## Problem Statement Satellites travel in a region where the mean free path of particles in the atmosphere is much greater than the characteristic size of the body. The pressure felt on the satellite during this rarefied gas condition is given as: \[ P = \frac{1}{3} n v \] where: - \( n \) = number density of particles in the atmosphere - \( v \) = mean speed of particles Given: - \( n \) = number density (particles per unit volume) - \( m \) = mass of each particle - \( v \) = mean speed of particles - \( T \) = temperature (implied in the context of particle velocity distribution) - \( k \) = Boltzmann constant --- ## Objective Show that the pressure on the satellite is approximately: \[ P \sim \frac{1}{3} n v \] and that the **pressure** is **no longer** proportional to \( n k T \) in this regime. --- ## Key Concepts and Derivation ### 1. Relationship between particle velocity and temperature For a gas in thermal equilibrium, the mean speed \( v \) of particles is related to temperature \( T \): \[ v \sim \sqrt{\frac{k T}{m}} \] ### 2. Number density \( n \) Number density: \[ n = \frac{\rho}{m} \] where \( \rho \) is the mass density. ### 3. Pressure in a rarefied gas In kinetic theory, the pressure exerted by particles is related to their momentum transfer upon collision with the surface. - The **average momentum transfer** per particle is proportional to \( m v \). - The **collision rate** is proportional to \( n v \). Thus, the pressure: \[ P \sim \text{(number of particles hitting per unit area per unit time)} \times \text{(momentum transfer per collision)} \sim n v \times m v = n m v^2 \] But the detailed kinetic theory shows that, for a Maxwellian distribution, the pressure is: \[ P = \frac{1}{3} n m v^2 \] --- ## **Final Result** Since: \[ v \sim \sqrt{\frac{k T}{m}} \] we find: \[ P \sim \frac{1}{3} n m \left( \frac{k T}{m} \right) = \frac{1}{3} n k T \] **However**, in the free molecular flow regime where the particles are rarefied, the pressure is more directly proportional to \( n v \): \[ \boxed{ P \sim \frac{1}{3} n v } \] and **not** necessarily proportional to \( n k T \), due to the non-continuum, free-molecular nature of the gas. --- ## **Summary** - The pressure felt by the satellite in a rarefied atmosphere is approximately proportional to \( n v \). - This deviates from the usual ideal gas law relationship \( P \sim n k T \), especially in the free molecular regime where particle collisions are infrequent.