Standard Planetary Information, Formulae and Constants

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TABLE OF CONTENTS

Summary of constants
Tables of atmospheric parameters for the planets
Adiabatic Lapse Rate
Auto-convective Lapse Rate
Amagats
Beers Law
Beta plane approximation
Brunt Vaisala or Buoyancy Frequency
Cloud clearing eta and N* methods
Cloud properties
Dobson Units
Double Linear Interpolation
Earth: molecular abundances
Earth: Molecular spectral regions
Earth: Oceans
Equations of Motion
FFT, Cosine Transform
Gas Constant
Geopotential
Geostrophic Approximation
Gravity
Heat Balance
Humidity
Hurricanes
Hydrostatic Equilibrium
Ideal Gas Law
IR filter bandpasses
Latitude: planeto-graphic and planeto-centric
Longitude Systems and Conversion
Orbits: Equations of motion, Period(altitude)
Planck Function
Planet Mission Summary
Potential Temperature
Potential Vorticity
Radiative time constant
Radiative Transfer Equation (IR)
Rayleigh Scattering: Optical Depth
Rayleigh Scattering: Phase Function
Richardson Number
Rossby Number
Saturation Vapor Pressure
Scale Height
Thermal Wind Equation
Velocity of Sound
Vorticity
Weighting Functions
Wind Measurement

Physical Constants

(ref. Physics Today, Aug. 1995 pg. BG9)

AU = astronomical unit = 1.4960 · 1013 cm
c = speed of light = 2.99792458 · 1010 cm/s
G = 6.67259(85) · 10-8 dyne cm2/gm2
h = Planck's constant = 6.626075(40) · 10-27 erg sec
k = Boltzmann's constant = 1.380658(12) · 10-16 erg/K
Na = Avogadro's number = 6.0221367(36) · 1023 gm/mole
sigma = Stefan-Boltzmann constant = 5.67051(19) · 10-5 erg/s/cm2/K4
S0 = 1.37 · 106 erg/cm2
Runiv = Na · k = 8.3143 · 107 erg/mole/K
<mw>H2O = 18.016 gm/mole
<mw>O3 = 47.9982 gm/mole

Earth Constants

Req = 6378.388 km
Mearth = 5.975 · 1027 gm
<mw>dry air = 28.964 gm/mole
Rg = 0.28705 · 107 erg/gm/K
N0 = Loschmidt's number = 2.686763(23) · 1019 molecules/cm3
gs = Standard surface gravity = 980.665 cm/s
Ps = Standard surface pressure = 1.01325 · 106 dyne/cm2
cp = 1.006 · 107 erg/gm/K at stp
cv = 0.718 · 107 erg/gm/K at stp
gamma = cp/cv = 1.401 at stp

Unit Conversion

1 knot = 1 nautical mile/hour = 1.151555 mile/hour = 1.853248 Km/hr
1 megaton = 4.2 · 1022 erg = 4.2 · 1015 Joule
1 AU = 1.495 · 108 Km = 8.31 light minutes

Tables of atmospheric parameters for the planets

Planet Size Information
area
Km2		volume
Km3		Mass
grams		Mass
Mearth		density
gm/cm3
Venus4.6011e+089.2805e+114.8850e+270.8175.264
Earth5.1006e+081.0832e+125.9790e+271.0005.520
Mars1.4416e+081.6275e+116.4570e+260.1083.967
Jupiter6.1469e+101.4313e+151.8986e+30317.5511.327
Saturn4.2694e+108.2713e+145.6854e+2995.0890.687
Uranus8.0840e+096.8336e+138.6825e+2814.5221.271
Neptune7.6280e+096.2637e+131.0310e+2917.2441.646
Titan8.3323e+077.1519e+101.3500e+260.0231.888

Planet Rotation Period
hourshr:mn:secsday/earth°/day
Venus-5833.2000000-5833:12: 0.0-243.050001.481177
Earth23.933333023:56: 0.00.99722361.002791
Mars24.633333024:38: 0.01.02639350.744254
Jupiter: SI..9.84166759:50:30.00.41007877.900010
Jupiter: SII.9.92795339:55:40.60.41366870.270008
Jupiter: SIII9.92491979:55:29.70.41354870.536010
Saturn: SI..10.233329010:14: 0.00.42639844.300032
Saturn: SIII10.656222010:39:22.40.44401810.793919
Uranus-17.2333330-17:14: 0.0-0.71806501.353975
Neptune16.110000016: 6:36.00.67125536.312849
Titan382.6799927382:40:48.015.9450022.577611

Orbital Parameters
a(AU)period(yrs)inclin(°)a.node(°)arg.perih(°)eccL(6430)(°)
Venus0.72330.623.3976.55131.250.006818267.689
Earth1.00001.000.000.00102.770.01670499.372
Mars1.52371.881.8549.45335.820.093329195.082
Jupiter5.202511.871.31100.3515.460.048075329.233
Saturn9.553129.532.49113.5592.200.051565238.772
Uranus19.264284.550.7773.99175.280.046227253.217
Neptune30.2337166.241.77131.827.600.007972274.526
Titan0.00820.040.330.000.000.0300000.000

Planet Pole
<r>(AU)inclin(°)RA(°)DEC(°)
Venus0.7233177.40272.7867.21
Earth1.000023.400.0090.00
Mars1.523725.20317.5757.52
Jupiter5.20253.10268.0464.49
Saturn9.553126.7039.3683.45
Uranus19.264297.90257.43-15.10
Neptune30.233729.00295.3340.65
Titan0.00820.000.000.00

Velocity
Vorbit
Km/s		Vescape
Km/s		Vequator
Km/s		Vsound
Km/s
Venus35.02210.378-0.0020.236
Earth29.78611.1920.4650.320
Mars24.1305.0450.2400.227
Jupiter13.05960.51112.5720.816
Saturn9.63736.3769.8710.716
Uranus6.78621.411-2.5890.543
Neptune5.41723.6702.6890.543
Titan5.5732.6450.0120.185

Effective (Teff,Peff) Values
Teff
K		Peff
mBar		rho
gm/cm3		H
Km		Vsound
m/s		taurad
years		phase
°
Venus229.020.0004.6230e-054.862235.80.0073.94
Earth255.0500.0006.8297e-047.471320.20.13139.47
Mars210.06.0001.5124e-0510.605226.60.0061.17
Jupiter124.0400.0008.6132e-0519.146816.34.53467.39
Saturn95.0300.0008.1128e-0536.975715.520.79477.27
Uranus59.0430.0002.0161e-0424.234543.4131.31184.15
Neptune59.0500.0002.3443e-0419.206543.4121.00877.67
Titan85.01.0004.0568e-0618.152184.70.05382.54

Stratopause Values
T
K		P
mBar		rho
gm/cm3		H
Km		Vsound
m/s		taurad
years		phase
°
Venus400.050.0006.6166e-058.492311.70.0031.85
Earth270.01.0001.2901e-067.911329.50.0000.08
Mars140.00.0051.8905e-087.070185.00.0000.00
Jupiter163.03.0004.9143e-0725.167935.90.0150.45
Saturn145.02.0003.5435e-0756.435884.00.0390.48
Uranus120.01.0002.3053e-0749.290775.00.0360.15
Neptune130.02.0004.2559e-0742.319806.70.0450.10
Titan170.01.0002.0284e-0636.304261.30.00743.69

Tropopause Values
T
K		P
mBar		rho
gm/cm3		H
Km		Vsound
m/s		taurad
years		phase
°
Venus250.0100.0002.1173e-045.307246.40.02614.83
Earth217.0100.0001.6051e-046.358295.40.04314.96
Mars140.00.0103.7809e-087.070185.00.0000.01
Jupiter110.0140.0003.3983e-0516.984768.82.27350.28
Saturn85.0100.0003.0224e-0533.083676.89.67764.10
Uranus53.0110.0005.7414e-0521.770515.146.34073.81
Neptune54.0200.0001.0246e-0417.579519.963.13267.26
Titan70.0100.0004.9261e-0414.949167.79.50689.96

