Astronomy 405 Worksheet 15 – Atmospheric Pressure 1. The equation of hydrostatic equilibrium is dp = −ρ g dz where ρ and p are the density and pressure at altitude z and g is the downward acceleration of gravity. This equation expresses the idea that the increase in pressure through a little vertical distance –dz is just enough to support the weight of that slab of material ρgdz. a) Integrate the above expression to derive a relation between pressure and depth (d = z2 – z1) for an ocean. You may assume an atmospheric pressure at sealevel of Patm= 1.01 × 105 Pa. b) Calculate the pressure at a depth of 800 m for an ocean (psaltwater=1025 kg/m3). c) Rewrite the equation of hydrostatic equilibrium for altitude in the Earth’s atmosphere (which doesn’t have a constant density). You should express the density of the atmosphere using the ideal gas law: ρ kT p= µ mµ d) Integrate the above expression and derive a formula relating p and z. e) Calculate the scale height of the Earth’s atmosphere (the height over which pressure or density falls off by a factor of 1/e). You should clearly state the assumptions you are making for T and μ. f) The variable h1/2 is also commonly used to express the altitude at which the pressure (and density) have decreased to half of their sealevel values. Calculate the value of h1/2 for the Earth’s atmosphere. g) Imagine that the density of the earth’s atmosphere was uniform having the same density that we experience at sealevel 1.29 kg/m3. Also assume that the atmosphere extended upward to a certain altitude and then abruptly the density decreased to zero. What would this altitude be to give us the observed sealevel pressure of 1.01 × 105 Pa?