Part 1: The GRACE satellite mission measures and observes subsurface aquifers
See slides 9-13 of the lecture given on October 28th for reference.
The force due to gravity (Fg) between two objects is governed by Newton’s Law of Universal Gravitation:
(1)
Where G is the gravitational constant, G = 6.674×10-11 N?•?(m/kg)2, m1 and m2 are the masses of the objects, and r is the distance between the objects. Let’s now assume m1 is a small object (say a person) on the surface of the earth and m2 is the mass of the earth. The force due to earth’s gravity is the weight, Fg = m*g, of the small object, where g is the acceleration due to earth’s gravity. So
, where m1 cancels and
(2)
Test this by plugging in the values for the earth:
Mass of the Earth = 6.00097 × 1024 kg
Radius of the Earth = 6,371 km
Plugging these values into equation 2 results in an answer of ~9.87 m/s2. You may recall that the acceleration due to Earth’s gravity is 9.81 m/s2. The difference between what you remember and what you calculated is because the earth is not a perfect sphere, as the calculations in this homework assume. Make sure you can use equation 2 above to get an answer of ~9.87 m/s2 before proceeding with the homework (watch your units- convert the radius of earth to meters from kilometers).
GRACE is a satellite that measures the strength of the Earth’s gravitational field at all points along its orbit. The orbit has a constant radius, so by equation 2 above, any variations in the strength of g must then be caused by variations in mass. So how can the mass of the earth change?
The answer is that the strength of g is based not on the total mass (which is constant), but on the density of the earth in a direct line between the satellite and the center of the earth. The density of various regions of the earth is not constant, so the strength of g varies from place to place. For example, oceans are less dense than land, so the pull of earth’s gravity is slightly less over the oceans compared to the land. So on what scale is GRACE able to detect variations in gravity? Lets find out.
1) GRACE orbits at an average distance of 500 km from the earth’s surface. What is the strength of earth’s gravity (i.e. solve for g) acting on GRACE? (Watch your units- you need to use meters and give your final answer in m/s2. Also don’t forget to add Earth’s radius to the height of GRACE’s orbit to get the total distance from GRACE to the center of the Earth). Show your work and put a box around your answer.
Next we have to recalculate the effective mass of the earth based on the density below GRACE. Let’s say that for the lower 6,368 km of earth’s radius, the density is uniform- it’s all crust, mantle, and core. This density = 5540 kg/m3. Above 6,368 km, the final 3 km, are objects with varying density. We can ignore the density and mass of the atmosphere and space between GRACE and the earth’s surface because it is so small compared to the mass of the earth. The average density (?) of an object with multiple components of different density along a given radius is:
(3)
Where r1, r2, r3, … , rn, is the thickness of each of the different components, ?1???2???3????????n ?is the density of each component, and r is the total radius (in this case, the radius of the earth). To get mass, you multiply volume by density. We know density now, and the volume of a sphere is:
(4)
2) If the upper 3 km are ocean, which has a density = 1030 kg/m3, what is the gravitational acceleration (g) detected by GRACE? As always, watch your units. Show your work and put a box around your answer. Hint: This question has 2 density components, so in equation 3 above, the earth is component 1 (radius = 6368 km and density = 5540 kg/m3) and component 2 is the ocean (radius = 3 km and density = 1030 kg/m3).
3) If the upper 3 km are rock, which has a density = 2700 kg/m3, what is the gravitational acceleration (g) detected by GRACE? Show your work and put a box around your answer.
4) If the upper 3 km is 0.5 km of freshwater aquifer, density = 1000 kg/m3, and 2.5 km rock, density = 2700 kg/m3, what is the gravitational acceleration (g) detected by GRACE? Show your work and put a box around your answer. Hint: Now there are 3 components for equation 3.
5) What difference does the presence of an aquifer 500 meters thick cause in the measured gravitational field strength compared to solid rock? In other words, compare your answers from questions 3 and 4. Answer in m/s2. Show your work and put a box around your answer
6) Based on your answer to #5, to what sensitivity is GRACE able to detect variations in the Earth’s gravitational field? Report your answer as a percent of the strength of Earth’s gravity at GRACE’s orbit (your answer to Question #1). Show your work and put a box around your answer. Hint: IF your difference (answer to #5) is 0.10 m/s2, and the gravity at GRACE (answer to #1) is 8.00, that is a sensitivity of 1.25%
Part 2: Aquifer Recharge
See slides 24-25 of the lecture given on October 28th for reference.
One of the resource issues related to aquifers is that humans are using the water faster than it can be replaced through natural processes. This causes the aquifer to dry up. The following questions will give you an idea of these timescales. The numbers are based on data about the Ogallala aquifer, which underlies most of the western Great Plains and is a crucial water source for agriculture in the region.
The Ogallala aquifer underlies a surface area of 450,000 km2. On average, it is found at a depth of 15 to 90 meters (a thickness of 75 meters). Its average recharge rate is 21.59 mm/year. A few decades ago during the peak of human pumping, the average rate of withdrawal was as high as 1 meter/year (1000 mm/year). These rates mean that the thickness of the aquifer increases or decreases by those distances every year. For the purposes of this assignment, we assume that these rates affect the aquifer uniformly. This is not actually the case- pumping varies from region to region, but these averages give us a basic idea of the processes involved.
7) At the peak rate of pumping, what was the annual rate of overdraft of the aquifer? (Overdraft is the rate by which the withdrawal exceeds the recharge). Watch your units! Give your answer in mm/year. Show your work and put a box around your answer.
8) At this rate, how many years would it take for human actions to drain the aquifer? Show your work and put a box around your answer.
9) Recently, conservation efforts have reduced overdraft of the aquifer to only 55.86 mm/year. At this rate, how many years will it take for human actions to drain the aquifer? Show your work and put a box around your answer.
10) If the aquifer is completely depleted, how long will it take to refill? Show your work and put a box around your answer.