Catastrophe Theory
•
Consider a potential function
V (x) = x3 + ax.
•
When a < 0 there is both a
stable minimum (dots) and an
unstable maximum in the potential.
•
As a is slowly increased, the
equilibrium system moves smoothly
p
to smaller x: xeq = −a/3.
•
When a = 0, the stable minimum
disappears, and for a infinitesimally
positive, the system moves abruptly to a
new 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.
•
•
•
Over 99% of all species that have ever existed are now extinct.
Extinction was established as a fact by Georges Cuvier in 1796, and was critical
for the spread of uniformitarinism in spite of the fact that Cuvier viewed extinctions
as evidence in favor of catastrophism and opposed Lamarckian evolution theories.
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 are
marked by extinctions and originations.
Cambrian-Ordovician transition 488 Myr A series of extinctions that eliminated many
brachiopods and tribolites.
Ordovician-Silurian transition 444 Myr Second largest extinction.
Late Devonian 360 Myr A series of extinctions near the Devonian-Carboniferous
transition, lasted 20 Myr with several pulses.
Permian-Triassic extintinction 251 Myr Earth’s largest extinction killed 96% of all
marine species and 70% of land species (plants, insects, vertebrates). Ended
dominance 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 mass
extinction, due to human intervention. Not universally accepted as many
Lattimer, AST 301, Lecture cat – p.2/36
argue there are significant differences from extinctions.
Major Extinctions
Rohde & Muller, Nature, 2005
Lattimer, AST 301, Lecture cat – p.3/36
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 recorded
11 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 producing craters less than 100 km wide have been recorded
over 50 times, but most not connected to any extinctions.
•
Other astronomical events, like supernovae and gamma ray bursts, could irradiate
the Earth and destroy the ozone layer among other things. It’s been suggested a
supernova was connected with the Ordovician-Silurian transition, but this is only
weakly supported by evidence.
•
Other terrestrial events connected with global warming or cooling, like Snowball
Earth or continental drift.
Asteroid impacts producting craters over 100 km wide have been recorded once
and associated with a mass extinction. These could produce intense volcanic
activity as well as prolong heating or cooling due to deposition of soot and
aerosols into atmosphere.
Lattimer, AST 301, Lecture cat – p.4/36
The Cretaceous-Tertiary Boundary Layer
Raton Pass, NM
Gubbio, Italy
Lattimer, AST 301, Lecture cat – p.5/36
Detail of K-T boundary layer
Lattimer, AST 301, Lecture cat – p.6/36
Cretacious-Tertiary Extinction
Available evidence points strongly to an asteroid impact:
In 1980, Alvarez, Alvarez, Asaro and Michels discovered that
sedimentary layers at the K-T boundary contain concentrations of iridium and
other rare-earth elements between 30 and 130 times normal. These elements are
rare in the Earth’s crust, having been drained to Earth’s core when the Earth was
molten as part of the Earth’s differentiation. But the same elements are
corresondingly abundant in asteroids and comets. The amount of the excess
iridium suggested an impactor of 10-15 km diameter.
Iridium enhancements
2
2
d)1/3
d = 4πR3 /3 ⇒ R = (3R⊕
4πR⊕
d = 0.1 cm, R⊕ = 6373 km ⇒ R = 5 km
Spherules
Droplets of rock melted by high temperatures
Shocked quartz
Soot
Formed under high pressure conditions
Ashes in amounts consistent with burning most of Earth’s biota
Worldwide distribution
A 10-km body will impact the Earth about once per 100 million years, and the
K-T extinction occurred 65 Myr ago.
Liklihood
Several craters with ages of 65 Myrs found: Chicxulub, Mexico; Boltysh crater in
Ukraine; Silverpit crater in the North Sea
Craters
An impactor produces a crater 20 times its own size, so a 10-km impactor
makes a 200 km crater. Chicxulub crater is about 180 km in diameter.
Crater size
Lattimer, AST 301, Lecture cat – p.7/36
Chicxulub Crater
www.student.oulu.fi/ jkorteni/space/boundary
Lattimer, AST 301, Lecture cat – p.8/36
Spherule Distribution
Lattimer, AST 301, Lecture cat – p.9/36
Alternative Extinction Theories
Lattimer, AST 301, Lecture cat – p.10/36
Crater Formation
Craters are always circular.
