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The published magnitudes of deep sky objects (nebulae, galaxies and star clusters) can be quite misleading when it comes to indicating observability. With a little experience, the beginner deep sky observer soon becomes aware that the visibility of such objects cannot be determined by published magnitude alone. At a visual magnitude of 6.5, for example, one might be led to believe that NGC 7293 (the Helix Nebula) would be readily visible in an 8 inch telescope since such an instrument has a limiting magnitude of about 13.5. Experience proves otherwise. Although its published magnitude indicates that this object is bright, it is in fact one of the more difficult deep sky objects, comparable to the Owl Nebula, requiring the best of observing conditions and perhaps a nebula filter to render it visible. Much "fainter" objects such as M27 (the Dumbell Nebula) at magnitude 8 and M57 (the Ring Nebula) at magnitude 9 are spectacular in an 8 inch while the "brighter" Helix might be invisible. Why is it that a telescope capable of seeing a 13th magnitude star might have difficulty showing a 6.5 magnitude nebula? The answer is in the size of the object.

The published magnitudes of deep sky objects indicate integrated brightness, or the total brightness of the object squeezed down to a point. If one could squeeze the light of the Helix, which is 16 arc minutes, or 16 x 60 = 960 arc seconds, in diameter (almost one half the apparent size of the moon) down to a point, it would be concentrated to the brightness of a 6.5 magnitude star. Conversely, if one was to defocus a 6.5 magnitude star to an image size of 960", the defocused disk would hav e an apparent brightness equal to that of the Helix. The reason, therefore, that this nebula appears so dim, is that its available light is spread over a large area. Although many other readily visible objects might be fainter, they are generally s maller, so their total available light is more concentrated resulting in a higher surface brightness. The Ring Nebula, for example is 2.5 magnitudes fainter than the Helix, but at a diameter of approximately 70" is about 14 times smaller so its avai lable brightness is considerably more concentrated resulting in a higher surface brightness.

The surface brightness of an object is therefore directly related to two factors, (1) its size, or more correctly, its area, and (2) its total available light. If two objects were equally bright (same visual magnitude) but one was twice as large, the larger would appear four times fainter since the same amount of light is spread over an area four times larger. Note that area is proportional to the square of the diameter. An object three times larger would appear nine times fainter, and so forth.

If the objects being compared do not have the same total available brightness, as is usually the case, the contribution of area to apparent brightness must be adjusted by the difference in total available light. As an example, suppose object A is four times "brighter" than object B (published magnitude indicates four times more total brightness), and is also ten times larger than B. Which will appear brighter, and by how much? Since A is ten times the size of B, it has 10² = 100 times the area of B, and would therefore appear 100 times fainter than B if both were of equal magnitude. Since A has four times the total brightness, however, the dimming effect of its greater area is somewhat offset and A would actually appear only 100/4 = 25 times fainter than B (even though its published magnitude is brighter). If, on the other hand, B had four times the total brightness of A, the dimming effect of the greater area would be enhanced and A would appear 100 x 4 = 400 times fainter tha n B. This analysis assumes that the objects are uniformly bright over their respective areas, which is not usually the case with real deep sky objects. Nevertheless it allows us to make some interesting comparisons of many deep sky objects as shown in the accompanying table.

In the table below, M57 (the Ring Nebula) is used as a standard against which other objects are compared. The values under "mv" and "d" are the published visual magnitude and size of the object. "Bar" and "Bmr" are "brightness factors" indicati ng how much the object's area (Bar) and total brightness (Bmr) relative to M57 contribute to its apparent brightness. For example, at 150" diameter, M97 has 4.6 times more area than M57 ((150/70)² = 4.6) and would appear 1/4.6 = 0.22 times as bright if both were of equal magnitude (Bar = 0.22). At a total visual magnitude of 11.2, M97 is 2.2 magnitudes fainter than M57 which corresponds to a factor of 0.13 in brightness (Bmr = 0.13: see note). Multiplying these two brightness factors gives th e surface brightness factor, Bsr, which indicates how bright the object appears relative to M57 (its surface brightness relative to M57). The table shows that M97 has only 0.03 times as much surface brightness as M57, corresponding to a magnitude di fference, or relative surfact magnitude mSR, of -3.9. This means that M97 appears 3.9 magnitudes fainter than M57 even though it is only 2.8 magnitudes dimmer in total magnitude. Using the total visual magnitude of M57 (9.0) as a standard, M97 would appear as a 9 + 3.9 = 12.9 magnitude object as indicated in the last column, me, which is the effective visual magnitude relative to M57.

Returning to the original example, the table shows that NGC 7293 appears only 0.05 times as bright (1/0.05 = 20 times fainter) as M57, corresponding to a visual magnitude difference 3.2. Hence, if M57 appears as magnitude 9.0, NGC 7293 appears as magnitude 12.2. NGC 7009 (the Saturn Nebula), on the other hand, appears 14.9 times brighter than M57 (2.9 mag), so relative to M57 appears as magnitude 6.1 object. This type of comparison may seem a bit confusing at first, but after some thought, helps to explain why objects like the Helix can be difficult despite an impressive total visual magnitude while "fainter" objects like the Saturn Nebula can appear quite bright.

The table shows that large objects such as the galaxy M33 and NGC 2237 (the Rosetter Nebula) can appear especially faint. It is no coincidence that the Rosette is not only the largest, but also the faintest appearing object in the table. Extreme ly large objects usually have very low surface brightness making them difficult to detect in a telescope. Observing with very low magnifications, binoculars or even naked eye enhances the visibility of these objects by concentrating the available li ght onto a smaller area on the eye's retina.

The brightest objects shown are the small planetary nebulae NGC 2392, 6210, 7009 and 7662. Small planetaries like these usually appear bright and are easy to see, but since they are almost stellar at low magnifications, can be difficult to find. Sometimes only their generally greenish hue makes them distinguishable from stars at low power. They usually take magnification well since there is a great deal of brightness available in a concentrated area. Finally, the table shows that among th e Messier objects, M33, M74, and M101 are about tied for faintest in appearance. Many observing guides rank M74 as the most difficult M object, but I always thought M101 was more difficult. The table shows this as being the case.

As stated earlier, this comparison is not perfect, since all objects are not uniformly bright. The magnitude of star clusters, for example, can be dominated by a handful of bright stars, and nebulae often contain bright knots or filaments. In ge neral, this method does not work well with star clusters even if their star distribution is fairly uniform. This may be due to the large amount of dark sky between stars in the clusters. In the case of most globular clusters, a gradual dropoff in s tar density near the edges makes estimating size difficult and leads to error. Another error results from the fact that the published dimensions of many objects are based on photographic images which include faint outer edges that are not visible vi sually. Additionally, the shapes of many objects are elongated or irregular making it difficult to compare areas, and published magnitudes vary based on the method of measurement. (Some fairly wide variations on size and magnitude for the same obje ct can often be found when various sources are consulted). Even with all these limitations in mind, however, some interesting comparisons are still possible, resulting in a better idea of how deep sky objects might appear in a telescope.

Note: The relationship between magnitude and brightness is a little confusing due to the nature of the magnitude scale which is "logarithmic" with a log base of about 2.51. This means that each increase in one magnitude corresponds to a brightness change of 2.51. Two magnitudes corresponds to a factor of 2.51 squared, three magnitudes 2.51 cubed, and so on. In the example, 2.2 magnitudes corresponds to a decrease in brightness of 2.51 raised to the 2.2 power, or 7.6, which means the object has 1/7.6 = 0.13 times the total brightness of M57.

© 1992 Frank Loso. All rights reserved. This page updated 25 Feb 1996