Encounter with an Asteroid

There are many asteroids whose orbits around the sun bring them close to the earth. What happens if one of these should
collide with the earth?
Disaster?
Extinction of the human
race?
End of the earth?
Certainly if an asteroid the size of Eros were to collide
with the earth, major disaster would be the result. But Eros is one
of the largest asteroids that comes anywhere close to the earth.
Most are much smaller. Many disintegrate or are melted away
as they are heated by friction as they move through the earth's atmosphere. This process is called ablation.
Only if the original asteroid is large enough will part of it get all the
way to the surface of the earth. The part of it that does get to
the earth earth is then called a meteorite.
In this module our goal is to seehow to estimate how large an asteroid must be in order for a large portion of it to reach the surface of the earth, and if it does reach the earth how much mass remains.
There are a lot of questions that need to be answered
along the way. The questions can be answered using our understanding of work and energy, momentum and impulse, and thermodynamics. 

This picture of Eros, the first of an asteroid taken from an orbiting
spacecraft, was obtained by NEAR Shoemaker on February 14, 2000 shortly
after it was placed in orbit about the asteroid. One can estimate
the size of the asteroid from the information that the large crater near
the center of the photograph is about 6 kilometers (4 miles) in diameter.
The surface of Eros is heavily cratered as the result of collisions with
many much smaller asteroids.
NEAR Shoemaker began orbiting asteroid 433 Eros February 14, 2000 and landed on the asteroid February 12,having completed the most detailed profile ever of a small celestial body. For more information, including an animated calculation of Eros's orbit
showing how close its orbit comes to the earth, visit the NEAR Web site:
http://near.jhuapl.edu. 


Why
does an asteroid make a bright streak of light as it moves at high speed
through the atmosphere?
As the asteroid moves through the air at high speed, it collides with the molecules of air in its path. When it first enters the earth's atmosphere the asteroid is moving with a speed that is much greater than the speed of the air molecules. As the air molecules bounce off of the asteroid they speed up, gaining kinetic energy. Conversely, these impacts make the asteroid slow down, losing kinetic energy.
The molecules bouncing off the asteroid have lots of kinetic energy, but they are moving randomly in all directionsthis corresponds to an increase in temperature of the air along the path followed by the meteor. The atoms at the surface of the asteroid are also bounced around randomlyresulting the temperature of the asteroid increasing as well. In fact, both the air and the asteroid become very hotreaching temperatures of several thousand degrees. At such high temperatures both the asteroid and the air left in its path emit light, just like the hot filament of a light bulb. The light emitted by the trail of hot air behind the asteroid is the streak of light that can be seen in the sky that we call a shooting star or meteor.
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This brilliant Leonid meteor is seen lighting up the dark desert skies of the California desert area of Joshua Tree National Park. This meteor is seen near the Pleiades. It was one of thousands seen with 23 meteors being seen per second at the peak of the show. Photo taken with a 35mm camera with a 50mm lens on a tripod. Copyright by Wally Pacholka / wally@AstroPics.com / more of Pacholka's photos are at http://www.AstroPics.com.

