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Does the sky fall often over our head?

The fate of the Earth is related to irregular major events such as impact of meteorites.

This phenomenon by irreversible ecological disturbance which it engenders could be at the origin of the big evolutions of the alive.

Meteorites have two origins: asteroids and comets, first category being pre-eminent.

On this base, this statement aims only at describing various methods to establish approximately the frequency of impact following the size of objects. It is more about a means of discovery of asteroids by various approaches than of answering in impossible question: in when the next end of dinosaurs?

A) What is the problem ?
B) Method of estimation on a geologic and  observational bases
C) Method of estimation according to the number of craters
D) Method of estimation on a base of observational classification
E) Estimation on the basis of the encounters close in the time
F) Method of estimation on a base of calculation of disturbance
G) In summary

A) What is the problem ?

On 2001 YB5, this asteroid 300 metres long is crossed on January 7th in 830 000 km of the Earth. Its speed:  30 km/s, its capacity of destruction: a crater of one km deep and 6 km in diameter. It was discovered only 10 days before its passage to the highlight.

Another missed meeting had already taken place in 1989 for a celestial body of size and comparable distance. A question comes naturally to the spirit: is there a significant risk that a consequent impact occurs?

There are 2 types of celestial bodies in the potentially unstable trajectories: comets and asteroids.

It is considered that there are four times more asteroids approaching the Earth that of comets.

Comets can have an orbit close to the ecliptic. In that case their source is probably the belt of Kuiper situated beyond Neptune. Their orbit is sometimes very elliptic, the period of which stages from some years to several hundreds of years. So the historic comet of Halley has a period of 76 years.

The life expectancy of the comets which approach the Sun is rather short astronomicaly speaking because comets are formed by "dirty" snow melt under the influence of the solar wind. It is likely that comets at the end of life become small asteroids.

The orbits of comets can be also except the plan of the ecliptic and in that case result from the cloud of Oort in the approximately a year sunlight. This cloud is formed about 1000 billion comets, count naturally estimated according to the number of those who, by dropping out under the influence of the close stars, cross the solar system. The period of these comets as that of Hall-Bopp can be several thousand years.

Asteroids can consist of chondres (those are chondrites) formed during the sedimentation of the diverse particles forming the essential disc on the ecliptic of the Sun, there is 4,54 billion years.

Others are formed essentially by light cliffs or conversely by iron and by Nickel. They are the fragments of big asteroids having undergone under the influence of their own gravity a differentiation of materials.

Asteroids today already recorded are among 30 000. They are classified in 3 categories: Aten (their orbit is shorter than that of the Earth), Apollo their orbit is comparable to that of the Earth and Amor their orbit  is generally situated beyond Mars.

Asteroids passing in the neighborhood of the Earth are NEO (Near Earth Object) if they cross unless 0,3 astronomic - unit (distance Earth-Sun) means  45 million km and the PHA (Potentially Hazardous Asteroids) if they pass unless 0,05 A.-U. that is 7,5 million km and what they possess a magnitude absolved in 1 A.-U. from less than 22 (body of at least hundred metres). So the asteroid 2001 YB5 should return to 27 million km in 2052 remaining in the category of the NEO.

B) Method of estimation on a geologic and  observational bases

We observe that the number of these asteroids  varies in inverse function of the square of their biggest diameter: 10 times as big 100 times less numerous, which amounts to saying than the mass is accreted 10 times more in the 1 km bodies than in the 100 metre bodies.

The craters of impacts are called astroblemes. There is there approximately 150 recorded such Meteor Crater, Arizona (1,2 km, 50 000 years), Kara-kul, Tadjikistan (45 km, 10 million years), the crater of Ries, Bavaria (25 km, 15 million years), of Haughton, Island of Devon, Nunavut (20 km, 23 million years), or Chicxulub, bay of Yucatan (170 km, 65 million years), of Manicouagan, Quebec (100 km, 212 million years) and to show that there is everywhere a crater of Rochechouart, Poitou (25 km, 200 million years). And it is necessary to go back up(to raise) to 2 billion years to find huge craters  such those of Vredefort, in South Africa (300 km) and of Sudbury, Ontario, Canada (250 km) Only the biggest provoked hydrothermal drainages allowing deposits of ores.

