Kinetic Theory 1 Particle Distribution Function 2 Density

Kinetic Theory 1

Particle Distribution Function 2

Density Function ns Number density, mass density, charge density 3

The Boltzmann Equation: time variation of a distribution function 4

Total Derivative in Phase Space 5

Homework Problem 3. 6, 3. 7, 3. 8 6

Examples of Distribution Functions 7

Maxwellian Distribution Functions 8

One-dimensional Maxwellian, cont’d 9

Thermal speed 10

Equilibrium Distribution charge x voltage = energy 11

Velocity Moments 12

Bulk flow velocity 13

Pressure tensor 14

Scalar pressure 15

Heat Flux 16

Continuity equation: 0 th moment of B-equation 17

18

19

Momentum Equation: 1 st moment of B-equation 20

21

22

23

The momentum equation 24

Pressure Gradient Force 25

Pressure as Thermal Energy Density 26

Charge and Current Densities 27

Moments Measurements in Space Physics • Particle detector measures electric current produced by the charged particles • The electric field current, count-rate, is converted to number of density flux according to calibration • The flux can be converted to distribution function • The detector can select the energy range with energy resolution, as well as type of the particle species • Each detector has a finite field angle which determines the spatial resolution • The detector(s) scan over directions • Fluxes from all energy channels and all looking directions can be added together. • Moment factor can be added in the integration process to obtain moments, which are used as macroscopic quantities, such as density, velocity, temperature, (temperature) anisotropy • Satellite charge is a major problem in deriving correct distribution function • In the existence of multiple populations, “moments” may mean different things. Distribution functions have to be examined in this case. 28