Common formula

gravity

  1. Newton’s gravitational law

    symbol

    description

    unit

    \(g\)

    normal gravity at sea level

    \(\frac{m}{s^2}\)

    \(g_{h}\)

    gravity at specific height

    \(\frac{m}{s^2}\)

    \(z\)

    altitude

    \(m\)

    \(R\)

    local earth curvature radius

    \(m\)

    \[g_{h} = g\left(\frac{R}{R + z}\right)^2\]
  2. Normal gravity at sea level ellipsoid

    This is the WGS84 ellipsoidal gravity formula as taken from NIMA TR8350.2

    symbol

    name

    unit

    \(a\)

    WGS84 semi-major axis

    \(m\)

    \(b\)

    WGS84 semi-minor axis

    \(m\)

    \(e\)

    eccentricity

    \(m\)

    \(g\)

    normal gravity at sea level

    \(\frac{m}{s^2}\)

    \(g_{e}\)

    gravity at equator

    \(\frac{m}{s^2}\)

    \(g_{p}\)

    gravity at poles

    \(\frac{m}{s^2}\)

    \(\phi\)

    latitude

    \(degN\)

    \begin{eqnarray} e^2 & = & \frac{a^2-b^2}{a^2} \\ k & = & \frac{bg_{p} - ag_{e}}{ag_{e}} \\ g & = & g_{e}\frac{1 + k {\sin}^2(\frac{\pi}{180}\phi)}{\sqrt{1 - e^2{\sin}^2(\frac{\pi}{180}\phi)}} \\ g & = & 9.7803253359 \frac{1 + 0.00193185265241{\sin}^2(\frac{\pi}{180}\phi)} {\sqrt{1 - 0.00669437999013{\sin}^2(\frac{\pi}{180}\phi)}} \end{eqnarray}
  3. Gravity at specific altitude

    This is the WGS84 ellipsoidal gravity formula as taken from NIMA TR8350.2

    symbol

    name

    unit

    \(a\)

    WGS84 semi-major axis

    \(m\)

    \(b\)

    WGS84 semi-minor axis

    \(m\)

    \(f\)

    WGS84 flattening

    \(m\)

    \(g\)

    normal gravity at sea level

    \(\frac{m}{s^2}\)

    \(g_{h}\)

    gravity at specific height

    \(\frac{m}{s^2}\)

    \(GM\)

    WGS84 earth’s gravitational constant

    \(\frac{m^3}{s^2}\)

    \(z\)

    altitude

    \(m\)

    \(\phi\)

    latitude

    \(degN\)

    \(\omega\)

    WGS84 earth angular velocity

    \(rad/s\)

    The formula used is the one based on the truncated Taylor series expansion:

    \begin{eqnarray} m & = & \frac{\omega^2a^2b}{GM} \\ g_{h} & = & g \left[ 1 - \frac{2}{a}\left(1+f+m-2f{\sin}^2(\frac{\pi}{180}\phi)\right)z + \frac{3}{a^2}z^2 \right] \\ \end{eqnarray}

geopotential height

symbol

description

unit

\(g\)

normal gravity at sea level

\(\frac{m}{s^2}\)

\(g_{0}\)

mean earth gravity

\(\frac{m}{s^2}\)

\(g_{h}\)

gravity at specific height

\(\frac{m}{s^2}\)

\(p\)

pressure

\(Pa\)

\(R\)

local earth curvature radius

\(m\)

\(z\)

altitude

\(m\)

\(z_{g}\)

geopotential height

\(m\)

\(\phi\)

latitude

\(degN\)

\(\rho\)

mass density

\(\frac{kg}{m^3}\)

The geopotential height allows the gravity in the hydrostatic equation

\[dp = - \rho g_{h} dz\]

to be replaced by a constant gravity

\[dp = - \rho g_{0} dz_{g}\]

providing

\[dz_{g} = \frac{g_{h}}{g_{0}}dz\]

