TABLE OF CONSTANTS

CONSTANT SYMBOL VALUE

speed of light in a vacuum c 3.00 × 108 m s-1
gravitational constant G 6.673 × 10-11 N m2 kg-2
orbital constant G ME 3.986 × 1014 m3 s-2
standard gravitational acceleration g 9.81 m s-1
Planck's constant h 6.626 × 10-34 J s
Boltzmann's constant k 1.381 × 10-23 J K-1
first radiation constant c1 1.191 × 10-16 W m2 sr-1
second radiation constant c2 1.439 × 10-2 m K
Stefan-Boltzmann constant σ 5.670 × 10-8 W m-2 K-4
Wien's displacement law constant none 2.8979 × 10-3 m K
Avogadro's number NA 6.022 × 1023 molecules mole-1
Dobson unit DU 1 DU = 2.687 × 1016 molecules/cm2
molar gas constant R 8.3143 J mole-1 K-1
angular velocity of Earth dΩE / dt 7.292 × 10-5 rad s-1
solar day Tsolar 86,400 s
sidereal day Tsidereal 86,164.1 s
mean radius of Earth RE 6.371 × 106 m
mass of Earth ME 5.9737 × 1024 kg
standard pressure po 1013.25 mbar = 1.01325 × 105 N m-2
standard temperature To 273.15 K
scale height of Earth's atmosphere H = RT/Mg ~ 7 km
radius of Sun (visible disk) Rsun 6.96 × 108 m
mean Earth-Sun distance dsun 1.50 × 1011 m
solar constant Ssun 1368 W m-2
molecular mass of CO2 M 44 g/mole = 7.3065×10-26 kg/molecule
molecular mass of H2O M 18 g/mole = 2.9890×10-26 kg/molecule
molecular mass of O2 M 32 g/mole = 5.3138×10-26 kg/molecule
 LIST OF EQUATIONS – 1 of 2

Gm 1 m 2 r3 2 1 a (1 − ε 2 )
F= T = 2π v=Gm E  −  r=
r2 Gm E r a 1 + ε cos θ
cos e − ε cos θ + ε ~ 2π
M = e − ε sin e = n ( t − t p ) cos θ = cos e = T=
1 − ε cos e 1 + ε cos θ dω
n+
dt
ψ ψN ∆ψ   
LT = UT + ECT = UT +
= 12 + S = c 2
ε E ×H
15  15  15 
0

∞ ∞ ∞
2hc 2 λ−5 c1 λ−5
I = ∫ I λ dλ = ∫ I ν dν = ∫ I ν dν Bλ = =
 hc  c 
0 0 0
exp  − 1 exp 2  − 1
 λkT   λT 
2897.9 cT
λ max (µm) = M BB = σT 4 Bλ ≅ 1 4 α λ + R λ + τλ = 1 αλ = ελ
T(K ) c 2λ
dI λ = −σ a (λ )I λ ds = −ρk a (λ )I λ ds = − k a (λ )I λ du
 z 2 k (λ )ρ   u2 
τ λ (z 1 , z 2 ) = exp  − ∫ a dz  = exp − ∫ k a (λ )du 
 z1 µ   u1 
2π 1
µ dI λ ~ B (T ) + ω ~ < I '> 1
I λ (µ ′, φ′)p(ψ S )dµ ′dφ′
4π ∫0 −∫1
= − I λ (θ, φ) + α λ λ λ λ < I λ ' >=
ρk e (λ ) dz

~ = σ a (λ )
α ~ = σ s (λ )
ω
µ dI λ
= − I λ (µ, φ) + B λ (T )
λ λ
σ e (λ ) σ e (λ ) ρk e (λ) dz
S αL p To
k aL = α L (T, p) = α L (To , p o )
π (ν − ν o ) 2 + α L 2 po T
S  (ν − ν o ) 2  2kT ν o
k aD = exp  −  α D ( T, ν o ) =
αD π αD
2
  M c

