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Arc Minutes to Estimate Distance... ?
Zeromus
Registered User regular
Registered User regular
So, I realize that there's a fine line in this forum about asking for help and asking for answers, so hopefully this will fall into the former category and I can get some pointers in regards to this ridiculous physics class that I have to take to fulfill a general requirement at my school. I am seriously in over my head here and have already spent a good deal of time flipping through my textbook to try and get pertinent info, but no dice so far...
Basically, I need to answer the following question:
To be completely honest, I'm not really even sure where to start. In terms of angles, I know that 1 arc minute is 1/60th of a degree. The only equation I can find that really seems like it pertains is:
diameter = distance x (angular diameter / 57.3 degrees)
... But I have two unknown variables which means I can't proceed from here.
Clearly I'm missing something, but I'm totally lost. Can anyone help me understand how to do this?
Basically, I need to answer the following question:
Tycho Brahe's observations of the stars and planets were accurate to about 1 arc minute (1'). To what distance does this angle correspond at the distance of (a) the Moon; (b) the Sun; and (c) Saturn (at closest approach)?
To be completely honest, I'm not really even sure where to start. In terms of angles, I know that 1 arc minute is 1/60th of a degree. The only equation I can find that really seems like it pertains is:
diameter = distance x (angular diameter / 57.3 degrees)
... But I have two unknown variables which means I can't proceed from here.
Clearly I'm missing something, but I'm totally lost. Can anyone help me understand how to do this?
Zeromus on
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Posts
So it's just an arc length problem.
Example:
a 1 degree arc on a circle that happens to be 360 feet in circumference is 1 foot in length.
So you'll basically need to be able to look up the distances that are involved. Since the arc is so short there's probably not much difference the orbits in question (which are pretty much ellipses) and circles but I'd make sure of that if ellipses are mentioned in proximity to that problem.
Clear?
Yes, it's looking for the length of an arc. But, it's actually simpler than that, because your dealing with astronomy and astronomers are lazy. If you're dealing with really small angles and/or really large circles, you don't care about the curve of the arc, you can treat it as a straight line. (Do you notice the curvature of the earth when you're looking at a one foot square?). Luckily, you are dealing with really big circles and really tiny angles.
So, I have a circle, that has some big radius. And I care about a tiny angle. And that tiny angle, projected out to that big circle, will be a straight line. Lemmie illustrate it for you:
(me)_______________________________________________| <- length i care about
really big distance
You know what that also looks like? A right triangle. If you decide that (really big distance) is the adjacent side, then you wind up using tangent:
tangent(tiny angle)= (length i care about)/(really big distance)
If you decide that (really big distance) is the hypotonuse, then you use sine:
sine(tiny angle)=(length i care about)/(really big distance)
Both work.
"But wait! Sine and Tangent are different, how could it be both?", you say? Well yes, but remember, astronomers are lazy, and this time the math is on their side. You see, tangent(really tiny angle) ~ sine(really tiny angle) ~ (really tiny angle). Try it, just remember to have your calculator in radians, not degrees. ( 1 radian is 57.2958 degrees).
After all that, I'm left with:
(tiny angle in degrees)/57.3 = (length i care about) / (really big distance)
or, rearranging it:
(length i care about)= (really big distance) * (tiny angle in degrees)/57.3
Look familiar?
You have the angle, just look up the distances.
You are being asked to plug in three known distances (Earth to Moon, Earth to Sun, and Earth to Saturn). From those you get your 1' arc lengths.
You're halfway there.
You're not looking for distance of these bodies by the angular measurement, but by the length of the arc minute at the distance.
Think of it this way: "X is off by Y(Angular measurement). Determine Y(Distance) of Z" Is what you've got. You're just reading it in a way that is asking for Z, but it's not, it's asking for Y. For that matter, the beginning isn't even important. Rewritten the question is: "Determine the length of 1 arc minute at the distances of the moon, the sun, and saturn"
So basically, it's plug+chug from here on. Distances from the earth to the moon, sun, and saturn, then just throw it in the formula to find the length.
So, judging by the answers in the back of the book, I seem to have gotten the right value for the Moon by calculating:
tan (1/60 [the conversion from arc minutes to degrees]) * 384,403 km [distance from Earth to Moon] = about 112 km
But as soon as I try that out for the Sun like so...
tan (1/60) * 93,000,000
I get 27050 km, when the book is telling me I should be getting 44,000 km. Similarly, I'm not getting the right distance for Saturn. What am I doing wrong?
God I hate this class.
Edit: Actually, I got a value for Saturn that is probably close enough to correct (book says 370,000 km, I got 350,000 km, so perhaps there is just a discrepancy in the distances used; I can't seem to find a table in the textbook that lists distances and had to resort to Google). Still not sure why I'm not getting the right value for the Sun, though.
Edit 2: Oh, I got my units confused on the Sun problem. Fixing that got me the right answer. Phew!
Thanks for all the help, guys.