Astronomy Formulas

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Introduction

Here is a set of formulas and numbers that I have always wished was in one place. Now they are. Enjoy.

f ratio (or focal ratio) = primary mirror focal length / primary mirror diameter

This number tells you how "fast" or "slow" the telescope is.
Example: DS-3 is as follows: 60 inch / 12 inch = f5. (Or, 1520mm / 305mm = f5)
Example: DS-4 is as follows: 72 inch / 16 inch = f4.5. (Or, 1828mm / 406mm = f4.5)

Magnification = focal length of primary / focal length of the eyepiece

This number represents how many times the view that you are looking at has been magnified.
Example: 22 mm SuperWide Lanthanum on DS-3 would have the following magnification: 1520 / 22 = magnification of 69
Example: 22 mm SuperWide Lanthanum on DS-4 would have the following magnification: 1828 / 22 = magnification of 83

True field of view = Eyepiece apparent field of view / Magnification

This represents how much of the sky you can see at a time. See Handy Sky Distance Measures below.
Example: 22 mm 65° apparent field SuperWide Lanthanum on DS-3 would have a true field of view of: 65 / 69 = 0.94°. This eyepiece is basically looking at an area of the sky the size of my little finger nail.
Example: 22 mm 65° apparent field SuperWide Lanthanum on DS-4 would have a true field of view of: 65 / 83 = 0.78°. This eyepiece is still looking at an area of the sky the size of my little finger nail.

Eyepiece exit pupil diameter = eyepiece focal length / telescope f ratio

This represents the diameter of the light cone that enters the eye. A young person has about a 7mm pupil and an older person will have a pupil diameter of 6mm or smaller. Since having an eyepiece exit pupil larger than the eye pupil will waste light, it is generally accepted that an exit pupil larger than the eye pupil is not a good idea. A large exit pupil will also facilitate the user starting to see a black center to the field - the secondary mirror is starting to be come visible.
Example: DS-3, using a 22mm eyepiece has an eyepiece exit pupil diameter as follows: 22/5 = 4.4mm.
Example: DS-4, using a 22mm eyepiece has an eyepiece exit pupil diameter as follows: 22/4.5 = 4.9mm.

Lowest power useful eyepiece = focal ratio * user's eye pupil diameter

This represents the widest field of view that you can get, assuming you don't want to loose light hitting your cornea. In theory, pupil sizes are 7mm for younger people, 6mm or smaller for older folks.
In practice, if you slightly bend the rules, the eyepieces will work fine. Things just won't be a bright for older folks as younger folks. If you break the rules by too much, you will start to see a black center to the field.
Example: DS-3 is as follows: 5.0 * 6 = 30mm. So, for my eyes, I shouldn't go above a 30mm eyepiece.

Most detail eyepiece =~ 2mm exit pupil =~ focal ratio * 2

This represents the eyepiece that should give you the best detail when looking at objects. If seeing is great, more power will work. If seeing isn't so great, less power will be better. But, this is where you should probably buy your first expensive eyepiece.
Example: DS-3 is as follows: 5.0 * 2 = 10mm. So, I will probably get a lot of use out of a 10mm. I have an 8mm and a 13mm, and use them all of the time. If seeing is clear, I use the 8mm. Otherwise, I use the 13mm.
Example: DS-4 is as follows: 4.5 * 2 = 9mm. Once again I will probably use the 8 and 13mm.

Dawes limit =~ 4.56 arc seconds / primary mirror diameter

This represents the theoretical resolving ability of a telescope. It was derived trying to resolve double stars of similar magnitude by a guy named Dawes.
Example: DS-3 has a Dawes limit of 4.56 / 12 = 0.38 arc seconds. In practice, DS-3 has resolved double stars down to about 1.1 arc seconds. This was the double-double in Lira. I need to try resolving tighter stars, but haven't yet.
Example: DS-4 has a Dawes limit of 4.56 / 16 = 0.29 arc seconds.

Telescope Magnification limit =~ 40 to 50 * primary mirror diameter

This represents the theoretical useful power of a telescope. This represents very, very good seeing.
This limit also has a cutoff at about 400X to 500X. This also represents very, very good seeing. This is due to turbulence in the atmosphere.
Surprisingly, this is true of very large telescopes also.
Example: DS-3 has a theoretical limit of about 600X. Then, you clamp it down to about 400X to 500X as stated above. However, the best views that I have ever had of Saturn was about 380X, with very good skies, and a mirror that had stabilized temperature. Anything more would have severely pushed the optics.

Handy Sky Distance Measures

Extend your hand out as far as your arm will go. The number of degrees across the sky can be approximated as follows:
1° - Width of your little finger.
5° - Width of the first three fingers of your hand, held together.
10° - Width of your fist.
15° - Width of the outside edges of your two outside fingers, after opening up your hand as much as possible.
25° - Distance between the end of your thumb and small finger, after opening up your hand as much as possible.

25° - Width of the big dipper.
10° - Width of the cup of the big dipper.
5° - Height of the cup of the big dipper.
28° - Distance from the end of the big dipper to Polaris, the north star.
1/2° - Width of the full moon.

Magnification of different binoculars and telescopes

Viewing tools	Magnification of	Light gathering	Max practical	Resolution limit	Resolution limit	Faintest star (visual)
	an optimal eyepiece	vs normal eye	magnification	Theoretical (arc sec)	Practical (arc sec)	(magnitude)
Eyes	1x	1x				5.5
8X50 binoculars	8x	51x	100			10.2
10X50 binoculars	10x	51x	100			10.2
4" telescope f10 20mm eyepc	50x	211x	200	1.1	2.0	11.8
6" telescope f8 16mm eyepc	76x	474x	300	0.8	1.4	12.7
8" telescope f6 12mm eyepc	101x	843x	400	0.6	1.0	13.3
10" telescope f5 10mm eyepc	127x	1317x	500	0.5	0.8	13.8
12" telescope f5 10mm eyepc	152x	1899x	500	0.38	0.7	14.2
15" telescope f5 10mm eyepc	190x	2962x	500	0.3	0.5	14.7
16" telescope f4.5 9mm eyepiece	203x			0.29
18" telescope f4.5 9mm eyepc	228x	4266x	500	0.3	0.5	15.1
20" telescope f5 10mm eyepc	254x	5267x	500	0.2	0.5	15.3
200" 5m hale telescope f3.3	2357x ⁽¹⁾	510204x	500	0.02⁽²⁾	0.5⁽²⁾	20.3⁽⁴⁾
10m Keck telescope	n/a	2040816x	500	?⁽²⁾	?⁽²⁾	21.8⁽⁴⁾
2.4m Hubble f10	n/a	117511x	4,700⁽³⁾	0.05⁽³⁾	0.05⁽³⁾	?⁽⁴⁾

(1) The hale telescope is not set up for visual work. These numbers are my best guess for comparison.
(2) Notice what the atmosphere does to the resolving ability of large telescopes. However, these calculations are NOT correct for telescopes such as the Keck. The Keck uses adaptive optics that partially compensate for atmospheric turbulence.
(3) The Hubble telescope is not degraded at all by the atmosphere. This is the reason that it is the standard that all other telescopes are compared against.
(4) By using a CCD camera, it is possible to gather light over a long period of time. This allows fainter stars to be "seen" in pictures than can be seen by the human eye. Thus, the magnitudes are significantly higher for CCD astronomy than visual astronomy.