Question 1174508: Astronomers often measure large distances using astronomical units (AU) where 1 AU is the average distance from Earth to the Sun. In the image, d represents the distance from a star to the Sun. Using a technique called “stellar parallax,” astronomers determined \large \theta is 0.00001389 degrees. How far away is the star from the Sun in astronomical units? Show your reasoning.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the distance to the star using stellar parallax:
**Understanding Stellar Parallax**
* Stellar parallax is the apparent shift in the position of a star when viewed from different points in Earth's orbit around the Sun.
* This shift is very small and is measured as an angle (θ).
* The distance to the star is inversely proportional to the parallax angle.
**Calculations**
1. **Relating Parallax to Distance:**
* When the parallax angle (θ) is very small, we can use the approximation tan(θ) ≈ θ (in radians).
* The relationship between distance (d), the baseline (1 AU), and the parallax angle (θ) is: tan(θ) = (1 AU) / d.
* Therefore, d = 1 / tan(θ) AU.
2. **Convert Degrees to Radians:**
* θ = 0.00001389 degrees
* To convert to radians, multiply by π / 180:
* θ (radians) = 0.00001389 * (π / 180) ≈ 2.422 × 10⁻⁷ radians.
3. **Calculate Distance:**
* d = 1 / tan(2.422 × 10⁻⁷)
* Since the angle is so small, tan(θ) is very close to θ.
* d = 1 / (2.422 × 10⁻⁷) AU
* d ≈ 4,128,819 AU.
**Therefore, the star is approximately 4,128,819 astronomical units away from the Sun.**
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