2πr ks ∞
[
8π 3 n o ( λ ) 2 − 1 ]
2

χ= Qs = σs = ∫ πr Q s N(r )dr σ s (λ ) ≅ f (δ )
2

λ πr 2 0 3λ4 Ns
2

2π π / 2
m = n − i n' I r (θ r , ϕ r ) = ∫ ∫ I (θ , ϕ ) γ
0 0
i i i r (θ r , ϕ r ; θ i , ϕ i ) cos θ i sin θ i dθ i dϕ i

I r (θ r , φ r ) I
γ r (θ r , φ r ; θ sun , φ sun ) = = r
I sun Ω sun cos θ sun E sun
2π
M
A=
E
= ∫γ
0
r (θ r , ϕ r ; θ sun , ϕ sun ) cos θ r dΩ r

τ =1 satellite
I λ (z sat ) = I λ (0)τ(0, z sat ) + ∫ B λ (T)dτ λ = I λ (0)τ(0, z sat ) +
τ ( 0, , z sa )
∫B
surface
λ (T )K λ ( y)dy
 LIST OF EQUATIONS – 2 of 2

Mp ( z )  (z − z o ) 
dp = − g ρ air dz ρ(z) = p ( z ) = p ( z o ) exp  − 
RT  H 
∞
d τ λ ( x , z tangent ) ~
∫ B [T ( x ) ]
k +1 I
I λ ( z tangent ) = λ dx B i (T i ) = B i ( Ti ) k i k
k

−∞
dx I i ( Ti )
B i (T k +1
ij
~
[
) = B i ( T jk ) + I i − I ik ( T jk ) ] y = F(x, b ) + ε
dT dI dF
xˆ = x o + S o K T ( K S o K T + S y ) − 1 ( y − K x o ) A = = = DK
dx dy dx
I λ = E sun ( λ ) [τ ozone ( λ ) ] R ( θ sun , θ satellite ; R surface , R air )
x

 I(λ ) 
N ( λ ) = − 100 log 10   = − 100 log 10 [R ( λ ) exp {− k a ( λ ) Ux }]
 sun
E ( λ ) 
∞
I λ (h i )  
τ λ (h i ) =
Iλ
o
= exp  −

∫σ
−∞
e,λ ( s ) ds 


I ( λ , z tan ) = E o ( λ ) ∫τ
l−o−s
in ( λ , ∞ : z )S ( λ , z , θ ) τ out ( λ , z : ∞ ) dz

I ( λ ) = (1 − N ) I clear ( λ ) + N ε I cloud ( λ ) + N (1 − ε ) I clear ( λ )

µ  To − TB 
k LU L + k VU V = − ln  
2  {1 − ε ( µ )}T o 
~
ω 1
satellite
I aerosol ≅ o E sun p ( ψ sun )
4π µ ∫σ
surface
aerosol dz

∆ R β(λ )  R

Pr ( λ , R ) = P t ( λ ) C ( λ ) 2 exp  − 2 ∫ σ ext ( λ , r ) dr 
R 4π  0 

[ ] 1 dβ  P1 ( R )  R

 ≅ − 2 ∫ ρ [k 1 − k 2 ]dr
dS
S ( R ) = ln R P r ( R ) 2
= − 2 σ ext ln 
dR β dR  P2 ( R )  0

 P (R + ∆R )   P1 ( R ) 
ln  1  − ln  
 P2 ( R + ∆ R )   P2 ( R )  G λ2 G 2 λ2
ρ (R ) = − Ae = PR = P t σ
2(k 1 − k 2 )∆R 4π (4 π )3 R 4
π2G 2 K
2
π5 m2 − 1
σ = 4 K D6 K = 2 PR = Pt ∑D
2 6

λ m +2 64λ R
2 4
pulse volume

∞ 2
K
Z= ∑D
unit volume
6
= ∫ N(D) D dD
6
PR = C
R2
Z
0

dl dθ Hλ
a (sin i + sin θ) = mλ h ' = ha ' (H − h ) =f Ra =
dλ dλ L cos θ