With Newton’s gravitational law this becomes

\[dz_{g} = \frac{g}{g_{0}}\left(\frac{R}{R + z}\right)^2dz\]

And integrating this, considering that \(z=0\) and \(z_{g}=0\) at sea level, results in

\[z_{g} = \frac{g}{g_{0}}\frac{Rz}{R + z}\]
\[z = \frac{g_{0}Rz_{g}}{gR-g_{0}z_{g}}\]

gas constant

symbol

name

unit

\(k\)

Boltzmann constant

\(\frac{kg m^2}{K s^2}\)

\(N_A\)

Avogadro constant

\(\frac{1}{mol}\)

\(R\)

universal gas constant

\(\frac{kg m^2}{K mol s^2}\)

Relation between Boltzmann constant, universal gas constant, and Avogadro constant:

\[k = \frac{R}{N_A}\]

ideal gas law

symbol

name

unit

\(k\)

Boltzmann constant

\(\frac{kg m^2}{K s^2}\)

\(N\)

amount of substance

\(molec\)

\(p\)

pressure

\(Pa\)

\(R\)

universal gas constant

\(\frac{kg m^2}{K mol s^2}\)

\(T\)

temperature

\(K\)

\(V\)

volume

\(m^3\)

\[pV = \frac{NRT}{N_{A}} = NkT\]

barometric formula

symbol

name

unit

\(g_{0}\)

mean earth gravity

\(\frac{m}{s^2}\)

\(g_{h}\)

gravity at specific height

\(\frac{m}{s^2}\)

\(k\)

Boltzmann constant

\(\frac{kg m^2}{K s^2}\)

\(M_{air}\)

molar mass of total air

\(\frac{g}{mol}\)

\(N\)

amount of substance

\(molec\)

\(N_A\)

Avogadro constant

\(\frac{1}{mol}\)

\(p\)

pressure

\(Pa\)

\(R\)

universal gas constant

\(\frac{kg m^2}{K mol s^2}\)

\(T\)

temperature

\(K\)

\(V\)

volume

\(m^3\)

\(z\)

altitude

\(m\)

\(z_{g}\)

geopotential height

\(m\)

\(\phi\)

latitude

\(degN\)

\(\rho\)

mass density

\(\frac{kg}{m^3}\)

From the ideal gas law we have:

\[p = \frac{NkT}{V} = \frac{10^{-3}NM_{air}}{VN_{a}}\frac{kTN_{a}}{10^{-3}M_{air}} = \rho\frac{RT}{10^{-3}M_{air}}\]

And from the hydrostatic assumption we get:

\[dp = - \rho g_{h} dz\]

Dividing \(dp\) by p we get:

\[\frac{dp}{p} = -\frac{10^{-3}M_{air}\rho g_{h} dz}{\rho RT} = -\frac{10^{-3}M_{air}g_{h}dz}{RT}\]

Integrating this expression from one pressure level to the next we get:

\[p(i+1) = p(i)e^{-\int^{z(i+1)}_{z(i)}\frac{10^{-3}M_{air}g_{h}}{RT}dz}\]

We can approximate this further by using an average value of the height dependent quantities \(M_{air}\), \(g_{h}\) and \(T\) for the integration over the range \([z(i),z(i+1)]\). This gives:

\begin{eqnarray} g' & = & g_{h}(\phi,\frac{z(i)+z(i+1)}{2}) \\ p(i+1) & = & p(i)e^{-10^{-3}\frac{M_{air}(i)+M_{air}(i+1)}{2}\frac{2}{T(i)+T(i+1)}\frac{g'}{R}\left(z(i+1)-z(i)\right)} \\ & = & p(i)e^{-10^{-3}\frac{M_{air}(i)+M_{air}(i+1)}{T(i)+T(i+1)}\frac{g'}{R}\left(z(i+1)-z(i)\right)} \end{eqnarray}