Let us consider Meteor Crater with its 1,2 km in diameters and 180 metres deep probably owed to a body of in most 50 m which collided the Earth here is 50 000 years. It is the only known recent crater.

By considering that approximately 20 % of the Earth is formed by accessible lands (the rest is the oceans, the seas, the lakes, icepack or dense  forest), and that more than staying 95 % are rather quickly affected, it would fall an asteroid of 50 m on an accessible zone on average every 2500 years, and on the whole surface of the Earth everything on the 500 years. By taking into account the diminution of the visible proportion of asteroids this comes back to a 300 metre asteroid every 18 000 years and from 1 km every 200 000 years.

Let us consider on the contrary Kara-Kul in Tajikistan, old  of 5 million years. With its 52 km in diameters and with the same report from 1 to 20, the asteroid had to be 2600 metres. The craters of this importance are too deep so that the erosion made its work. By considering the same ratio of 20 % of detectable craters and the same law of diminution we obtain a 1 km asteroid every 148 000 years.

In fact also by proceeding with 15 main craters of less than 200 million years, we see that the estimated frequency increases when we consider more recent craters, when the ancient craters disappeared by the erosion. The estimation seems to converge on this order of height.

C) Method of estimation according to the number of craters

Another method rests(bases) on a precise estimation of the number of craters of impact. If there is there approximately 150 recorded, among which 15 of more than 50 km. It is necessary to take into account the fact that, over a period of several dozens million years, same the continental blocks are heightened by the tectonics of patches and the mountainous formings erase holes formed by the craters of impact. The only method of effective estimation bases on the study of an ancient continental zone and in a geographic zone where there are enough specialists to be interested in the question. We are indeed surprised noticing that whole zones in Africa and in Oriental Asia seems exempt from craters of impact. A region corresponding to our criteria is big Quebec as 3 times France, with 9 listed craters of impact. We can consider that the erosion remains weak enough in this region so that all the biggest craters (due to asteroids 10 km in diameter) do not disappear within 200 million years. There are few tectonic movements on the other hand since 2 or 3 million years the erosion has of to accelerate under the influence of ices in this region. This surface represents 0,3 % of the surface of the Earth and allows an extrapolation.

On this base there would be 9 x 332 ~ 3000 craters of impact which can go back up to 200 million years and which would be potentially visible on Earth if most of them were not covered with water, with forest or with ice or not transformed by the tectonics.

Now the erosion depends on the depth of the proportional crater in the diameter of the asteroid. Knowing that for a given size the frequency of fall of asteroids is constant and that the number of astéroids vary in the opposite of the square of its size, the number of craters over a given period has to vary simply contrary to the size of asteroids. Example: all the million years if we obtain 100 asteroids of 100 metres, we find 10 of 1 km. To find the number of impact of crater by size from the number of still visible impact, it is necessary to correct this way.

Also let us consider that only asteroids beyond 30 to 50 metres can supply craters; indeed the asteroids of some metres in some dozens metres melt under the influence of the heat and tend to explode in the atmosphere.

If we distribute the 3000cratères of impact  from  50 metres to 10 km of diameter by slice of 50 metres we obtain 650 from it engendered by 50 metres asteroids and approximately 32 craters for asteroids of 1 km in diameter (number = 508 / P with P by slice of 50 metres with S p=1 in 200 508 / p = 508* 5,87 = 2988).

In fact it is necessary to increase this figure of a factor a for the asteroids of 50 m and a / p for the asteroids of p time 50 m. If we consider that the craters of impact due to 10 km asteroids are again all visible, it comes a/200 =1.

There would have been thus on 200 million years 650*200 130 000 craters due to 50 metre asteroids (1 every 1500 years), 32* 20 =640 be owed to 1 km objects (1 every 312000 years) and 2 from 3 to 10 km (1 all 65 in 100 million years).

D) Method of estimation on a base of observational classification

The current location supplies a figure about 350 asteroids of at least 100 m orbits of which are potentially dangerous. In view of the growth of the discoveries and the very late discovery of objects of this size, it is not unreasonable to estimate their actual number in 3500 and to extrapolate in this number the current proportion of 3 classes of geocruisers.

