The Ultimate CSIR NET Earth Sciences Formula Atlas

The Ultimate CSIR NET Earth Sciences Formula Atlas

Scope: A high-yield formula atlas for CSIR NET Earth, Atmospheric, Ocean and Planetary Sciences Part B and Part C. Formulas are grouped by exam domain and written in standard notation used in Earth-science textbooks and competitive examination problems.

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1. Geophysics

P-wave velocity

VP=K+43μρV_P=\sqrt{\frac{K+\frac{4}{3}\mu}{\rho}}

  • VPV_P = P-wave velocity, m s−1^{-1}
  • KK = bulk modulus, Pa
  • μ\mu = shear modulus, Pa
  • ρ\rho = density, kg m−3^{-3}

Exam Application: Use when elastic moduli and density are given for seismic velocity.

S-wave velocity

VS=μρV_S=\sqrt{\frac{\mu}{\rho}}

  • VSV_S = S-wave velocity, m s−1^{-1}
  • μ\mu = shear modulus, Pa
  • ρ\rho = density, kg m−3^{-3}

Exam Application: Use to show that S-waves cannot propagate through fluids where μ=0\mu=0.

Seismic impedance

Z=ρVZ=\rho V

  • ZZ = seismic impedance, kg m−2^{-2} s−1^{-1}
  • ρ\rho = density, kg m−3^{-3}
  • VV = wave velocity, m s−1^{-1}

Exam Application: Use for reflection/transmission at layer boundaries.

Normal-incidence reflection coefficient

R=Z2−Z1Z2+Z1R=\frac{Z_2-Z_1}{Z_2+Z_1}

  • RR = reflection coefficient, dimensionless
  • Z1,Z2Z_1,Z_2 = seismic impedances of upper and lower media, kg m−2^{-2} s−1^{-1}

Exam Application: Use to determine sign and strength of reflection from impedance contrast.

Seismic travel time

t=xVt=\frac{x}{V}

  • tt = travel time, s
  • xx = path length, m
  • VV = velocity, m s−1^{-1}

Exam Application: Use for direct travel-time, delay and residual questions.

Free-air correction

ΔgFA≈0.3086h\Delta g_{FA}\approx0.3086h

  • ΔgFA\Delta g_{FA} = free-air correction, mGal
  • hh = elevation above datum, m

Exam Application: Use to correct gravity for height above sea level.

Bouguer slab correction

ΔgB=2πGρh\Delta g_B=2\pi G\rho h

  • ΔgB\Delta g_B = Bouguer correction, m s−2^{-2} or mGal
  • GG = gravitational constant, m3^3 kg−1^{-1} s−2^{-2}
  • ρ\rho = slab density, kg m−3^{-3}
  • hh = slab thickness/elevation, m

Exam Application: Use in Bouguer anomaly reduction problems.

Simple Bouguer anomaly

ΔgBA=gobs−gtheoretical+ΔgFA−ΔgB\Delta g_{BA}=g_{obs}-g_{theoretical}+\Delta g_{FA}-\Delta g_B

  • ΔgBA\Delta g_{BA} = Bouguer anomaly, mGal
  • gobsg_{obs} = observed gravity, mGal
  • gtheoreticalg_{theoretical} = normal gravity, mGal
  • ΔgFA\Delta g_{FA} = free-air correction, mGal
  • ΔgB\Delta g_B = Bouguer correction, mGal

Exam Application: Use to calculate reduced gravity anomaly from field gravity.

Horizontal-cylinder gravity anomaly ratio

ΔgxΔgmax=z2x2+z2\frac{\Delta g_x}{\Delta g_{max}}=\frac{z^2}{x^2+z^2}

  • Δgx\Delta g_x = anomaly at horizontal distance xx, mGal
  • Δgmax\Delta g_{max} = maximum anomaly, mGal
  • zz = depth to cylinder axis, m
  • xx = horizontal offset, m

Exam Application: Use for depth estimation of buried cylindrical bodies.

Airy isostatic root thickness

r=ρchρm−ρcr=\frac{\rho_c h}{\rho_m-\rho_c}

  • rr = root thickness, m
  • ρc\rho_c = crustal density, kg m−3^{-3}
  • ρm\rho_m = mantle density, kg m−3^{-3}
  • hh = topographic elevation, m

Exam Application: Use for mountain-root compensation under Airy isostasy.