When using geopotential height the formula is the same except that \(g=g_{0}\) at all levels:

\[p(i+1) = p(i)e^{-10^{-3}\frac{M_{air}(i)+M_{air}(i+1)}{T(i)+T(i+1)}\frac{g_{0}}{R}\left(z_{g}(i+1)-z_{g}(i)\right)}\]

mass density

symbol

name

unit

\(N\)

amount of substance

\(molec\)

\(N_A\)

Avogadro constant

\(\frac{1}{mol}\)

\(M_{air}\)

molar mass of total air

\(\frac{g}{mol}\)

\(V\)

volume

\(m^3\)

\(\rho\)

mass density

\(\frac{kg}{m^3}\)

\[\rho = \frac{10^{-3}NM_{air}}{VN_{a}}\]

number density

symbol

name

unit

\(n\)

number density

\(\frac{molec}{m^3}\)

\(N\)

amount of substance

\(molec\)

\(V\)

volume

\(m^3\)

\[n = \frac{N}{V}\]

dry air vs. total air

symbol

name

unit

\(n\)

number density of total air

\(\frac{molec}{m^3}\)

\(n_{dry\_air}\)

number density of dry air

\(\frac{molec}{m^3}\)

\(n_{H_{2}O}\)

number density of H2O

\(\frac{molec}{m^3}\)

\(M_{air}\)

molar mass of total air

\(\frac{g}{mol}\)

\(M_{dry\_air}\)

molar mass of dry air

\(\frac{g}{mol}\)

\(M_{H_{2}O}\)

molar mass of H2O

\(\frac{g}{mol}\)

\(\rho\)

mass density of total air

\(\frac{kg}{m^3}\)

\(\rho_{dry\_air}\)

mass density of dry air

\(\frac{kg}{m^3}\)

\(\rho_{H_{2}O}\)

mass density of H2O

\(\frac{kg}{m^3}\)

\begin{eqnarray} n & = & n_{dry\_air} + n_{H_{2}O} \\ M_{air}n & = & M_{dry\_air}n_{dry\_air} + M_{H_{2}O}n_{H_{2}O} \\ \rho & = & \rho_{dry\_air} + \rho_{H_{2}O} \\ \end{eqnarray}

virtual temperature

symbol

name

unit

\(k\)

Boltzmann constant

\(\frac{kg m^2}{K s^2}\)

\(M_{air}\)

molar mass of total air

\(\frac{g}{mol}\)

\(M_{dry\_air}\)

molar mass of dry air

\(\frac{g}{mol}\)

\(M_{H_{2}O}\)

molar mass of H2O

\(\frac{g}{mol}\)

\(N\)

amount of substance

\(molec\)

\(N_A\)

Avogadro constant

\(\frac{1}{mol}\)

\(p\)

pressure

\(Pa\)

\(p_{dry\_air}\)

dry air partial pressure

\(Pa\)

\(p_{H_{2}O}\)

H2O partial pressure

\(Pa\)

\(R\)

universal gas constant

\(\frac{kg m^2}{K mol s^2}\)

\(T\)

temperature

\(K\)

\(T_{v}\)

virtual temperature

\(K\)

\(V\)

volume

\(m^3\)

From the ideal gas law we have:

\[p = \frac{NkT}{V} = \frac{10^{-3}NM_{air}}{VN_{a}}\frac{kTN_{a}}{10^{-3}M_{air}} = \rho \frac{RT}{10^{-3}M_{air}}\]

The virtual temperature allows us to use the dry air molar mass in this equation:

\[p = \rho\frac{RT_{v}}{10^{-3}M_{dry\_air}}\]

This gives:

\[T_{v} = \frac{M_{dry\_air}}{M_{air}}T\]

volume mixing ratio

symbol

name

unit

\(n\)

number density of total air

\(\frac{molec}{m^3}\)

\(n_{dry\_air}\)

number density of dry air

\(\frac{molec}{m^3}\)

\(n_{H_{2}O}\)

number density of H2O

\(\frac{molec}{m^3}\)

