10 Seconds of Arc

" 10 Arc Seconds translates to ±1/16" at 100'.... "....or 1/8"

A circle can be divided into 360 degrees. Each degree is divided into 60 arc minutes, and each arc minute is divided into 60 arc seconds.

http://www.1728.org/angsize.htm

Angular size in degrees = (size * 57.29) / distance

...let us take the tangent of 1 degree which is .017455 which means that when an object's size is .017455 times its distance, it has an angular size of 1 degree. OR to put it another way: When an object's distance is 57.29 times its size, it has an angular size of 1 degree.

Multiplying 57.29 by 60 minutes per degree we get 3,437.4 which means that an object at a distance of 3,437.4 times its size will have an angular size of 1 minute.
Multiplying 57.29 * 3,600 seconds per degree we get 206,244 meaning an object at a distance of 206,244 times its size displays an angular size of 1 second.
We can generate another simple formula:

so only 100 feet away the scale would be correct?

That sound a LOT better! he he he.

Thanks Again,

You are a huge help on this place!

Being it's a scientific magazine, I would suspect the distance might be 200 meters.

My calculations for 200 meters comes out as 0.3817 inches.

You need to have very long arms...

That very well could be!

Joe

Using a handy free screen pixel ruler: Jruler

I make that 883/(2*1164) of an inch = 0.3793 inches = .009634 0f a meter

And that divided by 200 and "sin^-1" = .0027624° = 9.936 seconds

So if you now correct for the "measurement line" being off horizontal I reckon you're right.

Looks good to me. Another way to do it would be:

A pretty good first order approximation:

tan (10 arcsec) ≈ 0.375 ÷ x

x ≈ 0.375 ÷ tan (10 arcsec)

x ≈ 7,734.930228205942 inches

x ≈ 644.577519017161 feet

The more accurate way would be to divide the object size in half, and divide the angle in half and do it again.

tan (5 arcsec) = 0.1875 ÷ x

x = 0.1875 ÷ tan (5 arcsec)

x = 7,734.930232751070 inches

x = 644.577519395922 feet

There is a difference of 4.545 microinches between the two methods.

Or x = 196.467227911877 meters

Off topic of course. I am amazed at the accuracy that astronomers achieve when measuring when all objects are moving in 3 dimensions.

Who's going to tell them they're wrong?

"Who's going to tell them they're wrong?"

The younger (next generation?) astronomers who have more precise telescopes and electronics.

What about the astronauts who don't land/dock where they expect to?

I thought, since we're talking about arc-sec and astronomy, and, when I think about arc seconds I think about circles and arc-lengths, I'd throw my 2-cents in using a slightly different approach which, fortunately for me, agrees with you folk who have provided the sine/cosine/tangent approach but requires me to use tables (yes, I'm that old), or scientific calculators. So, here goes: converting 10 sec or arc into degrees is 10/3600 of a degree, converting to radians is 10/3600 * Pi/180 = 4.85E-5 radians. Using r*θ=.375 we get r=7,732 inches or 644 ft. I did round off but it's close enough for most engineers, although the mathematician's in the group are probably more finicky.

I'm not a mathematician but I did stay at a Holiday Inn Express last night.

Actually, a mathematician competent in physics (or any applied science) should insist on rounding off. No calculated value can be more precise than the least precise original information. The precision of the "10 arc-seconds" is not at all clear, so I think the thee digits of precision you presented is appropriate!

But what's the fun in that?

Microsoft Calculator gives us 30 orders of useless precision after the decimal point.

This is very true as I've had teachers who would mark you down if you were excessive in your precision for that very reason... I do wonder at times how astronomers seem to offer calculations on objects so distant and are so sure of their conclusions...

Quit thinking logically and follow the science.

Yours is not to question, but to obey.

Sorry! I forgot "resistance is futile."... follow the science... no matter where it doesn't lead you?

<sigh> . . . yeah.