Catastrophe Theory

• Consider a potential functionV (x) = x3 + ax.

• When a < 0 there is both astable minimum (dots) and anunstable maximum in the potential.

• As a is slowly increased, theequilibrium system moves smoothly

to smaller x: xeq =p

−a/3.

• When a = 0, the stable minimumdisappears, and for a infinitesimallypositive, the system moves abruptly to anew equilibrium very far removed from the old.

Lattimer, AST 301, Lecture cat – p.1/36

Catastrophes and Evolution• Extinction was not widely accepted before 1800.

• Extinction was established as a fact by Georges Cuvier in 1796, and was criticalfor the spread of uniformitarinism in spite of the fact that Cuvier viewed extinctionsas evidence in favor of catastrophism and opposed Lamarckian evolution theories.

• Over 99% of all species that have ever existed are now extinct.

• Extinctions occur at an uneven rate.• There have been 6 major and many minor (up to 20) extinctions observed in the

fossil record. Nearly all divisions among geological eras, eons and periods aremarked by extinctions and originations.

Cambrian-Ordovician transition 488 Myr A series of extinctions that eliminated manybrachiopods and tribolites.

Ordovician-Silurian transition 444 Myr Second largest extinction.

Late Devonian 360 Myr A series of extinctions near the Devonian-Carboniferoustransition, lasted 20 Myr with several pulses.

Permian-Triassic extintinction 251 Myr Earth’s largest extinction killed 96% of allmarine species and 70% of land species (plants, insects, vertebrates). Endeddominance of mammal-like reptiles and led to dinosaur dominance.

Triassic-Jurassic extinction 200 Myr Last of large amphibians extinguished.

Cretacious-Tertiary extinction 65 Myr About 50% of all species, including dinosaurs,were extinguished.

Holocene extinction, now About 70% of biologists view the present as a massextinction, due to human intervention. Not universally accepted as manyargue there are significant differences from extinctions. Lattimer, AST 301, Lecture cat – p.2/36

Major Extinctions

Rohde & Muller, Nature, 2005

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Theories of Extinctions• A theory should

• explain all the losses, not just a few groups like dinosaurs• explain why some organisms survived and others didn’t• be based on events that actually happened.

• Flood basalt events, triggered by extensive volcanic activity, have been recorded11 times and each is connected to an extinction event.

• Sea level falls have been recorded 12 times and 7 are associated with extinctions.• Asteroid impacts producting craters over 100 km wide have been recorded once

and associated with a mass extinction. These could produce intense volcanicactivity as well as prolong heating or cooling due to deposition of soot andaerosols into atmosphere.

• Asteroid impacts producing craters less than 100 km wide have been recordedover 50 times, but most not connected to any extinctions.

• Other astronomical events, like supernovae and gamma ray bursts, could irradiatethe Earth and destroy the ozone layer among other things. It’s been suggested asupernova was connected with the Ordovician-Silurian transition, but this is onlyweakly supported by evidence.

• Other terrestrial events connected with global warming or cooling, like SnowballEarth or continental drift.

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The Cretaceous-Tertiary Boundary Layer

Raton Pass, NM

Gubbio, Italy

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Detail of K-T boundary layer

Meteor Crater Field Trip Field Image Gallery Image Gallery Than’s Web Site Than’s CU Page

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Cretacious-Tertiary ExtinctionAvailable evidence points strongly to an asteroid impact:Iridium enhancements In 1980, Alvarez, Alvarez, Asaro and Michels discovered that

sedimentary layers at the K-T boundary contain concentrations of iridium andother rare-earth elements between 30 and 130 times normal. These elements arerare in the Earth’s crust, having been drained to Earth’s core when the Earth wasmolten as part of the Earth’s differentiation. But the same elements arecorresondingly abundant in asteroids and comets. The amount of the excessiridium suggested an impactor of 10-15 km diameter.

4πR2⊕d = 4πR3/3 ⇒ R = (3R2

⊕d)1/3

d = 0.1 cm, R⊕ = 6373 km ⇒ R = 5 kmSpherules Droplets of rock melted by high temperatures

Shocked quartz Formed under high pressure conditions

Soot Ashes in amounts consistent with burning most of Earth’s biota

Worldwide distribution

Liklihood A 10-km body will impact the Earth about once per 100 million years, and theK-T extinction occurred 65 Myr ago.

Craters Several craters with ages of 65 Myrs found: Chicxulub, Mexico; Boltysh crater inUkraine; Silverpit crater in the North Sea

Crater size An impactor produces a crater 20 times its own size, so a 10-km impactormakes a 200 km crater. Chicxulub crater is about 180 km in diameter.

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Chicxulub Crater

ww

w.student.oulu.fi/jkorteni/space/boundary

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Spherule Distribution

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Alternative Extinction Theories

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Crater Formation

Craters are always circular.

Energy comes from kinetic energy ofimpactor and is much larger than theequivalent release of chemical energy (TNT).

Impact speed is the Earth’s escapespeed plus the original space velocity.

