If i know the radius r of a star in cm, how can I convert it to arcseconds?
For example, if I have a star with r = 3.18e13 cm, and distance to the star d = 220 parsecs, what is the relation to convert the radius from cm to arcseconds.
Thank you.
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Sign up to join this communityIf i know the radius r of a star in cm, how can I convert it to arcseconds?
For example, if I have a star with r = 3.18e13 cm, and distance to the star d = 220 parsecs, what is the relation to convert the radius from cm to arcseconds.
Thank you.
Using basic circular maths: where $d$ is the distance of the star from the observer in AU, and $r$ is the star's radius in AU, and $a$ is the angle encompassed by the radius of the star in degrees:
$ r = \frac{a}{360}2\pi{d} $
Now rearrange it to make $a$ the subject:
$ a = \frac{180r}{\pi{d}} $
To get $a$ in arcseconds, you need to multiply the result by 3600 (because there are 3600 arcseconds in a degree):
$ a = \frac{648000r}{\pi{d}} $
Now, convert your numbers from cm and parsecs to astronomical units:
$r$ = 3.18e13cm = 2.126 AU
$d$ = 220pc = 4.538e7 AU
Put them into the equation:
$ a = \frac{648000\times2.126}{{4.538\times10^7}\pi} $
To get:
$ a = 0.00966327′′ $
Hope that helps!
Edit:
As Mike has pointed out, the final equation I've ended with can be further simplified if you use different units, to:
$ a = \frac{r}{d} $
where $r$ is in AU and $d$ is in parsecs, giving an answer $a$ in arcseconds.
This is because the equation $a = \frac{648000r}{\pi{d}}$ has $\frac{648000}{\pi}$ in it - which is the definition of a parsec in AU. Therefore, multiplying $d$ by this number to convert it from AU to parsecs, we get $a = \frac{648000r}{648000/\pi\times\pi{d}}$. Cancel out the pi on the bottom, and cancel the ${648000}$ on the top and bottom, and you're left with $a = \frac{r}{d}$ where $r$ is in au and $d$ is in pc.
+1
Working things though step-by-step and explaining along the way is a great way to answer, and can be very helpful for the OP or for future readers.
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– uhoh
Mar 5 '18 at 10:43
2 * arctan(r / d)
Use Wolfram Alpha, it takes free-form units like cm, parsecs, etc and usually does the right thing. And you can specify the unit for the answer you're looking for (e.g. arcseconds).
http://www.wolframalpha.com/input/?i=2+*+arctan(3.18e13+cm+%2F+220+parsecs)+in+arcseconds
In this case, the answer is 0.019 arc seconds
2 arctan(r/d)
since arctan(r/d)
is half the angle? Of course, in this case, the angles would be virtually identical.
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– user21
Mar 4 '18 at 23:07
arctan(2r/d)
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– user21
Mar 5 '18 at 15:14