Astronomy 241: Problem Set #5 Due September 30, 2016 Solve The

Astronomy 241: Problem Set #5 due September 30, 2016

Solve the problems listed below, and write up your answers clearly and completely. Do not turn in a rough draft – make a clean copy after checking your calculations. Use brief sentences and phrases to explain your solution and introduce symbols and key equations. Show your work!

1. In some respects, Earth and Venus are “twin planets”; for example, Earth has mass 24 6 24 ME 6.0 10 kg and radius RE 6.4 10 m, while Venus has mass MV 4.9 10 kg ≃ × 6 ≃ × ≃ × and radius RV 6.1 10 m. However, they have very diﬀerent atmospheres. The surface ≃ × 5 1 2 pressure on Earth is P0,E 1.01 10 kg m− s− , while the surface pressure on Venus is 6 1 2≃ × P 9.2 10 kg m− s− . 0,V ≃ × (a) Using the masses, radii, and surface pressures given above, calculate the mass of each planet’s atmosphere in (i) kilograms and (ii) as a fraction of the planet’s mass. (Hint: pressure is weight of atmosphere per unit surface area, and weight is gravitational acceleration times mass.) (b) Earth’s oceans have an average depth of 4kmandcover70%ofEarth’ssurface;the 3 ∼ density of seawater is 1025 kg m− .CalculatethemassofEarth’soceansin(i)kilograms and (ii) as a fraction of Earth’s mass. (c) Suppose Earth’s oceans were vaporized. Estimate the surface pressure.

2. The mean molecular weight, µ, of a mixture is the average mass per molecule in units of the mass of a hydrogen atom, mH.Inotherwords,ifm is the average mass per molecule, then µ = m/mH.

(a) The atmosphere of Venus is almost pure CO2. Given that mC =12mH and mO =16mH, calculate µ for the atmosphere of Venus.

(b) The atmospheres of Jupiter and Saturn are basically a mixture of H2 and He, with H2 making up 75% of the mass, and He making up the remaining 25%. Given that mHe =4mH, calculate µ for this mixture.

(c) The Earth’s atmosphere is basically a mixture of N2 and O2.Byvolume,aircontains roughly 80% N2 and 20% O2. Given that mN =14mH, calculate µ for air.

3. For an isothermal (constant temperature) atmosphere, the pressure P at a distance z above the planet’s surface is

z/H kBT P (z)=P0 e− , where H = . (1) µmHg

Here P0 is the pressure at the surface, and the scale height H depends on the temperature T , mean molecular weight µ,andtheplanet’sgravitationalaccelerationg;theBoltzmann 23 2 2 1 27 constant is k =1.38 10− kg m s− K− ,andm =1.67 10− kg. B × H × (a) Verify that H uas units of length.

1 (b) Substitute this equation for P (z) into the equation for hydrostatic equilibrium,

dP µm g = H P, (2) dz − kBT and verify that it is a solution. (c) Given that Venus has a atmospheric temperature T =737K,usetheinformationprovided in questions 1 and 2 to calculate the scale height of its atmosphere.