Breaking rocks: Doomsday

 

The sun rises in the east and it sets in the west, a simple sketch will then show the earth turns on it's axis from west to east, anticlockwise seen from the north pole. In order to understand the following ideas a few rough sketches will be helpful.

 

It is easy to work out approximately the speed at with the earth surface is moving, spinning. You would use the circumference depending then on what latitude you are, and the length of one day since earth takes one day for a revolution. As you can imagine it is actually going very fast. The highest surface speed is at the equator. Taking the radius as 6873km the circumference is 2π.6873km and divided by 24 hours gives a surface speed of 1799km/h. The answer is quite reliable but not terribly accurate and actually a sidereal day of 23h56, i.e. a day relative to the stars should be used.

 

For these reasons a rocket powered spacecraft would be launched sharp direction east so that you can get a kind of slingshot effect. The contribution is significant. When ballistic missiles and unpropelled projectiles are launched as when you fire a cannon or launch a long range missile such as an ICBM you want the opposite effect. You would like to shoot it toward the west. If it is travelling toward the west it means as it flies earth is rotating underneath and in effect bringing your target closer and closer.

 

In both these examples I'd think you would also like to be close to or on the equator because the earth surface speed is highest there. This you can see easily from the geometry and the fact that the earth is a sphere. Finding trajectories of short range projectiles in constant gravity and the earth as “locally” flat ground is a simple matter it is a parabolic curve. However for the kinds of long range ballistic trajectories the story is more complicated. Amongst others the Globe's curvature would play a role and possibly even the atmosphere's rotation together with earth.

 

Problems such as for long distance projectiles require much more advanced calculations. Since gravity weakens with a larger distances from earth, its height, an object's weight must also decrease as it goes higher, and in turn gravitational acceleration decreases with increasing height. It means it cannot be approximated by a constant. In addition the atmosphere also has an influence as well as the ground i.e. surface curvature.

 

Still, near to earth’s surface for small heights the altitude has a negligible effect and gravitational acceleration g ≈ 9.8m/s² is constant for all practical purposes.

 

 

There has been much talk of the possibility of a gigantic meteorite, an asteroid or a comet colliding with earth. This anticipation has now been to the extent that protection of planet earth is actively promoted with funding for research experiments and space probes to develop the strategy and means. There would certainly be very valuable discoveries in technology and new inventions in such an undertaking but not much to do with the actual original motive.

 

The feasibility of pro-active asteroid and comet interception then is to be determined. The idea is to plant nuclear bombs to deviate the course of such a solar body on collision course. It is a plot apparently very popular especially in the films. Now people are taking it so serious as even to start space programmes with intention protecting earth for such an eventuality with intercepting space vehicles massive explosives tests etc. All exciting and impressive. However there are fundamental considerations of linear momentum.

 

That is if you do have a right even just to do such a thing out of your own, I wonder. Legally I don’t think you have.

 

 

These definitions in mechanics are just for sake of completeness of the text no-one has to read it.

 

A physical body or any collection of matter may be regarded as and called a system of particles. It could be fluid, gas, solid, fragments granules etc. The particles could even be the individual atoms. For each particle let the mass be m, its position in an inertial coordinate system r and the velocity v.

 

The concept of a centre of mass is very close to our everyday common understanding of the word.

 

The centre of mass of a system of particles is formally defined as the point given by the total vector sum of all the products mr divided by their total mass M, which of course is the sum of all the m. The momentum of an individual particle is defined as its mass multiplied by its velocity, mv, and the vector summed total over the entire system is the total momentum.

 

 

The centre of mass always moves as if the total mass is located, and the resultant external force is applied at this point. If the resultant external force acting on a system of particles is zero then the total momentum always remains constant, it is conserved. In such a case the centre of mass is either at rest or in motion with constant velocity.

 

The centre of mass in our case is that of the rock together with the bomb. Clearly the physical bomb itself‘s contribution is negligible. One cannot alter the course of the centre of mass of a body or bodies, or fragments in space with an explosion. The total momentum before and after an intercepting nuclear explosion would be the same since there is essentially no resultant external force.

 

The fact is that even if you should then blast the rock to stones gravel and to dust it would not alter the centre of mass's course and theoretical point of impact by a centimetre. What is more, it does not matter where you want to propose to plant the bomb, at one side, or the other, in front of the rock or behind. It wouldn’t make any difference.

 

Debris will however be spread and dispersed as a result of the explosion, how fast and to what extent one would not really know. One interesting side effect is you will increase the heat and total kinetic energy of the total of the parts even more, the fragments by as much as the energy released in the explosion. And with an atom bomb things will get very hot indeed. Honestly it is a stupid thing to want to do.

 

I found it hard to get suitable data. No doubt the reason is that the ranges are very wide. Actually the very large asteroids are extremely scarce. The fact is that the larger the body the less likely a collision is for the simple reason smaller bodies are in much greater abundance.

 

However for typical values we take a “very large” body of 10km in diameter. That is either a very big asteroid or a comet. These are comparatively large values and are more likely to result in terrible disaster. The only practical difference really is their orbit and an asteroid has higher density it’s about three times more. Most meteorites’ density is around 3gram/cm³ and the usual speed relative to us is about 15km/s when it enters our atmosphere, which then increases until when it crashes.

 

 

The energy with which a meteorite strikes the earth is very relevant and is of great importance. Appropriate values for a comet but without considerations of the atmosphere are described now. The ideas used for an asteroid are very similar. The potential devastation of the Comet will now be estimated with realistic data.

 

Comets consist mainly of more volatile low density matter. I take the density of a comet nucleus 1g/cm³ compared with water as 1g/ml. A comet should impact at the higher speed but for the same diameter has a mass the third of the norm for asteroids. The numbers appear to balance out so that the earth-impact energies are of similar order.