1 BAR Values
T
K		P
BAR		rho
gm/cm3		H
Km		Vsound
m/s		taurad
years		phase
°
Venus360.01.0001.4704e-037.643295.70.08741.57
Earth288.01.0001.2094e-038.438340.30.18248.82
Mars140.00.0072.6466e-057.070185.00.0244.59
Jupiter165.01.0001.6182e-0425.476941.64.81168.57
Saturn134.01.0001.9172e-0452.154849.824.69979.23
Uranus76.01.0003.6399e-0431.217616.8142.87384.62
Neptune76.01.0003.6399e-0424.740616.8113.23076.85
Titan86.01.0004.0096e-0318.66185.851.26589.99

Surface (H2O) Values
T
K		P
BAR		rho
gm/cm3		H
Km		Vsound
m/s		taurad
years		phase
°
Venus731.092.0006.6619e-0215.519421.40.95484.14
Earth288.01.0131.2252e-038.438340.30.18449.19
Mars214.00.0071.7314e-0510.807228.70.0071.29
Jupiter294.07.0006.3574e-0445.3941256.95.95372.40
Saturn313.021.0001.7237e-03121.8231298.840.69883.41
Uranus366.0260.0001.9651e-02150.3361353.5332.59887.68
Neptune360.0283.0002.1746e-02117.1921342.3301.49584.98
Titan94.01.5005.5026e-0320.074194.358.88789.99

Adiabatic Lapse Rate, Dry

from the first law of thermodynamics

dQ = dU + deltaW = n · cv dT + P dV = 0
where cv is given in units of erg/K/mole and n is the number of moles. The derivative of the ideal gas law, P · V = n R T, is
V dP + P dV = n R dT
equating P dV and noting that R = cp - cv yields
dQ = n cv dT - V dP + n (cp-cv) dT
dQ = n cp dT - V dP = 0 for adiabatic
Cp = cp/<mw> and rho = n · <mw>/V so that
dT/dP = V/(n · cp) = 1 /(Cp · rho)
From hydrostatic equilibrium and the gas law we can convert from pressure to height coordinates:
dP = - g rho dz
dT/dz|a = -g/Cp
Gammaa ident -dT/dz|a = g/Cp


<g>
cm/s2
Cp
J/gm/K		Adiabatic
Lapse Rate
K/Km
Rg
J/gm/K		Autoconvective
Lapse Rate
K/Km
Venus889.890.850110.4680.1889247.104
Earth979.861.00409.7600.2871034.130
Mars374.100.83124.5000.1889219.802
Jupiter2425.6112.35911.9633.745186.477
Saturn1000.0914.01290.7143.892462.569
Uranus880.0713.01370.6763.614912.435
Neptune1110.4613.01370.8533.614913.072
Titan135.801.04401.3010.290004.683

Autoconvective Lapse Rate

From hydrostatic equilibrium and the ideal gas law, P = Rg rho(z) T(z) we have

rho = -(1/g) · (dP/dz) = -(Rg/g) · [ T(z) drho/dz + rho(z) dT/dz ]
dT/dz = - g/Rg - [T(z)/rho(z) · drho/dz]
But T(z) and rho(z) are always positive so that when
Gamma ident -dT/dz geq g/Rg
then drho/dz geq 0. Thus, the autoconvective criteria is that when the density increases with altitude the atmosphere will be forced to convectively adjust and this condition is met when
Gammaauto ident g/Rg

Amagats

From the ideal gas law we can define the number of particles at standard temperature (273.15 K) and standard pressure (1 atm = 1013.25 milli-bars). The ideal gas law can be written many ways, here we will adopt the notation(s)
P = N k T = rho Na k T/<mw> = rho Rg T
where, N is the number of molecules per unit volume, Boltzmann's constant k = 1.3806 · 10-16 erg/K, the gas constant
Rg = Na · k/<mw>, Avogadro's number, Na = 6.02 · 1023 particles per mole, <mw> is the molecular weight in grams/mole.
This number of molecules at STP is called Loschmidt's number and has the value,
N0 = Pstp/(k · Tstp) = 2.687 · 1019 cm-3
The number of amagats of a gas is given as a ratio of the actual number of particles at the given temperature and pressure to Loschmidt's number. In planetary atmospheres the number density is a function of height. We can calculate the thickness of an equivalent atmospheric column at standard temperature and pressure. This is denoted as cm-amagats or Km-amagats and is given by
Z = (1/N0) integ Ni(z) · dz
Hydrostatic equilibrium relates the pressure and vertical coordinates
dP ident - rho g dz, or dz = -dP/(rho g),
and the number density, Ni can be written as a function of the density,
Ni = qi · rho Na/<mw>, molecules/cm3
where, qi is the volumetric fraction of species i.
Z = Na/(g N0) integ qi/<mw> · dP approx Na/(g N0) · ( qi/<mw>) · P = H0 · P
H0 = 10 · Na qi/(g N0 <mw>) Km-amagat/Bar

EarthMarsJupiterSaturnTitanUranusNeptune
g981.0374.12425.31000.0136.0880.11110.5 cm/s
q(H2)1.001.000.900.961.000.850.85
<mw>28.9744.012.222.1428.002.302.30 gm/mole
H07.8813.6137.45100.5058.8594.0874.56 Km-am/Bar
1000/H0126.973.4926.709.9516.9910.6313.41 mB/Km-am

Beer's Law

taunuu = e-sec(theta) integ kappanu rho dz
kappanu = sumkappanu(CO2,CO,CH4,N2O) + kappanu(H2O) + kappanu(O3)

Transmittance between the top of the atmosphere and level at z is a function of the quantity and absorption of all gases.

The taller the glass, the darker the brew,
The less the amount of light that comes through.

Beta Plane Approximation

f = 2 · Omega · sin(phi) approx f0 + f' · phi
beta = f' = (2 · Omega · cos(phi))/R(phi) = beta0 · (Req/R(phi)) · cos(phi),   beta0 ident 2 Omega/Req

Omega
rad/s		f(45)
s-1		f(60)
s-1		f/fearthbeta0
(cm*s)-1
Venus-2.9921e-07-4.2314e-07-4.2314e-07-0.004103-9.889e-16
Earth7.2925e-051.0313e-041.0313e-041.0000002.287e-13
Mars7.0852e-051.0020e-041.0020e-040.9715834.176e-13
Jupiter1.7585e-042.4869e-042.4869e-042.4114384.920e-14
Saturn1.6378e-042.3163e-042.3163e-042.2459495.435e-14
Uranus-1.0128e-04-1.4323e-04-1.4323e-04-1.388781-7.925e-14
Neptune1.0834e-041.5321e-041.5321e-041.4856208.730e-14
Titan4.5608e-066.4500e-066.4500e-060.0625413.542e-14

Brunt Vaisala or Buoyancy Frequency

N2 = (Rg/H(z)) · [ dT0/dz + (Rg/Cp) · (T0(z)/H(z)) ] = (Rg/H(z)) · [ dT0/dz + g/Cp ] = (Rg/H(z)) · sigma
where, T0(z) = <T(z,phi,lambda,t)>, i.e., temperature averaged over longitude lambda, latitude phi, and time t.
The static stability, sigma, is related to the buoyancy frequency as
sigma ident dT0/dz + g/Cp
The static stability is positive for stable conditions and zero or negative for unstable conditions.

Cloud clearing, eta method

Given the clear radiance, Rclr and the fraction of clouds, alphai in scene i.

Ri = (1-alphai) · Rclr + alphai · Rcld
Rclr = R1 + eta · ( R1 - R2 )
eta ident alpha1/(alpha2 - alpha1)

Chahine, M.T. 1974, J. Atmos. Sci. 31 p.233

Susskind, J., J. Joiner and C.D. Barnet 1995. Determination of atmospheric and surface parameters from simulated AIRS/AMSU sounding data. I. Retrieval methodology. submitted to J. Atmos. Oceanic Tech. (JAOT) Sep. 1995.

