LAB 10: GRAVITY ANOMALIES AND CORRECTIONS

Transcription

1 NAME: LAB TIME: LAB 10: GRAVITY ANOMALIES AND CORRECTIONS The following lab will introduce you to the basic concepts of gravitational forces, accelerations, and using variations in gravity to understand the subsurface structure of the Earth. Be careful to use proper units! Some of the questions here are based on materials provided by Anna Courtier (James Madison) and Derek Schutt (Colorado State). Part I: Newton s Law of Universal Gravitation 1) The acceleration due to gravity on Earth is approximately 9.81 m/s 2. What is g on Mars? a. Calculate the acceleration due to gravity, g, on Mars to the nearest 0.01 m/s 2. Assume: mass of Mars is 6.42x10 23 kg, Mars radius is 3397 km, and the universal gravitational constant is 6.67x10-11 m 3 kg -1 s -2. Make sure that you are clear in showing why the acceleration due to gravity only depends on the mass of Mars and not any object s mass. b. Is gravity on Mars more or less than on Earth? c. What two parameters control the acceleration due to gravity on a planet? Page 1 of 7

2 d. Hypothetical planet A has a radius of 2000 km and a mass of 5x10 23 kg while planet B has double the mass and radius. Exactly how much different (in percent) is the acceleration due to gravity on planet B compared to planet A? Hint: you do not need to actually calculate each planets gravity. Just look at the gravity equations and calculate what the change would be if both mass and radius were doubled. 2) You are sitting on the couch meters away from your 7.00 kg cat, Fluffy (she s big boned). You recall our discussion of gravitational forces in class and become curious about the gravitational forces between you and Fluffy relative to the other planets in the Solar System. Calculate the magnitudes of the gravitational forces between you and Fluffy relative to the maximum gravitational force exerted on you by Mars and Jupiter? I.e., F calculate cat. Use the data in table below as needed. To make this simple, let s assume that all of the planets F planet are organized in a straight line relative to the Sun. Planet Distance From Sun (km) Mass (kg) Earth 1.50E E+24 Mars 2.28E E+23 Jupiter 7.79E E+27 a. For this particular question you are not interested in the actual forces between you and these objects, just how the force between you and Fluffy compares to the force between you and the planets (i.e. the ratios). Before you calculate anything, show your setup of how you would calculate the relative force between two different objects. Use the following variables in your setup: Mplanet, Myou, Mcat, are the masses of the three objects in question, Rplanet and Rcat are the distances from you to the planet in question and fluffy, respectively, and G is the universal gravitational constant. Hint: If you set this up properly, you will find that you don t need to know G or your mass. What the final simplified equation (no numbers, just variables) for calculating the relative gravitational forces? Make sure that you algebraically simplify the final equation so it is not two fractions divided by each other. This will save you time later. Page 2 of 7

3 b. Now, using your equation from part a, calculate the relative forces (i.e. the force ratios) exerted by each planet (excluding Earth) on you relative to the cat s force. What are the units? c. Explain in your own words what these results mean. ( 3 sentences) d. How far (in meters) from you would fluffy have to be from you for her gravitational force exerted on you to be equal to the force exerted by Mars on you? Be clear and show your work carefully. Page 3 of 7

4 Part II: Gravity Anomalies 1) You run into your good friend Bob, an exploration geophysicist, who happens to be very excited about some gravity data he has just collected. He explains to you that his gravity data proves that there is a fairly large body of a dense material outside of Shenandoah National Park in Virginia. In fact, he goes on to say the material has a density of 33.0 g/cm 3, the body is roughly spherical with a radius of 862 m, and it is buried to a depth of 5000 m (to its center). a. Based on what you have learned in Geophysics, you wonder if Bob isn't horribly mistaken, and you ask if you can see his data. What is one clue that Bob might be wrong? ( 3 sentences) Follow the instructions below and use the Excel file provided on the course website Fill in the cells for the instrumental drift corrected data. Do this by assuming that the data drifted linearly during the time interval between the two measurements at the base station (Distance=0). To see if your instrumental drift correction worked properly, make a plot (scatter plot with straight line connected symbols) of the raw gravity data and your drift corrected data on the same plot. Fill in the cells for the free air and drift corrected data. Use the simplified form (i.e. the linear equation) of the free air correction given in the textbook and the course lecture notes. To see if your free air correction worked, add the free air corrected data to your plot. Your final corrected data should look something like the buried sphere anomalies shown in the lecture notes. Fill in the cells for Bob s model. To do this, use the analytical solution for the gravity anomaly due to a buried sphere and plug in Bob s values from above. Note that the analytical solution assumes that the buried sphere is at x=0, which is clearly not the case with Bob s data. Determine the x location to the nearest 1000 m by translating the analytical solution until it best fits the data. Assume that G = 6.67x10-11 m 3 kg -1 s -2 and that the rock surrounding the buried sphere is limestone (ρ = 2.7 g/cm 3 ) for all models. Make a new plot similar to your previous plot with the corrected data and Bob s model results. Hint: Bob s model should fit the data relatively well, so if your calculation of Bob s model doesn t fit the data well, you may want to double check all of your units. Is Bob s proposed model unique? To address this question fill in the two Excel columns for your two models. Use z = 5000 m and R = 4000 m for both models, but choose two geologically reasonable values for Δρ given the range of rock densities shown in Table below. Once you have created your two models add these data to your plot along with Bob s model. Make sure to add a clear legend to both of your plots, so I will know which curve is which. Common Rock Types in the Shenandoah Region Rock Type Approximate Density (g/cm 3 ) Limestone Sandstone Shale Diabase Page 4 of 7

5 b. What to hand in? You should include two plots at the end of this lab with brief captions. Figure 1: The first plot will show the raw gravity data, the drift corrected data, and the free air corrected data (i.e. the final corrected data). Figure 2: The second plot will show your final corrected data, Bob s model, and your two models. c. At what x location is the buried spherical object? Fill in the value below. As instructed above, you only need to estimate the location to the nearest 1km. Buried object is at X = meters d. Based on your plot of Bob s models and your two models, can you tell which model is right? Can you tell if a model is wrong? Why? ( 4 sentences) e. Do you think that there are other models that could also fit this data? Can geological or mathematical reasoning eliminate any of the models? Be clear in your explanation. ( 5 sentences) Page 5 of 7

6 2) Your textbook states that the free air correction is approximately mgal/m of elevation change. This suggests a linear relationship between the rate of change in gravity with respect to elevation. Wait a minute rate of change that sounds like a calculus problem! a. Calculate the equation for the rate of change in gravity (in mgal/m) with respect to distance from the center of the Earth. Is this a linear relationship? Why or why not? b. What is the rate of change in g, in mgal/m, at sea level? (Assume that the radius of the Earth is 6371 km at sea level). Page 6 of 7

7 c. What is the rate of change in g, in mgal/m, at 8 km elevation in the Himalayas (i.e. radius = 6379 km)? d. Do you think that the textbook s simplification of this correction is justified? Why or why not? ( 3 sentences) Page 7 of 7