1. Convert Units:
* Angular size: 0.044 arcseconds. We'll need to convert this to radians:
* 1 arcsecond = 4.84813681 × 10⁻⁶ radians
* 0.044 arcseconds = 0.044 * (4.84813681 × 10⁻⁶) radians ≈ 2.13 × 10⁻⁷ radians
* Distance: 427 light-years. We'll need to convert this to meters:
* 1 light-year ≈ 9.461 × 10¹⁵ meters
* 427 light-years ≈ 427 * (9.461 × 10¹⁵) meters ≈ 4.04 × 10¹⁸ meters
2. Use the Small Angle Approximation:
For small angles (like this one), we can use the small angle approximation:
* θ ≈ (d / D)
* Where:
* θ is the angular size in radians
* d is the actual diameter of the star
* D is the distance to the star
3. Solve for Diameter (d):
* d = θ * D
* d ≈ (2.13 × 10⁻⁷ radians) * (4.04 × 10¹⁸ meters)
* d ≈ 8.60 × 10¹¹ meters
4. Convert to a More Convenient Unit:
* Let's convert the diameter from meters to solar radii:
* 1 solar radius ≈ 6.957 × 10⁸ meters
* d ≈ (8.60 × 10¹¹ meters) / (6.957 × 10⁸ meters/solar radius)
* d ≈ 1237 solar radii
Therefore, the diameter of the star is approximately 1237 times the radius of the Sun.