Cloud clearing, N* method

N* ident alpha1/alpha2 = (R1 - Rclr)/(R2 - Rclr)
Rclr = (R1 - N* R2)/(1 - N*)
eta = N* / (1 - N*)

McMillin, L.M. and C. Dean 1982. Evaluation of a new operational technique for producing clear radiances. J. Appl. Meteor. 21 p.1005-1014.

Smith, W.L. 1968. An improved method for calculating tropospheric temperature and moisture from satellite radiometer measurements. Monthly Weather Review 96 p.387-396.

Clouds, Properties

from NOAA chart

low clouds:

up to 6500 ft
up to 2 Km
up to 800 mBar (275 K)
Sc,St,Cu,Cb (Stratocumulus,Stratus,Cumulus,Cumulonimbus)

middle clouds:

6500 ft to 23,000 ft
2 Km to 7 Km
800 mBar to 400 mBar (240 K)
Ac,As,Ns (Altocumulus, Altostratus, Nimbostratus)

high clouds:

16,500 to 45,000 ft
5 Km to 13.75 Km
530 mBar (255 K) to 140 mBar (216 K)
Ci,Cc,Cs (Cirrus, Cirrocumulus, Cirrostratus)

ReffVeff (Hansen, 1971)
Fair Weather cumulus5.560.111
Altostratus7.010.113
Stratus11.190.193
Cumulus congestus10.480.147
Stratocumulus5.330.118
Nimbostratus10.810.143

Earth is typically covered about 50%
average precipitable water is about 2.5 cm
average time aloft is 9 days (from 100 cm of rain/year)
in thunderstorm approximately 10% of water is released

suspended droplets 1 µm to 100 µm
cloud over continents have smaller droplets

Dobson Units

DOBSON ident 10-3 cm-amagat
given the number density, N, in molecules per cm3 then
rhox = (<mw>x · Nx(z))/Na grams/cm2
C(L) ident integ N(z) dz appeq <N(L)> · Delta z = <N(i)> · Delta P/(rhot g) = q · Na · Delta P/(<mw> g) molecules/cm2
Note that rho = q · rhot = N · <mw> /Na where q is the volumetric mixing ratio of the species being measured.
To convert to mass column density, M, in grams per cm2
M(L) = (<mw>O3 /Na) · C(L) = Delta P/ g
DOBSON = 1000 * Z = 1000 * sumC(L) / N0
where N0 is Loschmidt's number.

The atmosphere consists of fixed gases (e.g., CO2, N2O, CO), water, and ozone so the pressure within any level is given as

Delta p(L) ident integ rhot g dz = g integ (rhof + rhow + rhoo) dz
so that
Delta p(L) · Na / g = <mw>f · Cf(L) + <mw>w · Cw(L) + <mw>o · Co(L)
Cf(L) = (Delta p(L) · Na)/(<mw>f · g) - (<mw>w/<mw>f) · Cw(L) - (<mw>o/<mw>f) · Co(L)
we can also define the total column density as
Ct(L) ident (Delta p(L) · Na) / (<mw>(L) · g)
and if we require
Ct(L) = Cf(L) + Cw(L) + Co(L)
then,
<mw>(L) = (<mw>f · Cf(L) + <mw>w · Cw(L) + <mw>o · Co(L)) / Ct(L)
Ct(L) in eqn 6 and <mw> can be solved for iteratively with an initial guess of <mw> = <mw>f
The volumetric mixing ratio, i.e., molecules of species x to the total number of molecules, is given by
fo(L) = Co(L) / Ct(L) = Co(L) / (Delta P(L) · Na) · <mw>(L) · g

Double Linear Interpolation

Delta x = s - s0,       0 leq Delta x leq 1
Delta y = l - l0,       0 leq Delta y leq 1
s1 = s0 + 1,       l1 = l0 + 1
DN(s,l0) = DN(s0,l0) + Delta x · (DN(s1,l0)-DN(s0,l0))
DN(s,l1) = DN(s0,l1) + Delta x · (DN(s1,l1)-DN(s0,l1))
DN(s,l) = DN(s,l0) + Delta y · (DN(s,l1) - DN(s,l0))
or
DN(s,l) = DN(s0,l0)

+ Delta x · (DN(s1,l0)-DN(s0,l0))
+ Delta y · (DN(s0,l1)-DN(s0,l0))
+ Delta x Delta y · (DN(s1,l1) + DN(s0,l0) - DN(s0,l1) - DN(s1,l0))

Earth: molecular abundances

From US Standard Atmosphere 1976, Table 3 and page 33 (Trace Constituents)
Gas<mw>
gm/mole
H1.00794
C12.011
N14.00674
O15.9994
S32.066

Gas<mw>
gm/mole		volumetric
fi
N228.01340.78084
O231.99880.209476
Ar39.9480.00934
CO244.00995314 ppmv
Ne20.18318.18 ppmv
He4.00265.24 ppmv
Kr83.801.14 ppmv
Xe131.300.087 ppmv
CH416.043031.5 ppmv
H22.015940.5 ppmv
<<mw>>28.9644
H2O18.016f approx 0.005 at 1 km (900 mB)
O347.9982f approx 20 ppmv at 36 km (5 mB)
N2O44.0129270 ppbv
NO30.00610.5 ppbv
NO246.00551 ppbv
H2S34.08190.05 ppbv
NH317.03064 ppbv
SO264.06481 ppbv
CO28.0104190 ppbv

Methane, CH4
1992 1.7086 ppm
1984 1.6180 ppm --> +0.0113 ppm/year = +0.7 %/year
0.02 ppm peak-to-peak seasonal variation = ± 0.01 ppm = ± 0.62 %
qCH4 = 1.618 + 0.0113 · (t-1984.0) ± 0.01 ppm

Carbon Dioxide, CO2
1995 361.0 ppm
1980 336.8 ppm --> +1.6 ppm/year = +0.475 %/year
0.105 ppm peak-to-peak seasonal variation = ± 0.052 ppm = ± 0.015 %
qCO2 = 337 + 1.6 · (t-1980.0) ± 0.052 ppm

C12O16O18 = 4.08 · 10-3 · C12O16O16
C12O16O17 = 7.42 · 10-4 · C12O16O16     Kiehl & Ramanathan 1983. JGR 88 p.5191.
C13O16O16 = 1.12 · 10-2 · C12O16O16

Earth: molecular spectral regions

CO2 15 µm spectral region
nu2 (667.381 bending mode) vibrational-rotational band
nu1 (1388.23 symmetric stretch mode) fermi-resonance stimulated by nu2 (618.029, 720.805)
B(CO2) = 0.39 cm-1
Q-branch at 667.5 (2 mb)
line wings have 1.5 scale height wide weighting functions
line centers have 2.5 scale height weighting functions

CO2 band correlation parameters at 180 K, Cess & Ramanathan, 1972
band
cm-1		S at 180 K
cm-2 amagat		A0
cm-1		gamma0
atm-1 cm-1		d
cm-1
66754917.30.097(300/T)(2/3)1.56
2350450515.40.097(300/T)(2/3)??
371510232.20.097(300/T)(2/3)??

