I’m doing homework for my Astronomy class and I’m stuck on a couple of questions, and I would like some help.
The first problem is:
At the present time, the North Celestial Pole of the Earth is pointed approximately toward the star Polaris, the star whose arc is the very bright one, closest to the center in the above photograph. Suppose there is a star named HD358036 which is located 171.0 degrees from Polaris in a COUNTER-CLOCKWISE direction along the North Celestial Path, as viewed from above the North Pole. How long, in years, will it take before HD358036 becomes our North Star?
Hint: The Earth’s rotational axis precesses through 360 degrees in a CLOCKWISE direction, as viewed from above Earth’s North Pole, approximately every 26,000 years. See Fig. 3-22 and pp. 95-96 of Kuhn & Koupelis for additional discussion of precession.
My question is how do I create a formula to solve this problem.
Second question:
If you held a grain of sand at arm’s length, what angle of sky would it cover? Suppose you pick up a grain of sand and it is spherical with a diameter of 0.4-mm. Suppose further that when your arm is outstretched, the distance from the tips of your fingers to your eyes is 1.04-yards. What angle of sky would this grain of sand cover if you held it at arms length?
In this problem you are trying to find the angle subtended by your grain of sand. The diameter of the grain of sand is the value of “s” and the distance from your fingers to your eyes is “r”. In the equation s = ®(theta), “s” and “r” must both be in the same units and the computed value of theta will be in radians.
My question is, since I have the formula, I do know that .04mm=1.04yds(theta). I don’t know which to convert and how to finish the problem.
That’s it…any help is greatly appreciated! Just looking for explanations and clarifications, NOT the answer. TIA!
I’ll give you a hint for the first one. Once you figure out how many degrees per year (or century, or whatever you feel is convenient), and how many degrees CW HD358036 is from Polaris, the problem becomes trivial.
It’s interesting that these are not astronomy problems at all - they are simple math problems.
For the second problem: the second paragraph says that s and r should be expressed in the same units. It doesn’t matter what the units are - furlongs, parsecs, nanometers will all produce the same answer. For convenience, use millimeters.
Thanks for the tip, if I understand your hint correctly, the degree is 343…so the formula now is .04m=1.04yds(343 degrees)…now, maybe I’m going in the wrong direction, but wouldn’t you then divide 3.14/180 degrees…then multiply that by 343?
ARRRRGGGGGGGGGHHHHHHHHHHHHHHH!!!
The board ate my reply.
Okay here goes again.
In answer to the second part of your question, the important thing is to be consistent in your units of measurement. So let’s convert that 1.04 yads into millimeters. Here’s a great converter for that: http://www.1728.com/convleng.htm Yes, it is My site
Anyway, we find that 1.04 yards equals 950.98 mm.
Dividing .4mm (you say .04mm later on, but I’ll say it is .4 here) divided by 950.98 mm = 0.000420619.
If you look up the arctangent of that value it will be the angle you need.
The units it will be in (Angles, radians, etc) will depend on the way you have set your calculator.
Just as a hint, if the sand is .4mm, then the angle subtended should be roughly between 80 and 90 minutes of arc. (I’ll leave you to get the exact answer).
If you need to convert angles to different units then go to http://www.1728.com/angles.htm
(yep my site again)
Good luck.
It’s not 343, unless you meant to type 17.0 degrees, instead of 171.0. You’re on the right track, but be sure you’re using the correct numbers. You need now to figure out how to calculate the rate of precession, given 20,000 years to complete one full circle of 360 degrees.
Nah, I meant to write 171.0 degrees. Also, reading my book In Quest of the Universe by Kark F. Kuhn and Theo Koupelis say the precession rate is 26,000 years instead of 20. Does that make a huge difference?
Q.E.D. Thanks a bunch, I figured it out…13,756 since half of a 360 degree circle is 13,000 years, I just had to get the last 9 degrees by dividing 26000 by 189…
**
Wolfmeister**, still working on the first question, I do appreciate the help though!
Not quite. But very close. Think about what you need your equations to give you, then think about how to put the equations together to get that result. You’re on the right track.
**Q.E.D., ** it seems to have been the answer in my online homework (give or take 2%). Another question…correct me if I’m wrong, to find the seconds of arc in a degree, you must divide the number by 60 (the first 60 being minutes of arc) then divide that number by 60 again (that being the 60 seconds of arc)?
Wolfmeister, thanks alot for your website, it saved me time!
[QUOTE=JohanDane]
Another question…correct me if I’m wrong, to find the seconds of arc in a degree, you must divide the number by 60 (the first 60 being minutes of arc) then divide that number by 60 again (that being the 60 seconds of arc)?
[QUOTE]
Yep, that’s correct.
I get 13,650 years, given 26,000 years to complete one rotation of precession, which I suppose is within 2% of your figure. But you should be getting precisely my result, asuming we are using the same numbers. Could you post your work here?
[QUOTE=JohanDane]
Another question…correct me if I’m wrong, to find the seconds of arc in a degree, you must divide the number by 60 (the first 60 being minutes of arc) then divide that number by 60 again (that being the 60 seconds of arc)?
Actually, yours is the precise answer, but my test gives or takes a few. What I did was subtract 360 degrees from 170 = 189, then divided that into 26000. I technically got 137.56. Based off of my calculations, half of 26k is 13k (obviously), and another 10% roughly was the remaining number.
That’s not quite the right track, then. Let’s think for a moment about what your equation gives us. First of all, you’re 100% correct with 189 degrees. So now, you’re dividing 26,000 years by 189 degrees, which gives you an answer in years per degree. This isn’t quite what we want, however. We want an amswer simply in years. Therefore we want degrees in both the numerator and the denominator, so that degrees cancel out. If we multiply some number of years times some number of degrees, and then divide this by some other number in degrees, the degrees will cancel out of the equation, leaving only a value in years. So, our equation takes the form (Years x Angle[sub]1[/sub]) / Angle[sub]2[/sub]. Now, plug the correct values into their corresponding variables in the equation.
The answer definitely is the arctangent of .000421
However, what you did to “convert” this answer was to divide it by the number of degrees in a radian 57.29577951… and you got 7.348e-6 which is NOT the correct answer. Before you convert the .000421, you must first find its arctangent.
Time to explain arctangent I think.
Okay, the tangent of 1 degree is .017455…
So, we can say the arctangent of .017455… is one degree.
The arctangent of 0.424474816… is 23 degrees.
On a calculator, the arctangent key is usually shown as tan[sup]-1[/sup]
To make sure you are finding the arctangent correctly, use the 2 examples I just gave.
Now to get the answer you need, you have to look up the arctangent of 0.000420619.
Since I have made so many postings, I think it is only fair that I say what the answer is.
The arctangent of .000421 is 0.024121522 degrees.
You can use the converter to see what this is in radians, minutes seconds, etc.
To answer your last posting, to convert degrees to minures, multiply by 60.
Example, .5 degrees = .560 minutes = 30 minutes.
To convert degrees to seconds, multiply by 3,600.
Example, .5 degrees = .53600 seconds = 1800 seconds.
Hope that clears things up. I apologize again for posting a few incorrect calculations.
So I was on the right track at first. 360-171=189 (Angle1). If I multiply that times 26000 (years), then divide the answer by 360 (Angle 2, being the whole circle), and get 13650. Correct?
