Hydrodynamics Continuity equation Notation: Lagrangian derivative Mass conservation: Time rate of change of mass density must balance mass flux into/out of a volume, hence the divergence of v in the Eulerian case

Hydrodynamics Euler’s equation (equation of motion) Time rate of change of velocity at a point plus change in velocity between two points separated by ds = total change in the Lagrangian velocity which must = the sum of forces on a fluid element

Hydrodynamics Energy Equation Time rate of change of kinetic + internal energy must balance divergence of mass flux carrying temperature or enthalpy change

Hydrodynamics Sound waves Pressure and density are perturbed in sound waves such that P = P0+P’ ;  = 0+ ’ ; P’ or ’ << P0 or 0 Evaluate the hydro eqns neglecting small quantities of 2nd order

Hydrodynamics Sound waves Assuming adiabaticity we get with the above eqns and defining we can construct a dispersion relation and eqn of motion

Hydrodynamics Sound waves So Mach number of flow corresponds to compressibility of fluid

Hydrostatic Equilibrium HSE and nuclear burning responsible for stars as stable and persistent objects HSE is a feedback process PT, so as compression increases T, P increases, countering gravity, with the converse also true

Hydrostatic Equilibrium Start from hydro eqn of motion Forces from pressure gradient and gravity equal & opposite If gravity and pressure are not in equilibrium, there are accelerations Lagrangian coordinates

Virial Theorem The virial theorem describes the balance between internal energy and gravitational potential energy, whether internal energy is microscopic motions of fluid particles or orbital motions of galaxies in a cluster HSE is a special case of the virial theorem, so we can use it to study the stability of stars

Virial Theorem

Virial Theorem For ideal gas  = 5/3 For radiation gas  = 4/3 Gravitationally bound Half of potential energy into L, half into heating For radiation gas  = 4/3 W = 0 Unbound

Understanding the Mass-Luminosity Relation

Understanding the Mass-Luminosity Relation How do we make sense of stellar lifetimes? t ~ Enuc/L Enuc M easy so complexities enter into L(M) M 0.01 1 40 150 t(yr) 1012 1010 3x106

Understanding the Mass-Luminosity Relation Relation of pressure to luminosity At low masses ~1 HSE requires fg=-fp T doubling M requires doubling T, so L16L LM4

Understanding the Mass-Luminosity Relation Relation of pressure to luminosity At high masses 0 HSE requires fg=-fp T4 doubling M requires doubling P, T21/4T L2L LM tL/M t M-3 at low mass and t  const at high mass