ψ ϵ dψ dϵ

F_{1} = F_{2} = \frac{Gm_{1}m_{2}}{r^{2}}  (1)

F (r) = \frac{GM_{E}m_{2}}{r^{2}}  (2)

T = 2π\sqrt{\frac{r^{3}}{GM_{E}}}  (3)

d⃗f(x⃗) = \frac{Gmdm′(x⃗ − x′)}{||x⃗ − x′||^{3}}  (4)

d⃗g(x⃗) = \frac{df(x⃗)}{m} = \frac{Gdm′(x⃗ − x′)}{||x⃗ − x′||^{2}}  (5)

g⃗(x⃗) = ∫_{mass} \frac{Gρ(x′)(x′ − x⃗)}{||x⃗ − x′||^{3}}d^{3}x′  (6)

∇Φ(x⃗) = ∂_{x}Φe⃗_{x} + ∂_{y}Φe⃗_{y} + ∂_{z}Φe⃗_{z}  (7)

∇Φ(x⃗) = g⃗(x⃗)  (8)

Φ(x⃗) = ∫_{mass} \frac{Gρ(x′)}{||x⃗ − x′||}d^{3}x′  (9)

Φ(x⃗) = \frac{GM}{d}  (10)

∇^{2}Φ(x⃗) = ∂_{x}^{2}Φ + ∂_{y}^{2}Φ + ∂_{z}^{2}Φ  (11)

∇^{2}Φ(x⃗) = −4πGρ(x⃗)  (12)

∫_{S} n̂ · g⃗dS = −4πG ∫_{V} ρdV  (13)

\frac{GM}{||x⃗ − x′||}  (14)

∇^{2}Φ(x⃗) = 0  (15)

Φ_{1} = Φ_{2}  (16)

n̂ · ∇Φ_{2} − n̂ · ∇Φ = −4πGδρ  (17)

Φ(x⃗) = ∫_{mass} \frac{Gρ(x′)}{||x⃗ − x′||}d^{3}x′ + \frac{1}{2}(x^{2} + y^{2})ω^{2}  (18)

∇^{2}Φ(x⃗) = −4πGρ(x⃗) + 2ω^{2}  (19)

∇^{2}Φ(x⃗) = 2ω^{2}  (20)

x = r sin θ cos φ  (21)
y = r sin θ sin φ  (22)
z = r cos θ  (23)

∇^{2}Φ = \frac{1}{r}∂_{r}(r^{2}∂_{r}Φ) + \frac{1}{r^{2} sin θ}∂_{θ}(sin θ∂_{θ}Φ) + \frac{1}{r^{2} sin^{2} θ}∂_{φ}^{2}Φ  (24)

Φ(r, θ, φ) = T (θ)L(φ)R(r)  (25)

R(r) = {r^{l}r^{−(l+1}  (26)
L(φ) = {e^{imφ}e^{−imφ}  (27)
T (cos θ) = P_{l}(cos θ)  (28)

P_{l}: ”Legendre polynomials

Φ(r, θ, φ) = {r^{l}r^{−(l+1)}} × Y_{l}^{m}(θ, φ)  (29)

Y_{l}^{m}(θ, φ) = \sqrt{\frac{2l + 1}{4π} \frac{(l − m)!}{(l + m)!}}P_{l}^{m}(cos θ)e^{imφ}  (30)

Y_{l}^{m} ≡ (−1)^{m}Y_{l}^{(−m)∗}  (31)

ρ(x⃗′) = _{l′,m′}^{∑} ρ_{l′}^{m′}(r′)Y_{l′}^{m′}(θ′, φ′)  (32)

Φ(x⃗) = 4πG _{l,m}^{∑} \frac{Y_{l}^{m}(θ, φ)}{2l + 1} \frac{1}{r^{l+1}} [∫ (r′)^{l+2}ρ_{l}^{m}(r′)dr′]  (33)

Φ_{T} = [\frac{GM}{r} + \frac{1}{3}Ω^{2}r^{2}] − [\frac{1}{3}Ω^{2}r^{2} + \frac{GM}{r^{3}} a^{2}J_{2}] P_{2}(cos θ)  (34)
−\frac{GM}{r} ∞ ; ∑ ; l=2 l ; ∑ ; m=−l;(l,m)≠(2,0) Φ_{l}^{m}Y_{l}^{m}(θ, φ) (\frac{a}{r})^{l}  (35)

N = \frac{ΔΦ}{g_{T}}|_{r=a}  (36)

ΔΦ = −\frac{GM}{r} ∞ ; ∑ ; l=2 l ; ∑ ; m=−l;(l,m)≠(2,0) Φ_{l}^{m}Y_{l}^{m}(θ, φ) (\frac{a}{r})^{l}  (37)

N = −a _{l,m}^{∑} Φ_{l}^{m}Y_{l}^{m}  (38)

N = −\frac{a}{2πg_{T}|_{a}} ∫ Δg(θ′, φ′)f(γ) sin θ′dθ′dφ′  (39)

