The Stellar Magnitudes and the Limiting Visual Magnitude of a Telescope

Proxima Centauri: Image credit Esa/Nasa

The brightness of the stars is indicated with a scale, called the scale of magnitudes, which has the following rules:

  1. The more the star is shining, the smaller the number is. Thus a star of magnitude 1 is brighter than a star of magnitude 2.
  2. The scale is not linear, but logarithmic; between a value and the next one there is a value of about 2.5 times (the fifth root of 100, which is almost exactly 2,512). Thus a star of magnitude 1 is more than 2,512 times brighter than a star of magnitude 2.
  3. The brightest stars in the sky are called “first magnitude”, but in reality there are stars of magnitude 0 (like Vega) and 3 stars, Sirius, Canopus and Arturo, which have negative magnitudes. The Sun has an average magnitude of -26.73.
  4. The stellar magnitudes are understood to be apparent, because the brightness of a star also depends on its distance. There is a scale of magnitude that places the stars all at the same standard distance (1 parsec = 3.26 light years) and which is called Absolute Magnitude; this value expresses the real brightness possessed by the star.

The telescope is able to show, in the visual use, stars of magnitude weaker than that perceived by the naked eye, and the larger the telescope’s lens, the higher the magnitude of the high limit, or the weaker stars are seen, thus in larger quantities. As an example, a small 60 mm refracting telescope can show us stars of a magnitude of about 13.0 at high magnifications and in the best observational conditions. Many factors condition this limit value. The main factors are: light pollution, the transparency of the sky, the atmospheric turbulence, the height of the star observed on the horizon, the color (ie the “spectral type”), the magnifications and the optical quality of the instrument, if optics are clean or dirty, but also the experience, age and visual acuity of the observer.

Seeing weak stars is one of the factors that determines the satisfaction of a visual observation because it allows us, for example, to discover the nature of dense galactic clusters like M11 or globular clusters like M3, M13 and M22.

In the past many books have been published in which the magnitude limit of a telescope of a certain opening was decidedly underestimated, perhaps to prevent the “disappointment of expectation” by beginners or, more likely, because the authors did not personally experience the real limits of the tools they used, simply copying and pasting the data found in previously published texts.

One of the “classical” formula for determining the visual magnitude limit of a telescope is the following: ML = 9.5 + 5.0 * Log10 (D)

where D = aperture in inches of the telescope lens (1 inch = 25.4mm) Source:: “THE OBSERVATIONAL AMATEUR ASTRONOMER” by Patrick Moore.

Applying these old formulas that do not take into account other factors of great importance such as magnification are obtained, for a 60 mm aperture telescope, a magnitude gain of about 5-6 with respect to the night-time limit magnitude to the naked eye. So, in the case of a dark and transparent night, observing from a place not disturbed by light pollution, with magnitude limit to the naked eye equal to 6a, according to these old formulas the stellar magnitude limit of this small refractor would have been little more than 11a .

Practical experience, and perhaps even the noticeable advances in optics that have improved the quality of lenses and anti-reflective treatments, has taught me that the real limit is far higher. Under mountain skies I easily reached with my refractor ED 66/400 used at 100x, the visual magnitude of 13. As already mentioned, in addition to the obvious advantage given by the dark sky of the mountain site, with low light pollution, greater transparency and the choice of high stars on the horizon, played a key role in the use of high magnifications, which have the effect of darkening the sky background, improving sharply the contrast between the stars, which are point-like objects at any magnification, and the let us fuse it that instead it is an extended source; the contrast between the two subjects increases as magnification increases, always taking into account the limits represented by atmospheric turbulence and diffraction.
The color of the stars also plays a fundamental role, because the eye is less sensitive to red than to yellow-green, even when our eyes act in conditions of scotopic (ie nocturnal) vision so if a weak star is red or orange, you will see it less easily than a yellow or white star.

In the past years Professor Bradley Shaefer, a professor of astrophysics, has studied the problem in more depth, reformulating the calculation of the visual limit magnitude in a more articulated way and allowing a better description of reality.

On this site: there is an application that, using the Shaefer formula, allows to enter the parameters of the instrument, the observer and the quality of the sky, arriving to results which are very close to the real ones.

Trying to insert data in this application and trying to change some fundamental parameters, such as the type of telescope (refractor, reflector or catadioptric) the limit magnitude to the naked eye, the height on the horizon or the magnification, important differences between the results have been achieved.

For example, if you enter the data of a 10 × 50 pair of binoculars, without changing the other parameters you will find that the ML is, with good sky (SQM 20) equal to 10.6 while if you take a small telescope with the same aperture of 50mm but used at 50x (1 magnification per millimeter) the value of ML rises to 12.1.
As a second example, we take a 200mm Newton with a focal length of 1000mm, used with a 32mm eyepiece that many authors recommended, in the past, as an optimal eyepiece for observations of deep-sky objects. With this eyepiece you get only 31x and then an exit pupil of 6.4mm. By entering the data of this configuration into the application, a limit magnitude of 13.1 is obtained, a rather low value for a 200mm. However, if we enlarge at 200x (1 magnification per millimeter) the limit magnitude jumps to the value of 15, which allows us to distinguish, if we are expert observers, the stars of many globular clusters belonging to the Messier catalog.

It means that, if you are looking at the deep sky and you want to see weaker objects, especially if you prefer stars, but also galaxies and nebulae, you need to bring the magnification of the telescope to at least 1 magnification per millimeter of aperture. And if you want to frame large fields at the same time, take advantage of the availability of Ultragrandangular eyepieces like Nagler, Ethos or similar. For example, using the 10mm Ethos on a 200mm f / 5 Newtonian you get 100x and you set a field of 1 degree, it is enough to contain the vast majority of objects of the deep sky within a 200mm range.

I conclude by repeating one of my “fixed points”: never trust too much the values shown on tables, articles and books, always check yourself. It is the observation made with a method that makes data valid and which, above all, makes observational astronomy stimulating.

Plinio Camaiti

Telescope Doctor

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