OF THE ROGER N. CLARK

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VISUAL ASTRONOMY OF THE DEEP SKY
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Table F.2. The optimum magnified visual angle (in arc-minutes) for various surface brightnesses
S.B. OMVA Log(OMVA) S.B. OMVA Log(OMVA) S.B. OMVA
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Log(OMVA)
4.0 9 0.97 16.0 16 1.20 22.0 62 1.79
9.0 10 1.01 17.0 18 1.26 23.0 68 1.83
11.0 11 1.04 18.0 23 1.36 24.0 72 1.86
13.0 12 1.07 19.0 31 1.49 25.0 1.90
79
14.0 13 1.11 20.0 42 1.62 26.0
1.96
15.0 14 1.14 21.0 51 1. 71 27.0 117 2.07
S.B. is the surface brightness in magnitudes per square arc-second.
guess fo r the magnification was made. In this
case, 100x was selected (for no other reason
than it is a starting point). Now compute the
background surface brightness Eo through
the 8-inch telescope at 100x using equations
F.2 and F.3. We find that Eo is 27.5. Next
estimate the OMVA from Figure 2.7b or interpolate
it using Table F.2. In the table the
last entry is at 27, so the OMVA must be
found by extrapolation. After doing this we
find the OMVA should be 132.4 arc-minutes.
Using this guess for the optimum size,
compute the magnification that would give
that size using equation F.3. Because NGC
134 has a smaller dimension of 1.0 arcminutes
we find the optimum magnification
would be 132.4/1.0 or 132.4x. But we guessed
100 X. These do not match very well so
use 132.4x as a new guess and start over.
Examining each iteration in Table F.3 for the
8-inch we see that the guess and the computed
magnification are getting closer. By
iteration 9 the difference is less than I, so
186X is very close. At iteration 12 the difference
is only 0.1 X so 187X is certainly better
than one needs. In practice the final magnification
is only approximate for reasons discussed
below. Any value between 180 and
190 is accurate enough.
Once the two magnifications agree you
have converged on the ODM!
Try the same exercise using a 24-inch telescope.
Only 5 iterations are needed to converge
on a value of 76x. Because 81X is
already the lowest useful magnification on a
24-inch, a lower power could not be beneficially
used, so 81x should be considered the
OUM.
320
91
WHAT THE ODM MEANS
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=
By examining the entries in the catalog you
will note that the optimum magnification on
the large telescope is less than on the small
one! The reason is that the larger a telescope
is, the greater an object's surface brightness is
at a given magnification, so it can be detected
at a smaller apparent size.
Examining the solutions for NGC 134 in
the catalog, we see that the ODM on the
8-inch is 187x, and on the 24-inch 81x.
Notice that for very small telescopes, the
magnification must be higher still. A 4-inch
needs about twice the power of an 8-inch to
show NGC 134 best.
Once an object has been found, and ifit is
not at the very threshold of detection, try
raising the power to enlarge any internal d : ­
tail toward its ODM. Recall that the opnmum
magnifications in the catalog are only
approximate, because the entire object's
mean surface brightness is used. Man y'
objects
show considerable variation in brIghtness
across their surfaces.
Spiral galaxies, for instance, typically have
a bright central region and faint arm . The
fainter parts may not be seen at all, whl!e the
bright inner area is much smaller (tendm to
raise the ODM) and much brighter (tendmg
to lower it but usually not as much as e
increase cused by the smaller size). ThIS
. .
b· n't be
catalog is only a gmde; If an 0
ect ca . her
detected at the listed ODM, try both hlg
and lower powers.
DM
com-
In the catalog, when the 0 was f,
I
Puted to be less than the minimum us
, nuDlpower
of the telescope, the telescope s _
APPENDIX F: OPTIMUM DETECTION MAGNIFICATIONS FOR DEEP-SKY OBJECTS
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Iteration Telescope Guess
# size (inches) magnification
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8.0 100.0
2 8.0 132.4
3 8.0 154.6
4 8.0 168.3
5 8.0 176.3
6 8.0 180.9
7 8.0 183.5
8 8.0 184.9
9 8.0 185.7
10 8.0 186.2
11 8.0 186.4
12 8.0 186.5
I
1 24.0 100.0
2 24.0 80.4
3 24.0 76.7
4 24.0 75.9
5 24.0 75.8
mum was listed. If the total magnitude of the
object is simply too faint for the telescope
under any conditions whatsoever, the entry
for that telescope is left blank.
THE ROLE OF CONTRAST
Even where an optimum magnification is
listed, it does not necessarily mean the object
will be seen at that power. The contrast must
also be high enough, as described in Chapter
6. To check on the contrast, find the apparent
size of the object's minimum dimension at the
ODM, as well as its apparent surface brightness,
as des cri bed earlier.
For example: in a 12-inch telescope NGC 134
has an ODM of about 1 14 X , at which a dark
country sky background is reduced 2.63 magmtudes
to a value of 24.25 + 2.63 = 26.88
magnitudes per square arc-second. We can
round this off to 27. The size of NGC 134's
smaller dimension is 1.0 arc-minutes, which
when magnified 114X is 114 arc-minutes.
This is the object's OMV A.
The smallest detectable contrast can be
fOund from Figure 2.6 or interpolating the
values in Table F.4. The Table lists the log
threshold contrast for different values of
Table F.3. Sample iterative calculationjor NGC 134
321
Possible OMV A Computed
Bo (arc-min) magnification
27.5 132.4 132.4
28.1 154.6 154.6
28.4 168.3 168.3
28.6 176.3 176.3
28.7 180.9 180.9
28.8 183.5 183.5
28.8 184.9 184.9
28.8 185.7 185.7
28.8 186.2 186.2
28.8 186.4 186.4
28.8 186.5 186.5
28.8 186.6 186.6
25.1 80.4 80.4
24.6 76.7 76.7
24.5 75.9 75.9
24.5 75.8 75.8
24.5 75.8 75.8
background surface brightness and OMV A.
The entries were used to plot the curves in
Figure 2.6. Again you must use log contrast
values for any interpolations.
Figure 2.6 shows that, at an apparent size
of 114 arc-minutes on the curve for background
brightness 27, a contrast of about I is
needed for detection. This is a log contrast of
O.
The Cl column gives a log contrast for
NGC 134 of 1.0 (which equals 101 or a contrast
of 10), well above the threshold. Thus,
NGC 134 should be easily detectable. I have
used the logarithm of the contrast because, as
for brightness, the eye responds to contrast
on a logarithmic scale. Just think of the Cl
column as a contrast scale with a higher
number meaning more contrast.
The contrast of each object in the catalog
was evaluated for each telescope to determine
whether it is high enough for detection.
Where the contrast is too low, the letter "u"
(for undetectable) appears in front of the
magnification entry. Once again, this does
not necessarily mean all of the object is invisible,
because the calculation was done using
the mean surface brightness. If the object has
a brighter region, like the nucleus of a galaxy,