CO2 band correlation parameters, Kiehl & Ramanathan, 1983
band
cm-1		transition
nu1nu2nu3		S at 300 K
cm-2 amagat		Ej
cm-1		d
cm-1
667.3810000 --> 01101940.01.56
667.7510110 --> 022015667.3810.78
720.805(I) 0110 --> 10005.00667.3811.56
618.029(II) 0110 --> 10004.27667.3811.56
668.1070220 --> 03300.851335.1310.78
647.063(II) 1000 --> 11100.71285.4101.56
668.670(I) 1000 --> 11100.31388.1851.56

NOTE: for C12O16O17 and C12O16O18 the mean line spacing is 0.78 for all bands

gamma0 = 0.067 (300/T)(2/3) atm-1 cm-1
A0 = 22.18 (T/296)0.5 cm-1
S(T) appeq S(T0) · (T0/T) [ (1 - exp(-1.439 nu/T))3 ] / [(1 - exp(-1.439 nu/T0))3 ]

CO2 4.3 µm spectral region
nu3 (2349.16 asymmetric stretch mode) band, no Q-branch
high T dependence of high J lines makes narrower weighting functions free of isotopes and hot lines

H2O 6.3 µm spectral region}

bandH2OHDO
nu13657.052723.68
nu21594.751403.49
nu33755.933707.47

non-linear molecule with angle of 104.45° nu1 ~ nu3 ~ 2 · nu2

O3 9.6 µm spectral region
nu1 1110 cm-1
nu2 1045 cm-1
nu3 701 cm-1

CH4 spectrum
of the 9 modes, only nu1 through nu4 are independent, and nu3 and nu4 are triply degenerate.
13C is 1.108 % of all Carbon

The Strongest Bands of Methane (rel. to ground state) from Andrews et al.
band12CH413CH4
nu41310.761302.77
nu21533.37
nu33018.923009.53
2nu42612
nu2 + nu428302822
2nu2 + nu43062
nu1 + nu44223
nu3 + nu44340
nu2 + nu34540

CH4 band correlation parameters, Cess & Chen, 1975
band
cm-1		S
cm-2 amagat-1		A0
cm-1		gamma0
atm-1 cm-1		d
cm-1
130618552(T/300)0.50.075(300/T)0.55.3
3020320124(T/300)0.50.075(300/T)0.510.5
422020124(T/300)0.50.075(300/T)0.510.5
58613124(T/300)0.50.075(300/T)0.510.5

Earth: Oceans

Thermocline:
ocean layer in which temperature decreases rapidly with depth. From 200 to 1000 meters the temperature decreases by 15 degrees F. In summer the thermocline becomes thin and gradients of 1 degree per foot can be observed.

Oceans:
P = g*rho*Z

P(Bar) = 981 cm/s2 * 12*2.54 * Z(ft) * rho(gm/cm3) * 1.0E-6 Bar/(dyn/cm2)
P(Bar) = 31.0 * Z(1000 ft)

Z(ft)rho (gm/cm3)
01.028 (assume 35 parts-per-thousand salinity)
10001.033
20001.0375
200001.071

ZP
32010 Bar
3,200100 Bar
3,790120 Bar <Z> of all Oceans - depth of life Zone
4,690145 Bar depth of Mediterranean
4,900150 Bar maximum depth of life
7,282225 Bar depth of Gulf of Mexico
20,000600 Bar max. depth of Atlantic
35,6001100 Bar 1960 depth of Trieste into Chall.II (7 miles!!)

Note: ratio of Volume of Ocean to Land (above Z=0) = 11
ratio of Area of Ocean to Land:

N 61:39 = 3:2
S 81:19 = 4:1
all 142:58 = 5:2
source: 1982 Encyclopedia Brittanica

Equations of motion

Momentum

dV/dt = - 2 Omega · V - (1/rho) delP + k g + F
V ident i u + j v + k w
du/dt - 2 Omega v sin(phi) = (-1/rho) (partial P/partial x) + (u v tan(phi)/a) - (u w/a) - 2 Omega w cos(phi) + Fx
dv/dt + 2 Omega u sin(phi) = (-1/rho) (partial P/partial y) - (u2 tan(phi)/a) - (v w/a) + Fy
dw/dt - 2 Omega u cos(phi) = (-1/rho) (partial P/partial z) + ((u2 + v2)/a) - g + Fz

Continuity

(1/rho) (d rho/dt) + del · V = 0
when motions are on the order of a scale height or less
del · (rho0V) = 0

Energy

q'/T = cp (d loge(Theta)/dt) = (cp/T) (dT/dt) - (Runiv/P) (dP/dt)

FFT

Forward FFT

F(u) = FFT(f(x),-1) ident 1/m sumf(x) · e-i · u · (2 pi x/m)     for u = 0,m-1

Inverse FFT

f(x) = FFT(F(u),+1) ident sumF(u) · ei · x · (2 pi u/m)     for x = 0,m-1

Forward Cosine transform

F(u) ident (1/m) · integ f(x) · cos(u · pi x/m) · dx approx [(f(0) + f(m) · cos(pi u))/2 m] + 1/m sum f(x) · cos(u · pi x/m) = FFT(g(x),-1)

g(x) = f(x)     for x = 0,m
g(2m-x) = f(x)     for x = 1,m-1

Inverse Cosine Transform

f(x) = F(0) + F(m) · cos(pi x) + 2 · sum F(u) · cos(x · pi u/m) = FFT(G(u),+1)

G(x) = F(x)     for x = 0,m
G(2m-x) = F(x)     for x = 1,m-1

Gas Constant

Runiv = cp - cv = 8.3143 Joules/mole/K = 8.3143 · 107 erg/mole/K
<mw>air = 28.964 gm/mole
Rg(air) = Runiv/<mw>air = 287.05 Joule/Kg/K
Cp(air) = 1005 J/Kg/K
kappa ident Rg/Cp = Runiv/cp
gamma ident Cp/Cv = cp/cv

<mw>
gm/mole		Rgas
J/gm/K		Cp
J/gm/K		kappagamma
Venus44.010.188920.85010.22221.2857
Earth28.960.287101.00400.28601.4005
Mars44.010.188920.83120.22731.2941
Jupiter2.223.7451812.35910.30301.4348
Saturn2.143.8924614.01290.27781.3846
Uranus2.303.6149113.01370.27781.3846
Neptune2.303.6149113.01370.27781.3846
Titan28.670.290001.04400.27781.3846

Geopotential

dPhi = -g · dz
Phi = integdPhi

Geostrophic Approximation

Horizontal motions are in balance with the pressure gradient force. Friction is negligible ( i.e., far away from surface). Steady flow with small curvature ( i.e., dV/dt = 0 ). Works well for z geq 1 km, phi > 10°.

-rho · f · Vg = partial P/ partial x
rho · f · Ug = partial P/ partial y

Gravity

Gravitational Forces
<g>
cm/s2		<g>
gearth		g(0)
cm/s2		g(45)
cm/s2		g(90)
cm/s2
Venus889.890.9082889.89889.89889.89
Earth979.861.0000976.81981.29987.16
Mars374.100.3818372.40374.90378.10
Jupiter2425.612.47552256.642504.582833.44
Saturn1000.091.0206882.361055.281283.11
Uranus880.070.8982860.29889.23928.60
Neptune1110.461.13331087.161121.311167.07
Titan135.800.1386135.80135.80135.80

q = (omega2 · a3)/(G · M)

106*J2106*J4106*J6106*q
Venus
Earth
Mars1960.454
Jupiter14697-584.031.089180
Saturn16331-914.0108.0154766
Uranus3516-31.90.029513
Neptune40000.00.028960
Titan

Hubbard and Marley (1989)
Kieffer

Heat Balance

Fsun = S0/<rs>2     S0 ident 1.37 · 106 ergs/cm2
Pabs = (1 - Ab) · Fsun · sunlit area = (1 - Ab) · Fsun · pi · Req · Rpl
a = total area = 2 pi · R2eq + pi (R2pl/e) · loge [ (1 + e)/(1 - e)]
e = sqrt[R2eq-R2pl]/Req
Fabs = Pabs/a
Teq = [Pabs/(a · sigma)]0.25
If the eccentricity is equal to zero, e = 0, then
a = 4 pi · Req · Req
Fabs = (1 - Ab) · Fsun/4
Teq = [ (1 - Ab) · Fsun/(4 · sigma) ]0.25

bond
albedo		Fsun
erg/s/cm2		Fabs
erg/s/cm2		Pabs
erg/s		Teq
K
Venus0.7502618462.8163653.97.5299e+23231.89
Earth0.3061370000.0237414.71.2110e+24254.49
Mars0.250590102.3110447.61.5922e+23210.18
Jupiter0.34350616.08123.34.9933e+24109.45
Saturn0.34215011.72381.41.0167e+2480.54
Uranus0.3003691.7641.05.1818e+2258.01
Neptune0.2901498.8264.02.0141e+2246.47
Titan0.22015011.72927.32.4391e+2184.80

