Geography 3422A/B Study Guide - Comparative Advantage, Human Capital

Astrophysicists in the 1960s tried to explain the existence and thestructure of pulsars – extremely regular astronomical sources ofradio pulses whose periods ranged from seconds to milliseconds. Atone point, these radio sources were given the acronym LGM, standingfor “Little Green Men,” a reference to the idea that they might besignals of extraterrestrial civilizations. The explanation giventoday is no less interesting. Consider the following. Our Sun,which is a fairly typical star, has a mass of 1.99x10^30 kg and aradius of 6.69x10^8 m. Although it does not rotate uniformly,because it isn’t a solid body, its average rate of rotation isabout 1 rev / 25 days. Stars larger than the sun can end theirlives in spectacular explosions called supernovae, leaving behindcollapsed remnants of the stars called neutron stars. Neutron starshave masses comparable to the original masses of the stars, butradii of only a few kilometers. These stars emit beams of radiowaves. Because of the rapid angular speed of the stars, the beamssweep past Earth at regular, very short, intervals. To produce theobserved radio-wave pulses, the star has to rotate at rates fromabout 1 rev/s to 1000 rev/s.

a. Estimate the rotation rate of the Sun if it were to collapseinto a neutron star of radius 10 km. The Sun is not a uniformsphere of gas, and its moment of inertia is given by I = 0.059MR2.Assume that the neutron star is spherical and has a uniform massdistribution.
b. Is the rotational kinetic energy of our Sun greater of smallerafter the collapse? By what factor does it change, and where doesthe energy go to or come from?

3. 3 Stratosphere-troposphere exchange

The rate of exchange of air between the troposphere and thestratosphere is critical for determining the potential of variouspollutants emitted from the surface to reach the stratosphere andaffect the stratospheric ozone layer. One of the first estimates ofthis rate was made in the 1960s using measurements of strontium-90(90Sr) in the stratosphere. Strontium-90 is a radioactive isotope(half-life 28 years) produced in nuclear explosions. It has nonatural sources. Large amounts of 90Sr were injected into thestratosphere in the 1950s by above-ground nuclear tests. Thesetests were banned by international treaty in 1962. Following thetest ban the stratospheric concentrations of 90Sr began to decreaseas 90Sr was transferred to the troposphere. In the troposphere,90Sr is removed by wet deposition with a lifetime of 10 days (bycontrast there is no rain, and hence no wet deposition, in thestratosphere). An intensive stratospheric measurement network wasoperated in the 1960s to monitor the decay of 90Sr in thestratosphere. We interpret these observations here using a 2-boxmodel for stratosphere-troposphere exchange with transfer rateconstants kTS and kST (yr-1) between the tropospheric andstratospheric reservoirs. The reservoirs are assumed to beindividually well-mixed.



Let mS and mT represent the masses of 90Sr in the stratosphere andin the troposphere respectively. Observations of the decrease inthe stratospheric inventory for the period 1963-1967 can be fittedto an exponential mS(t) = mS(0)exp(-kt) where k = 0.77 yr-1.

1. Write mass balance equations for mS and mT in the 1963-1967period.

2. Assuming that transfer of 90Sr from the troposphere to thestratosphere is negligible (we will verify this assumption later),show that the residence time of air in the stratosphere is tS =1/kST = 1.3 years.

3. Let m'T and m'S represent the total masses of air in thetroposphere and the stratosphere, respectively. Show that theresidence time of air in the troposphere is tT = tS (m'T/m'S) = 7.4years. Conclude as to the validity of your assumption in question2.

4. Hydrochlorofluorocarbons (HCFCs) have been adopted asreplacement products for the chlorofluorocarbons (CFCs), which werebanned by the Montreal protocol because of their harmful effect onthe ozone layer. In contrast to the CFCs, the HCFCs can be oxidizedin the troposphere, and the oxidation products washed out byprecipitation, so that most of the HCFCs do not penetrate into thestratosphere to destroy ozone. Two common HCFCs have trade namesHCFC-123 and HCFC-124; their lifetimes against oxidation in thetroposphere are 1.4 years and 5.9 years, respectively. There are noother sinks for these species in the troposphere. Using our 2-boxmodel, determine what fractions of the emitted HCFC-123 andHCFC-124 penetrate the stratosphere.