ASTR 321-Spring 2018 Problem set #7
due thurs May 24
1) Estimate the pressure (in gigapascals) at the center of the Earth with the assumption that Earth’s radius is 6400km and the radius of its core is 3500 km. Use 11,000 and 5000 kg m-3 for the densities of the core and mantle. Assume that the total central pressure is the sum of the central pressure of just the core by itself (no mantle above) plus the pressure at the radius of the core-mantle boundary calculated for a full Earth with a just the density of the mantle.
2) Estimate the local gravitational constant (little g) at the core mantle boundary and compare it with the 9.9 ms-2 at Earth’s surface.
3) The phase diagram of water, a plot of temperature versus H2O vapor pressure, has a point called the triple point where vapor, ice and liquid can co-exist. At vapor pressures below the triple point (612 pascals or 0.006 atm) water can only exist either as vapor or solid ice. Estimate the smallest diameter body (in km) that could have a high enough internal pressure to have liquid water present at its center. Assume that the body is a sphere and has a uniform density of 500 kg m-3 and that the interior pressure is purely determined by hydrostatic equili
ium.
3) The pressure (P) in a constant temperature atmosphere varies with altitude (Z) as
P = Po e-Z/H
where Po is the surface pressure, Z is the altitude and H is the scale height (the change in altitude required for the pressure to decrease by e-1). As derived from hydrostatic equili
ium (dP/dr= -g) and the perfect gas law (P=kT/µMh) -
H =
where µ is the molecular weight (relative to hydrogen), Mh= hydrogen mass, K is the Boltzmann constant 1.38 × 10-23 m2 kg s-2 K-1, T is absolute temperature, and g is the local gravitational constant.
Assuming a pure N2 (molecular wt =28) atmosphere and a temperature of 300K calculate the pressure at the top of Mt Rainier (height defined here to be 4 km) relative to the pressure at sea level (the answer is a number than <1).
4) What would the pressure ratio (peak/sea level) be for Mt Rainier if we had a pure H2 atmosphere as may exist on some Super Earths?
5) Once global warming heats Earth and melts all the ice on Mount Rainier, will the pressure at the summit be higher or lower than it is today, assuming that sea level pressure remains constant and that the summit altitude stays the same.