Hydrodynamics

1
Hydrodynamics

• Notation Lagrangian derivative
• Continuity equation
• 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

2
Hydrodynamics

• Eulers 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

3
Hydrodynamics

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

4
Hydrodynamics

• Sound waves
• Pressure and density are perturbed in sound waves
  such that
• P P0P ? ?0 ? P or ? ltlt P0 or ?0
• Evaluate the hydro eqns neglecting small
  quantities of 2nd order

5
Hydrodynamics

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

6
Hydrodynamics

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

7
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

8
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
9
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

10
Virial Theorem
11
Virial Theorem

• For ideal gas ? 5/3
• Gravitationally
  bound
• Half of potential
  energy into L, half into heating
• For radiation gas ? 4/3
• W 0 Unbound

12
Understanding the Mass-Luminosity Relation
13
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 3x106
14
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

15
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