On 1000 asteroids, approximately 10 % would have orbits of duration shorter than that of the Earth, 25 % of the orbits of comparable duration and 65 % of longer lasted.

On 89 geocruisers of type NEO crossing in closer of the Earth, we have 1,8  year for the average duration of orbit.

Thus every year there would be on average 3500/1,8 = 1940 geocruisers of at least 100 m close to the Earth orbit. Let us consider that half of them crosses effectively the Earth orbit that is 970.

Over one year (3,1 107 Seconds), what is the total duration during which these bodies cross the Earth orbit?

It is necessary to cross 12000 km one diameter seen under an average angle of attack of 45 °. Hence 12700/Cos 45 ° = 17 900 km.

The speed of bodies is 30 km/s under an average angle of 45 °. The speed of the Earth is 25 km/s
Its relative speed with regard to the Earth is of V = ((25-30*cos45 °) ² + (30*sin45) ² )1/2  = 21,5 Km/s.

The average time of crossing is 17,9 106 /21,5 104 = 835 seconds.

Asteroids have independent orbits.

The total time of crossing is 970 asteroids 835 seconds 2 passages (down and upward) 1,6 106 seconds. 2 passages are supposed to be in the same year because the low part of the trajectory of the asteroid is a weak portion of the orbit and it is made in the highest speed.

The ratio on the duration is equal in:

Total duration of  independent passages of  geocruisers / average period of passage on all the geocruisers = 1,6 106/ 3,1 107 = 1/20

In other words on average every 20 970 seconds of the Earth on the orbit (approximately 5 hours), there is 1 passage of asteroid which could correspond to a collision if the Earth was on its trajectory.

The Earth orbit is of 2p * 150 109 m = 947 million km = 9,5 1011 m

Ratio in density is 970 objects on a length of 9,5 1011m  is 1 object every  9,7 108 m

It is the average space between 2 passages of geocruisers.

Where from a density given by Earth diameter / average spacing between 2 passages of geocruisers 1,2 107 /9,7 108 = 1/81

The duration means between 2 collisions is thus 2,3 *25* 81 = 4700 years.

What would be the size of this asteroid?

We estimate the total number of asteroids from 1 km a 2000, the very great majority not approaching the Earth

Where from n, the number of asteroids of diameter D, of value 2 109/D ²

The number falls to 1 when the diameter is equal to 45 km what corresponds approximately to the size of the biggest asteroid NEO (Ganymede=32km) but not in the biggest asteroid (Ceres 1000km). The estimate seems all the same acceptable for the Earth suburb.

The diameter average Dm is given by ò 2 109/D ² applied to asteroids approaching the Earth taken between 50 metres and 40 km and 100 metres.

Where from 1 / (4104) -1 / Dm = 1 / Dm -1 / 100, it comes Dm=200m

This asteroid would be on average 200 metres. The collision with a 50 metre asteroid would make every

4700 * (1/4) ² = 300 years and with a 1 km asteroid every 117 000 years.

E) Estimation on the basis of the encounters close in the time

From 6 asteroids listed by type PHA of an average diameter of 1,5 km should touch us in coming 30 years, as " 4179 Toutatis" with a average distance of passage of  107 Earth diameter .

To strike a target of a Earth diameter would thus be needed 107 ²*5 = 57000 years.

Now the size averages of recorded objects which come close  is 1,5 km (more small objects come close but we do not discover them). Where from a 1 km object every 57 000/1,5 ² = 25 500 years.

The frequency is raised but a simple test exists. Let us take recorded  asteroids and let us consider the one who in a near future is going to touch us closer. It is about WO107 who 1 December, 2140 will pass in 510-3 Astronomical unity = 75000 km or  6 Earth diameters.

The number of object and thus the probability of moved closer passage varies contrary to the square of the size.

Thus let us estimate its size. It should be (107/6) ² 5 1590 years old for a 1,5 km object. But since it is going to touch us in 140 years, its size is 1,5 / (1590/140)1/2 = 445 m.

