1. 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

2. 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

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 = P0+P’ ;  = 0+ ’ ; P’ or ’ << 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 For radiation gas  = 4/3
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

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