Assume that the nearest stars to us have an intrinsic luminosity about the same as the Sun’s. Their apparent brightness, however, is about 10¹¹ times fainter than the Sun. From this, estimate the distance to the nearest stars.

Verified step by step guidance

1. Identify the relationship between apparent brightness, intrinsic luminosity, and distance using the inverse square law for light: \( b = \frac{L}{4\pi d^2} \), where \( b \) is the apparent brightness, \( L \) is the intrinsic luminosity, and \( d \) is the distance.
2. Set up the ratio of the apparent brightness of the Sun to the apparent brightness of the nearest stars. Given that the stars are 10¹¹ times fainter, the ratio can be expressed as \( \frac{b_{\text{sun}}}{b_{\text{stars}}} = 10^{11} \).
3. Express the brightness of the Sun in terms of its distance (1 AU) using the inverse square law: \( b_{\text{sun}} = \frac{L_{\text{sun}}}{4\pi (1 \text{ AU})^2} \).
4. Substitute the expressions for \( b_{\text{sun}} \) and \( b_{\text{stars}} \) into the ratio and solve for \( d \), the distance to the stars: \( \frac{\frac{L_{\text{sun}}}{4\pi (1 \text{ AU})^2}}{\frac{L_{\text{sun}}}{4\pi d^2}} = 10^{11} \).
5. Simplify and solve the equation for \( d \), the distance to the nearest stars, which will involve taking the square root of the ratio multiplied by 1 AU.