In 2014 I published a paper in Monthly Notices of the Royal Astronomical Society called "Human Contrast Threshold and Astronomical Visibility". It investigated the limits of what can be observed visually, with naked-eye or through a telescope, when seeking out stars and deep-sky objects. I did this using existing laboratory data on visibility which I modelled mathematically. I tested my model against existing field data from visual astronomers, used it to make predictions about telescope performance under different conditions (aperture, magnification, light pollution), and made proposals about dark-sky classification. I presented my results at the ALAN (Artificial Light at Night) conference in Leicester in 2014. Astronomical Visibility
Andrew Crumey
My paper is quite technical, but I also wrote a more general outline of my ideas for the journal of the Webb Deep-Sky Society, and I gave a talk at their annual meeting in Cambridge in 2015. I gave a similar talk at the Abingdon Astronomical Society in the same year. The following links are to the two journal articles and the PowerPoint presentations of my talks.
Human Contrast Threshold and Astronomical Visibility [MNRAS 442, 2600-2619 (2014)] Modelling the Visibility of Deep-Sky Objects [The Deep-Sky Observer 171, 7-12 (2016)]
Human Contrast Threshold and Astronomical Visibility (De Montfort University, Leicester, 4th September 2014)
What Will I See? - Modelling the Visibility of Deep-Sky Objects (Abingdon Astronomical Society, 12th January 2015)
Modelling the Visibility of Deep-Sky Objects (Institute of Astronomy, Cambridge, 20th June 2015)
Further work by other researchers (testing, applying or critiquing my model):
What is the probability of seeing any particular astronomical object through a telescope? by John Barentine
Confirmation methodology for a lunar crescent sighting report by Faid et al
Realistic telescopic limiting magnitudes (a discussion thread at the Cloudy Nights forum).
Here are some main points arising from my paper
1. A good definition of what counts as a "dark sky" for visual astronomy is one where the Milky Way is visible to the naked eye (assuming it's far enough above horizon).2. The traditional method of using naked-eye stellar magnitude limit as a subjective measure of sky quality is good. Sky Quality meters do something similar but are more objective. Assessment scales, such as the Bortle Scale, are interesting but problematic. Naked-eye limiting magnitude or Sky Quality meter reading are to be preferred.
3. Magnitude (abbreviation mag) is a unit of illumination, equivalent to the SI unit lux. Surface brightness (which astronomers measure in mag per square arcsec) is equivalent to a quantity called luminance (whose SI unit is candelas per square metre). A Sky Quality meter measures surface brightness in units almost identical to the astronomical ones. For simplicity, all surface brightness units will be abbreviated here as SQ.
4. Earth's sky is never completely black, due to natural skyglow. The very darkest skies have a surface brightness of approximately 22SQ, and a normally sighted person will see stars down to about 6 mag. If there were no atmosphere (e.g. on the Moon), the same person would see stars to about 8 mag, assuming their helmet visor was clear enough.
5. Naked-eye stellar magnitude limits can easily be over-estimated, because faint stars can be momentarily brightened by scintillation, where Earth's atmosphere effectively focuses a star's light for a fraction of a second (as discussed in Section 1.6.2. of my paper). For instance a 7 mag star might momentarily brighten to 6 mag, and be seen for an instant by someone who then reports the site as having a naked-eye limit of 7. Reliable sky quality measures should be based on definite visibility for a matter of seconds, not momentary glimpses.
6.1. Magnitude (illumination) is a function of distance; surface brightness (luminance) is independent of distance. Move a lamp further away and it will give you less illumination, but the bulb won't look dimmer. Stars are effectively point sources, so a telescope doesn't make them look closer, but does increase the illumination they shed on the eye.
6.2. Point a telescope at the Moon, put your hand under the eyepiece, and you will see a patch of moonlight brighter than the surrounding. The telescope has improved the Moon's illumination (i.e. magnitude), but not its surface brightness when you look through the telescope - though your dark-adapted eye may be dazzled. Look at the Moon in daylight, with and without telescope, and you will see that although a telescope boosts magnitude, it does not boost surface brightness.
6.3. In fact a telescope weakens surface brightness, making extended objects dimmer. It may make a distant lamp look closer, but doesn't actually bring it closer. Instead, magnification spreads the image over a larger angular area, weakening surface brightness (i.e. increasing the value in SQ units).
6.4. A telescope magnifies and darkens the background sky (i.e. weakens its surface brightness) while enhancing the illumination of stars. This is why we see more stars through a telescope than with the naked eye. If magnification is high enough, the background sky becomes effectively black, and there can be no further boost to stellar visibility. Then the telescope's limiting magnitude has been reached.
