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Geography & Earth

Distances, maps, coordinates and the planet

Haversine Distance

Advanced
d=2rarcsinsin2Δϕ2+cosϕ1cosϕ2sin2Δλ2d = 2r\arcsin\sqrt{\sin^2\frac{\Delta\phi}{2} + \cos\phi_1\cos\phi_2\sin^2\frac{\Delta\lambda}{2}}

Great-circle distance between two lat/long points.

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Map Scale

Basic
Scale=map distanceground distance\text{Scale} = \frac{\text{map distance}}{\text{ground distance}}

Ratio of map distance to real-world distance.

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Slope / Gradient

Basic
gradient=riserun×100%\text{gradient} = \frac{\text{rise}}{\text{run}} \times 100\%

Steepness of terrain as a percentage.

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Earth Curvature Drop

Advanced
hd22Rh \approx \frac{d^2}{2R}

Approximate drop of the horizon over distance d.

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Barometric Formula

Advanced
P=P0eMghRTP = P_0 e^{-\frac{Mgh}{RT}}

Atmospheric pressure as a function of altitude.

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Degrees to Distance

Basic
1lat111 km1^\circ \text{lat} \approx 111\ \text{km}

Approximate ground distance per degree of latitude.

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Richter Magnitude

Intermediate
M=log10AA0M = \log_{10}\frac{A}{A_0}

Earthquake magnitude from seismograph amplitude.

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Population Density

Basic
D=PAD = \frac{P}{A}

Number of people per unit area.

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Time Zone Difference

Basic
Δt=Δlongitude15\Delta t = \frac{\Delta \text{longitude}}{15^\circ}

Each 15 degrees of longitude equals one hour.

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Earthquake Energy (Richter)

Advanced
log10E=4.8+1.5M\log_{10} E = 4.8 + 1.5\,M

Energy released by a quake of Richter magnitude M.

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Moment Magnitude

Advanced
Mw=23log10M010.7M_w = \tfrac{2}{3}\log_{10} M_0 - 10.7

Modern magnitude from the seismic moment of a quake.

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Seismic Wave Speed

Advanced
v=Eρv = \sqrt{\frac{E}{\rho}}

Speed of an elastic wave through rock.

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Lithostatic Pressure

Intermediate
P=ρghP = \rho g h

Pressure from the weight of overlying rock.

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Geothermal Gradient

Intermediate
ΔT=GTd\Delta T = G_T \, d

Temperature rise with depth into the Earth.

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Tsunami Wave Speed

Intermediate
v=gdv = \sqrt{g\,d}

Speed of a shallow-water wave across the ocean.

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Radiometric Dating Age

Advanced
t=1λln ⁣(1+DP)t = \frac{1}{\lambda}\ln\!\left(1 + \frac{D}{P}\right)

Age of a rock from parent and daughter isotopes.

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Schumann Resonance

Advanced
f1c2πREf_1 \approx \frac{c}{2\pi R_E}

Fundamental electromagnetic resonance of the Earth.

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Isostatic Root Depth

Advanced
r=ρchρmρcr = \frac{\rho_c\,h}{\rho_m - \rho_c}

Depth of a mountain's crustal root floating in the mantle.

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Darcy's Law

Advanced
Q=KAΔhLQ = -KA\frac{\Delta h}{L}

Rate of groundwater flow through porous rock.

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