Conductive heat flow

q=−kdTdzq=-k\frac{dT}{dz}

  • qq = heat flow, W m−2^{-2}
  • kk = thermal conductivity, W m−1^{-1} K−1^{-1}
  • dT/dzdT/dz = geothermal gradient, K m−1^{-1}

Exam Application: Use when conductivity and geothermal gradient are given.

Magnetic inclination-paleolatitude relation

tan⁡I=2tan⁡λ\tan I=2\tan\lambda

  • II = magnetic inclination, degrees or radians
  • λ\lambda = magnetic paleolatitude, degrees or radians

Exam Application: Use to estimate paleolatitude from remanent inclination for a geocentric axial dipole.

Paleolatitude from inclination

λ=tan⁡−1(12tan⁡I)\lambda=\tan^{-1}\left(\frac{1}{2}\tan I\right)

  • λ\lambda = paleolatitude, degrees or radians
  • II = inclination, degrees or radians

Exam Application: Use when CSIR gives inclination and asks paleolatitude.

Virtual geomagnetic pole angular distance

p=tan⁡−1(2tan⁡I)p=\tan^{-1}\left(\frac{2}{\tan I}\right)

  • pp = angular distance from site to VGP, degrees or radians
  • II = magnetic inclination, degrees or radians

Exam Application: Use as the first step in paleopole/VGP calculations from inclination.

2. Physical Oceanography & Climatology

Coriolis parameter

f=2Ωsin⁡ϕf=2\Omega\sin\phi

  • ff = Coriolis parameter, s−1^{-1}
  • Ω\Omega = Earth angular velocity, rad s−1^{-1}
  • ϕ\phi = latitude

Exam Application: Use in rotating-fluid, geostrophic, Ekman and Rossby-wave problems.

Beta parameter

β=∂f∂y=2Ωcos⁡ϕa\beta=\frac{\partial f}{\partial y}=\frac{2\Omega\cos\phi}{a}

  • β\beta = meridional gradient of ff, m−1^{-1} s−1^{-1}
  • aa = Earth radius, m

Exam Application: Use in Rossby wave and planetary-vorticity questions.

Ocean geostrophic current

ug=−gf∂η∂y,vg=gf∂η∂xu_g=-\frac{g}{f}\frac{\partial \eta}{\partial y},\quad v_g=\frac{g}{f}\frac{\partial \eta}{\partial x}

  • ug,vgu_g,v_g = geostrophic current components, m s−1^{-1}
  • gg = gravitational acceleration, m s−2^{-2}
  • ff = Coriolis parameter, s−1^{-1}
  • η\eta = sea-surface height, m

Exam Application: Use to compute geostrophic current from sea-surface slope.

Geostrophic wind

ug=−1ρf∂p∂y,vg=1ρf∂p∂xu_g=-\frac{1}{\rho f}\frac{\partial p}{\partial y},\quad v_g=\frac{1}{\rho f}\frac{\partial p}{\partial x}

  • ug,vgu_g,v_g = geostrophic wind components, m s−1^{-1}
  • ρ\rho = air density, kg m−3^{-3}
  • pp = pressure, Pa

Exam Application: Use where pressure-gradient force balances Coriolis force.

Ekman transport per unit width

ME=τρwfM_E=\frac{\tau}{\rho_w f}

  • MEM_E = Ekman transport per unit width, m2^2 s−1^{-1}
  • τ\tau = wind stress, N m−2^{-2}
  • ρw\rho_w = seawater density, kg m−3^{-3}

Exam Application: Use for wind-driven ocean transport questions.

Wind stress

τ=ρaCDU102\tau=\rho_a C_D U_{10}^2

  • τ\tau = wind stress, N m−2^{-2}
  • ρa\rho_a = air density, kg m−3^{-3}
  • CDC_D = drag coefficient, dimensionless
  • U10U_{10} = wind speed at 10 m, m s−1^{-1}

Exam Application: Use before Ekman transport when wind speed is supplied instead of stress.

