Observation Some Usual Formulae for Instruments

note that the millimeter is an unit often used in this tutorial; 1 millimeter is 0.03937". 1 inch is 2.54 mm

Barlow Lens

A Barlow lens is an optical device -a tube and lenses- which is used to increase the magnifying power of an instrument. It is used intercalated between the eyepiece holder and the eyepiece. A Barlow divides by 2 (or another value depending on the concept of the Barlow) the focal length of the eyepiece, hence doubling the magnification. As an example, a 4"/1,000 mm reflector used with a 10 mm eyepiece yields a power of 100x. Used with a Barlow the eyepiece's focal turns into 5 mm, hence the power turns 200x. Another way of seing the effect of a Barlow is to say that is doubles (or multiply by another value) the focal length of the instrument. In the previous example, the focal of the instrument may be said multiplied by 2, becoming 2,000 mm (as the focal of the eyepiece remains 10 mm). Barlows were usually discarded by amateur astronomers as degrading the quality of the image. They have gained more consideration now as used for planetary purposes with short-focal/wide-field refractors

Binoculars Field of View

Binoculars field of view is characterized by two values. The angular, and the linear fields of view. The angular field of view is the width, in degrees, which you are seeing on the sky. The linear field of view if the width, in feet, which is seen at 1,000 yards. Both the angular and the linear field of view are usually found on the binoculars themselves or they may be found in the manufacturer specifications. The conversion between both fields of view is obtained like: angular field of view = linear field of view/52.5 (result in °). Linear field of view = angular field of view x 52.5 (result in ft at 1,000 yards). Example: a binocular with an angular field of view of 7° has a linear field of view of 7 x 52.5 = 367.5 ft at 1,000 yards. More generally, binoculars of the Porro prisms type have their two optical axis more distant from each other than the observer's eyes, which yields a better assessment of relief and of the distance of distant objecs

Chromatic Aberration

The chromatic aberration (which affects only refractors and not reflectors) is due to a different refraction index through a optical lens. Each wavelength of white light is refracted differently. Hence, the entire spectrum is not focused at the same focal length. The larger the diameter of a refractor lens, the greater the focal ratio must be, to ensure a effective correction of chromatism

Exit Pupil

Exit pupil (in mm) = instrument aperture in mm / magnification; or: exit pupil (in mm) = eyepiece focal length in mm / instrument f ratio. The exit pupil is the circular beam of light which is coming out of the eyepiece; the question of the exit pupil is related to not wasting the beam of light coming out of the instrument. The exit pupil must not exceed the observer's eye pupil's diameter so no part of light is wasted (it is also said that a 3mm exit pupil is enough for daytime use, a 5mm is suitable for daytime and twilight, a 7mm is desirable for maximum nighttime or astronomy use). An observer's pupil is varying according to the pupil's accomodation to night and to the observer's age. A young person's pupil may be 7 mm wide when dark adapted, middle-aged adult pupil are 6, elder people 5 mm (note that these figures vary according to sources). The concept of eye pupil leads to the definition of a minimum magnification for the instrument. This one is practical and differs from the theoretical minimum magnification of the instrument (see below). The eye pupil-related minimum magnification is the one which does not yield an exit pupil larger than the observer's pupil diameter. It's computed like: intrument aperture in mm / person's pupil in mm. Going below the eye-pupil related minimum magnification may be a trouble with telescopes with a secondary mirror as it may eventually lead to forbid any beam coming out of the instrument due to interactions between eye, primary and secondary mirrors. As far as a refractor is concerned, this does not yield any issue except that a part of the light beam is lost. It seems that this concept of eye-pupil-related minimum magnification is mainly used for the observation of deep-sky objects. Such objects really need that the maximum of light be gathered and transmitted to the observer's eye. The issue seem less important for planetary observation, as the domain requires important magnifications, mostly yielding, in most of the cases, small exit pupils

Eye Relief

Eye relief is the distance from the eye to the eypiece lens when the image is in focus and when one sees the whole field. The concept is useful for glass wearers. Modern design eyepieces are providing long eye relief regardless of the eyepiece's focal. In classical eyepieces, eye relief is proportional to the eyepiece's focal length. The shorter the focal length, the shorter the eye relief (eye will have to be nearer to the eyepiece to see the entire field). Note that people wearing glasses may take them off when observing (it's the focusing of the scope which will compensate for their optical defect), except people suffering of astigmatism, who have to keep them, mainly for small magnifications, as they should be able to take them off for high magnifications

Eyepiece Barrel Sizes

Usual eyepiece barrels sizes (that is the diameter used to insert the eyepiece into the eyepiece holder) are: .965" or .96" (24.5 mm; these are now low-end barrels as they originally were a Japanese standard), 1.25" (31.75 mm; most instruments accomodate this size; a U.S. standard), 2" (50.8 mm; found with higher end instruments; they are yielding an increased field of view and brighter images; a U.S. standard too)

Focal Length

The focal length of an instrument is the distance from the primary objective to the focal point, this point where the object observed is in focus. The focal length is expressed in mm. It's usually given by the manufacturer. Example: "this 100 mm refractor has a 800 mm focal length". Optically speaking, the larger the focal length, the larger the size of the image at the focal point