Now its estimated size is 540 metres...very close;

Hence this method seems with a  40 % margin  for durations..

F) Method of estimation on a base of calculation of disturbance

We are going to estimatea maximal value from a speed of gap of orbits.

Let us remain within the framework of average duration estimated of 1,8 years of a geocruiser asteroid .

The disturbance is essentially due to Jupiter orbiting over 11, 8 years. Over 6 periods of rotation of the asteroid that is 10,8 years,  Jupiter, late of one year, moved of an angle of 2p1/10,8 = 0,53 radians.

In fact there is no reason that the main line of the orbit of the asteroid (and thus the perihelion) is towards Jupiter with regard to the Sun. We can consider the estimate on a long duration which the distance averages between Jupiter and the asteroid is equal to the distance Jupiter - Sun

The mass of Jupiter is approximately 1/1050 of that of the Sun, the report of attraction Jupiter on asteroid / attraction Sun is on average 1/1050 under this hypothesis.

The lateral projection of the force of attraction of Jupiter is of  cos 0,53/1050 = 8,2 104 Time the attraction of the Sun.

What is the distance between Jupiter and the asteroid? Let us call up has the main line of the orbit

Relation of Kepler Aast3/Tast² = A 3Jupiter /T ² Jupiter Where from A ast = A Jupiter ( Tast/T Jupiter)2/3 = 778 * (1,8 / 11,8)2/3 = 221 million km.

The asteroid is moved thus about 221*109*arc tan (8,2 104) = 182000 km ( 14,3 Earth diameters every 6 rotations) where from 28 seconds of bow in every passage.

To note that the border of 45 million km of the Earth orbit, sharing the asteroids NEO of the others is about the distance where the effect of the Earth' attraction is superior to the attraction of Jupiter The influence of Jupiter with the Earth balances itself for a distance (778-d ² / 318=d ²/1 where from d = 41 million km.

Considering that geocruisers is in a ring from 130 to 260 milions of km of the Sun, we see that the Earth' attraction will thus play only marginal way on these bodies with regard to the attraction of Jupiter.

In a network in n crossings (crossing between 2 perpendicular roads with an unpredictable choice of direction), the maximal frequency of passage in 1 point aims towards logn  ² / p (the demonstration dates the end of 2001 applies to the drunkard who in his erratic walking in a statistical tendency to be crossed in certain points).

Let us consider that asteroids navigate in a ring between 130 and 260 million km or between  10 000 and 20 000 Earth diameters. We shall also consider that Jupiter provokes only a dispersal of trajectories in the initial plan of the asteroid as if Jupiter and the asteroid were in the same plan, what is not rigorous, and we shall neglect the slope on the ecliptic, thus the thickness of the ring.

The unity is here  19 Earth diameters. The number of crossing in a square of side 2*R is of 2R ² and in a disc of  radius R of p/4 (2*R ² = p ( R ) ² and the number of maximal passage in 1 point is of (log (p R ²) ² / p

The number of crossing is p (20 000/19) ² - p (10 000/19) ² = p 830 000 And the number of maximal passage of (log (p 830 000) ² / p in the ring.

The Earth would be collided for most 1 time every p 830000 / (Log (p * 830000) ² = 12 000 disturbances of the asteroid by Jupiter. The  19 Earth diameter periodicity of the gap being of 11,5 years, this critical crossing would be crossed in most every 130 000 years. The covering zone 19 ² Earth diameters, this asteroid will collide the Earth in most every 47 million years.

If we resume the number of 2000 asteroids of 1 km (see previous method), we would have a 1 km object all 47 106 /2000 = 24 000 years, at most.

Methods Lasted between 2 impacts of object of 100 m Lasted between 2 impacts of 1 km object
Geologic base (extrapolation from a very reduced number of recent craters) 1480 148 000
Number of craters (extrapolation from a characteristic zone but over a reduced surface) 3120 312 000
Observational classification  (number of crossing by estimation of the number of geocruisers from the observed number) 1170 117 000
Estimation from the  encounter , close in the time 250 25 000
Maximal value from a calculation of disturbances 240 24 000