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- Planets were formed in an accretion disk orbiting the Sun in our early Solar System evolution. There is still a lot of debris left over and these fragments continue to orbit between the planets.
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- Some are referred to as “Apollo asteroids“. This is a group of near Earth asteroids that have elliptical orbits that cross from inside to outside to inside Earth's orbit. They are a potential threat to our planet. The 2013 Chelyabinsk meteor that exploded over Russia is one of these Apollo class asteroids.
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- To learn what our chances are being hit by a meteor we first collect data on all size meteors, zodiac light, which is space dust, meteorites, lunar craters, Apollo asteroids, and comets. Next we plot the data: the number of impacts, “y”, as a function of the mass, “x”, in grams. Number of impacts are measured in square kilometers per year.
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-We get an enormous range of masses and ranges. To get these all on the graph use a
log-log plot. The resulting curve is nearly a straight line.
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- When meteors impact the surface of the earth they are called “meteorites“. Fundamentally, they are all comets, or, parts of comets. They are the debris left over from the accretion disk that formed the planets. Comets tend to carry frozen gas and water. Asteroids are the rocky cores after most of volatiles have burned off.
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-The impacts of zodiac dust numbers in 100,000,000’s particles impacting per square kilometer per year.
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- The size of the space dust is 0.000001 grams to 0.0001 grams.
- Meteors are the size of 10 grams
- Meteorites range from 10 grams to 1,000,000 grams
- Lunar craters range from 1 gram to 100,000,000,000,000 grams.
- Apollo Comets from 10^8 to 10^20 grams
- Comets to10^20 grams.
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- The range is so great. It makes sense to condense the range by taking the logarithms of each number then plotting the data on a log-log graph.
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- How often does the Earth get hit with a 1 ½ inch diameter meteor, weighing about 100 grams?
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- The surface of the Earth, counting the atmosphere ,which but a thin skin over the ground, is included in counting the impacts. The surface area has a radius of 6,378 kilometers.
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--------------------------- Area = 4 * pi * radius^2
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-------------------------- Area = 6.28 * (6.378*10^3)^2 = 5.1 * 10*8 km^2
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---------------------------- Area = 510,000,000 square kilometers.
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- A meteor that is 4 centimeters in diameter, 1 ½ inches. With a density of 3 grams per cubic centimeter. Its mass is 100 grams:
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------------------------------ Mass = Density * Volume
------------------------------ Volume = 4 * pi * r^3
------------------------------ Volume = 4 * (3.14) * 2^3
------------------------------ Mass = 3 grams / cm^2 * 33.4 cm^3
------------------------------ Mass = 100 grams
-------------------------------log Mass = log 100 = 2
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- If we create an equation from the plot of the number of impacts as a function of the mass of the meteors the best fit equation is:
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---------------------- Number of impacts = 0.025 ( mass ) ^-0.9
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----------------------- The graph is a log-log plot. The log of 100 is 2, that is 10^2 = 100
-------------------- Number of impacts = 0.025 ( 100 ) ^-0.9
--------------------- Number of impacts = 0.025 / 63.1
-------------------- Number of impacts = 0.000396 per km^2 per year.
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- Multiply the Number of impacts by the surface area of the Earth:
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------------------- Number of impacts on Earth = (0.000396 ) * ( 5.1*10^8)
----------------- Number of impacts on Earth = 203,000 impacts per year
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----------------- At 100 grams per impact these 1 ½ diameter meteors alone account for 2,030 kilograms of mass hitting the Earth each year, equal to 2 tons per year.
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------------------ 1 metric ton = 1000 kilograms
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- Looking at the log-log plot of this data you observe a diagonal line that is nearly a straight line. The linear equation for a straight line is:-
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------------------------ y = m * x + b
----------------------- “m” is the slope of the line
---------------------- “b” is the “y” intercept when “x” is zero
---------------------- b = -1.6
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----------------------- The slope is constant, so the ratio of rise over run is constant :
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----------------------- y - y1 / x - x1 = y2 - y1 / x2 - x1
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- From the graph the ( x , y ) coordinates at the end points of the straight line are (-4,+2) and (16,-16).
