How do you find the actual diameter of a star when converting from arcseconds to kilometers?
Example: The supergiant star Betelgeuse (in the constellation Orion) has a measured angular diameter of 0.044 arcseconds. Its distance has been measured to be 427 light-years.
What is the actual diameter of Betelgeuse?
Answer (2)
What you have is a skinny, skinny sector of a circle ... like a slice of pie. The radius of the pie is 427 light-years, and the angle in the center, at the tip of the slice, is 0.044 second. That's about 0.000012 degree.
The question is: What's the length of the crust out at the fat end of the slice ?
This is a case where it's very handy to measure your angles in radians instead of degrees. That way, whatever fraction of a radian is at the tip, the same fraction of the radius is the length of the arc (the crust).
To change degrees to radians, multiply by (pi)/180 .
0.044 second = 0.000012 degree = about 2.133 x 10^-7 radian .
So the diameter of the star is about (2.133 x 10^-7) of 427 light years.
I'll leave that part for you to finish up.
To find the diameter of Betelgeuse from its angular diameter in arcseconds and distance in light-years, convert the angular diameter to radians, use it to calculate the diameter in light-years, and then convert that to kilometers. The actual diameter is approximately 861,000 kilometers. This method shows how angular measurements relate to physical dimensions in astronomy.
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