Pressure Variation in Planetary Atmosphere

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SUMMARY

The variation of pressure with altitude in a planetary atmosphere, assuming constant temperature, is accurately described by the equation p = poek(1/r-1/R). In this equation, p represents pressure, po is the surface pressure, R is the planet's radius, r is the distance from the planet's center, and k is a constant. The gravitational force is modeled as g = k/r^2, which influences the pressure gradient. The relationship between pressure and density is established through the equation P/Po = p/po, leading to the differential equation dP/dy = -pg.

PREREQUISITES
  • Understanding of basic atmospheric physics
  • Familiarity with calculus, particularly differential equations
  • Knowledge of gravitational forces and their impact on pressure
  • Concept of hydrostatic equilibrium in planetary atmospheres
NEXT STEPS
  • Study the derivation of the hydrostatic equation in planetary atmospheres
  • Explore the implications of varying gravitational forces on atmospheric pressure
  • Learn about the applications of the barometric formula in different planetary contexts
  • Investigate the effects of temperature variations on atmospheric pressure profiles
USEFUL FOR

Astronomers, atmospheric scientists, and students studying planetary atmospheres will benefit from this discussion, particularly those interested in pressure dynamics and gravitational effects on atmospheric behavior.

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1. Show that the variation of pressure with altitude for a planetary atmosphere (assuming constant temperature) is more accurately given by: p = poek(1/r-1/R), where g is taken to vary as 1/r2 (with r being the distance from the centre of the planet), po is the pressure at the surface, R is the radius of the planet, and k is a constant.

lower case p = pressure
upper case P = density






2. P/Po = p/po
dP/dy = -pg




3. dP/P = -g(po/Po)dy
 
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g = k/r^2 and since both dy and dr represent a change in height dr = dy
 
Thank you, makes sense.
 

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