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----------------------- (y - 2) / ( x + 4) = ( -16 - 2 0 / (16 + 4 )
--------------------- ( y - 2 ) * 10 = -9 * ( x + 4)
--------------------- 10 y - 20 = -9 x - 36
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--------------------- y = -0.9 x - 1.6
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- To get the log for this linear equation:
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--------------------- Log impacts = -0.9 ( log mass ) - 1.6
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--------------------- impacts = 10^-.9x * 10^-1.6
--------------------- impacts = 10^-9x (0.025)
---------------------- impacts = 0.025 * (mass ) ^-0.9
----------------------impacts = 0.025 * (100)^-.9 = 0.025 / 63.1
--------------------- impacts = 0.000396 impacts per km^2 per year, the same answer previously calculated.
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- Now we have 2 equations to calculate meteor impacts as a function of the size of the meteor.
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---------------------- Log impacts = -0.9 ( log mass ) - 1.6
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--------------------- Impacts = 0.025 * (mass ) ^-0.9
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- Meteors range from dust to massive comets. Adding them all up, how many tons of meteors strike surface each year?
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- What is the total mass of all the sizes of meteors that impact the Earth? The sizes range from 1 gram to 10^20 grams. If we integrate the area under the curve we get the number of impacts times the size of each impact.
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--------------------------- Impacts = 0.025 ( mass ) ^-0.9
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---------------------Integral of total impacts from 0.1 grams to 10^20 grams
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--------------------- Integral of total Impacts = 0.025 * (-0.9) ( 1^0.1 - 10^20 ^0.1 )
--------------------- Integral of total Impacts = 0.025 * (0.9) ( 100 )
-------------------- Integral of total Impacts = 2.25 grams per km^2 per year
-------------------- Integral of total Impacts = 2.25 ( 5.1*10^8 km^2)
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---------------------Integral of total Impacts = 1.15* 10^6 kilograms
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- 1,150 metric tons of meteors hit the Earth every year.
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-The Chelyabinsk meteor that exploded over Russia February 14, 2013 was 10,000 tons, 56 feet in diameter. It hit the atmosphere at 40,000 mph.
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- 1 ton of TNT explodes with 4.2 * 10^9 Joules of Energy.
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- How many tons of TNT were in this comet impact?
-------------- 1 ton equals 1000 kg
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--------------- meteor equals 10 ^7 kilograms
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------------------ speed equals 18 kilometers per second
--------------- Kinetic Energy + ½ mass * ( velocity)^2
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--------------- Kinetic Energy = 1.62*10^15 Joules
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---------------- Kinetic Energy = 386,000 tons of TNT
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- What are the odds that someone will see such an event as a meteor striking the atmosphere?
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--------------Earth is 72% ocean and 28% land.
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--------------3% of the 28% land is inhabited equals 0.8%.
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-If there was one event per year there would be 1/125 ,or, a 1 chance in 125 of someone seeing it. You would need two events per year to have a chance of someone seeing it in their lifetime of 62 years.
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- For yourself to see it you would need to be within an area of 100 km² of where it happened. The Earth surface is 500,000,000 km^2 time 1/125 = 40,00,000 km^2
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------------------ 100 km^2 / 40,000,000 km^2 = 1 chance in 40,000.
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- If 50,000 events happen in a year or 1/125 happen over inhabited areas then 1/125×50,000 = 400. So, you have 400 chances per year of someone seeing the event. For you to be in the right place, 400 / 40,000 = 1/100. You have one chance in 100.
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- You would have to watch the entire sky every night for 100 years.
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- So you can see that the fireball meteors are rare events. You have to be lucky to see one. Or, unlucky if it is too close for comfort.
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(1) My first Review done with Dragon. Does not work well with so much math. But, text is sweet. Saves on the typing.
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RSVP, with comments, suggestions and corrections. An index of all reviews is available. Some reviews are at: ------------------------------- http://jdetrick.blogspot.com
,or: email request for copies to: ------------------ jamesdetrick@comcast.net
or: ---- “Jim Detrick” ------- www.facebook.com, ------------- www.twitter.com
or cell: 707-536-3272 ------------------------------- Tuesday, December 3, 2013
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