Focal ratio
N = FD
N = FD
The smaller N, the “faster” the instrument for imaging and the wider its field.
Typical values: Newtonian f/4 to f/6; Schmidt-Cassegrain f/10; Maksutov f/12 to f/15.
Useful formulas
The amateur astronomer’s formulas on a single card. Each one works both ways: fill in every box but one, and the card works out the missing one, whichever it is.
Example: which eyepiece gives 100× on a telescope of 1000 mm focal length? In the magnification card, enter F = 1000 and M = 100, empty f: the card shows 10 mm. When every box is filled, it checks that the values agree. Decimals may be typed with a point or a comma.
The printed version keeps the formulas, their meaning and typical values, without the input boxes: two pages to keep by the telescope, plus a third for spherical trigonometry.
The telescope
Light grasp
L = (Dpœil)2
L = (Dpeye)2
How many times more light than the naked eye: the ratio of the areas of the aperture and the eye pupil.
Typical values: 114 mm: 265 times; 200 mm: 816 times. Eye pupil 7 mm when young, 5 to 6 mm past 50.
Limiting magnitude
mlim ≈ 2 + 5 log10 D
mlim ≈ 2 + 5 log10 D
The faintest stars visible in the eyepiece under a dark sky.
Typical values: 114 mm: 12.3; 200 mm: 13.5. A common approximation; others use 2.7 + 5 log D. Sky, experience and magnification shift the result by a magnitude.
Central obstruction
o = dsD
o = dsD
The fraction of the diameter hidden by the secondary mirror; the light lost is o² (25 % → 6 %).
Typical values: Visual Newtonian 15 to 25 %; Schmidt-Cassegrain 30 to 35 %. Above 25 %, contrast drops noticeably.
Eyepieces and magnification
Magnification
G = Ff
M = Ff
How many times bigger things look: the ratio of the two focal lengths, telescope over eyepiece.
Typical values: On 1000 mm: a 25 mm eyepiece gives 40×, a 10 mm 100×.
Magnification with a Barlow
G = B × Ff
M = B × Ff
A 2× Barlow doubles the focal length, hence the magnification; a focal reducer has a factor below 1.
Typical values: Barlow 2× or 2.5×; reducer 0.63×.
Exit pupil
p = DG
p = DM
The width of the beam leaving the eyepiece and entering the eye; it sets how bright the image looks.
Typical values: 5 to 7 mm: wide fields; 1 to 2 mm: galaxies, planets; below 0.5 mm the image turns soft.
Exit pupil from the eyepiece
p = fN
p = fN
The same pupil, from the eyepiece focal length and the focal ratio.
Typical values: At f/5: a 25 mm eyepiece gives 5 mm, a 10 mm 2 mm.
Minimum useful magnification
Gmin = Dpœil
Mmin = Dpeye
Below it, the exit pupil is wider than the eye’s and part of the light is lost.
Typical values: Eye pupil 7 mm (young) to 5 mm (past 50); 200 mm: 29× to 40×.
Maximum useful magnification
Gmax ≈ 2 × D
Mmax ≈ 2 × D
Beyond it the image gets bigger but shows no new detail (exit pupil of 0.5 mm).
Typical values: 2× per mm of aperture, 50× per inch, for perfect optics. The real quality of the optics and the collimation often limits it to 1× per mm (M = D), and seeing to 200 or 250×.
True field
Cr = CaG
TF = AFM
The patch of sky seen in the eyepiece: its apparent field divided by the magnification.
Typical values: Plössl 50 to 52°, wide field 68 to 82°, ultra-wide 100°. The full Moon spans 0.5°.
True field from the field stop
Cr = 57,3 × dcF
TF = 57.3 × dfsF
The most accurate method: the eyepiece field stop seen from the objective.
Typical values: 25 mm Plössl: stop of about 27 mm; at most 27 mm in a 1.25″ barrel, 46 mm in a 2″ barrel.
Resolving power
Resolving power (Dawes)
ρ = 116D
ρ = 116D
The closest two equal stars can be and still be seen apart.
Typical values: 114 mm: 1.0″; 200 mm: 0.58″. Seeing often limits it to 1 or 2″.
Resolving power (Rayleigh)
ρ = 138D
ρ = 138D
The theoretical diffraction criterion, 1.22 λ/D, in green light (550 nm); a little stricter than Dawes.
Typical values: 114 mm: 1.2″; 200 mm: 0.69″.
Astrophotography
Image scale at focus
E = 206 265F
E = 206 265F
The angle of sky covered by one millimetre at the focal plane.
Typical values: 1000 mm: 206″/mm; 2000 mm: 103″/mm.
Sampling
e = 206,265 × pxF
e = 206.265 × pxF
The angle of sky covered by one pixel.
Typical values: Deep sky: 1 to 2″/px depending on seeing; planets: 0.1 to 0.3″/px.
