Estimate the diameter of the supergiant star betelgeuse, using its angular diameter of about 0.05 arcsecond and distance of about 600 light-years.

To estimate the diameter of the supergiant star Betelgeuse, we can use the small-angle formula, which relates the physical diameter \( D \), angular diameter \( \theta \), and the distance \( d \):

\[
D = \theta \times d
\]

Here, we need to ensure that the units are consistent. Let's break it down:

### 1. Angular Diameter Conversion
The angular diameter \( \theta \) is given as 0.05 arcseconds. To convert this to radians:

\[
1 \text{ arcsecond} = \frac{1}{206265} \text{ radians}
\]

Thus:

\[
\theta = 0.05 \times \frac{1}{206265} = 2.424 \times 10^{-7} \text{ radians}
\]

### 2. Distance Conversion
The distance to Betelgeuse is approximately 600 light-years. To convert this into meters:

\[
1 \text{ light-year} \approx 9.461 \times 10^{15} \text{ meters}
\]

So:

\[
d = 600 \times 9.461 \times 10^{15} \text{ meters} = 5.6766 \times 10^{18} \text{ meters}
\]

### 3. Applying the Small-Angle Formula
Now we can estimate the diameter:

\[
D = \theta \times d = 2.424 \times 10^{-7} \times 5.6766 \times 10^{18}
\]

Let me compute the result.The estimated diameter of Betelgeuse is approximately \( 1.38 \times 10^{12} \) meters, or about 1.38 billion kilometers.

Would you like further details or clarification on any steps?

Here are 5 related questions:
1. How does Betelgeuse compare in size to our Sun?
2. What factors affect the accuracy of angular diameter measurements?
3. How does the small-angle formula work for distant astronomical objects?
4. How do astronomers measure the distance to stars like Betelgeuse?
5. What is the significance of Betelgeuse’s changing brightness?

**Tip:** When using the small-angle formula, ensure that both the angular diameter and the distance are in compatible units (radians and meters) for an accurate result.

Estimate the diameter of the supergiant star Betelgeuse, using its angular diameter of about 0.05 arcsecond and distance of about 600 light-years.

Learn how to estimate the diameter of the supergiant star Betelgeuse using the small-angle formula, with an angular diameter of 0.05 arcseconds and a distance of 600 light-years. This step-by-step guide explains key concepts in trigonometry and astronomy, including unit conversions and formula applications. Suitable for Grades 10-12.

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