Pint = Fint · a
Fout = Fint + Fabs
ratio = Fout/Fabs = (Fint + Fabs)/Fabs
Teff = (Fout/sigma)0.25

Fint
erg/s/cm2		Pint
erg/s		Fabs+Fint
erg/s/cm2		Teff
K		out/in
ratio
Venus0.00.0000e+00163653.9231.891.0000
Earth62.03.1624e+20237476.7254.511.0003
Mars0.00.0000e+00110447.6210.181.0000
Jupiter5440.03.3439e+2413563.3124.421.6697
Saturn2010.08.5815e+234391.493.851.8440
Uranus42.03.3953e+21683.058.941.0655
Neptune433.03.3029e+22697.059.242.6399
Titan0.00.0000e+002927.384.801.0000

Humidity

Absolute humidity

From the ideal gas law, the density of water, rhow is given by
rhow= ( <mw>w fw e)/(R* T)
where <mw>w is the molecular weight of water (18.016 gm/mole), fw is a correction factor for non-ideal behavior which can be taken as fw = 1, e is the partial pressure of water, R* is the universal gas constant, and T is the local temperature.

Specific humidity

The total density, rho, is given by the sum of dry and moist densities ( i.e., assumes all other species are included in ``dry'').
rho = rhod + rhow = [ <mw>d · (P - fw e)]/(R* T) + (<mw>w fw e)/(R* T) = [<mw>d · (P - 0.37803 fw e)]/(R* T)
where <mw>d is the molecular weight of dry air (28.966), <mw>w/<mw>d = 0.62197, and (<mw>d - <mw>w)/<mw>d = 0.37803. The specific humidity is the ratio of the moist to total density and is dimensionless, but is usually expressed in gm/Kg units.
q = rhow/rho = (0.62197 fw e) / (P - 0.37803 fw e) appeq 0.622 e/P
Note that
rhow = q · rho
and the volumetric mixing ratio, f, (i.e., ratio of water number density, Nw, to the total number density Nt) is given by
fw ident Nw/Nt = <mw>/<mw>w · q
where the average molecular weight, <mw>, is given by
<mw> = <mw>d · (1 - fw) + <mw>w · fw

Mass Mixing Ratio

r = rhow/rhod = 0.62197 · [(fw e)/(P - fw e)] appeq 0.62197 · e/(P - e) approx q

from US Standard Atmosphere 1976, Table 20, pg. 44, r(z) ppm by mass
Alt
(km)		P
(mB)		record
low		1%
low		midlat.
mean		1%
high		record
high
sfc1013.250.15.04,68630,00035,000
189024.027.03,70029,00031,000
279021.031.02,84324,00028,000
461016.024.01,26818,00022,000
64706.212.05547,7008,900
83506.16.12164,3004,700
102655.343.21,300
121951.211.3230
141401.53.348
161001.03.338

% Relative Humidity

U ident 100 · r/rs appeq 100 q/qs appeq 100 e/es
where rs ident r(e = es) and qs ident q(e = es).
The saturated vapor pressure is approximately given by
es appeq 6.11 · 10[(a T)/(b + T)] mBar
rs appeq 0.62197 · es/(P - es)   for es > P
rs appeq 0.62197   for es geq P
where T is given in °C and a and b are constants.

ab
above water7.5237.3
above ice9.5265.5

Dew Point Temperature

The Dew point is the temperature at which the partial pressure of water reaches the saturation value, that is
e = (rhow R* T)/(<mw>w fw) = es(Tdp)
The empirical expression for es can be used or a table lookup can be utilized after e is calculated.

Hurricanes

Tropical Classification
Gale force winds(> 15 m/s)
Tropical Depression(20-34kts and a closed circulation)
Tropical Storm (named)(35-64kts)
Hurricane(65+kts or 74+mph)

Saffir-Simpson Scale
knotsm/hPs,(mB)inch.Hg
Category 164- 8374- 95>980>28.94
Category 283- 9596-110965-97928.50-28.91
Category 396-113111-130945-96427.91-28.47
Category 4114-135131-155920-94427.17-27.88
Category 5>135>155<920<27.16

Note: 1 knot = 1 nautical mile/hour = 1.151555 mile/hour = 1.853248 Km/hr

StormDateCatPcRcwinds
NancySep.12,1961888185 kt
TipOct.12,197987085 m/s
Camille19695909165 kt
Allen1980165 kt
GilbertSep. 1988888
Hugo19894
Andrew19924922

Empirical formula for approximate energy, ergs, of a storm with a central pressure, Pc, given in mBar and a radius of nearest circular isobar, Rc, given in kilometers is

E = 0.712 · 1022 · (1010 - Pc) · (Rc/111)2 ergs

Hydrostatic Equilibrium

partial P ident -g · rho · dz = -g · n · <mw>/V · dz
If the Ideal gas law holds (note: Rg = Runiv/<mw>), then
partial P = -P · (g · <mw>)/ (Runiv T) · dz = -P · dz/H(z),   where H(z) = Rg(z) · T(z)/g(z)
If the scale height is constant, then
P = P0 · exp(-z/H0),   where H0 = Rg · T/g

Ideal Gas Law

P · V = N · k · T = n · R · T
where N is the total number of molecules, n is the total number of moles, Na = 6.02217 · 1023 molecules/mole, Runiv = 8.314 · 107 erg/K/mole, k = Runiv/Na = 1.380622 · 10-16 erg/K
P = (N/V) k T = (rho · Na/<mw>) k T
Rg ident Runiv/<mw> erg/K/gm
P = rho · (Runiv/<mw>) · T = rho Rg T

IR filter bandpasses

µmrangetransthick
J1.251.1 - 1.460%2.0mm
H1.651.5 - 1.860%2.0mm
K2.202.0 - 2.460%1.5mm
L3.823.5 - 4.1560%1.0mm
M4.704.4 - 5.060%1.0mm

Latitude: planeto-graphic and planeto-centric

Planeto-centric latitude, theta and the planeto-graphic latitude, phi

beta ident Req/Rpl
theta= tan-1 [ tan(phi)/beta2 ]
phi = tan-1 [ tan(theta) · beta2 ]
R(theta) = Req Rpl/sqrt[R2eq · sin2(theta) + R2pl · cos2(theta)]
f = (Req - Rpl)/Req
e = sqrt[ R2eq - R2pl]/Req = sqrt[ 1 - (Rpl/Req)2]
S = total area = 2 pi · R2eq + pi (R2pl/e) · loge [ (1 + e)/(1 - e) ]
V = total volume = 4/3 pi R2eq · Rpl
If e = 0 then
S = 4 pi · Req · Req

CRC Standard Mathematical Tables 26th edition, pg. 129. The volume of an oblate sphere is equal to a sphere with radius <R>.