Energy comes from kinetic energy of
impactor and is much larger than the
equivalent release of chemical energy (TNT).
Impact speed is the Earth’s escape
speed 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
GmM⊕
2
=
mvesc
⇒ vesc =
2
R⊕
s
2GM⊕
= 11 km s−1 ≃ 25, 000 mph
R⊕
In fact, an impactor is likely to crash with about 50% more velocity.
•
The kinetic energy per projectile mass m is
1 2
v = 1.3 · 1012 erg g−1 ≃ 100 × TNT.
2
•
A 1 km-diameter projectile has a mass
m=
4π
ρ(D/2)3 = 1.6 · 1015 g
3
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 is
about 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!
Comet Shoemaker-Levy 9, 1994
Peekskill, NY, 1992
Lattimer, AST 301, Lecture cat – p.13/36
Tunguska, Siberia, 1908
Lattimer, AST 301, Lecture cat – p.14/36
Impact Rate
Chapman and Morrison, Nature, 1994
Lattimer, AST 301, Lecture cat – p.15/36
Impact Rate
Chapman and Morrison, Nature, 1994
Lattimer, AST 301, Lecture cat – p.16/36
Fatalities Per Year
Chapman and Morrison, Nature, 1994
Lattimer, AST 301, Lecture cat – p.17/36
Impact Fatality Estimates
Simulation of 100,000 years duration, keeping worst event per decade
Asteroid/ comet
diameter (m)
13-99
100-199
200-499
500-999
All
∗ per
#
fatal events
949
124
29
3
1105
% Fatal
10
72
94
100
11
Average
yield (MT)∗
18
300
2000
35 000
170
Average
fatalities∗
43 000
280 000
700 000
13 million
120 000
Annual risk
of fatal event
1 in 100
1 in 800
1 in 3500
1 in 30 000
1 in 90
fatal event
Simulation
Asteroid/ comet
diameter (m)
13-99
100-199
200-499
500-999
1000-1999
2000+
All
∗ per
#
events
9792
173
31
3
9999
of
1
million
#
events
8563
1065
311
45
14
2
10 000
fatal event
years
#
fatal events
2619
736
295
44
14
2
3710
duration,
% Fatal
31
69
95
98
100
100
37
keeping
worst
Average
yield (MT)∗
30
300
3900
29 000
220 000
2 million
2 600
event
per
Average
fatalities∗
92 000
310 000
1.8 milion
14 million
180 million
1.7 billion
2 million
century
Annual risk
of fatal event
1 in 382
1 in 1359
1 in 3390
1 in 22 727
1 in 71 429
1 in 500 000
1 in 270
AST 301, Lecture cat – p.18/36
J. Lewis, U.Lattimer,
of Arizona
Fatality Odds For USA
Terrorism−→
C. R. Chapman
Lattimer, AST 301, Lecture cat – p.19/36
Fatality Rates
C. R. Chapman
Lattimer, AST 301, Lecture cat – p.20/36
Prediction Consequences
Mistaken or exaggerated media report of near-miss or near-term
predicted impact causes panic and demands for official action.
Nature of Problem
This has already occurred, and is likely to hapen again soon.
Most likely way of impact hazard becoming
of urgent concern to public officials.
Probability of Happening
Page-one stories develop in hours
with officials being totally surprised.
Warning Time
Public education in science and risk,
in particular. Development of critical
thinking concerning risk. Science education
and journalism need much improvement.
Mitigation
Examples
•
Near miss by 100-m asteroid 60,000 km
away. Will people believe official statements?
•
Mistaken famous astronomer predicts impact
in 10 years, but report not withdrawn for days.
•
Official IAU prediction of 1-ini-a-few-hundred
impact possibility later in century; not refined
for months.
•
Grotesque media hype of one of above cases.
From C. R. Chapman, Southwest Research Institute
Lattimer, AST 301, Lecture cat – p.21/36
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 to
pulverize it. Power of sunlight:
P = 4.3 · 1016
„
R
1 km
«2 „
d
1 AU
«−2
erg/s
Acceleration: a = 2P/(M c). Deflection distance: ∆ = at2 /2
P t2
= 1.4
∆=
Mc
„
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 at
a 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 of
about 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 damage
over a wider range with a cumulative effect at least as large if not larger than the
original impactor.