Rossby radius of deformation

LR=NHfL_R=\frac{NH}{f}

  • LRL_R = Rossby radius, m
  • NN = buoyancy frequency, s−1^{-1}
  • HH = vertical scale/depth, m

Exam Application: Use to estimate horizontal scale where rotation affects motion.

Brunt-Vaisala frequency

N2=−gρdρdzN^2=-\frac{g}{\rho}\frac{d\rho}{dz}

  • NN = buoyancy frequency, s−1^{-1}
  • ρ\rho = density, kg m−3^{-3}
  • dρ/dzd\rho/dz = vertical density gradient, kg m−4^{-4}

Exam Application: Use for static stability in atmosphere or ocean.

Thermal expansion sea-level rise

Δh=αΔTH\Delta h=\alpha\Delta T H

  • Δh\Delta h = sea-level rise, m
  • α\alpha = thermal expansion coefficient, K−1^{-1}
  • ΔT\Delta T = warming, K
  • HH = ocean-layer thickness, m

Exam Application: Use for sea-level-rise numerical problems.

Hydrostatic equation

dpdz=−ρg\frac{dp}{dz}=-\rho g

  • pp = pressure, Pa
  • zz = height, m
  • ρ\rho = density, kg m−3^{-3}

Exam Application: Use for pressure variation with height/depth.

Potential temperature

θ=T(p0p)R/cp\theta=T\left(\frac{p_0}{p}\right)^{R/c_p}

  • θ\theta = potential temperature, K
  • TT = temperature, K
  • p0p_0 = reference pressure, Pa or hPa
  • pp = pressure, same unit as p0p_0
  • RR = gas constant for dry air, J kg−1^{-1} K−1^{-1}
  • cpc_p = specific heat at constant pressure, J kg−1^{-1} K−1^{-1}

Exam Application: Use to compare air parcels at different pressures.

Dry adiabatic lapse rate

Γd=gcp≈9.8K km−1\Gamma_d=\frac{g}{c_p}\approx9.8 \text{K km}^{-1}

  • Γd\Gamma_d = dry adiabatic lapse rate, K m−1^{-1} or K km−1^{-1}

Exam Application: Use for atmospheric stability and parcel-lifting questions.

Clausius-Clapeyron equation

desdT=LvesRvT2\frac{de_s}{dT}=\frac{L_v e_s}{R_v T^2}

  • ese_s = saturation vapor pressure, Pa
  • LvL_v = latent heat of vaporization, J kg−1^{-1}
  • RvR_v = gas constant for water vapor, J kg−1^{-1} K−1^{-1}

Exam Application: Use for saturation vapor pressure and moisture thermodynamics.

Planetary albedo

α=SrSi\alpha=\frac{S_r}{S_i}

  • α\alpha = albedo, dimensionless
  • SrS_r = reflected solar radiation, W m−2^{-2}
  • SiS_i = incoming solar radiation, W m−2^{-2}

Exam Application: Use in radiation-budget and climate-feedback questions.

Stefan-Boltzmann law

E=σT4E=\sigma T^4

  • EE = emitted blackbody radiation, W m−2^{-2}
  • σ\sigma = Stefan-Boltzmann constant, W m−2^{-2} K−4^{-4}
  • TT = temperature, K

Exam Application: Use for planetary radiation and star/temperature brightness problems.

Rayleigh scattering wavelength dependence

I∝1λ4I\propto\frac{1}{\lambda^4}

  • II = scattered intensity, relative units
  • λ\lambda = wavelength, m

Exam Application: Use to explain why shorter wavelengths scatter more strongly.

3. Geochemistry & Isotope Geology

Radioactive decay equation

N=N0e−λtN=N_0e^{-\lambda t}

  • NN = parent atoms remaining
  • N0N_0 = initial parent atoms
  • λ\lambda = decay constant, yr−1^{-1} or s−1^{-1}
  • tt = time, yr or s

Exam Application: Use for remaining parent isotope or age calculation.

Daughter isotope growth

D=D0+N(eλt−1)D=D_0+N(e^{\lambda t}-1)

  • DD = present daughter abundance
  • D0D_0 = initial daughter abundance
  • NN = present parent abundance

Exam Application: Use for parent-daughter radiometric dating.