<R> = (R2eq · Rpl)(1/3)

Req
Km		Rpl
Km		Req/Rplfe<R>
Km
Venus6051.06051.01.0000000.0000000.0000006051.00
Earth6378.56356.01.0035400.0035270.0839206370.99
Mars3393.03375.01.0053330.0053050.1028693386.99
Jupiter71492.066854.01.0693750.0648740.35431669911.33
Saturn60268.054364.01.1086010.0979620.43165858232.01
Uranus25559.024973.01.0234650.0229270.21290625362.16
Neptune24820.024274.01.0224930.0219980.20859724636.66
Titan2575.02575.01.0000000.0000000.0000002575.00

Longitude Systems and conversion

P1 = reference rotational period of system #1
P2 = reference rotational period of system #2
L1(t-t0) is a known system #1 longitude relative to time t0
L2(t-t0) is the desired longitude value in system #2 relative to time t0

L2(t-t0) = L1(t-t0) + L21 + (t-t0) · [ (24 · 360/P2) - (24 · 360/P1) ]
L21 is a reference longitude. It is defined as:
L21= L2(t=t0) - L1(t=t0)

Jupiter

SI = 9h 50m 30.003s = 877.90°/24h. Used for equatorial motions.
SII = 9h 55m 40.632s = 870.27°/24h. Used for mid-latitudes and GRS.
SIII = 9h 55m 29.711s ± 0.04s = 9.9249197h ± 0.000011h = 870.536°/24h ± 0.000966°/24h
lambdaI = 67.10 + 877.900 · (t-t0)
lambdaII = 43.30 + 870.270 · (t-t0)
lambdaIII = 284.95 + 870.536 · (t-t0)
lambda(SIII) = lambda(SI) + 217.85 - 7.364 · (t-t0)
lambda(SIII) = lambda(SII) + 241.65 - 0.266 · (t-t0)
where t0 = 2451545.0 TDB or Jan. 1.5, 2000.
UIII(SI) = 106.35 m/s
UIII(SII) = -3.4 m/s

Saturn

SIII = 10h 39m 22.4s ± 7s = 10.656222h ± 0.00194h = 810.7939024°/24h ± 0.148°/24h
lambdaIII = 38.90 + 810.793902 · (t-t0)
lambdaI = 227.2037 + 844.3 · (t-t0)
lambda(SIII) = lambda(SI) + 171.6963 - 33.5061 · (t-t0)
UIII(SI) = 410.36 m/s

For the Voyager encounters the reference time should be set to the closest approach time. Other possibilities are:

P1P2FDS(t=0)L21
Neptune17.86616.1111200.0-334.5365

Orbital: Equations of Motion

<r> = a/[ 1 + (e2/2) ]

Orbits: Earth

V = sqrtt[ (G · Msmearth)/r ] cm/sec
where,
G = 6.67 · 10-8 dyne cm2/gm2
Msmearth = 5.975 · 1027 grams
r = 105 · (<Rsmearth> + z)
Rsmearth = 6374.87 Km
P = (2 · pi · r)/(V · 3600) hours
Vg = (2 · pi · 105 · Rsmearth)/(P · 3600) cm/sec

r(Km)P(hr)V(Km/s)Vg(Km/s)
200.01.4747.7867.549
400.01.5427.6707.217
700.01.6457.5056.763
900.01.7157.4016.486
35863.924.0003.0720.464

Planck function

The Planck function, Bnu(T), is given by

Bnu(T) = (alpha1 nu3)/[exp(alpha2 nu/T) - 1]
where,
nu = wavenumber in cm-1 = 104/lambda, lambda in µm.
T = temperature in degrees Kelvin
alpha1 = 2 h c2 = 1.191066 · 10-5 for radiance in units of mW · m-2 · steradian-1/cm-1.
alpha2 = h c / k = 1.438833 K/cm-1
h is Planck's constant (6.62620 · 10-34 Joule)
c is the speed of light (2.99793 · 108 m/second)
k is Boltzmann's constant (1.38062 · 10-23 Joule/K)
sigma is the Stefan-Boltzmann constant = 5.67 · 10-5 erg · s-1 · cm-2 · K-4 or mW · m-2 · K-4

Radiant Energy = A · sigma · T4

The brightness temperature, Tb, of a given radiance, Rnu, is found with the inverse of the Planck function.

Tb ident B-1nu(Rnu) = (alpha2 · nu)/ loge [ 1 + (alpha1 nu3/Rnu) ]

The derivative of the Planck function is given by

partial Bnu/partial T = (alpha1 alpha2 nu4/T2) · exp(alpha2 nu/T)/[exp(alpha2 nu/T) - 1]2

Infrared approximation: for nu geq 600 cm-1 and T leq 300 K.

Bnu(T) appeq alpha1 nu3 exp(-alpha2 nu/T)   mW · m-2 · steradian-1/cm-1
Tb appeq (-alpha2 · nu)/ loge [ (alpha1 nu3/Rnu) ]
partial Bnu/partial T approx (alpha1 alpha2 nu4)/T2 · exp[- alpha2 nu /T]

Microwave Rayleigh Jeans approximation: lambda(mm) = 300/f, nu = f/30 cm-1, f is in GHz

Wavenumbers, nu, are frequency units and are assumed to be in vacuum while wavelength is the wavelength specified within the medium. Typically, wavelengths are expressed as wavelength in air, lambdaa or wavelength in vacuum, lambdav.

nu ident 10000/lambdav
lambdav = na · lambdaa
nu = 10000/(na · lambdaa)     na = 1.00027

fcm-1µmmm
0.66 GHz0.02454545454.55P band (SAR)
1.25 GHz0.04240000240.00C band (SAR)
5.33 GHz0.185628556.29L band (SAR)
6.6 GHz0.224545445.45MIMR (surface)
23 GHz0.771304313.04AMSU-A
50 GHz1.6760006.00AMSU-A
60 GHz2.0050005.00AMSU-A
89 GHz2.9733713.37AMSU-C
118 GHz3.9325422.54MHS-X
183 GHz6.1016391.64AMSU-B

Bnu appeq alpha1/(alpha2 · c2 · 10-18) · f2 · T = 9.2105 · 10-9 · f2 · T   mW · m-2 · steradian-1 ·/cm-1
Tb appeq 1.0857 · 108 Rnu/f2
partial Bnu/partial T approx 2 · c · k · nu2 = alpha1/alpha2 · nu2 = alpha1/(alpha2 · c2 · 10-18) · f2

wavelength
mm		frequency
GHz		B(T=300K)
mW/m2/ster/cm-1		dB/dT
mW/m2/ster/cm-1/K		(1/B) · dB/dT
%/K
6.0500.0070.0000230.335
3.01000.0270.0000920.336
2.01500.0610.0002070.337
1.52000.1090.0003680.338

wavelength
µm		wavenumber
cm-1		B(T=300K)
mW/m2/ster/cm-1		dB/dT
mW/m2/ster/cm-1/K		(1/B) · dB/dT
%/K
16.7600153.381.5591.0
9.1110081.491.4411.8
6.2160022.690.5812.6
4.323002.350.0863.7
3.727000.560.0244.3
3.330000.180.0094.8

Planetary Missions

Mars

1965
Mariner 4 : 22 low resolution pictures
1969
Mariner 6 : flyby
1969
Mariner 7 : flyby
1971
Mariner 9 : orbiter (resol. = .1-1 km)
1976
Vikings : 55,000 images, resol. 150-300 meters
Viking 1 orbiter (4 yrs)
Viking 1 lander 22.27N 47.97W
       July 20, 1976 4:13 pm LMT
              T=250 K
Viking 2 orbiter
Viking 2 lander 47.57N 225.74W
       Sept. 3, 1976 9:49 am LMT
       T=150 K diurnal variation 35-50K
Martian Year :
N. polar cap H2O ice
S. polar cap CO2 ice
Olympus Mons 600 km diameter volcano, 26 km tall
       4 km high scarp (cliff) rings its base
``face'' on Mars is located in the Cydonia Mensae region at roughly 40.9 degrees North latitude and 9.45 degrees West longitude.