Φ = \frac{GM}{r} (1 + n_{max} ; ∑ ; n=2 (\frac{a}{r})^{n} n ; ∑ ; m=0 P_{nm}(sin ϕ) [C_{nm} cos mλ + S_{nm} sin mλ])  (40)

Effect  Source  Magnitude

(1 mGal = 10^{−5} m · s^{−2})

Main  Total mass of Earth, Oblate shape, Earth Rotation 980,000 - 990,000 mGal
Elevation Distance from centre of Earth  0.3 mGal/m
Crustal Density variations in rocks within the Earth’s crust <500 mGal [usually 10’s mGal]
Tides  Sun and moon  <1 mGal

a⃗_{0} = \frac{GM_{b}}{s^{2}} \frac{s⃗}{s}  (41)
a⃗_{0}: homogeneous centrifugal force in body 1 due to revolution around barycenter;
M_{b}: mass of the body 2;
s⃗: distance between the centers of the two bodies.
b⃗ = a⃗_{p} − a⃗_{0} = \frac{GM_{b}}{d^{2}} · \frac{d⃗}{d} − \frac{GM_{b}}{s^{2}} \frac{s⃗}{s}  (42)

d⃗: distance between point P and the center of body 2.

b⃗ = ∇V  (43)

∇V = \frac{∂f}{∂r} ˆ r⃗ + \frac{1}{r} \frac{∂V}{∂ϑ} ˆ ϑ⃗ + \frac{1}{r sin ϑ} \frac{∂V}{∂φ} ˆ φ⃗  (44)

V = GM_{b} (\frac{1}{d} − \frac{1}{s} − \frac{r cos ψ}{s^{2}} )  (45)

ψ: geocentric zenith angle

V = \frac{GM_{b}}{s} · ∞ ; ∑ ; l=2 (\frac{r}{s})^{l} P_{l}(cos ψ)  (46)
Body  r/s l_{max}
Moon 1.6 · 10^{−2} 6
Sun  4.5 · 10^{−5} 3
Planets  2

P_{l}(cos ψ) = \frac{1}{2l + 1} l ; ∑ ; m=0 P_{lm}(cos ϑ) · P_{lm}(cos ϑ_{b}) · cos(mφ − mφ_{b})  (47)
ϑ: geocentric spherical polar distance of the point;
ϑ_{b}: geocentric spherical polar distance of the second body;
φ: geocentric spherical longitude of the point;
φ_{b}: geocentric spherical longitude of the second body.
P_{lm}: Fully normalized spherical harmonics
V (t) = GM_{b} l_{max} ; ∑ ; l=2 \frac{r^{l}}{s^{l+1}(t)} \frac{1}{2l + 1} l ; ∑ ; m=0 P_{lm}(cos ϑ) · P_{lm}(cos ϑ_{b}(t)) · cos(mφ − mφ_{b}(t))  (48)

with s, ϑ_{b}, and φ_{b} time dependent.

V (t) = D l_{max} ; ∑ ; l=1 l ; ∑ ; m=0 (\frac{r}{a})^{l} Γ(ϑ) · P_{lm}(cos ϑ)  (49)
· _{i}^{∑}[C_{i}^{lm}(t) cos(α_{i}(t)) + S_{i}^{lm}(t) sin(α_{i}(t))]  (50)
D, Γ: Normalization constants
a:  semi-major axis of the reference ellipsoid
C_{i}, S_{i}: time dependent coefficients of the catalogue; given as

C_{i}^{lm}(t) = C0_{i}^{lm} + t · C1_{1}^{lm} and S_{i}^{lm}(t) = S0_{i}^{lm} + t · S1_{1}^{lm}.

α_{i}(t) = m · λ + j_{max} ; ∑ ; j=1 k_{ij} · arg_{j}(t); k_{i1} = m  (51)
k_{ij}:  integer coefficients, given in catalog
arg_{j}(t): astronomical arguments, can be computed from ephemerides (polynomials)

catalogue  no. of no. of max. truncation ephemerides
waves coeff. degree  [m^{2}/s^{2}]
Doodson (1921)  378  378 3  1.0 · 10^{−4} BN
Cartwright et al. (1971, 1973)  505 1,010 3  0.4 · 10^{−4} BN
Büllesfeld (1985)  656  656 4  0.2 · 10^{−4} BN
Tamura (1987)  1,200 1,326 4  0.4 · 10^{−5} DE118/LE62
Xi (1989)  2,934 2,934 4  0.9 · 10^{−6} BN
Tamura (1993)  2,060 3,046 4  0.4 · 10^{−5} DE200/LE200
Roosbeck (1996)  6,499 7,202 5  0.8 · 10^{−7} CT/BR
Hartmann & Wenzel (1995a,b) 12,935 19,271 6  0.1 · 10^{−9} DE200/LE200

δN = \frac{V}{g}  (52)
Love/ l = 2 definition

Shida

h  0.612 ratio of height of solid body tide to the deforming potential
k  0.303 ratio of additional tidal potential to the deforming potential

l 0.04 ratio of horizontal displacement of the crust to that of the equlibrium tide of a fluid Earth

x⃗(t) = x⃗_{0} + K ; ∑ ; k=1 δx⃗_{k}(t)  (53)

