Let N ( d ) be the number of asteroids of diameter ≤ d kilometers. Data suggest that the diameters are distributed according to a piecewise power law:

Question

Let N(d) be the number of asteroids of diameter ≤d kilometers. Data suggest that the diameters are distributed according to a piecewise power law:

𝑁′(𝑑)= 2×10^9𝑑^−2.4 for d<69

1.8×10^12x𝑑^−4 for d≥69

Answer 1

The problem provides a piecewise power law for the distribution of asteroids' diameters, where the number of asteroids N(d) depends on the diameter d:

1. For diameters d < 69 kilometers, the distribution is given by:

N'(d) = 2 × 10^9 * d^(-2.4)

2. For diameters d ≥ 69 kilometers, the distribution is given by:

N'(d) = 1.8 × 10^12 * d^(-4)

To find the total number of asteroids N(d) with diameters less than or equal to d, you need to integrate the distribution over the relevant range for each part.

For the first part, when d < 69 kilometers:

N(d) = ∫[2 × 10^9 * d^(-2.4)] from 0 to d

For the second part, when d ≥ 69 kilometers:

N(d) = ∫[1.8 × 10^12 * d^(-4)] from 69 to d

So, you have two separate equations to calculate N(d) for the two different ranges of diameters.