Outer Planet Mission Summary

PlanetCraftDateJulianLs
JupiterPioneer 1012/ 3/732442019.51.4
Pioneer 1112/ 2/742442383.533.9
Voyager I3/ 5/792443937.5170.4
Voyager II7/ 9/792444063.5180.4
SaturnPioneer 119/ 1/792444117.5354.1
Voyager I11/12/802444555.58.9
Voyager II8/25/812444841.518.4
UranusVoyager II1/24/862446454.5271.3
NeptuneVoyager II8/25/892447753.5243.3

Potential Temperature

Derived directly from integration of the 1st law of thermodynamics. It is the temperature a parcel of air at P and T would have if it were at Ps. It is conserved for adiabatic motions, ( i.e., dTheta/dt = 0).

dQ/(n · <mw> · T) = (Cp dT/T) - (Rg dP/P) = 0
Theta = T · (Ps/P)kappa = P/(rho Rg) [(Ps/P)kappa]     kappa = Rg/Cp
For earth kappa = 0.286 (<mw> = 28.96, Cp = 1.004 Joules/gram/K). Some authors write this equation with gamma = Cp/Cv = 1/(1-kappa)
partial P/partial z = - rho · g = (P · g)/(Rg · T) = - P/H(z)
partial log(P) = -H(z) · partial z
therefore,
P/Ps = e- integdz'/H(z')
Theta = T · e- integdz'/H(z')
if H(z) = H0
Theta = T · e-kappa · z/H0

Potential Vorticity

For adiabatic motion ( i.e., Theta is a constant) the density is rho = P(Cv/Cp). For motions between layers of constant Theta

q = (zeta + f) · partial Theta/partial p = constant
In the Shallow Water Model, where rho is a constant ( i.e., incompressible), the conserved quantity is usually written as:
q ident (zeta + f)/(g h)   seconds/meter2
example : flow over a mountain
<u> is eastward (westerly) then flow turns towards equator as it passes mountain, then oscillates around original latitude.
<u> is westward (easterly) then flow turns equatorward (note flow turning poleward `bucks' the flow causing it to turn equatorward. System is returned to original latitude after it passes mountain. No oscillations.

Radiative time constant

The volumetric heating rate, Q

partial Q/partial T ident 1/taur
Qir = sigma · T4 · partial epsilon/partial z
taur = (rho Cp)/(4 sigma T3 partial epsilon/partial z)
Harshvardan and Cess (1976, Tellus 28 p.1-10) give a value for Earth CO2 of partial epsilon/partial z = -0.0081 per Km and Cess and Khetan (1973, JQSRT 13 p.995-1009) give a value for Jupiter, Saturn, Neptune, and Uranus of partial epsilon/partial z = -0.0068, -0.0063, -0.0061, and -0.0058 per Km, respectively.
taur = (rho H Cp)/(4 epsilon sigma T3)
The phase shift, Phi, is the seasonal lag due to the radiative time constant. It arises from the 1st Fourier moment of the thermal response. A value of Phi=90° indicates a full seasonal shift ( e.g., the warm maximum expected at the summer solstice would occur at the autumnal equinox).
Phi = tan-1[ 2 pi · taur/tauo ]
where tauo is the orbital period. For epsilon = 0.3 the values are:

Teff
K		Peff
mBar		rho
gm/cm3		H
Km		taurad
years		phase
°
Venus229.020.0004.6230e-054.8620.0073.94
Earth255.0500.0006.8297e-047.4710.13139.47
Mars210.06.0001.5124e-0510.6050.0061.17
Jupiter124.0400.0008.6132e-0519.1464.53467.39
Saturn95.0300.0008.1128e-0536.97520.79477.27
Uranus59.0430.0002.0161e-0424.234131.31184.15
Neptune59.0500.0002.3443e-0419.206121.00877.67
Titan85.01.0004.0568e-0618.1520.05382.54

Radiative transfer Equation

Thetai,calc = B-1nu(Ri,calc)
Ri,calc = epsilon"i Bi(Ts) + rho'i Hi tau'i(ps) cos(theta0) + integ Bi(T(p)) (dtauui/dln p) (dln p) + rhoi tauui(ps) integ Bi(T(p)) (dtaudi/dln p) (dln p)

Bi(T) is the Planck function evaluated at the effective channel wavenumber.
Hi is the channel averaged solar irradiance at the top of the atmosphere.
theta0 is the local zenith angle of the Sun.
tau'i is the channel averaged two path transmittance from the Sun to the surface to the satellite.
tauui(p) is the atmospheric transmittance measured between p and the top of the atmosphere for channel i.
taudi(p) is the atmospheric transmittance between p and the surface for channel i.

Explicit retrieved parameters
epsiloni is the surface emissivity at nui.
Ts is the surface temperature.
T(p) is the surface temperature profile.
rhoi is the surface spectral reflectivity at nui.
rho'i is the surface spectral bidirectional reflectance of solar radiation at nui.

Implicit retrieved parameters ( i.e., within taui and tau'i).
CO2(p) is the carbon dioxide profile.
q(p) is the humidity (water) profile.
O3(p) is the ozone profile.
CO(p) is the carbon monoxide profile.
CH4(p) is the methane profile.
NO2(p) is the nitrogen dioxide profile.

Rayleigh Scattering: Optical Depth

The average scattering cross section per particle is given by

sigma(lambda) = (128 pi5 alpha2)/(3 lambda4) · [ (6 + 3 delta)/(6 - 7 delta) ]
alpha = (n - 1)/(2 pi N0) approx (n2 - 1)/(4 pi N0),   for n near unity
where, n is the index of refraction, N0 is the number of molecules per unit volume at standard temperature and pressure ( i.e., conditions of index of refraction), delta is the depolarization factor.

The optical depth can be related to the ``thickness'' of the atmosphere. If Z is the thickness in Km-amagats then:

tauray(lambda) = integ N(z) · sigma(lambda) dz approx sigma(lambda) · integ N(z) · dz = Z · N0 · sigma(lambda)
The index of refraction has a wavelength dependence. This is usually represented by two constants, A and B, as follows:
(n-1) = A · (1 + B/lambda2),   lambda in µm
(n-1)2 = A2 · (1 + 2 B/lambda2 + B2/lambda4)
tauray = Z a0/lambda4 · (1 + a1/lambda2 + a2/lambda4)   where,
a0 = (32 pi3)/(3 N02) · [(6 + 3 delta)/(6 - 7 delta) ] · sum qi · A2
a1 = sum qi · 2 B,   and   a2 = sum qi · B2
tauray(H2) = Z · 2.19 · 10-4/lambda4 · ( 1 + 0.0157248/lambda2 + 0.0001978/lambda4 ),   lambda in µm
The wavelength dependence in the VIAMP code is taken from Dalgarno and Williams (ApJ, 1962). This equation is similar to the equation above, however, it differs in the lambda8 term.
sigmaray(H2) = (8.14 · 10-13/lambda4 + 1.28 · 10-6/lambda6 + 1.61/lambda8) cm2,   lambda in Å
The optical depth per Km-amagat of hydrogen, denoted by tau1(H2), is then
tau1(H2) = 2.687 · 1024 · sigmaray(H2) = 2.687 (8.14 · 1011/lambda4 + 1.28 · 1018/lambda6 + 1.61 · 1024/lambda8),   lambda in Å
tau1(H2) = 2.687 (8.14 · 10-21 f4 + 1.28 · 10-30 f6 + 1.61 · 10-40 f8),   f = 108/lambda   and   lambda in Å
And the total optical depth for a mixture of gases is given as
tauray = tau1(H2) · sum Zi · [ (ni - 1)2 / (nH2 - 1)2 ]
Note that this formulation always uses the wavelength dependence of hydrogen, even if other gases are used.
(n-1) = A · (1 + B/lambda2),   lambda in µm
ray = (n-1)2/(nH2 - 1)2

from Allen (pg. 92)
raynA
(10-5)		B
(10-3)		alpha
(10-24)		deltaa0
µm4/Km
air4.44591.000291828.715.673.400.03110.7E-4
H21.00001.000138413.587.521.610.022.35E-4
He0.06411.00003503.482.30
O23.86341.00027226.635.070.054
N24.60351.00029729.067.73.440.03010.9E-4
H2O3.36901.000254
CO210.56111.000449843.96.40.09
CO5.82471.00033432.78.1
NH37.34271.00037537.012.0
NO4.60351.00029728.97.4
CH410.15091.000441

The depth of penetration of Rayleigh scattering is computed for various objects using the data above. A summary of the characteristics of the plots in Figure 9 is given below (g is gravity in cm/s2, µ = molecular weight in gm/mole, Z = KM-amagats per Bar).

