Limiting magnitude

Modelling magnitude limits

We see stars if they have sufficient contrast against the background sky. A star's brightness (more precisely its illuminance) must exceed the sky's surface brightness (i.e. luminance) by a sufficient amount. Earth's sky is never completely black – even in the absence of light pollution there is a natural airglow that limits what can be seen. The astronomer H.D. Curtis reported his naked-eye limit as 6.53, but by looking at stars through a hole in a black screen (i.e. against a totally dark background) was able to see one of magnitude 8.3, and possibly one of 8.9.

Naked-eye magnitude limits can be modelled theoretically using laboratory data on human contrast thresholds at various background brightness levels. Andrew Crumey has done this using data from experiments where subjects viewed artificial light sources under controlled conditions. Crumey showed that for a sky background with surface brightness ${\displaystyle {\mu }_{\text{sky}}>21\mathrm{m}\mathrm{a}\mathrm{g}/\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{e}{\mathrm{c}}^2}$, the visual limit ${\displaystyle m}$ could be expressed as: $${\displaystyle m=0.4260{\mu }_{\text{sky}}-2.3650-2.5\log F}$$ where ${\displaystyle F}$ is a "field factor" specific to the observer and viewing situation. The very darkest skies have a zenith surface brightness of approximately 22 mag/arcsec², so it can be seen from the equation that such a sky would be expected to show stars approximately 0.4 mag fainter than one with a surface brightness of 21 mag/arcsec². Crumey speculated that for most people ${\displaystyle F}$ will lie between about 1.4 and 2.4, with ${\displaystyle F=2}$ being typical. This would imply ${\displaystyle m=6.25}$ at the darkest sites, consistent with the traditionally accepted value, though substantially poorer than what is often claimed by modern amateur observers.

To explain the discrepancy, Crumey pointed out that his formula assumed sustained visibility rather than momentary glimpses. He reported that "scintillation can lead to sudden 'flashes' with a brightening of 1 to 2 mag lasting a hundredth of a second." He commented, "The activities of amateur astronomers can lie anywhere between science and recreational sport. If the latter, then the individual's concern with limiting magnitude may be to maximise it, whereas for science a main interest should be consistency of measurement." He recommended that "For the purposes of visibility recommendations aimed at the general public it is preferable to consider typical rather than exceptional performance... Stars should be continuously visible (with direct or averted vision) for some extended period (e.g. at least a second or two) rather than be seen to flash momentarily."

Crumey's formula, stated above, is an approximation to a more general one he obtained in photometric units. He obtained other approximations in astronomical units for skies ranging from moderately light polluted to truly dark. $${\displaystyle \begin{array}{ccc}m& =0.27{\mu }_{\text{sky}}+0.8-2.5\log F& (18<{\mu }_{\text{sky}}<20\mathrm{m}\mathrm{a}\mathrm{g}/\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{e}{\mathrm{c}}^2)\\ m& =0.383{\mu }_{\text{sky}}-1.44-2.5\log F& (20<{\mu }_{\text{sky}}<22\mathrm{m}\mathrm{a}\mathrm{g}/\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{e}{\mathrm{c}}^2)\end{array}}$$ If an observer knows their own SQM (i.e. sky brightness ${\displaystyle {\mu }_{sky}}$ measured by a sky quality meter), and establishes their actual limiting magnitude, they can work out their own ${\displaystyle F}$ from these formulae. Crumey recommended that for accurate results, the observer should ascertain the V-magnitude of the faintest steadily visible star to one decimal place, and for highest accuracy should also record the colour index and convert to a standard value. Crumey showed that if the limit is ${\displaystyle m_c}$ at colour index ${\displaystyle c}$, then the limit at colour index zero is approximately ${\displaystyle m_c+0.27c}$

Some sample values are tabulated below. The general result is that a gain of 1 SQM in sky darkness equates to a gain in magnitude limit of roughly 0.3 to 0.4.

