If we assume that a globular cluster 4 minutes of arc in diameter is actually 25 pc in diameter, how far away?

Question

why am i so stupid!! ok i've been trying to figure this out. it's not working

the small-angle formula is:

angular diameter/ 206,265"= linear diameter/distance

Can someone help me, not give me the answer, but explain this to me.. if you get it? thanks

Answer 1

D = X * d / 206,265
where D is the linear size of the cluster = 25 parsec
X is the angular size of the cluster = 240 arcseconds (4 arcminutes * 60 sec/min)
d is the distance to the cluster in parsecs

shifting your terms around you get:

d = 206,265 * 25 parsec / 240 arcseconds (I'll let you do the calculation)

It's an trigonometric approximation, based on the geometry of a right triangle, for relating angular (or apparent) size to linear (or actual) size and distance.

Think of it this way: a 50ft tree is still 50ft tall, no matter how far away it is. But it LOOKS smaller at a distance than up close. This simply relates the three things together : how tall it is (linear size), how tall it looks (angular size), and how far away it is (distance).

The confusing thing to people is often where the 206,265 number came from. It's approximately equal to the number of seconds of arc in a radian.

Answer 2

Are you perhaps not expressing the angular diameter in arcseconds?

The question says 4 arcminutes (=240 arcseconds) is 25 pc.

240/206265 = 0.00116355.

Distance is 25 pc / 0.00116355 = 21,486 parsecs.

For any right-angled triangle tan(angle)=opposite/adjacent, where the opposite and adjacent sides make the right angle. For small angles tan(angle) = angle, provided the angle is specified in radians. One arcsecond is 1/3600 degrees and one degree is pi/180 radians. One arcsecond is thus pi/648000 radians. Alternatively, if you have an angle in arcseconds you have to divide by 648000/pi = 206264.8 to get the answer in radians. The opposite side is the actual width of the cluster, the adjacent side is the distance to the cluster (provided both are in the same units.)