Half-life

t1/2=ln⁡2λ=0.693λt_{1/2}=\frac{\ln 2}{\lambda}=\frac{0.693}{\lambda}

  • t1/2t_{1/2} = half-life, yr or s
  • λ\lambda = decay constant, yr−1^{-1} or s−1^{-1}

Exam Application: Use to convert half-life and decay constant.

Mean life

τ=1λ=t1/20.693\tau=\frac{1}{\lambda}=\frac{t_{1/2}}{0.693}

  • τ\tau = mean life, yr or s

Exam Application: Use when CSIR asks mean life from half-life.

Rb-Sr isochron equation

(87Sr86Sr)=(87Sr86Sr)0+(87Rb86Sr)(eλt−1)\left(\frac{^{87}\text{Sr}}{^{86}\text{Sr}}\right)=\left(\frac{^{87}\text{Sr}}{^{86}\text{Sr}}\right)_0+\left(\frac{^{87}\text{Rb}}{^{86}\text{Sr}}\right)(e^{\lambda t}-1)

  • Ratios are isotope ratios, dimensionless
  • λ\lambda = decay constant of 87^{87}Rb, yr−1^{-1}

Exam Application: Use for Rb-Sr isochron age and initial ratio interpretation.

Isochron age from slope

t=1λln⁡(m+1)t=\frac{1}{\lambda}\ln(m+1)

  • tt = age, yr
  • mm = isochron slope, dimensionless

Exam Application: Use when slope of an isochron is given.

Stable isotope delta notation

δ18O=[(18O/16O)sample(18O/16O)standard−1]×1000\delta^{18}\text{O}=\left[\frac{(^{18}\text{O}/^{16}\text{O})_{sample}}{(^{18}\text{O}/^{16}\text{O})_{standard}}-1\right]\times1000

  • δ18O\delta^{18}\text{O} = isotope difference, per mil
  • Sample and standard ratios are dimensionless

Exam Application: Use for paleoclimate, ice-core and foraminifera isotope problems.

Fractionation factor

αA−B=RARB\alpha_{A-B}=\frac{R_A}{R_B}

  • αA−B\alpha_{A-B} = fractionation factor, dimensionless
  • RA,RBR_A,R_B = isotope ratios in phases A and B

Exam Application: Use to compare isotope partitioning between two phases.

Nernst partition coefficient

Di=CisolidCiliquidD_i=\frac{C_i^{solid}}{C_i^{liquid}}

  • DiD_i = partition coefficient, dimensionless
  • CisolidC_i^{solid} = concentration in solid, ppm or wt%
  • CiliquidC_i^{liquid} = concentration in liquid, ppm or wt%

Exam Application: Use for trace-element compatibility and magma differentiation.

Bulk distribution coefficient

D=∑XjDjD=\sum X_jD_j

  • DD = bulk distribution coefficient, dimensionless
  • XjX_j = modal mineral fraction, dimensionless
  • DjD_j = mineral/melt partition coefficient, dimensionless

Exam Application: Use when multiple minerals control trace-element behavior.

4. Mineralogy & Petrology

Gibbs phase rule

F=C−P+2F=C-P+2

  • FF = degrees of freedom
  • CC = number of components
  • PP = number of phases

Exam Application: Use when both pressure and temperature vary.

Condensed phase rule

F=C−P+1F=C-P+1

  • FF = degrees of freedom
  • CC = components
  • PP = phases

Exam Application: Use for fixed-pressure petrology phase diagrams.

Bragg’s law

nλ=2dsin⁡θn\lambda=2d\sin\theta

  • nn = diffraction order
  • λ\lambda = X-ray wavelength, m
  • dd = interplanar spacing, m
  • θ\theta = Bragg angle

Exam Application: Use for XRD mineral identification problems.

Crystal density from unit cell

ρ=ZMNAV\rho=\frac{ZM}{N_AV}

  • ρ\rho = density, kg m−3^{-3}
  • ZZ = formula units per unit cell
  • MM = molar mass, kg mol−1^{-1}
  • NAN_A = Avogadro constant, mol−1^{-1}
  • VV = unit-cell volume, m3^3

Exam Application: Use for crystallography density calculations.

