It is observed that the number of asteroids (or meteoroids) of a given diameter is roughly inversely proportional to the square of the diameter.

Part A

Assuming a density of 3000 kg/m3 and approximating the actual distribution of asteroids and meteoroids as a single 1000-km body (Ceres), one hundred 100-km bodies, 10 thousand 10-kmbodies, and so on, calculate the total mass in the form of 1000-km bodies.

Part B

Calculate the total mass in the form of 100-km bodies.

Part C

Calculate the total mass in the form of 10-km bodies.

Part D

Calculate the total mass in the form of 1-km bodies.

(a) M =  =1.57 *10²¹kg

(b) M =  =1.57 *10²⁰kg

(c) M =  =1.57 *10¹⁹kg

(d) M =  =1.57 *10¹⁸kg

Explanation:

(a) For 1000km body, diameter D = 1000km = 1,000,000 m

and N = (1000/D)² = (1000/1000km)² =  1

substituting into the formula, we have

M = 1 * 4/3 π*3000*[1000000/2]³

   =1.57 *10²¹kg

(b) For 100km body, diameter D = 100km = 100,000 m

and N = (1000/D)² = (1000/100km)² =  100

substituting into the formula, we have

M = 1 * 4/3 π*3000*[100000/2]³

   =1.57 *10²⁰kg

(C) For 10km body, diameter D = 10km = 10,000 m

and N = (1000/D)² = (1000/10km)² =  100

substituting into the formula, we have

M = 1 * 4/3 π*3000*[10000/2]³

   =1.57 *10¹⁹kg

(C) For 1km body, diameter D = 1km = 1,000 m

and N = (1000/D)² = (1000/1km)² =  100

substituting into the formula, we have

M = 1 * 4/3 π*3000*[1000/2]³

   =1.57 *10¹⁸kg

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