Lattimer, AST 301, Lecture cat – p.11/36

Impact Effects• Equate kinetic energy of impactor with gravitational potential energy. Establishes

an estimate of impact velocity that is equivalent to the Earth’s escape velocity.

1

2mv2

esc =GmM⊕

R⊕

⇒ vesc =

s

2GM⊕

R⊕

= 11 km s−1≃ 25, 000 mph

In fact, an impactor is likely to crash with about 50% more velocity.

• The kinetic energy per projectile mass m is

1

2v2 = 1.3 · 1012 erg g−1

≃ 100 × TNT.

• A 1 km-diameter projectile has a mass

m =4π

3ρ(D/2)3 = 1.6 · 1015 g

which strikes with an energy equivalent to 1.6 · 105 MT of TNT.

• A general rule of thumb is that, on the Earth, a meteoroid will create a crater that isabout 20 times bigger than itself.

• On the Moon, the crater will be about 12 times the size of the impactor.Lattimer, AST 301, Lecture cat – p.12/36

Impacts Do Happen!

Peekskill, NY, 1992

Comet Shoemaker-Levy 9, 1994

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Tunguska, Siberia, 1908

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Impact Rate

Chapman and Morrison, Nature, 1994

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Impact Rate

Chapman and Morrison, Nature, 1994

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Fatalities Per Year

Chapman and Morrison, Nature, 1994Lattimer, AST 301, Lecture cat – p.17/36

Impact Fatality EstimatesSimulation of 100,000 years duration, keeping worst event per decade

Asteroid/ comet # # % Fatal Average Average Annual riskdiameter (m) events fatal events yield (MT)∗ fatalities∗ of fatal event

13-99 9792 949 10 18 43 000 1 in 100100-199 173 124 72 300 280 000 1 in 800200-499 31 29 94 2000 700 000 1 in 3500500-999 3 3 100 35 000 13 million 1 in 30 000

All 9999 1105 11 170 120 000 1 in 90∗per fatal event

Simulation of 1 million years duration, keeping worst event per century

Asteroid/ comet # # % Fatal Average Average Annual riskdiameter (m) events fatal events yield (MT)∗ fatalities∗ of fatal event

13-99 8563 2619 31 30 92 000 1 in 382100-199 1065 736 69 300 310 000 1 in 1359200-499 311 295 95 3900 1.8 milion 1 in 3390500-999 45 44 98 29 000 14 million 1 in 22 727

1000-1999 14 14 100 220 000 180 million 1 in 71 4292000+ 2 2 100 2 million 1.7 billion 1 in 500 000

All 10 000 3710 37 2 600 2 million 1 in 270∗per fatal event J. Lewis, U. of ArizonaLattimer, AST 301, Lecture cat – p.18/36

Fatality Odds For USA

C. R. Chapman

Terrorism−→

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Fatality Rates

C. R. ChapmanLattimer, AST 301, Lecture cat – p.20/36

Prediction ConsequencesNature of Problem Mistaken or exaggerated media report of near-miss or near-term

predicted impact causes panic and demands for official action.

Probability of Happening This has already occurred, and is likely to hapen again soon.Most likely way of impact hazard becomingof urgent concern to public officials.

Warning Time Page-one stories develop in hourswith officials being totally surprised.

Mitigation Public education in science and risk,in particular. Development of criticalthinking concerning risk. Science educationand journalism need much improvement.

Examples• Near miss by 100-m asteroid 60,000 km

away. Will people believe official statements?

• Mistaken famous astronomer predicts impactin 10 years, but report not withdrawn for days.

• Official IAU prediction of 1-ini-a-few-hundredimpact possibility later in century; not refinedfor months.

• Grotesque media hype of one of above cases.From C. R. Chapman, Southwest Research Institute

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Mitigation• The energy needed to pulverize an impactor that has a kinetic energy of 1 MT is

about 2 MT.• Energy needed to break apart same impactor is about 0.001 MT.

• The energy needed to deflect an impactor is about 3 · 10−8 of that needed topulverize it. Power of sunlight:

P = 4.3 · 1016

„

R

1 km

«2 „

d

1 AU

«−2

erg/s

Acceleration: a = 2P/(Mc). Deflection distance: ∆ = at2/2

∆ =Pt2

Mc= 1.4

„

d

1 AU

«−2 „

1 km

r

« „

t

1 yr

«2

km

• For ∆ ≃ 8000 km, the size of the Earth, t ≃ 76 yrs. is needed if the impactor is ata mean distance of 1 AU from the Sun and is about 1 km in radius.

• The space shuttle main engines could deflect a 1-km body with a lead time ofabout 30 years.

• Delta 2 1st stage rocket could deflect 100-m body with lead time of 6 months.

• Danger of a bomb attack is that it could lead to many impactors producing damageover a wider range with a cumulative effect at least as large if not larger than theoriginal impactor.

