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Craters

	The sky is falling, the sky is falling!
	-- Chicken Little

Overview

• Evaluate parameters affecting crater formation.
• Find the size of the asteroid/comet that killed the dinosaurs.
• Make your own series of craters, to observe the "geological" results.

Introduction

One look at the surface of the Moon should convince you that "empty space" is not so empty after all. There is actually a wide range of objects floating between the planets, from tiny particles to asteroids that can be a hundred miles across, debris left behind when the planets were formed. These objects can be perturbed from their orbits (by a close passage by a planet, a passing star, any number of things) and onto paths that cross ours -- or any other planet or moon. When that happens, a collision occurs and an impact crater is formed.

The size and shape of the crater depend on the impactor: its size, shape, speed, and the angle is hits the ground with. Specifically, the size of the crater depends on the energy of the impactor. However, the relationship is not linear, but rather is a power law:

where D is the diameter, E is the energy of the impactor when it hits the ground, n is the power, and k is a constant. In part one of this lab, you will use a model to determine what k and n are.

The energy when the impactor strikes the ground is all kinetic,

where m is the mass of the impactor, and v is the speed it's going when it hits the sand. Unfortunatly, v is inconvenient to measure in our classroom. Fortunatly, energy is conserved, so we can give the impactor a known energy and know it will hit the sand with that amount of energy. The total energy of a falling object is the sum of the kinetic and potential energy. If you drop the impactor so it starts with v = 0, the total energy is just potential (called gravitational potential energy):

where m is still the mass, g is the acceleration due to gravity = 9.81m/s² at the surface of the Earth, and h is the height above the ground.

To demonstrait cratering, you'll have a box of sand. There should be a thick base of white sand which you'll add a thin regolith of colored sand to. Get a box and some colored sand from your instructor (note if the sand in the box isn't white, you should get colored sand with a good contrast). Be sure to make the layer of colored sand very thin, since you want to see the pattern of ejecta when you make the crater.

Part 1: the Power Law

In this part you'll be figuring out the values of k and n for your model.

Find the mass of your ball bearing and fill in its value on the worksheet (note you can do this at any time during data collection) To use the scale, turn it on using the power button on the far right side, place the tray on it with the arrow pointing to the back, and push "TARE". When it reads "0.000 kg" (zero kg) place the ball bearing in the tray and take the reading when the numbers stabalize. If the display has the wrong unit, push the unit button until "0.000 kg" is displayed.

Your GSI will give you two heights to drop the ball bearing from. Smooth the sand and sprinkle a thin layer of colored sand on top. Do not knock the boxes on the floor to smooth the sand! Drop the ball bearing from the first height into the sand and measure the diameter of the crater from rim to rim. If the crater isn't round (especially if it lands too close to the side of the box) measure the longest axis. Enter the height and diameter in table 1. Pay attention to the shape of the crater and the pattern made by the ejecta.

Retrieve the ball bearing and drop it again from the other height into another part of the container. Record the height and crater diameter in table 1.

Record your crater diameters in the table on the chalkboard. Note there may be more than one group with the same height as you. Record all the other measurements from the chalkboard. If there is more than one value, write both values down, then calculate the average. Use the average in your calculations.

Get the broom and dustpan from your GSI and clean up any spilled sand (the custodians will not clean up messes made by lab equipment!)

You may want to skip to part 2 while waiting for the table to be finished.

Recall from the introduction that the impact energy is equal to the energy you gave the ball bearing. Write the equation you'll use to calculate the impact energy below table 1 and show the calculation for the first row (watch your units!) Check it with your GSI, then fill in the rest of the column.

Since the relationship between D and E is a power law, a graph of D vs. E would be curvy, and it would be very hard to figure out what n and k are. However, n is the slope of a log-log graph (see your GSI if this doesn't make sense to you). Calculate logD and logE and fill in those values in table 1. Do the graph of logD vs. logE. Be sure to include the graph with your lab.

Draw a "best-fit" line through the data points on the graph. Remember, a best-fit line is a straight line that comes close to, but probably not through, as many points as possible. Mark the two places (not data points) on your line and find the slope. Record this as n on the worksheet. If you use a computer program or graphing calculator to find the slope, include the equation of the best-fit line on your graph.

To find k, you will use one of your data points. Pick a data point that is close to your best-fit line (one that you think is a good value), and plug D, E, and your value for n into the equation D = k Eⁿ. Now you can solve that equation for the value of k. For example, suppose you dropped a 0.03kg ball bearing from a height of 2m. This give you an impact energy of E = mgh = (0.03kg) * (9.81m/s^2) * (2m) = 0.59Joules. And suppose that when you dropped the ball bearing, you measured the crater diameter to be 5cm = 0.05m, and also your graph had given you a slope = n = .25. Then,

plugged into the equation D = k Eⁿ

Now, go and put all this knowledge to work in the Activity #1 Questions!

Part 2: multiple craters and odd shapes.

Smooth the sand and apply a thin layer of colored sand. Throw the ball bearing in sideways and observe the crater and ejecta pattern. If you want to retrieve it be careful not to destroy your crater. Drop some of the non-spheroidal objects and observe the crater and ejecta patterns. Drop the ball bearing in from a couple different heights and observe what the surface looks like, then answer the questions at the end.

Data

Table 1: Cratering parameters and calculations

Equation and sample calculation of the impact energy (check this with your GSI before filling out the table).

Your values:

• n =
  
  
  
  
• k =
  
  
  
  
  

Questions:

1. Does the shape of the crater and the pattern of ejecta depend more on the shape of the impactor, or the angle it hits the ground at? How do you know?
   
   
   
   
   
   
   
   
   
   
2. If you looked at someone else's box, could you tell which craters were formed first? How?
   
   
   
   
   
   
   
   
   
   
3. A quick glance at the Moon through a telescope shows you a surface covered with craters. What does this tell you about the age of the Moon's surface compared to the age of the Earth's surface? Explain.
   
   
   
   
   
   
   
   
   
   
4. One of the most easily recognizable lunar craters is Copernicus: a round crater with bright rays of ejecta around it. What does this tell you about the object that made Copernicus?
   
   
   
   
   
   
   
   
   
   
5. The diameter of the crater from the asteroid that probably killed the dinosaurs is about 180 km. According to your model, how much kinetic energy did that asteroid have when it hit the Earth? Show your work.
   
   
   
   
   
   
   
   
   
   
6. A good estimate for the velocity (based on observations of other imapcts) of this asteroid when it hit the Earth would be about 20 km/s. What was the mass of the asteroid? Show your work.
   
   
   
   
   
   
   
   
   
   
7. Assuming a density of about 3 gm/cm^3 = 3000 kg/m^3 (an average density for asteroids) and a roughly spherical shape, what was the radius of the asteroid? (density of a sphere = ).
   
   
   
   
   
   
   
   
   
   
   
   
8. Does this size seem reasonable for a dinosaur-killer asteroid (both in terms of the crater size and what is present in the asteroid belt)? Explain.
   
   
   
   
   
   
   
   
   
   

Updated 9/15/06 by SAM