Geoid is defined as the surface

r = a + δr(θ, ϕ)  (54)

with a radius of the Earth,

\overline{δr(θ, ϕ)} = 0  (55)

and

V_{T}(r = a + δr, θ, ϕ) = const.  (56)

ΔV = V_{T} − allP_{2}−terms  (57)

N = \frac{ΔV}{g_{T}} |_{r=a}  (58)

Stoke’s formula:

N(θ, ϕ) = −\frac{a}{2πg_{T}|_{a}} ∫ Δg(θ′, ϕ′)f(γ) sin θ′dθ′dϕ′  (59)

with γ angle between (θ,ϕ) and (θ′,ϕ′) and

f(γ) = [\frac{1}{2}csc\frac{γ}{2} − 1 − cos γ]  (60)
+3 [1 − cos γ − 2 sin \frac{γ}{2} − cos γ ln \{sin \frac{γ}{2} + sin^{2} \frac{γ}{2} \}]  (61)

σ = ( \begin{matrix}σ_{xx} & σ_{xy} & σ_{xz} \\ σ_{yx} & σ_{yy} & σ_{yz} \\ σ_{zx} & σ_{zy} & σ_{zz} \end{matrix} )  (62)

σ_{ij}(ξ_{k}, t) = f_{ij}(x_{k}(η_{l}, t − s); ξ_{k})|_{s∈[0,∞] η_{l}∈B}.  (63)

B: body σ_{ij}: stress tensor f_{ij}: time-independent functional; ξ_{k}, η_{l}: positions; x_{k} : displacement; t: time

Elastic, isotrop, linear material (Hooke’s law):

σ_{ij} = 2μϵ_{ij} + λϵ_{kk}δ_{ij}  (64)

ϵ_{ij}: strain tensor; μ, λ: Lame’s constants.

Linear, visco-elastic body:

σ_{ij}(t) = 2μ(0)ϵ_{ij}(t) + λ(0)ϵ_{kk}(t)δ_{ij} + ∫_{0}^{∞} {2μ(s)ϵ_{ij}(t − s) + λ(s)ϵ_{kk}(t − s)δ_{ij}} ds  (65)

ϵ = ( \begin{matrix}ϵ_{xx} & ϵ_{xy} & ϵ_{xz} \\ ϵ_{yx} & ϵ_{yy} & ϵ_{yz} \\ ϵ_{zx} & ϵ_{zy} & ϵ_{zz} \end{matrix} )  (66)

ϵ_{ij} = \frac{1}{2} (\frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} )  (67)

u_{i}: displacement; x_{j}: position.

Volume force F_{i}:

F_{i} = lim_{ΔV →0} \frac{ΔF_{i}^{∗}}{ΔV}  (68)

F_{i}: volume force; ΔF_{i}^{∗}: total force acting on volume ΔV.

Surface force vector (p_{ni})dS (with index n indicating that the surface element dS has the outward pointing normal n_{i}):

p_{ni} = lim_{ΔS→0} \frac{p_{ni}^{∗}}{ΔS}  (69)

p_{ni}^{∗}: components of the total force acting on area ΔS. p_{ni} has the unit force/area (stress).

Momentum balance:

ρ (\frac{∂v_{i}}{∂x_{k}}v_{k} + \frac{∂v_{i}}{∂t}) = \frac{∂σ_{ij}}{∂x_{j}} + f_{i}.  (70)

ρ: density; v_{i}: velocity; t: time; f_{i}: forces.

Small gradients of velocity:

ρ\frac{∂^{2}u_{i}}{∂t^{2}} = \frac{∂σ_{ij}}{∂x_{j}} + f_{i}  (71)

u_{i}: displacement.

Boundary condition:

p_{i} = σ_{ji}n_{j}.  (72)

G_{r}(a, ϑ) = \frac{a}{M} ∞ ; ∑ ; n=0 h′_{n}P_{n}(cos ϑ)  (73)
G_{ϑ}(a, ϑ) = \frac{a}{M} ∞ ; ∑ ; n=0 ℓ′_{n} \frac{∂}{∂ϑ}P_{n}(cos ϑ)  (74)
G_{φ}(a, ϑ) = \frac{G}{a} ∞ ; ∑ ; n=0(k′_{n} + 1)P_{n}(cos ϑ)  (75)

Δt = ∫_{S} \frac{1}{v}ds − ∫_{G} \frac{1}{c}dg  (76)

ℓ = ∫_{S} \frac{c}{v}ds − ∫_{G} dg  (77)

ℓ = ∫_{S} \frac{c}{v}ds − ∫_{S} ds + ∫_{S} ds − ∫_{G} dg  (78)

ℓ = ∫_{S}(n − 1)ds + S − G  (79)
Δt: time delay
v: propagation velocity
c: speed of light in vacuum
S: actual path
G: straight line; path in vacuum
ℓ: excess path length
n: refractive index in the atmosphere