Question

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?

Answer 1

`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.

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

`.044 " arc seconds" = 2.13 xx 10^-7 " radians"`

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

`r = d sin(theta/2) ~~ d theta/2`

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

`D/2 = d theta/2`

Canceling the `2`s leaves;

`D = d theta`

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

`D = (427 " light years")(2.13 xx 10^-7)`

`D = 9.10 xx 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 xx 10^-5 " light years")(9.46 xx 10^12 " km / light year")`

`D = 8.61 xx 10^8 " km"`

This is about 600 times the diameter of the sun!