 

The volume of a spherical body of 10km diameter is found, with a density as that of water i.e. 1000kg/m³, its mass is 5.24x10^13 kg.

 

The mechanics’ reasoning is founded on the principle of the conservation of energy. The object’s kinetic energy when crossing the orbit of the earth is the same as the potential energy due to the sun. That is 4.66 x 10 ^23 Joule.

 

Similarly the earth’s gravity will contribute acceleration, as if the body just fell from standstill from far outer space. The comet would thus be attracted, and just ordinarily “fall” under influence of gravity, the effect at 3.28 x 10^22 J.

 

Both are converted to kinetic energy, and we may add them. The total kinetic energy at impact is 4.99 x10^23 J. From this, the velocity at impact can be found and is 43.6km/s.

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As an approach to the Asteroid’s calculations, that looks different but in principle is the same, I work the other way round. Seeing as the escape velocity must give the same as the potential energy due to gravity, I may use the value of 11.2km/s (from tables). It also seems reasonable enough to use the similar value 15km/s as the contribution of other relative motion. This is consistent with literature.

 

The total kinetic energy of such an asteroid collision then works out to 2.75 x10^23 J and the final speed would then be 18.7km/s.

 

Here follows a brief summary of the information that was used. M represents the known the earth or sun mass appropriate to the context ; r is the comet radius and R the earth equatorial radius or orbit radius depending. For a comet mass m with R and ρ given, the formulas are:

 

V = ⅓4πr³ ; m = ρV ; KE = ½mv² and PE = GMm/R

 

V is volume ; density is ρ ; KE kinetic energy, and PE potential energy. All are in SI Units. For the constants one can consult relevant astronomical tables. These are standard fairly routine calculations all this really is high school work.

 

 

A ton of TNT is a measure of energy and is equal to 4.184 Gigajoules = 4.184 x 10^9 J which is the approximate amount of energy released by the detonation of one ton. It is used as a standard measure for powerful explosions, as for the effect of large meteorites.

 

For interest’s sake I give the theoretical kinetic energy blast for our comet striking the earth surface in megatons of TNT. The value as obtained is 119 million megatons. For the asteroid it is of the same order and it is 65.9 million megatons.

 

One may compare these figures to the atom bombs America used against Japan in August 1945, the first of 15 kilotons and the other of 21 kilotons with a combined energy of 36 kilotons of TNT. The most powerful nuclear weapon, a hydrogen bomb at “a mere” 20 megatons or 20 thousand kilotons, already is 556 times more powerful.

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All said I do wonder if a giant asteroid meteorite impact could really be so terrifically devastating globally, they don’t move that fast and they’re not much bigger even then a comet. Recently we’ve had a bit of hysteria and panic on “close passes” of asteroids. If you talk of such an an apocalyptic event I think an asteroid should enter our atmosphere at around 20m/s or more and be more than 10km in diameter, that is compared to the largest of recent near-miss asteroids as only a hundred meters odd.

 

In my opinion the chances are not good with existing technology, any plan for such an interception would be useless.

 

How much debris hits the earth in the end depends on how quickly the pieces scatter as well as how fast the asteroid body’s centre of mass is approaching. If they are very quickly dispersed many of the fragments would miss earth right around. There would however still be a significant amount of impacts spread over the globe.

 

The only way out I can see is that by some great good fortune, the possibility exists that the body could hit the moon first. For instance, a comet has to approach either from the direction of the sun, or along the same line but falling from outer space and hurtling toward it. That is when taking in account the very typical extremely elongated shape, an extremely eccentric elliptic path and extraordinary size of a comet’s orbit. This in turn explains how fast it is travelling.

 

A comet thus has to come either from the stars, or from the sun, hopefully then with full or with new moon. All three bodies move in approximately the same plane, as almost all bodies in the solar planetary system do. In that case it might be stopped by the moon if aligned. This is not impossible. However with your own intervention you would have caused a terrific meteorite storm and fireworks as was ever ever. A meteor shower making a New-Year’s eve in San Francisco look like a child’s sparkler. A hard rain.

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They spoke on the radio the other night of the “Tunguska event” which was a most extraordinary occurrence, a terrific explosion in remote Siberia in 1908 of the magnitude of a 15 megaton bomb 5–10km above the ground and visible from 800km’s away according to the Britannica. A thousand times more powerful than the atomic bomb dropped on Hiroshima. Around the epicentre radially for some 15–30km everything had been devastated and scorched, fortunately it was over very sparsely inhabited land. In highly populated urban areas or over the sea this had the potential of causing unequalled natural disaster, massive physical and total infrastructure destruction. Of Old Testament proportions.

 

On the radio they told of the possibility of such an event again in the near future, they believe to be related to some type of violent solar activity and its influence on the ionosphere. However it is so that the cause, and what actually happened in Siberia really still is unknown. It was then speculated that such an event could result in fatal damage to power networks worldwide, which in turn, one would imagine, eventually be the end of civilization, and for the few survivors an aftermath of a sort of “beyond the thunder-dome” scenario.

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The terrifically greater threat of all is the nuclear weapons by their thousands, our gravest and most dire peril, those bombs are already here, on earth. Ready. At the drop of a hat. Oblivion. All of civilization and humanity utterly destroyed. And a hard rain.

 

As an extremely unlikely event “Saving the World” in this blowing up rocks in outer space kind of fashion really would be a waste of time and effort.

 

Please be assured, when we see the end coming, it is already here.

 

~

 

Winston Churchill– “This is not the end. It is not even the beginning of the end. But it is, perhaps, the end of the beginning.”

 

Why should it be the end? Why should it be the beginning of the end? Why should it not be the beginning? Why not, the beginning of the beginning?

 

~

 

Dedicated to the memory of Selby Logan