Lattimer, AST 301, Lecture cat – p.22/36
Mars
Mars Orbiter Laser Altimeter, NASA
Lattimer, AST 301, Lecture cat – p.23/36
Moon
Clementine Orbiter, NASA
Lattimer, 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 shattered
Mimas; an equivalently-sized crater on the Earth would be as wide as the U.S.
Lattimer, AST 301, Lecture cat – p.25/36
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 thick
atmosphere, and spins backwards.
•
One explanation is that two large proto-planetary bodies collided head-on and
merged. This led to the loss of all water and other volatile materials.
•
A clue is the lack of 40 Ar, produced by the
decay of 40 K. This indicates that most of this
gas has not yet been vented into the
atmosphere. An early loss of water prevents
plate tectonics from occurring, and water
in the interior does not outgas.
•
However, another clue is the high abundance
of deuterium (D), 150 times as much per H
atom as on Earth. This can be explained by
water’s photo-dissociation by sunlight and
preferential 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 destroyed
angular momentum and made a moon’s
formation 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.
Lattimer, AST 301, Lecture cat – p.27/36
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 data
from 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 day
period.
•
Fekel and Williamson determined the stars’ chemical composition using
spectroscopy and dated the stellar system to be several billion years old.
•
Dust indicates the presence
of planets: this would be the
first known case of planets in
a close binary system.
•
Dust should have been swept
out by stellar radiation billions
of years ago: therefore, it has
recently appeared (last few
hundred thousand years at
most). There must have been
a planetary collision.
Lattimer, AST 301, Lecture cat – p.28/36
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 from
Jupiter and passed near the Earth, changed its orbit and axis, and caused
innumerable catastrophes, explaining many mythologies and religious documents.
52 years later, another encounter stopped the Earth’s rotation and caused more
catastrophes. Finally, by 700 BC, it dislodged Mars from its orbit which passed
close 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 petroleum
gases and hydrocarbons, and has an abnormal orbit.
• He stated Jupiter emits radio noises. The Earth’s
magnetosphere reaches to the moon, the Sun has an
electrostatic potential of 1019 V, and Earth’s rotation can be
affected by electromagnetic fields.
• The theory has been summarily rejected by scientists. Simply,
it violates Newtonian mechanics. Sagan pointed out that it
was already known Venus was hot because of the greenhouse
effect, and its atmosphere is primarily CO2 . Also, the moon’s
orbit is extremely stable; the Jewish calendar is based on the
lunar month and hasn’t changed in 5800 years! Ice core
studies rule out the possibility of large-scale catastrophes
during the last 10,000 years. Finally, mythologists dispute
Velikovsky’s versions of mythologies.
Lattimer, AST 301, Lecture cat – p.29/36
Lattimer, AST 301, Lecture cat – p.30/36
Are Impacts Periodic?
•
•
Many publications have suggested a likely impact periodicity of about 26 ± 2 Myr.
•
•
Assume a model of impact rate Ṅ = y[1 + sin(ωt + θ)].
•
•
•
•
A simple exercise is to take the 50 or so dated impacts on the Earth in the last 250
million years and run a periodicity study.
ω = 2π/P where P is the period. θ is the phase.
RT
Determine the amplitude y by requiring 0 Ṅ dt = N , where N is the number of
craters in the study and T = 250 Myr.
ˆ
˜−1
y = N T + ω −1 (cos(θ) − cos(ωT + θ))
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.
R ti
Ṅ dt.
The model is that the number of impacts in bin i is Mi = t
i−1
ˆ
˜
Mi = y ∆ + ω −1 (cos(ωti−1 + θ) − cos(ωti + θ))
•
For a fixed bin size ∆, minimize χ2 =
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 ∆.
Lattimer, AST 301, Lecture cat – p.31/36
P
i (Mi
− Hi )2 /(M − 2) where M = T /∆
Cratering Data
Lattimer, AST 301, Lecture cat – p.32/36
Computational Results
Lattimer, AST 301, Lecture cat – p.33/36
Planetary Migration
Lattimer, AST 301, Lecture cat – p.34/36
Planetary Migration
Lattimer, AST 301, Lecture cat – p.35/36
Lattimer, AST 301, Lecture cat – p.36/36