Visual magnitude limit with a telescope

The aperture (or more formally entrance pupil) of a telescope is larger than the human eye pupil, so collects more light, concentrating it at the exit pupil where the observer's own pupil is (usually) placed. The result is increased illuminance – stars are effectively brightened. At the same time, magnification darkens the background sky (i.e. reduces its luminance). Therefore stars normally invisible to the naked eye become visible in the telescope. Further increasing the magnification makes the sky look even darker in the eyepiece, but there is a limit to how far this can be taken. One reason is that as magnification increases, the exit pupil gets smaller, resulting in a poorer image – an effect that can be seen by looking through a small pinhole in daylight. Another reason is that star images are not perfect points of light; atmospheric turbulence creates a blurring effect referred to as seeing. A third reason is that if magnification can be pushed sufficiently high, the sky background will become effectively black, and cannot be darkened any further. This happens at a background surface brightness of approximately 25 mag/arcsec², where only 'dark light' (neural noise) is perceived. Point sources such as a star will be easier to detect than a diffuse object such as a galaxy or comet.

Various authors have stated the limiting magnitude of a telescope with entrance pupil ${\displaystyle D}$ centimetres to be of the form $${\displaystyle m=5\log D+N}$$ with suggested values for the constant ${\displaystyle N}$ ranging from 6.8 to 8.7. Crumey obtained a formula for ${\displaystyle N}$ as a function of the sky surface brightness, telescope magnification, observer's eye pupil diameter and other parameters including the personal factor ${\displaystyle F}$ discussed above. Choosing parameter values thought typical of normal dark-site observations (e.g. eye pupil 0.7 cm and ${\displaystyle F=2}$) he found ${\displaystyle N=7.69}$.

Crumey obtained his formula as an approximation to one he derived in photometric units from his general model of human contrast threshold. As an illustration, he calculated limiting magnitude as a function of sky brightness for a 100mm telescope at magnifications ranging from ×25 to ×200 (with other parameters given typical real-world values). Crumey found that a maximum of 12.7 mag could be achieved if magnification was sufficiently high and the sky sufficiently dark, so that the background in the eyepiece was effectively black. That limit corresponds to ${\displaystyle N}$ = 7.7 in the formula above.

More generally, for situations where it is possible to raise a telescope's magnification high enough to make the sky background effectively black, the limiting magnitude is approximated by $${\displaystyle m=5\log D+8-2.5\log (p^{2}F/T)}$$ where ${\displaystyle D}$ and ${\displaystyle F}$ are as stated above, ${\displaystyle p}$ is the observer's pupil diameter in centimetres, and ${\displaystyle T}$ is the telescope transmittance (e.g. 0.75 for a typical reflector).

Telescopic limiting magnitudes were investigated empirically by I.S. Bowen at Mount Wilson Observatory in 1947, and Crumey was able to use Bowen's data as a test of the theoretical model. Bowen did not record parameters such as his eye pupil diameter, naked-eye magnitude limit, or the extent of light loss in his telescopes; but because he made observations at a range of magnifications using three telescopes (with apertures 0.33 inch, 6 inch and 60 inch), Crumey was able to construct a system of simultaneous equations from which the remaining parameters could be deduced. Because Crumey used astronomical-unit approximations, and plotted on log axes, the limit "curve" for each telescope consisted of three straight sections, corresponding to exit pupil larger than eye pupil, exit pupil smaller, and sky background effectively black. Bowen's anomalous limit at highest magnification with the 60-inch telescope was due to poor seeing. As well as vindicating the theoretical model, Crumey was able to show from this analysis that the sky brightness at the time of Bowen's observations was approximately 21.27 mag/arcsec², highlighting the rapid growth of light pollution at Mount Wilson in the second half of the twentieth century.

Large observatories

Telescopes at large observatories are typically located at sites selected for dark skies. They also increase the limiting magnitude by using long integration times on the detector, and by using image-processing techniques to increase the signal to noise ratio. Most 8 to 10 meter class telescopes can detect sources with a visual magnitude of about 27 using a one-hour integration time.

Automated astronomical surveys are often limited to around magnitude 20 because of the short exposure time that allows covering a large part of the sky in a night. In a 30 second exposure the 0.7-meter telescope at the Catalina Sky Survey has a limiting magnitude of 19.5. The Zwicky Transient Facility has a limiting magnitude of 20.5, and Pan-STARRS has a limiting magnitude of 24.

Even higher limiting magnitudes can be achieved for telescopes above the Earth's atmosphere, such as the Hubble Space Telescope, where the sky brightness due to the atmosphere is not relevant. For orbital telescopes, the background sky brightness is set by the zodiacal light. The Hubble telescope can detect objects as faint as a magnitude of +31.5, and the James Webb Space Telescope (operating in the infrared spectrum) is expected to exceed that.