with each successive jump in aperture.
In other words, there is a law of diminishing returns at play in the aperture game. As aperture increases,
threshold contrast is reduced but, with each step in aperture, the gains to be enjoyed become less
and less.
But the critical lesson to take from Matrix 2 is that threshold contrast can be reduced by
increasing aperture. This is how aperture allows us to see fainter and fainter objects without actually
making these objects brighter. They just look brighter because they are pushed farther and farther
from the threshold contrast level at which objects appear.
Matrixes 4 and 5 illustrate an aspect of extended object visibility that is similar in nature
to point source visibility. In Matrix 4, sky SB and object SB are held constant. Aperture and object
magnitude are held constant in Matrix 5. These scenarios produce results in which the "Log Obj. Cont."
and "Log Thresh. Cont." are held constant. This has the effect of allowing aperture and sky surface
brightness to impact extended object visibility in much the same manner as point source visibility is
impacted.
The key to this strategy--whether talking about a range of apertures under the same sky or
the same aperture under a range of sky brightnesses--is object size. As aperture or sky brightness
increase, look for smaller objects. This strategy allows the observer to enjoy the full benefits of aperture
under a dark sky, or to counter the impact of light pollution in their favorite telescope.
In Matrix 4, the extended object limiting magnitude (EOLM) changes by one full magnitude
with each step in aperture. Each step represents a 2.5-times change in light-gathering or a full
magnitude change in brightness. Applying the inverse-square relationship between apparent brightness
and distance, each step in aperture also represents a 1.58-times change in distance. In other words,
if you collect 2.5-times as much light, you should be able to see objects 0.4-times as bright. One way
of describing this would by to say we'd still be able to see a threshold galaxy if it were moved to 1.58-
times its current distance. At that distance, it would appear 63% as large and 40% as bright. ODM
predicts that increasing aperture from 5- to 7.9-inches would allow an observer to detect a galaxy
one magnitude fainter and 0.63-times the size. And each subsequent change in light-gathering yields
a similar result.
In Matrix 5, the same aperture is used under increasingly bright skies. The same strategy of
looking for smaller and smaller objects allows the observer to maintain the same EOLM.
The last three matrixes are concerned with sky brightness and its impact on our ability to
detect extended objects under different sky conditions. In Matrix 6, ODM predicts that threshold
contrast improves as the sky gets brighter. When I first saw these results, I thought there had to be
a mistake. But after giving the issue some thought, it does make sense. The human eye is better
able to resolve low contrast features in a brightly lit environment. As sky SB increases, the eye's
performance shifts from scotopic (darkness) to photopic (daylight). The threshold contrast at which
an extended object can be detected is lowered.
This might seem light great news for the deep-sky enthusiast. But the countering influence
of sky brightness on object contrast effectively negates any advantage…and then some.
In Matrix 7, object surface brightness remains constant even as sky SB gets brighter. Here,
we're comparing views in same-aperture telescopes under increasingly bright skies. By keeping
object surface brightness constant, we're effectively forcing objects deeper and deeper into the light
pollution. The only way to overcome this is to seek out larger and larger objects bright enough to
punch a hole through the murk.
In effect, Matrix 7 shows what would happen if we could move galaxies closer to overcome
light pollution. We begin by observing  a 1' diameter galaxy as a threshold object under pristine skies.
Relocating to a typical dark country sky with a SB of 21.0 MPSAS, the observer must move the galaxy
to 64% its initial distance. This makes it appear 1.58-times the size and 2.43-times brighter, which
corresponds to a 0.97-magnitude gain in integrated brightness.
Moving to a rural sky with a SB of 20.0 MPSAS, our deep-sky hunter has to move the galaxy
to 56% its distance, making it appear 2.79' in size and 3.2-times (1.26-magnitude) brighter. Finally,
from a suburban (19.0 MPSAS) site, the galaxy must be moved to 44% its previous distance. It has
an apparent size of 6.3' and is 5.1-times (1.77-magnitude) brighter.
Our observer loses about one magnitude going from a pristine to a dark country sky. But
moving to rural and suburban sites exacts increasing harsh tolls. Losses of 1.3- and 1.8-magnitude
in EOLM are experienced. The indication is that the scale describing the impact of increasing light
is not logarithmic. Matrix 8 confirms this.
If aperture and sky brightness have an equivalent impact on the visibility of extended objects,
then a one-magnitude change in sky brightness should be countered by an equivalent change in light-
gathering power. However, we see that this is not the case under moderate to heavy light-pollution.
Matrix 8 compares views in four different telescopes, all pointed at the same galaxy but under
very different sky conditions. A 5-inch aperture is predicted to discern our 1' diameter, 13.35-magnitude
metropolis under a pristine sky. The observer setup under a typical dark country sky needs a 7.7-inch
aperture to detect the same galaxy. From a rural site, a 13.9-inch aperture is needed. And in urban
conditions, nearly 32-inches of light-gathering oomph are required to see the same galaxy.
Although the sky brightness has increased by 3-magnitude, the aperture increase to maintain
detectability represents a 4-magnitude gain in light-gathering power.
Conclusions
ODM is a useful tool for exploring the nature of visual deep-sky observing with astronomical
telescopes. Its predictions suggest that increasing aperture lowers the threshold contrast at which
extended objects become visible. In this way, aperture brings fainter objects within reach and makes
threshold objects "look" brighter by raising them above the threshold of detectability.
ODM also suggests that sky brightness has the greatest impact on extended object visibility.
As sky SB increases, some objects are moved beyond the reach of any aperture. Those objects that
can be reclaimed are seen only by increasing aperture at a rate that outpaces the change in sky
surface brightness.
Anyway, that's how I interpret the results. What do you think?
line
Layout, design & revisions © W. D. Ferris
ODM © Mel Bartels
Comments and Suggestions: BillFerris@aol.com
Revised: June 26, 2004 [WDF]