\(n_{x}\)

number density of quantity x

\(\frac{molec}{m^3}\)

\(\nu_{x}\)

volume mixing ratio of quantity x with regard to total air

\(ppv\)

\(\bar{\nu}_{x}\)

volume mixing ratio of quantity x with regard to dry air

\(ppv\)

\begin{eqnarray} \nu_{x} & = & \frac{n_{x}}{n} \\ \bar{\nu}_{x} & = & \frac{n_{x}}{n_{dry\_air}} \\ \nu_{dry\_air} & = & \frac{n_{dry\_air}}{n} = \frac{n - n_{H_{2}O}}{n} = 1 - \nu_{H_{2}O} \\ \nu_{air} & = & \frac{n}{n} = 1 \\ \bar{\nu}_{dry\_air} & = & \frac{n_{dry\_air}}{n_{dry\_air}} = 1 \\ \bar{\nu}_{H_{2}O} & = & \frac{n_{H_{2}O}}{n_{dry\_air}} = \frac{\nu_{H_{2}O}}{\nu_{dry\_air}} = \frac{\nu_{H_{2}O}}{1 - \nu_{H_{2}O}} \\ \nu_{H_{2}O} & = & \frac{\bar{\nu}_{H_{2}O}}{1 + \bar{\nu}_{H_{2}O}} \end{eqnarray}

mass mixing ratio

symbol

name

unit

\(M_{air}\)

molar mass of total air

\(\frac{g}{mol}\)

\(M_{dry\_air}\)

molar mass of dry air

\(\frac{g}{mol}\)

\(M_{x}\)

molar mass of quantity x

\(\frac{g}{mol}\)

\(n\)

number density of total air

\(\frac{molec}{m^3}\)

\(n_{dry\_air}\)

number density of dry air

\(\frac{molec}{m^3}\)

\(n_{H_{2}O}\)

number density of H2O

\(\frac{molec}{m^3}\)

\(n_{x}\)

number density of quantity x

\(\frac{molec}{m^3}\)

\(q_{x}\)

mass mixing ratio of quantity x with regard to total air

\(\frac{kg}{kg}\)

\(\bar{q}_{x}\)

mass mixing ratio of quantity x with regard to dry air

\(\frac{kg}{kg}\)

\(\nu_{x}\)

volume mixing ratio of quantity x with regard to total air

\(ppv\)

\(\bar{\nu}_{x}\)

volume mixing ratio of quantity x with regard to dry air

\(ppv\)

\begin{eqnarray} q_{x} & = & \frac{n_{x}M_{x}}{nM_{air}} = \nu_{x}\frac{M_{x}}{M_{air}} \\ \bar{q}_{x} & = & \frac{n_{x}M_{x}}{n_{dry\_air}M_{dry\_air}} = \bar{\nu}_{x}\frac{M_{x}}{M_{dry\_air}} \\ q_{dry\_air} & = & \frac{n_{dry\_air}M_{dry\_air}}{nM_{air}} = \frac{nM_{air} - n_{H_{2}O}M_{H_{2}O}}{nM_{air}} = 1 - q_{H_{2}O} \\ q_{air} & = & \frac{nM_{air}}{nM_{air}} = 1 \\ \bar{q}_{dry\_air} & = & \frac{n_{dry\_air}M_{dry\_air}}{n_{dry\_air}M_{dry\_air}} = 1 \\ \bar{q}_{H_{2}O} & = & \frac{n_{H_{2}O}M_{H_{2}O}}{n_{dry\_air}M_{dry\_air}} = \frac{q_{H_{2}O}}{q_{dry\_air}} = \frac{q_{H_{2}O}}{1 - q_{H_{2}O}} \\ q_{H_{2}O} & = & \frac{\bar{q}_{H_{2}O}}{1 + \bar{q}_{H_{2}O}} \end{eqnarray}

molar mass of total air

  1. molar mass of total air from H2O volume mixing ratio

    symbol

    name

    unit

    \(M_{air}\)

    molar mass of total air

    \(\frac{g}{mol}\)