Focal Ratio (or f/D)

f/D = focal length in mm / aperture in mm. Example: a 6" instrument, having a focal length of 1300 mm has a focal ratio of 8.7 (f/8.7). The focal ratio of an instrument is the ratio of the focal length of the instrument in mm to the aperture in mm. The reverse formula is useful to find the focal length when only aperture and the f ratio are known (focal length = aperture x f ratio). Instruments with a f/D of /3.5 to f/6 are said fast focal ratio; instruments with a f/D of f/7 to f/11 are said medium; instruments with a f/D of f/12 and beyond are said slow

Limiting Magnitude

Limiting magnitude = 7.5 + 5 LOG aperture in cm. The limiting magnitude is the faintest star which can be seen with excellent atmosphere conditions. The concept of limiting magnitude applies to telescopes like to naked-eye observation and relates to visual observation only. The photographic limit magnitude is about 2 or more magnitudes fainter. Example: a 8" Schmidt-Cassegrain has a 14 limiting magnitude (7.5 + 5 LOG 20.32 = 7.5 + 5 x 1.3 = 14) that is it can be used visually to see stars or objects down to this magnitude. Used for astrophotography, such an instrument will allow to objects of magnitude 12 or brighter. The limiting magnitude is usually given by the instrument's manufacturer. for more about limiting magnitude see the tutorial "Limiting Magnitude"

Magnification

Magnification -or power (in times) = instrument focal length in mm / eyepiece focal length in mm. Example: a 100 mm refractor with a focal length of 900 mm used with a 12 mm eyepiece yields a power of 75x. A reverse formula is useful to find what eyepiece is to be used to have such power (eyepiece focal length to use = instrument focal length in mm / power); example: to get a 180x power with this same 100/900 refractor, one has to use a 5 mm eyepiece. Generally speaking, doubling the power using a different eyepiece just gives ¼ of the previous image brightness and reduces by ½ the sharpness

Maximum Magnification Possible for an Instrument

Instrument aperture in inches x 60, or: instrument aperture in mm x 2 or 2.4. The lowest power possible is 3 to 4 times the aperture in inches for night observing and 8 to 10 times for terrestrial viewing. Example: a 8" reflector has a maximum magnification ability of 480x and a minimum 24-32x for night observing (64-80x for daytime). One has to be aware that observing conditions, like atmospheric turbulence or types of objects observed, are always limitating the practical magnification to use, even with high-end, high-aperture instruments. This limit might be around between 200-400x due to atmospheric turbulence, and observing deep-sky objects may require mean or low magnifications. Thus, multiplying the aperture in inches by 6-25 times, instead of the values quoted above, may be a way to correctly assess the maximal magnification of a telescope. A other formula when the observation of the Moon or planets are concerned may be: power (in times) = aperture in centimeters x 15. Note too that planetary observation, on another hand, always needs high power (about 100x seems a minimum). An additional concept used in some countries like in Europe is the "resolving magnification". It's the power with which the resolving power of the scope is reached (no additional details are seen beyond this power). The formula is: radius of the instrument in mm x 3, or resolving magnification = diameter of the instrument in mm (useful too: the focal of the eyepiece to reach this magnification is equal to the focal ratio (f/D) of the instrument)

Objects Apparent Diameter at an Instrument Focal Point

It equals the object's apparent diameter in radians x instrument focal length in mm. Result is in mm. 1 radian is 206,264.8", 1" is 0.000004848 radian; 1' is 0.00029 radian. This formula is mostly useful for astrophotography. The radian is the other unit for measuring angles: there are 2 x pi (that is 6.2831852) radians in 360 degrees. Example: assuming observing a 35" Jupiter with a 4"/1,000 mm instrument, Jupiter's image at the focal point is 0.17 mm ((35 x 0.000004848) x 1,000). Moon (30' of apparent diameter) is 8.7 mm ((30 x 0.00029) x 1,000) at the focal point with the same telescope

Resolving Power

4.56 / aperture in inches, result in arcseconds. Example: a 4" refractor has a resolving power of 1.14". The resolving power is the smallest distance the primary optics of a telescope may separate. Originally the resolving aperture of an instrument is its ability to separe two close binaries. Resolving power depends on the aperture. Once the object observed further magnified by the eyepiece, the resolving power is multiplied by the magnification (example: if the lens or the mirror of a telescope has a resolving power of 1" and the telescope is used at a power of 120x, the smallest detail seeable is 2'-wide (1" x 120 = 120" = 2')). The human eye has an average resolving power of 1' only. Resolving power is a concept also used about binaries. In that case, once the telescope's resolving power determined, a formula may be used to determine what the minimal magnification is to observe a given binary: magnification = 240/S, S the separation of binary's components in arcsecond. The magnification, in terms of resolving power, generally, must not exceed twice the diameter of the telescope in millimeters (e.g. for a 114-mm Newtonian, the maximum magnification is about 230x)

True Field

true field = apparent field / magnification. The true field, or real field, is the area of sky which is seen through an eyepiece when attached to a telescope. The apparent field is the circle of light the eye sees through the eyepiece when held separately. The apparent field is given by manufacturers. Most of modern eyepiece have an apparent field of about 50°; an apparent field of 25 to 30° is considered narrow, of 80° or more is considered extra-wide. Example: a 16 mm eyepiece with a 52° apparent field, used on a 6"/1500 refractor, thus giving a 94x power, yields a 0.55 ° field (about 33')

Useful Magnification

The concept of 'useful magnification' is the magnification yielding the best comprise between contrast, luminosity and resolution. useful magnification = D (telescope's diameter in mm) / 1 to 1.4