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# Write down an expression for the variation of gas density units of mass per volume with height z above the Earths surface assuming the gas is at constant temperature T and in hydrostatic

Write down an expression for the variation of gas density  (units of mass per volume) with height z above the Earth’s surface, assuming the gas is at constant temperature T and in hydrostatic equilibrium. Take the gas to be made of a single kind of molecule of weight µ (units of mass). Express in terms of the density at ground level 0  (z = 0) and the density scale height h  kT/(µg), where g is the gravitational acceleration. Neglect variations of g with z. (b) Utterly neglecting collisions with other molecules, what is the maximum height a molecule would attain if launched from z = 0 with a typical thermal velocity? Is this close to h? This calculation is misleading insofar as it ignores collisions between molecules, which are crucial for describing air as a continuum fluid. Nevertheless, it provides a mnemonic for remembering the scale height, and it illustrates the sometimes surprisingly close connection between fluid mechanics and particle mechanics (kinetic theory). (c) At what height z would you expect the formula in (a), which depends on the con- tinuum hypothesis, to fail? This height marks the location of the exobase in planetary atmospheres. Just consider how intermolecular collisions validate the continuum ap- proximation. (d) Write down an expression for the hydrostatic variation of gas density  with height z above the midplane of a circumstellar disk at radius r. As in (a), assume constant T and µ. Take the gravitational field to be that from the star alone (ignore the self-gravity of the disk). Work in the limit that z  r. Express in terms of the density at the midplane 0 and the density scale height h  (kT/µ)1/2 -1, where is the Keplerian orbital angular frequency. The height h is often written h = cs/ , where cs = (kT/µ)1/2 is the speed of sound waves in gas that behaves isothermally [by behaving isothermally, we mean that the gas keeps the same temperature regardless of how it is compressed or expanded by the sound wave. This problem does not concern a sound wave per se, but people talk about the sound speed anyway because sound waves (pressure disturbances) are the means by which a medium establishes hydrostatic equilibrium.] (e) As in (b), calculate the maximum height that a gas molecule would attain if launched upwards from the midplane with a typical thermal speed, ignoring collisions. Compare 1 to h. (f) For the disk to be geometrically “thin” (h/r  1), what must be true about cs/vK? Here vK is the Keplerian orbital velocity.

Jun 19 2020 View more View Less

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