Question

On an asteroid, the density of dust particles at a height of 3 cm is ~30% of its value just above the surface of the asteroid. Assuming the temperature is 20 K and the average dust-particle mass is 10-19 kg. Estimate the gravitational acceleration on the asteroid.a. 2 m/s^2b. 10 m/s^2c. 0.1m/s^2

Answer 1

From the law of atmosphere

$N_v(y) = n_0*e^{-\frac{mgy}{Kb*T}}$

Where

$n_0$ = constant and is number density where the height y = 0cm

$n_V$ = Number density at height y=3cm

Kb = Boltzmann constant $= 1.38*10^{-23}J/K$

$T=20K$

$m = 10^{-19}kg$

Re-arranging the equation to have the value of the gravity,

$\frac{N_v(y)}{n_0} = e^{-\frac{mghy}{KbT}}$

$ln(\frac{N_v(y)}{n_0}) = -\frac{mgy}{KbT}$

Since it is 30% of value above surface, therefore $N_v = 0.3n_0$

$ln(\frac{0.3n_0}{n_0}) = -\frac{mgy}{KbT}$

$g = -\frac{KbT ln(0.3)}{my}$

$g = -\frac{(1.38*10^{-23}J/K)(20K)(Ln(0.3))}{10^{-19}(3*10^{-2})}$

$g = \frac{1.38*2*ln(0.3)*10^{-22}}{3*10^{-4}}$

$g = 1.104*10^{-1}m/s^2$

$g = 0.1m/s^2$

Therefore the correct answer is C.