Last night, I posted a method for converting between linear FOV and angular FOV. I have since simplified the procedure greatly. Here is the bottom line:

Let L represent linear FOV, that is, number of feet per 1000 yards (L=305 for my 9x63 binoculars).

Let A represent angular FOV in degrees (A=5.8 for my 9x63 binoculars).

Then:

A = 0.0191 × L

and

L = 52.4 × A

(I hope the "×" symbol works on all browsers--it is supposed to be the "x", the multiplication symbol).



Ok--for those interested, here is how to "derive" those formulas (which are approximations, but good ones).

Recall that, assuming you have an arctan() function that takes a ratio and returns an angle in radians, the following formula gives you the angular FOV exactly:

A = 2*arctan((L/2)/3000)*180/pi

where pi is 3.1415926535....

The Taylor Series for arctan is:

arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + x^9/9 - ....

Note that because x is very small (typically less than 0.1), x^3/3 (and the higher-order terms) are nearly zero. Thus, arctan(x) is approximately equal to x radians in this situation.

So, the formula becomes:

A = 2*((L/2)/3000)*180/3.1415926535

which simplifies to approximately 0.0191*L.

Now, to get the formula for L given A, just take the reciprocal of 0.0191 to get approximately 52.4.

So, we now see that Freshman Calculus was useful after all!

The night wasn't great, but it was good (especially when I quit, around 1am--by then, the Milky Way was visible in Cygnus in binoculars, which was near the zenith at the time).   I sketched the Sadr (Gamma Cygni) area, and the upper part of Lyra.  The former isn't perfect--I realized after the fact that a few stars were misplaced (I am getting better at this, but still mis-estimate where to plot the stars--part of the problem is it's not completely obvious which way, precisely, is "up" in binoculars, surprisingly).   

 

While I was outside, I decided to measure my 5.8 degree FOV binoculars.  I went to bright Deneb, and the "northwest" optical double Omicron Cygni.  I could fit them in the field of view, but if I move the binoculars just slightly, one or the other would exit the field of view.  This showed the FOV was just over the angular distance (5.1 degrees) between Sadr and Omicron Cygni.  5.8 is not "just" over, but 5.3 degrees appears close to the actual field of view.

 

Actually, the advertised field of view is 305 feet at 1000 yards.  Let's convert that to an angular field of view.

 

First, 1000 yards is 3000 feet.  Then, imagine looking at a target 305 feet high 3000 feet away, that just fits the field of view.  3000 feet is the distance to the center of the target, so 152.5 feet of the target is above the line of sight, and 152.5 feet is below.  So, we have a long right triangle with legs 3000 feet and 170.5 feet.  The acute angle (at the binoculars) is then arctan(152.5/3000).  But the target is both above and below the center line, so we double it.  Finally, my software works in radians, so I convert to degrees by multiplying by 180/pi

 

FOV = 2 * arctan(152.5/3000) * 180 / 3.1415926535

 

= about  5.82 degrees.

 

Yep--actual field of view is a little off!

 

 

again, nontelescopic.  They are on my photo page with the other photos.  Added are Lyra and Hercules.  I can use GIMP on the Hercules photo to bring out the globular cluster M13, but only by ruining the rest of the image.  Note in Lyra that Epsilon Lyrae (the double double) is split (once).

 

In other news, I liked the Van Gogh so much I made it my desktop background at home.  (At work, I use two computers--one has The Milky Way, and the other, the Eagle Nebula Pillars of Creation).  Unfortunately, I cannot remember where I got the pictures.  But--here are a couple similar ones:

 

 

 

 

This is a photo I took of the big dipper.  More photos are here.

 

Here is a practice drawing from Starry Night.

 

Here is Van Gogh's impression of the Big Dipper.  One of the first asterisms many people learn to recognize is the seven-star Big Dipper, consisting of the seven brightest stars of Ursa Major, the great bear.  These stars are, starting with the end of the dipper's handle (or end of the bear's tail), Alkaid, Mizar (with its fainter optical companion Alcor), Alioth, and then clockwise around the bowl starting with where the handle joins it, Megrez, Phecda, Merak, and Dubhe.  In addition to the Big Dipper asterism are less-familiar portions of the bear.  Three medium-bright stars form the bear's nose, and a pair of triangles of stars and a third very skewed triangle (nine total) form three visible paws.  The bowl of the dipper looks like a saddle on the bear's back.  Here is a map of Ursa Major.

 

 

 

The bear seems to be chased by Canes Venatici (hunting dogs--directly below Alkaid if viewing the dipper so the bowl faces up--see the bright star bottom center of the photo at the beginning of the post).  It seems to be walking on the little lion, Leo Minor.  It is sniffing out the hard-to-see lynx.  The dragon (Draco) winds between the big and little dippers.  Bootes the herdsman seems to be chasing the bear away.

 

 

Ursa Major is home to many galaxies, it being well outside the galactic plane (visible as the Milky Way) so one has a clear view out of our galaxy. M81 (Bode's Galaxy) is one of the brighter galaxies in the sky, just visible to the naked eye under very good conditions.  It is next to M82, the Cigar Galaxy, and in fact, the two recently had a close encounter that disrupted both galaxies.  In addition, M101 (Pinwheel Galaxy) is near Alkaid (so is the Whirlpool Galaxy, M51, though technically, it is in Canes Venatici).  Other galaxies in or near Ursa Major include M97, M108, M109, and nearby M63, M94, and M106 in Canes Venatici.

 

 

Ursa Major may be one of the oldest constellations.  American Indians, Australian Aborigines, and ancient Romans all saw it as a bear,  and the three cultures had not interacted for thousands of years.  It may be 50,000 years old even (to a time when humans worshiped bears).  See AAVSO's notes on the subject. 

 

In any case, the Greek myth was that Zeus fell in love with Callisto.  Zeus's jealous wife turned Callisto into a bear (Ursa Major) out of revenge.  Later, Arcas, Callisto's son by Zeus, nearly killed her in a hunt (unknowingly) but Zeus intervened by turning Arcas into a bear (Ursa Minor) and placed them both in the sky.

 

See Wikipedia's article for more information.

 

 

I have added a new photo of the moon to the astrophotography page.  I took this with the camera on automatic shutter/F settings, focus to infinity, and telephoto zoomed to full.  I have no photo or drawing, but in the telescope, the moon was particularly interesting.  Toward the top, right on the terminator, were three craters.  Because of the way the sunlight was hitting them, the were particularly beautiful.  I tried to figure out which craters they were by checking with Starry Night, but I can't seem to find them!  This is partly because Starry Night used a full moon to model the moon, and just drew the terminator on.  However, features change with the angle of the sun from what they look like during full moon.  This makes it hard to match features from a full moon photo to those of a crescent moon photo.

 

In addition, I sketched Jupiter again (not on this site, it isn't much different from the sketch on this page, and shows even less detail.  But, it includes the four Gallilean moons and two stars.  From East to West, we have Callisto, Ganymeade, Jupiter, Io, TYC5575-473-1, Europa, with HIP 70714 south of TYC5575-473-1.  So it appeared as if Jupiter was with six moons.  The moons and TYC5575-473-1 all had a strong yellow coloration, and HIP 70714 was white with a yellowish tint.  Jupiter was yellowish with tow dark reddish-brown bands.

 

I saw M81 in my telescope but didn't sketch it before losing it.  (The seeing and transparency were ok, not great, despite predictions).

 

I took other astrophotos, but the moon photo I posted was the only one that really turned out well.