Rayleigh fractional crystallization

Cl=C0F(D−1)C_l=C_0F^{(D-1)}

  • ClC_l = concentration in residual liquid
  • C0C_0 = initial concentration
  • FF = melt fraction remaining
  • DD = bulk partition coefficient

Exam Application: Use for trace-element enrichment during fractional crystallization.

Batch melting equation

Cl=C0D+F(1−D)C_l=\frac{C_0}{D+F(1-D)}

  • ClC_l = concentration in melt
  • C0C_0 = source concentration
  • DD = bulk distribution coefficient
  • FF = melt fraction

Exam Application: Use for partial-melting trace-element problems.

Clapeyron equation

dPdT=ΔSΔV\frac{dP}{dT}=\frac{\Delta S}{\Delta V}

  • dP/dTdP/dT = phase-boundary slope, Pa K−1^{-1}
  • ΔS\Delta S = entropy change, J mol−1^{-1} K−1^{-1}
  • ΔV\Delta V = volume change, m3^3 mol−1^{-1}

Exam Application: Use to interpret metamorphic reaction slopes.

5. Remote Sensing & GIS

Map scale

S=dmdgS=\frac{d_m}{d_g}

  • SS = representative fraction, dimensionless
  • dmd_m = map distance
  • dgd_g = ground distance in same unit

Exam Application: Use to convert between map/image and ground distances.

Aerial photograph scale

S=fHS=\frac{f}{H}

  • SS = photo scale
  • ff = focal length, m
  • HH = flying height above ground, m

Exam Application: Use for vertical aerial photo scale problems.

Relief displacement

d=rhHd=\frac{rh}{H}

  • dd = relief displacement on photograph
  • rr = radial distance from principal point
  • hh = object height
  • HH = flying height

Exam Application: Use for height/displacement calculations in aerial photographs.

Ground sampling distance

GSD=HpfGSD=\frac{H p}{f}

  • GSDGSD = ground sampling distance, m pixel−1^{-1}
  • HH = altitude above ground, m
  • pp = detector pixel size, m
  • ff = focal length, m

Exam Application: Use for spatial resolution and pixel-ground size problems.

NDVI

NDVI=NIR−RedNIR+RedNDVI=\frac{NIR-Red}{NIR+Red}

  • NDVINDVI = normalized difference vegetation index
  • NIRNIR = near-infrared reflectance
  • RedRed = red-band reflectance

Exam Application: Use for vegetation detection and health interpretation.

NDWI

NDWI=Green−NIRGreen+NIRNDWI=\frac{Green-NIR}{Green+NIR}

  • NDWINDWI = normalized difference water index
  • GreenGreen = green-band reflectance
  • NIRNIR = near-infrared reflectance

Exam Application: Use for surface-water detection.

Radiance from digital number

Lλ=MLQcal+ALL_\lambda=M_LQ_{cal}+A_L

  • LλL_\lambda = spectral radiance, W m−2^{-2} sr−1^{-1} um−1^{-1}
  • MLM_L = multiplicative rescaling factor
  • QcalQ_{cal} = digital number
  • ALA_L = additive rescaling factor

Exam Application: Use for satellite DN-to-radiance conversion.

Brightness temperature

TB=K2ln⁡(K1Lλ+1)T_B=\frac{K_2}{\ln\left(\frac{K_1}{L_\lambda}+1\right)}

  • TBT_B = brightness temperature, K
  • K1,K2K_1,K_2 = thermal calibration constants
  • LλL_\lambda = spectral radiance

Exam Application: Use for thermal remote-sensing temperature retrieval.

Raster area from pixel count

Atotal=nrxryA_{total}=nr_xr_y

  • AtotalA_{total} = mapped area, m2^2
  • nn = number of pixels
  • rx,ryr_x,r_y = pixel dimensions, m

Exam Application: Use for GIS land-cover area calculation.

Reference Backbone

  • CSIR-HRDG Earth, Atmospheric, Ocean and Planetary Sciences syllabus.
  • Introductory seismology, exploration geophysics, isotope geology, physical oceanography and remote-sensing formula conventions used in standard university texts.
  • NASA/USGS Landsat radiance and brightness-temperature formula convention for remote sensing.
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