How much heat energy is produced, and what does the heat do?
A meteor enters the earth's atmosphere with very high velocity. While the actual velocity is different for different meteors, one can get an idea how large the typical velocity would be by noting that it will have to be larger than the escape velocity from the earth. This would be the velocity that an object would have to have to get away from the pull of earth's gravity, and can be easily calculated using the law of conservation of energy. Another module can show you how to do this important calculation. Typical meteor speeds are 12,000 to 80,000m/s. Due to this high speed the atmosphere exerts a large drag force on the meteor. It loses kinetic energy as the drag force acts against the meteor's motion. Hence a lot of heat energy is produced and the meteor, as well as the air around it, gets very hot. One consequence of this is that the meteor and the trail of air behind it become so hot that they emit visible light, the "shooting star" we see as the meteor descends through the atmosphere. The other consequence is that the meteor begins to melt. The more heat produced, the greater the amount of mass that will be melted away.
In the following exercises, we will see how to estimate how much mass will be left when the meteor reaches the earth's surface. We will consider the asteroid to be metallic, made up mostly of iron. The large meteorites that make it to the surface of the earth have this composition. Other types of meteorites are more common, but usually break up in the earth's atmosphere, so that only small pieces reach the earth.
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How much is the meteor heated, and how much melts as it slows down?
Suppose a 100 metric ton (100,000 kg) meteor enters the earth's atmosphere with a speed of 15,000 m/s, and is slowed down to a speed of 14,800 m/s. How much kinetic energy has been lost?
Now suppose 20% of this energy goes into heating up the surface of the meteor. (The rest heats up the surrounding air.) How much mass of the meteor will be melted? How much mass reaches the earth's surface? What fraction is this of the original mass?
Here is some data about the meteor:
Since we are assuming the meteor is made of iron, the specific heat is 470 J/kg^{o}C.
The melting point of iron is 1535^{o}C, and we will take the initial temperature of the meteor to be 253^{o}C (20^{o}K, it's cold out there in space.)
Latent heat of fusion of iron = 270,000 J/kg.
Now suppose the meteor enters the earth's atmosphere with a speed of 15,000 m/s, and is slowed down to a speed of 14,700 m/s. Does any of the meteor make it to the earth's surface?
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A more accurate calculation of the amount of mechanical energy converted to heat.
The calculation you just did gives you an idea of how we can find the amount of the meteor that melts. But it is not a complete calculation you were given the starting and ending velocities, so you could readily find the amount of energy lost. But we do not know beforehand how much the meteor slows down. That depends on how large the force of air resistance is. The calculation is complicated because the force ofair resistance changes it depends on the velocity of the meteor, on the size of the meteor, and the density of the air. All of these things are changing. This makes it very difficult to calculate the changes in the velocity accurately, so that taking the difference of the final and initial kinetic energies is very inaccurate.


But we can still get a reasonably accurate calculation by making some reasonable calculations. A big simplification can be made by observing that the velocity of the meteor does not change very much before the meteor all melts away. In fact, because typical speeds of meteors are so fast, they spend so little time travelling through the atmosphere that their velocity changes by only a few per cent.
In order to calculate the change in mechanical energy, we can calculate the work done by the force of air resistance. This force is given by the equation:
.
r represents the density of air.
A is the cross section area of the meteor. We will assume that the meteor is approximately spherical in shape, so the area is pr ^{2}. You will be given the mass of the meteor, and you can find the volume by dividing the mass by the density of iron, which is 7800kg/m ^{3}. Using the formula for the volume of a sphere you can calculate the radius.
v is the velocity, which we will approximate as a constant 15,000 m/s.
C is called the drag coefficient. For a sphere it is approximately 1.
To take into account the fact that the density of the air changes as the meteor gets closert to the earth, we can calculate how much energy gets converted to heat in several segments. The table at the right gives you the average density of air in each of the intervals shown. Above 30 kilometers, the air density is so small that only a negligible amount of the meteor is melted.
Now start with a meteor whose initial mass is 1000 metric tons (1,000,000 kg). Calculate the amount of work done against air drag as the meteor travels from 30km to 20km (30,000m to 20,000m). Assume 20% of this work went into heating the meteor, and calculate how much of the meteor was melted away using the method of calulation you used in the previous activity. Use this mass to get the new radius, and calculate the work done as the meteor falls from 20km to 10 km and again find how much mass was melted away. If there is any mass left, repeat this for the interval from 10km to 5km, and from 5km to the ground.
You should find this meteor does not get anywhere close. Repeat the calculations with initial masses of 10,000 metric tons, and 100,000 metric tons to give yourself an idea of how big the original asteroid needs to be for a large chunk to get to the earth.
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