Math 2.0

Geophysics Atmospheric dynamics and meteorology Proposition

Barometric formula

Builds on Scale heightLapse rateExponential functionOrdinary differential equation

Statement

Let a column of ideal gas of molar mass MM have pressure p(z)p(z), density ρ(z)\rho(z) and absolute temperature T(z)T(z) at height zz. Assume hydrostatic balance dp/dz=−ρgdp/dz = -\rho g with constant gravitational acceleration gg. Assume the ideal gas law p=ρRT/Mp = \rho R T/M, where RR is the molar gas constant. Let p0=p(0)p_0 = p(0).

  1. If T(z)=T0T(z) = T_0 is constant, then p(z)=p0exp⁡(−z/H)p(z) = p_0 \exp(-z/H), where H=RT0/(Mg)H = R T_0/(M g) is the scale height.
  2. If T(z)=T0−LzT(z) = T_0 - L z with a constant lapse rate L>0L > 0, then for 0≤z<T0/L0 \leq z < T_0/L, p(z)=p0(1−LzT0)gM/(RL).p(z) = p_0 \left( 1 - \frac{L z}{T_0} \right)^{g M/(R L)}.

Remarks

Proof: the two laws give the ordinary differential equation dp/p=−(Mg/(RT(z))) dzdp/p = -\big( M g / (R T(z)) \big) \, dz; integrate from 00 to zz. For constant TT this gives an exponential; for T=T0−LzT = T_0 - Lz it gives ln⁡(p/p0)=(gM/(RL))ln⁡(1−Lz/T0)\ln(p/p_0) = (gM/(RL)) \ln(1 - Lz/T_0). As L→0L \to 0 the second formula tends to the first, since (1−Lz/T0)gM/(RL)→e−gMz/(RT0)(1 - Lz/T_0)^{gM/(RL)} \to e^{-gMz/(RT_0)}. Example: with T0=288T_0 = 288 K, H≈8.4H \approx 8.4 km, so in an isothermal model the pressure at 5.55.5 km is about p0e−0.65≈0.52 p0p_0 e^{-0.65} \approx 0.52 \, p_0, close to the observed half of sea-level pressure. Without constant gg the formula fails at heights comparable with the planet's radius, where gg decreases.

Going deeper

For a student

The formula combines the two basic laws of a resting atmosphere: hydrostatic balance and the ideal gas law. The standard atmosphere used in aviation is built from pieces of form 2, with different lapse rates in different layers. The density obeys the same laws, with exponent gM/(RL)−1gM/(RL) - 1 in case 2. The hypsometric equation is the general form for any temperature profile.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Exponential function, concept, no evidence levelExponential functionconceptLapse rate, concept, no evidence levelLapse rateconceptOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptScale height, concept, no evidence levelScale heightconceptBarometric formula, statement, no evidence levelBarometric formulastatement
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  1. Exponential function concept
  2. Lapse rate concept
  3. Ordinary differential equation concept
  4. Scale height concept
  5. Barometric formula statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Barometric_formula
About this page
TagV3MXN0MARH
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Kindstatement, proposition
WritingW0 stub
MSC86A10
StatusThis copy of the site has no build data. Run m2 status V3MXN0MARH.
AboutScale height
UsesScale height, Lapse rate, Exponential function, Ordinary differential equation
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Changed3 November 2026

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