The supergiant star Betelgeuse has a measured angular diameter of 0.044 arcsecond. Its distance has been measured to be 427 light-years. What is the actual diameter of Betelgeuse?

1 Answer
Zack M.
Dec 13, 2015

861,000,000 km

Explanation:

This is a pretty straight forward trigonometry problem. We can set up a diagram showing that the distance to Betelgeuse and the radius of Betelgeuse make a right angle.

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Therefore, we can use the sin function to find the radius of Betelgeuse. Since θ is very small, we can use the small angle approximation, sin(θ)≈θ if we convert θ to radians.

.044 arc seconds=2.13×10−7 radians

Since θ is the total diameter of Betelgeuse, we want to use sin(θ/2) to calculate the radius.

r=dsin(θ2)≈dθ2

But the radius is 1/2 of the diameter, D, so we have;

D2=dθ2

Canceling the 2s leaves;

D=dθ

Now we have an expression for the diameter, we can plug in what we know.

D=(427 light years)(2.13×10−7)

D=9.10×10−5 light years

Light years are not the most practical units for measuring the diameter of a star, however, so lets convert to km instead.

D=(9.10×10−5 light years)(9.46×1012 km / light year)

D=8.61×108 km

This is about 600 times the diameter of the sun!

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