Home / Questions / Write down an expression for the variation of gas density units of mass per volume with h...
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
Jupiter stores had a Quarter 2 beginning cash balance of $430. Sales for Quarters 1 through 3 are estimated at $600, $800, and $900, respectively. The cost of goods sold ...
Apr 02 2020Guilt in decision making. The effect of guilt emotion on how a decision maker focuses on the problem was investigated in the Jan. 2007 issue of the Journal of Behavioral ...
Aug 07 20204. ABC company suffered inventory loss from market declines in April 2017. As a result, the company wrote off $7,000 cost of inventory. However, the price of the same inv...
Sep 12 2020Northern Outfitters’ invoice to Rico’s Menswear for $2463.80 was dated October 22 with terms 2/10, n/30. Late payments are charged a 1% penalty on the overdue balance. Ri...
May 22 2020One reason for implementing global training programmes is the need to avoid business losses due to cultural insensitivity >
Jul 08 2020b. Suppose you buy a 6-month put option with a strike price of $50. What is the payoff in 6 months at the same prices for the underlying asset? c. Comparing the payoff...
Nov 24 2019explain pricing policies methods cost plus pricing going rate pricing rate of return pricing
May 07 2020If a company wants to increase its ROA through higher profit margin while also having a high asset turnover, which of the following will make it harder for the firm to ac...
Aug 31 2020the market has an expected rate of return of 12.8% The long termgovernment bond is expected to yield4.5% and the US T bill isexpected to yield 3.4%. The inflation rate is...
Nov 30 2019If one firm in a perfectly competitive market doubled its output, marketa.demand would doubleb.demand would increase by 50 percentc.supply would doubled.supply would in...
Dec 12 2019