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
why can mountains be bigger on smaller planets
g scales with r and h max is inversly proportinal to g
as long as h max is still much smaller than r all good
mass radius relation for uncompressed rock
m ~ R^3
gamma law equation of state
P=(γ-1)ρu_m=(γ-1)u_v
polytropic equation of state
P=kρ^((n+1)/n)
ideal gas equation of state
PV=nRT or P=ρR_*T
viral theorem
the ke of a gravitationaly bound system is minus a half fof the gpe
internal potential energy of a gas
3/2nRT=U
degeneracy pressure scaling law
P~ρ^5/3
flux of energy due to radiation
F=-4acT^3/(3σn)dt/dx =-4acT^3/(3κρ)dt/dx
whats not a fluid
Enough collisions to isotropize quantities and lead to continuous density, velocity, etc. “Smoothing out” over inherent granularity of matter
Typically means we are looking at scales»_space; mfp
escape velocity
v=sqrt(2GM/r)
body force
acts via center of mass
categories of planetary materials
rocks/metals ices gasses
rossby term
u/2omegaL
small if rotation is small compared to dynamical time
coriolis forces have time to influence
ie small if influenced by rotation
geostrophic balance
state, in which horizontal pressure gradients are balancing Coriolis forces
flow along isobars
flow is clockwise around
areas of high pressure
geostrophic balance is applicaple when
rossby and ekman number small
presssure balance coriolis in horizontal
taylor proudman constraint
flow constant on lines parallel to rotation axis - for barytropic fluid- obsticales project into atmosphere
equation linking the orbital period and masses of two orbiting bodies
a_s/a_p=M_p/M_s
kepplers thrird law
P^2/a^3 = 4π^2/G(Ms +Mp)
astrometry equation
β = a_s/d = a_p/d(Mp/M)
with beta being the observed angular change in arcseconds
astrometry
detecting the star’s reflex motion by simply observing its apparent position on the sky, and seeing whether this changes periodically over time
radial velocity method
measure reflex motion of star via doppler shift
star’s reflex velocity equation
vs/vp = −Mp/(Mp +Ms)
transit method equation
∆F/F ≈(Rp/Rs)^2
.
transit method
look for dipped luminosity
what are the axis for a cratering diagram
no of craters per unit area and crater size
saturation equilibrium
max number of craters per unit area - new impact kills old impact crater
effect of atmosphere on cratering
small impactors are screened off by atmosphere
planet resurfacing effect on cratering
younger surface will have fewer craters
what will slow a metorite
traveling through a collum of atmosphere roughly its own mass
conductive transport equation
F=-kdT/dr
possible heating sources for planets
Gravitational (accretion) energy Differentiation (grav PE) Latent heat/condensation effects Radioactivity \+ (of course) stellar insolation
Bond albedo
the fraction of power in the total electromagnetic radiation incident on an astronomical body that is scattered back out into space
planet cooling mechanisms
radiation conduction convetion and erruprion
diffusion equation
dT/dt=k/ρc ∇^2T
thermal diffusion time
t ∼ L^2/κ ,
equation for how flux and luminosity are related
F=L/4pid^2
energy absorbed by planet from star
E_in = (1 −a)πR^2F(d)
energy radiatied out by planet
E_out = 4πR^2pεσT^4
planet’s “radiative equilibriun temperature”
Te =((F(1 −a))/4εσ)^1/4
what happens to the radiative equilibrium surface as the atmo gets thicker
moves higher up
3 sources / types of atmos
primary - acreted from nebula
secondary - outgassing
tertiary- deliverd via impactors
how to determine type of atmo
composition - primary will be similar composition to central star
could earths atmosphere be primitive
no - consider neon in atmosphere - neon us heavy and primitive what we have is what had but extrapolataing back get only 0.9% of current atmo
4 ways the atmosphere is lost
thermal - jeans escape , dissociation via uv photons, impacts, loss to interior
exobase
mean free path > scale height - region in atmosphere where easy for molecules to escape
mean free path equation
1/n x cross sectional area
ice albedo feedback
positive feedback climate process where a change in the area of ice caps, glaciers, and sea ice alters the albedo and surface temperature of a planet
optically thick limit
I_v(D)=S_v
optically thin limit
I_v(D)=I_v(0)+[S_v-I_v(0)]T_v(D)
energy density of the radiation
U = aT^4 (with a = 4σ/c)
optical depth equation
dτ_v=s(α)ds=nσds=ds/mfp
optical deptha and pressure equation
P/g=τ/κ
lapse rate
rate of change in temperature observed while moving upward through the atmosphere. The lapse rate is considered positive when the temperature decreases with elevation
derivation of dry adiabatic lapse rate
this state has dθ/dz=0
use chain rule dT/dz=dT/dP dP/dz +dθ/dz dT/dθ
and ideal gas equation
typical structure of atmospheres
near surface dT/dz close to adiabatic laps rate then sabtle stratifications
dry adiabatic lapse rate final solution
dT/dz=-g/c_p
Boussinesq approximation
we assume that density fluctuations are negligible except where ρ is multiplied by g density fluctuations are linked only to temperature fluctuation
size of the perturbations relative to the mean in the Boussinesq approximation
ρ′/ρ = −αT ′
The non-intuitive result is that a horizontal density/temperature gradient
leads to vertical shear, that is to
variations with height of the horizontal wind
Rayleigh number in convection
Ra = gαΓL^4/νκ ∼ buoyancy driving/dissipation
with Γ = dT /dz = ∆T /L
this must hit a crit number b4 convection happnes
he temperature gradient approaches
the adiabatic value for efficient convection
velocity of heat flux
v ≈(F/ρ)^1/3
magnetic diffusivity
η = c^2/4πσ
