◄ 1.3. Optical system of a telescope   ▐    2.2. Telescope resolution ►

 

The purpose of a telescope is to help form a larger, brighter and more detailed image of distant objects on our retina. The more light it gathers, the finer details it resolves, the higher magnification makes possible - the better. Those three main functions - light gathering, resolution and magnification - are the measure of telescope efficiency. All three are related to some extent, but also have their individual characteristics and limits.

The fourth, informal telescope function is inspiring humbling awe by opening seemingly endless depth of space to the eye.

2.1. Light gathering power

Light-gathering power of a telescope mainly depends on its aperture diameter. However, it is the system light transmission that determines how much of the light that entered the telescope actually arrives at the final focus. Transmission losses are due to reflection, scattering and absorption of light, as well as due to opaque elements in the light path.

Aperture diameter determines a limit to the area receiving flux of light. Therefore, light-gathering gain of a telescope vs. naked eye comes mainly from its larger aperture. Naked eye pupil opening, at its widest, ranges anywhere from ~4mm to ~8mm in diameter, with 6mm being the most often cited average. Thus - neglecting for the moment transmission and possible obstructions - telescope of aperture D in mm will gather (D/6)2 times more light than an average eye.

Transmission losses at the mirror surface range from ~2% to ~20%, or more, depending on type and state of the coating, as well as the wavelength of light. Fresh aluminum coating reflection loss for the peak visible spectrum is ~12%, which can be nearly cut in half with a special, or "enhanced" aluminum. Dielectric reflective coatings reduce the loss to 2%-3%.

Lens elements lose light due to reflection from lens surface and absorption by the glass. Reflection from uncoated lens surface is ~4% for typical glasses and near-normal incidence. However, even simple anti-reflection coating, such as MgFl (magnesium fluoride), reduces reflectivity to ~1%, and more advanced coatings have it nearly eliminated. Unfortunately, nothing can be done about in-glass light absorption; it is at ~4% per inch of in-glass light travel for typical optical crown, and somewhat more for typical flint, averaged over the entire visual range (400nm-700nm). Losses due to in-glass absorption are roughly doubled in the blue/violet part of visible spectrum, compared to the green/red.

Light loss in glass elements, therefore, increases with the number of uncoated surfaces and the in-glass path length. For uncoated doublet objective, it is about 15% due to reflections, plus nearly 1% per inch of aperture due to in-glass absorption. For coated doublets, it is about 4% plus the absorption loss. The eyepieces are these days usually multicoated and, unless of exceptional size, have up to a few percent total light loss.

Finally, most reflecting telescopes have central portion of their main mirror obscured by a smaller secondary mirror - so called central obstruction. Size of central obstruction is usually between ~0.15D to ~0.4D, resulting in ~2% -16% light loss.

The true light-gathering power of a telescope is given by the product of its aperture area and transmission coefficient. At a rough average, light transmission is about 80% for amateur telescopes, although there are systems as low as ~60%, and those as high as ~95%. Often times, light-gathering power of a telescope is expressed in terms of limiting stellar magnitude detectable. The stellar magnitude - usually denoted by m - is a measure of apparent brightness, with the difference of 5 magnitudes corresponding to 100 times difference in brightness; thus the difference in one magnitude implies difference in brightness of 2.512 (from 1001/5), and any nominal difference in magnitude "x" implies 2.512x difference in brightness.

Taking that one magnitude implies the absolute difference in brightness by a factor of 2.512, limiting magnitude of a telescope resulting from its nominal light-gathering power is given by

                                                            mt' = me+ 5log(√Dτ/P)                                      (2)

for the aperture diameter D, with τ being the transmission coefficient, me the naked eye limiting magnitude and P the eye pupil diameter. Actual limiting magnitude achievable is up to about four magnitudes higher, mostly due to darkening of the background at high magnifications, moments of exceptional seeing and use of averted vision. Gain in the limiting magnitude due to magnification factor can be approximated by logM, M being the telescope magnification. Thus, more complete formula for the limiting magnitude of a telescope can be written as:

                                                      mt = me+ 5log(√Dτ/P) + logM                                (2.1)

It indicates gain in magnitude with respect to a star as seen by the naked eye, which is affected by atmospheric absorption. It is on average 0.25m at zenith, and increases approximately with 0.17z/(90-z) in magnitudes, z being the zenith distance in degrees. Other factors also influence limiting magnitude of the telescope, most of them being included in this neat limiting magnitude calculator.

Eye transmittance, which is incorporated into the naked eye magnitude, is a relatively significant factor influencing the limiting magnitude in a telescope. As the graph implies, average eye transmittance over the visual range can be anywhere from ~2/3 at the high end, to ~1/3 of the input flux at the low end. Thus, in otherwise identical conditions, it can vary up to 0.8 magnitudes - possibly more - from one individual to another. In other words, what is 5-magnitude sky for you, may be little better than 4-magnitude sky for someone else - or the other way around.

Stellar magnitude is the basis for integrated magnitude, which represents total brightness over an area expressed in terms of stellar magnitude. Thus, for given magnitude, an extended object is always fainter than a star (that is, with lower surface brightness), and limiting magnitude for extended objects is generally brighter than limiting stellar magnitude (it also varies depending on the intensity distribution over the object). Considering that surface brightness of the telescopic image diminishes with the square of magnification, it is always lower - at least for the transmission loss at the minimum useful magnification - in the image seen through a telescope, than in the image seen by the naked eye. However, telescopic image on the retina is much larger, and the total of light energy - or total brightness - much higher. That is why the Moon appears so much brighter in a telescope, despite its surface brightness being normally (due to magnification) much lower.

◄ 1.3. Optical system of a telescope   ▐    2.2. Telescope resolution ►

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