Finding the radius of a star in arcsec

Question

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.

Answer 1

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.

Answer 2

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