Let's break down the problem: **Given:** - Mass of the star, \( M = 20\,M_\odot \), where \( M_\odot \) (solar mass) \( \approx 2 \times 10^{30} \) kg. We are to **find the density** the star must have to be compressed within its **event horizon** (i.e., become a black hole). ### 1. **Event Horizon (Schwarzschild Radius)** The Schwarzschild radius is: \[ R_s = \frac{2GM}{c^2} \] where: - \( G = 6.674 \times 10^{-11} \) m³/kg/s² (gravitational constant) - \( c = 3.00 \times 10^8 \) m/s (speed of light) ### 2. **Volume Inside Event Horizon** The star must be compressed within a sphere of radius \( R_s \). \[ V = \frac{4}{3}\pi R_s^3 \] ### 3. **Density** Density \( \rho \) is: \[ \rho = \frac{M}{V} \] --- ## **Step-by-Step Solution** ### **Step 1: Calculate the Schwarzschild radius** \[ M = 20\,M_\odot = 20 \times 2 \times 10^{30} = 4.0 \times 10^{31} \text{ kg} \] \[ R_s = \frac{2GM}{c^2} = \frac{2 \times (6.674 \times 10^{-11}) \times 4.0 \times 10^{31}}{(3.00 \times 10^8)^2} \] Calculate numerator: \[ 2 \times 6.674 \times 10^{-11} \times 4.0 \times 10^{31} = 53.392 \times 10^{20} = 5.3392 \times 10^{21} \] Calculate denominator: \[ (3.00 \times 10^8)^2 = 9.00 \times 10^{16} \] So, \[ R_s = \frac{5.3392 \times 10^{21}}{9.00 \times 10^{16}} = 0.593 \times 10^{5} = 5.93 \times 10^4 \text{ m} \] --- ### **Step 2: Calculate the volume** \[ V = \frac{4}{3}\pi R_s^3 \] \[ R_s^3 = (5.93 \times 10^4)^3 = (5.93)^3 \times 10^{12} \approx 208.5 \times 10^{12} = 2.085 \times 10^{14} \] So, \[ V = \frac{4}{3}\pi \times 2.085 \times 10^{14} \approx 4.18879 \times 2.085 \times 10^{14} \approx 8.718 \times 10^{14} \text{ m}^3 \] --- ### **Step 3: Calculate the density** \[ \rho = \frac{M}{V} = \frac{4.0 \times 10^{31} \text{ kg}}{8.718 \times 10^{14} \text{ m}^3} \] \[ = \frac{4.0}{8.718} \times 10^{31 - 14} \text{ kg/m}^3 = 0.459 \times 10^{17} \text{ kg/m}^3 = 4.59 \times 10^{16} \text{ kg/m}^3 \] --- ## **Final Answer** \[ \boxed{4.6 \times 10^{16} \ \text{kg/m}^3} \] This is the density to which a **20-solar-mass star** must be compressed to fit within its own event horizon and become a black hole.