Earth P0 = 1013.25, g= 981, µ=28.97, Z=7.88
wave(µm):0.10000.20000.26400.30000.40000.5000
tau(P0):209.576.952.051.190.360.14
P(tau=1):4.8145.7493.2851.02834.77094.5

Jupiter P0 = 1000.00, g=2425.3, µ=2.22, Z=37.45/0.9
wave(µm):0.10000.20000.26400.30000.40000.5000
tau(P0):299.828.622.471.420.420.17
P(tau=1):3.3116.1405.2706.32390.86030.4

Saturn P0 = 1000.00, g= 1000, µ = 2.14, Z=100.5/0.96
wave(µm):0.10000.20000.26400.30000.40000.5000
tau(P0):754.3121.686.213.561.050.42
P(tau=1):1.346.1161.0280.7950.32396.9

Titan P0 = 1500.00, g= 136, µ= 28.0, Z=58.85
wave(µm):0.10000.20000.26400.30000.40000.5000
tau(P0):5378.65152.5943.5824.987.372.92
P(tau=1):0.39.834.460.1203.6513.9

The gas continuum absorption contains a term which is supposed to represent the effect of Raman scattering due to the nu1 vibration of H2 at 4161 cm-1.
Ref.: Belton et al. (1971). Atm. of Uranus. ApJ. 164, 191-209

tauraman = 0.0208 · tauray

Rayleigh Scattering: Phase Function

Definitions of Angles
See Figure 11. The angle between the incident flux and the local normal is given by alpha and µ0 ident cos(alpha). The angle between the local normal and the observer is given by epsilon and µ ident cos(epsilon). The phase angle, theta, is the angle between the incident and emission through the origin. The azimuthal angle, Deltapsi, is the projection of the solar incidence and emission directions onto the local horizon. Using spherical trigonometry cosine laws (e.g., see CRC pg. 146) we can easily obtain the value of Deltapsi) given µ0, µ, and theta. For solar scattering to an observer above the atmosphere the angles are related by:

cos(theta) = sqrt[(1-µ02)(1-µ2)] · cos(Deltapsi) -/+ µ µ0   (S = +, T = -)

This is the method employed by Chandrasekar, Liou, and others. We could also write the equations in terms of the scattering angles. The scattering angle between incidence and emission, Theta, which is related to the phase function, is given by Theta = pi - theta. The scattering angle in the horizontal plane, Deltaphi is related to the azimuthal angle, and is given by Deltaphi= pi - Deltapsi. This definition is used by Hansen, Tomasko, and Danielson. It is the method employed within the VIAMP programs. Note the sign change between the two methods.

cos(Theta) = sqrt[ (1-µ02)(1-µ2)] · cos(Deltaphi) ± µ µ0   (T = +, S = -)

The Rayleigh phase function is given in terms of scattering angles or phase angles by

Pray(Theta) = (3/4) (1 + cos2(Theta)) = (3/4)(1 + cos2(theta)) = Pray(theta)
The cos2 term can be expressed in terms of µ, µ0, and Deltaphi as
cos2(Theta) = (1-µ02)(1-µ2) · cos2(Deltaphi) ± 2µµ0 sqrt[ (1-µ02)(1-µ2) ] · cos(Deltaphi) + (µ2 µ02)
cos(2Deltaphi) = 2 cos2(phi) - 1   or,
cos2(phi) = 1/2 cos(2Deltaphi) + 1/2
cos2(Theta) = 1/2 (1-µ02)(1-µ2) · cos(2Deltaphi) ± 2 µ µ0 sqrt[ (1-µ02)(1-µ2) ] · cos(Deltaphi) + µ2 µ02 + 1/2 (1-µ02)(1-µ2)

So the Rayleigh phase function can be written in terms of 3 Fourier moments:

Pray(Theta) = P1 + P2 cos(Deltaphi) + P3 cos(2Deltaphi)
where the 3 coefficients are equal to
P0 = 3/4 [ 1 + µ2 µ02 + 1/2 (1-µ02)(1-µ2) ] = 3/8 ( 3 + 3 µ2 µ02 -µ02 -µ2 )
P1 = ± 3/2 ( µ µ0 sqrt[(1-µ02)(1-µ2)] )   (T = +, S = -)
P2 = 3/8 (1-µ02)(1-µ2)

Richardson Number

Dynamically significant stability indicator (similar to Burger number). As Ri --> 0 convection becomes strong.

Ri/Ro = Effect of vertical motions / Effect of Non-linear terms
when the Rossby Number, Ro geq 1 then the static stability buoyancy frequency, N = Ri/Ro. When Ro ll 1, N = Bo/Ro
Ri = (g · d Theta/dz)/ [T0 · (dU/dz)2] = N2/(dU/dz)2
Stone 1972 JAS 29 p. 405. For Jupiter: d Theta/dz = 6.0 · 10-10 K/cm and du/dz = 3.8 · 10-4 cm/s, then Ri = 3.7 · 10-3

Rossby Number

A small Rossby number indicates that the geostrophic approximation is valid ( i.e., du/dt = dv/dt = 0) and that the Coriolis acceleration, v · f, is greater than the horizontal fluid acceleration, du/dt.

Ro = (du/dt)/(v f) approx (v2/r)/(v f) = (v/r)/f = U/(f0 · L)
For the GRS, hurricanes, etc.
Ro = (Vt a/b2/eta3)/f
eta2 = [ cos2(theta) + (a4/b4) · sin2(theta) ] / [ cos2(theta) + (a2/b2) · sin2(theta) ]
where Vt is the tangential velocity, a is the semimajor axis, b is the semiminor axis, and theta is the angle between the semi-major axis and the direction of motion. Mitchell et al. 1981 JGR 86 p.8751.

terrestrialRed SpotFA/BC/DEsmall spots
Vt110 m/s120 m/s
a1.1E7 m4.9E6 m
Ro(0).36.36
Ro(90).04.08
g.lat22 S33 S40.5 S

Great Red Spot
40000 x 13000 km (1879-1882)
20-60 m/s velocity around spot, 6 day period
bounded by Jets at -19.5 (-60 m/s) and -27.0 (+50 m/s) graphic latitude
long drift = -3.4 m/s
± 1/2 degree longitude oscillation with 90 day period
anticyclonic (counterclockwise in southern hemisphere)

Saturation Vapor Pressure

Given T in Kelvin these equations will give es in milliBar.
from Fleagle and Businger, Vol.5, pg. 62 (QC880.F59)
The first law can be written as

L = T · (S2-S1) = U2 - U1 + Ps · (alpha2 - alpha1)
where alpha1 ident 1/rho1 and index 1 refers to the liquid phase and index 2 refers to the gas phase.

For an isothermal change of phase, the Clausius-Clapeyron equation has the form

d Ps/d T = L/[T · (alpha2-alpha1)]
Water vapor behaves like an ideal gas and alpha2 gg alpha1 for a change in state.
L approx 2.5 · 103 Joules/gm
L approx 2.824 · 103 Joules/gm over ice
Rw = R*/<mw>w = 8.3143/18.016 = 0.4615 Joules/gm/K
Ps = rho2 · Rw T
d Ps/d T = L/[T · (alpha2-alpha1)] appeq (L · rho2)/T = (L · Ps) /(Rw · T2)
d log(es) = d Ps/Ps = (L/Rw) · (dT/T2)
loge(es) = integ (L/Rw) · (dT/T2) = -(L/Rw)|T->T0 + C = -L/(Rw · T0) + L/(Rw· T) + C
at triple point all 3 phases can exist in equilibrium, 0.0098° C and Ps = 6.1