Schwarzschild Radius of a Black Hole with 4 Million Solar Masses
Question:
What is the Schwarzschild radius for the black hole at the center of our galaxy if it has the mass of 4 million solar masses?
Provide the final answer in astronomical units (AU).
Answer:
The Schwarzschild radius (rs) is the radius of the event horizon of a non-rotating black hole and is given by:
rs = (2 × G × M) / c²
Where:
- G = gravitational constant = 6.67430 × 10⁻¹¹ m³·kg⁻¹·s⁻²
- c = speed of light = 2.99792458 × 10⁸ m/s
- M = mass of black hole = 4 × 10⁶ solar masses
- Mass of Sun (M☉) = 1.989 × 10³⁰ kg
Step 1: Convert mass to kilograms
M = 4 × 10⁶ × 1.989 × 10³⁰ = 7.956 × 10³⁶ kg
Step 2: Plug values into the Schwarzschild formula
rs = (2 × 6.67430 × 10⁻¹¹ × 7.956 × 10³⁶) / (2.99792458 × 10⁸)²
Numerator: 2 × 6.67430 × 10⁻¹¹ × 7.956 × 10³⁶ ≈ 1.063 × 10²⁷ m³/s²
Denominator: (2.99792458 × 10⁸)² ≈ 8.98755 × 10¹⁶ m²/s²
rs ≈ (1.063 × 10²⁷) / (8.98755 × 10¹⁶) ≈ 1.183 × 10¹⁰ meters
Step 3: Convert to astronomical units (AU)
1 AU = 1.496 × 10¹¹ meters
rs = 1.183 × 10¹⁰ / 1.496 × 10¹¹ ≈ 0.079 AU
Final Answer:
The Schwarzschild radius for the black hole at the center of our galaxy is approximately:
rs ≈ 0.08 AU