Field of a sensor
θ = 57,3 × hF
θ = 57.3 × hF
The angle covered by one side of the sensor (small-angle approximation, within 1 % below 20°).
Typical values: APS-C 23.5 × 15.6 mm; full frame 36 × 24 mm; small planetary camera 11 × 6 mm.
Image size at focus
d = F × θrad
d = F × θrad
The diameter of the Moon or a planet on the sensor.
Typical values: Moon (31′) at 1000 mm: 9 mm; Jupiter (45″): 0.22 mm.
500 rule
t = 500k × F
t = 500k × F
The longest exposure without star trails on a fixed tripod. Simple, but too optimistic with today’s sensors.
Typical values: 24 mm on full frame: 21 s; 18 mm on APS-C (k = 1.5): 19 s.
NPF rule
t = 35 N + 30 pxF
t = 35 N + 30 pxF
The longest exposure without trails, taking the aperture and pixel size into account (simplified form by Frédéric Michaud, SAF).
Typical values: 14 mm at f/2.8, 4.3 µm pixels: 16 s.
Critical focus zone
Z = 4,88 λ N2
Z = 4.88 λ N2
The depth within which the image stays sharp: the smaller N, the harder focusing gets.
Typical values: λ = 0.55 µm: f/4, 43 µm; f/10, 268 µm.
The sky in numbers
Angular size
θrad = sd
θrad = sd
The angle an object spans, from its actual size and its distance.
Typical values: Moon: 3474 km at 384,400 km, 31′; Sun: 32′.
Distance modulus
m − M = 5 log10 (d10 pc)
m − M = 5 log10 (d10 pc)
Links the apparent magnitude m, the absolute magnitude M (brightness seen from 10 parsecs) and the distance.
Typical values: Vega: M = 0.6 at 7.68 pc, m = 0.03. Sun: M = 4.83.
Light travel time
t = dc
t = dc
How long light takes to reach us: we see the object as it was.
Typical values: Moon: 1.3 s; Sun: 8 min 19 s; Proxima Centauri: 4.2 years.
Telescope making
Sagitta of a mirror
s = D216 F
s = D216 F
The depth of the hollow at the centre of the mirror, the target of rough grinding.
Typical values: 200 mm at 1000 mm: 2.5 mm; 150 mm at 1200 mm: 1.2 mm.
Spherical trigonometry
On the celestial sphere, three great circles that cross one another outline a spherical triangle. Its sides are arcs, measured as angles seen from the centre of the sphere; its angles are those the great circles make with each other. Sides and angles run from 0 to 180°. The cosine rule for sides was already known in other forms, to astronomers of the Islamic world and later to Regiomontanus; François Viète is credited with publishing its dual form, for angles, in 1593, and both often bear his name. Angles are entered in degrees, arcminutes or arcseconds. When a value comes out of an arcsine or arccosine, two answers may fit: the card gives the main one and mentions the other.
Cosine rule for sides
cos a = cos b · cos c + sin b · sin c · cos A
cos a = cos b · cos c + sin b · sin c · cos A
Gives a side from the other two and the angle between them: the workhorse of positional astronomy.
Typical values: b = 60°, c = 70°, A = 80°: a ≈ 71.8°. When b or c is the unknown, two values may fit.
Cosine rule for angles
cos A = −cos B · cos C + sin B · sin C · cos a
cos A = −cos B · cos C + sin B · sin C · cos a
The counterpart of the previous one: an angle from the other two and the side joining them.
Typical values: B = 60°, C = 70°, a = 50°: A ≈ 69.4°. In a spherical triangle the angles always add up to more than 180°.
Sine rule
sin asin A = sin bsin B
sin asin A = sin bsin B
The sine of a side divided by the sine of the opposite angle gives the same number for all three pairs.
Typical values: A = 60°, b = 50°, B = 70°: a ≈ 44.9°. A sine cannot tell x from 180° − x: the card mentions the second solution when it is still possible, and the figure tells which to keep.
Altitude of a star
sin Hc = sin φ · sin δ + cos φ · cos δ · cos AHL
sin Hc = sin φ · sin δ + cos φ · cos δ · cos LHA
The cosine rule in the position triangle (pole, zenith, star): the computed altitude Hc from the latitude φ, the declination δ and the local hour angle LHA.
Typical values: Paris (φ = 48.85°), Vega (δ = 38.78°), LHA = 30°: Hc ≈ 66.3°. For the LHA, x and 360° − x give the same altitude.
Azimuth of a star
cos Z = sin δ − sin φ · sin Hccos φ · cos Hc
cos Z = sin δ − sin φ · sin Hccos φ · cos Hc
From the same triangle, the angle Z counted from north, 0 to 180°, towards the east or the west. If LHA < 180°, the star is in the west and the azimuth is 360° − Z; otherwise it is Z. The card solves Z between 0 and 180°.
Typical values: Same example: Z ≈ 103.8° and LHA = 30°, so the azimuth is ≈ 256.2°, in the west.