7.1. Since everyone's vision is different, and everyone's sky is different, the limiting magnitude of a telescope is specific to the observer and observing condition. What can be stated more generally is the boost that the telescope gives, i.e. the difference between the observer's naked-eye magnitude limit, and the limit for stars seen through the telescope by the same person under the same conditions.
7.2. In my paper, I wrap up all the variable conditions in a number called the field factor, F, and show that for an observer in dark-sky conditions F is typically in the range 1.4 to 2.4. Using a mathematical model of human vision (based on experimental data) I find an equation (Eq. 51) from which I estimate the dark site naked-eye limit to be 6.93 - 2.5 log F. Taking F = 2 gives a typical dark-sky limit of 6.18 mag (see Equation 51 and remarks following). Some people will do better, some worse. I show from historical records that some professional astronomers had naked-eye limits of 6.5 or better, and also show that William Herschel's limit (at age 44, with skies completely free of light pollution) was 6.1.
7.3. For stars seen through a telescope, I find the magnitude limit to be given by a formula involving F (the field factor), D (the telescope's aperture, or more accurately, "entrance pupil", in cm), and some other factors that take into account, for instance the telescope's light transmittance. As an example, taking the transmittance to be 75 per cent gives the limiting magnitude as 5 log D + 8.45 - 2.5 log F (Eq. 69).
7.4. Comparing the formulae in paragraphs 7.2. and 7.3 above, we see that in this case the telescope offers a magnitude gain of 5 log D + 8.45 - 6.93. For a 15cm (six-inch) refractor, this is 7.4 mag. An observer seeing stars to 6.1 mag with the naked eye should see to 13.5 mag with the telescope.
8.1. Objects are only visible if they have sufficient contrast against their surroundings. Crucial parameters are the target object's size, and the difference in surface brightness between target and surround. In normal daylight, we can read the small print in a newspaper. In moonlight we might only manage the headlines. The contrast of black print on white paper has not changed – what has changed is the degree of contrast our eyes need for letters of a given size to be visible.
8.2. It is possible to construct a mathematical "contrast threshold curve" showing the degree of contrast needed for targets of a given size to be visible, in conditions ranging from normal daylight to total darkness. These curves were plotted in the 1940s using experimental data. In my paper I showed how they can be modelled by mathematical formulae. Unlike previous attempts, mine did not rely on parameter tuning, but was instead based on an underlying linearity (Figure 5 of the paper).
8.3. The contrast threshold curve extends across all sizes, down to zero. So the visibility of point sources (stars) is an indicator of the likely visibility of extended objects (galaxies) and vice versa. This is why stellar magnitude limit, or the visibility of extended targets such as the Milky Way, are equally good as indicators of sky quality.
8.4. I show in my paper how, for extended objects, it is possible to estimate an equivalent point-source limit (which is not the same as the object's integrated magnitude). For instance, the galaxy M33 has a total magnitude of 5.8, from which we might suppose that if we can see stars down to 6.0, then we should be able to see the galaxy. However, using my model, I predict that M33 should not be visible to the naked eye unless the observer is seeing stars down to about 6.6 mag. This latter number can be considered the galaxy's "effective" visual magnitude.
8.5. A similar calculation could be done for any DSO (for instance, I show in my paper that M31 should be visible to the naked eye in any sky darker than 19.2 SQ, assuming it's sufficiently well placed). The caveat is that the experimental data was for uniform circular targets against uniform backgrounds, and for DSOs this is only an approximation.
9.1. A telescope makes galaxies and nebulae look closer, but can't improve their surface brightness. Instead it weakens their surface brightness, while also making the background sky darker. Making them look larger makes them easier to see, but making them too large will fade them too much in relation to the background, so that they disappear. There's a Goldilocks region in between, and my paper shows how to calculate it.
9.2. For an observer looking through a telescope, visibility is determined by a "telescopic threshold curve", presented in Section 6 of the paper. The axes are size and surface brightness, and for an object to be predicted as visible, its size and surface brightness should place it below the curve. As magnification increases (with all other parameters fixed) the curve flattens, meaning that magnitude limit increases while surface brightness limit decreases. Figure 17 of the paper shows curves for a 100mm telescope at a dark sky, viewing a hypothetical DSO at various powers, predicting visibility at x75 but not at x20 or x200.
9.3. From studying William Herschel's records of nebular sweeps I was able to calculate his telescopic threshold curve (Eq. 85). I also constructed curves for various instruments and viewing situations, and plotted Messier galaxies by size and surface brightness, to see if they were predicted to be visible (Fig 18). My calculations gave quantitative support to what is well known from experience: a 6-inch scope at a dark site will show more DSOs than a 16-inch at a light-polluted one.
10. My paper focussed on the astronomical application, however the model constructed in Sections 2 and 3 of the paper is applicable to visibility problems at any light level, including normal daylight.
Home Page