    \(M_{dry\_air}\)

    molar mass of dry air

    \(\frac{g}{mol}\)

    \(M_{H_{2}O}\)

    molar mass of H2O

    \(\frac{g}{mol}\)

    \(n\)

    number density of total air

    \(\frac{molec}{m^3}\)

    \(n_{dry\_air}\)

    number density of dry air

    \(\frac{molec}{m^3}\)

    \(n_{H_{2}O}\)

    number density of H2O

    \(\frac{molec}{m^3}\)

    \(\nu_{H_{2}O}\)

    volume mixing ratio of H2O

    \(ppv\)

    \begin{eqnarray} M_{air} & = & \frac{M_{dry\_air}n_{dry\_air} + M_{H_{2}O}n_{H_{2}O}}{n} \\ & = & M_{dry\_air}\left(1 - \nu_{H_{2}O}\right) + M_{H_{2}O}\nu_{H_{2}O} \end{eqnarray}
  2. molar mass of total air from H2O mass mixing ratio

    symbol

    name

    unit

    \(M_{air}\)

    molar mass of total air

    \(\frac{g}{mol}\)

    \(M_{dry\_air}\)

    molar mass of dry air

    \(\frac{g}{mol}\)

    \(M_{H_{2}O}\)

    molar mass of H2O

    \(\frac{g}{mol}\)

    \(n\)

    number density of total air

    \(\frac{molec}{m^3}\)

    \(n_{dry\_air}\)

    number density of dry air

    \(\frac{molec}{m^3}\)

    \(n_{H_{2}O}\)

    number density of H2O

    \(\frac{molec}{m^3}\)

    \(q_{H_{2}O}\)

    mass mixing ratio of H2O

    \(\frac{kg}{kg}\)

    \(\nu_{H_{2}O}\)

    volume mixing ratio of H2O

    \(\frac{kg}{kg}\)

    \begin{eqnarray} M_{air} & = & M_{dry\_air}\left(1 - \nu_{H_{2}O}\right) + M_{H_{2}O}\nu_{H_{2}O} \\ & = & M_{dry\_air}\left(1 - \frac{M_{air}}{M_{H_{2}O}}q_{H_{2}O}\right) + M_{air}q_{H_{2}O} \\ & = & \frac{M_{dry\_air}}{1 + \frac{M_{dry\_air}}{M_{H_{2}O}}q_{H_{2}O} - q_{H_{2}O}} \\ & = & \frac{M_{H_{2}O}M_{dry\_air}}{M_{H_{2}O} + M_{dry\_air}q_{H_{2}O} - M_{H_{2}O}q_{H_{2}O}} \\ & = & \frac{M_{H_{2}O}M_{dry\_air}}{\left(1-q_{H_{2}O}\right)M_{H_{2}O} + q_{H_{2}O}M_{dry\_air}} \\ \end{eqnarray}

partial pressure

symbol

name

unit

\(p\)

total pressure

\(Pa\)

\(p_{x}\)

partial pressure of quantity

\(Pa\)

\(\nu_{x}\)

volume mixing ratio of quantity x with regard to total air

\(ppv\)

\(\bar{\nu}_{x}\)

volume mixing ratio of quantity x with regard to dry air

\(ppv\)

\begin{eqnarray} p_{x} & = & \nu_{x}p \\ p_{x} & = & \bar{\nu}_{x}p_{dry\_air} \\ p_{x} & = & N_{x}kT \end{eqnarray}

saturated water vapor pressure

symbol

name

unit

\(e_{w}\)

saturated water vapor pressure

\(Pa\)

\(T\)

temperature

\(K\)

This is the August-Roche-Magnus formula for the saturated water vapour pressure

\[e_{w} = 610.94e^{\frac{17.625(T-273.15)}{(T-273.15)+243.04}}\]