1 Flashcards
bulk modulus of material equation
K = −V ∂P/∂V = ρ ∂P/∂ρ
V , P, and ρ are volume, pressure, and density respectively
what does a large bulk modulus mean
resist compression better than materials with a lower bulk modulus
when does “something interesting” happen if you keep adding mass to a pile
the material will begin to “fail” ( if this applied pressure is too great P ≈B
when the self-gravity of the pile becomes “important,”
expression of hydrostatic equilibrium
∇P = ρg
pressure at the center of a spherical body of uniform density
P = Cρ^2R^2
how to determine at what size self gravity is important
look at ratio between energy of bonds between atoms (1 eV works) and gravitational energy per particle
equation of hydrostatic equilibrium
∇P = ρg
polytropic equation of state
P = Kρ^( (n+1)/n)
gamma-law equation of state
P = (γ −1)ρu_m = (γ −1)u_v
um is the total internal energy (per unit mass) arising from microscopic processes, uv is the same quantity perunit volume, and γ is a constant
pressure scale height of the gas
HP = R_∗T/g
virial theorem
2U = −Ω
U = internal energy
Ω = −qGM^2/R is the gravitational binding energy
Degeneracy pressure in planet interiors
leads to a different equation of state (and in turn a different mass(radius) relationship). materials resist compression – that is, they have pressure – not just because of thermal pressure, but because of quantum mechanics - degenreacy pressure
pressure density relation for a non relativistic degenertate gas
P ∝ ρ^5/3
mass radius relation for a non relativistic degenertate gas
R ∝ M^−1/3,
Reynolds number
helps predict flow patterns in different fluid flow situations. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers flows tend to be turbulent.
uL/v=rho u L/ mu
inertial/ viscous term
rossby number
inertial/coriolis term = u/2ΩL^2
low if rapidly rotating
ekman number
viscous/ coriolis term = u/2ΩL
low if rapidly rotating
Very steep dT/dz
profiles are
unstable to
convection
“momentum equation,” aka“Navier-Stokes equation”
∂u/∂t +u.∇ u=-1/ρ∇P+ g -2sΩ x u + v∇^2u
Continuity equation (mass conservation)
∂ρ/∂t + ∇.(ρu)=0
for incompressible fluid
∇.u=0
eddy diffusivity
“eddy diffusivity” ~ velocity * length
eddy diffusion time
“eddy diffusion time” (~ dynamical time) L^2/velocity
pressure at center of planet
P~GM^2/R^4~Gρ^2R^2
how to find height of tallest mountain
bulk moduls = ρgh - when the pressure at the base of the mountain crushes the material