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Mars

Mars Orbiter Laser Altimeter, NASALattimer, AST 301, Lecture cat – p.23/36

Moon

Clementine Orbiter, NASALattimer, AST 301, Lecture cat – p.24/36

Mimas• Mimas clears the material from the Cassini Division, the gap between Saturn’s two

widest rings, because that location is in a 2:1 orbital resonance with Mimas

• Impact that produced the large crater (Herschel) almost completely shatteredMimas; an equivalently-sized crater on the Earth would be as wide as the U.S.

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Venus• Venus has the same mass and radius of the Earth, and has the same gross

chemical composition.

• But Venus is bone-dry, hot enough to melt lead, has an extremely thickatmosphere, and spins backwards.

• One explanation is that two large proto-planetary bodies collided head-on andmerged. This led to the loss of all water and other volatile materials.

• A clue is the lack of 40Ar, produced by thedecay of 40K. This indicates that most of thisgas has not yet been vented into theatmosphere. An early loss of water preventsplate tectonics from occurring, and waterin the interior does not outgas.

• However, another clue is the high abundanceof deuterium (D), 150 times as much per Hatom as on Earth. This can be explained bywater’s photo-dissociation by sunlight andpreferential escape of H relative to D. If 90%of the D has been lost, about 99.9% of the H(and therefore water) has been lost.

• So loss of water doesn’t implicate an impact.

• A head-on impact would have destroyedangular momentum and made a moon’sformation from debris all but impossible. Lattimer, AST 301, Lecture cat – p.26/36

Uranus• It is often proposed a massive collision knocked Uranus on its side.

• However, Uranus has an almost circular orbit and a regular moon system.

• It has a magnetic axis 60◦ tipped from its rotational axis.

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Worlds in Collision?• The star BD+20 307 in the constellation Aries has been found to have about a

million times as much dust as is orbiting our sun. Dust was found using X-ray datafrom the Chandra X-ray Observatory (Zuckerman, Henry and Muno)

• Weinberger has found that BD+20 307 is actually a close binary with a 3.42 dayperiod.

• Fekel and Williamson determined the stars’ chemical composition usingspectroscopy and dated the stellar system to be several billion years old.

• Dust indicates the presenceof planets: this would be thefirst known case of planets ina close binary system.

• Dust should have been sweptout by stellar radiation billionsof years ago: therefore, it hasrecently appeared (last fewhundred thousand years atmost). There must have beena planetary collision.

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Worlds in Collision• Best selling book (1950) by Immanuel Velikovsky describes celestial wars among

planets in historical times. Around 1500 BC a comet (Venus) was ejected fromJupiter and passed near the Earth, changed its orbit and axis, and causedinnumerable catastrophes, explaining many mythologies and religious documents.52 years later, another encounter stopped the Earth’s rotation and caused morecatastrophes. Finally, by 700 BC, it dislodged Mars from its orbit which passedclose to the Earth and causes a whole new set of catastrophes.He predicted Venus must still be hot as befits a young planet. It is rich in petroleumgases and hydrocarbons, and has an abnormal orbit.

•• He stated Jupiter emits radio noises. The Earth’smagnetosphere reaches to the moon, the Sun has anelectrostatic potential of 1019 V, and Earth’s rotation can beaffected by electromagnetic fields.

• The theory has been summarily rejected by scientists. Simply,it violates Newtonian mechanics. Sagan pointed out that itwas already known Venus was hot because of the greenhouseeffect, and its atmosphere is primarily CO2. Also, the moon’sorbit is extremely stable; the Jewish calendar is based on thelunar month and hasn’t changed in 5800 years! Ice corestudies rule out the possibility of large-scale catastrophesduring the last 10,000 years. Finally, mythologists disputeVelikovsky’s versions of mythologies.

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Are Impacts Periodic?• Many publications have suggested a likely impact periodicity of about 26 ± 2 Myr.

• A simple exercise is to take the 50 or so dated impacts on the Earth in the last 250million years and run a periodicity study.

• Assume a model of impact rate N = y[1 + sin(ωt + θ)].

• ω = 2π/P where P is the period. θ is the phase.

• Determine the amplitude y by requiringR T

0Ndt = N , where N is the number of

craters in the study and T = 250 Myr.

y = Nˆ

T + ω−1(cos(θ) − cos(ωT + θ))˜−1

• Bin the cratering data into bins of time width ∆.

• Make a histogram of the cratering data. Hi is the number of craters in bin i.

• The model is that the number of impacts in bin i is Mi =R ti

ti−1

Ndt.

Mi = yˆ

∆ + ω−1(cos(ωti−1 + θ) − cos(ωti + θ))˜

• For a fixed bin size ∆, minimize χ2 =P

i(Mi − Hi)2/(M − 2) where M = T/∆

is the number of bins.• To be significant, the minimum value of χ2 should be smaller than 1, the difference

between the maximum and minimum values of χ2 should be much larger than 1,and the determined period should not be very sensitive to ∆.

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Cratering Data

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Computational Results

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Planetary Migration

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Planetary Migration

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