X and the City: Modeling Aspects of Urban Life (73 page)

Hollywood is very fond of making disaster movies about volcanic eruptions, earthquakes, meteoric impacts, and alien invasions, among other threats to us Earthlings. In July 1994 there was an alien invasion of sorts—on the planet Jupiter. The following news flash [
41
] can be found on California Institute of Technology’s Jet Propulsion Laboratory (JPL) website:

In light of the impact(s) of ex-comet Shoemaker-Levy on Jupiter’s outer atmosphere the question has been raised: could it happen here on earth? As opposed to a cometary encounter, we hear more these days about the possibility of an
asteroid
colliding with the Earth. Such a collision could be globally disastrous of, course, particularly if the rocky body is of the order of a kilometer in size or more.

On May 15 1996, two University of Florida students in the astronomy graduate program discovered an asteroid headed in the Earth’s direction at about 58,000 km/hr. As is standard procedure (and extremely sensible), they immediately reported their observations to the Harvard-Smithsonian Center for Astrophysics. After confirming the details—location and projected trajectory—of the object, the Center posted the relevant information about it (by then designated 1996 AJ1) on the World Wide Web. At 4:34 p.m. (GMT) on May 19, AJ1 reached its point of closest approach to our planet, 450,000 km, just beyond the orbit of the moon (400,000 km). We were safe! More recently, however, an even closer encounter occurred. On November 9, 2011, the popular site
Astronomy Picture of the Day
(
http://apod.nasa.gov/apod/astropix.html
) posted a blurry picture of an asteroid with the following description:

“Asteroid 2005 YU55 passed by the Earth yesterday, posing no danger. The space rock, estimated to be about 400 meters across, coasted by just inside the orbit of Earth’s Moon. Although the passing of smaller rocks near the Earth is not very unusual—in fact small rocks from space strike Earth daily—a rock this large hasn’t passed this close since 1976. Were YU55 to have struck land, it might have caused a magnitude seven earthquake and left a city-sized crater. A perhaps larger danger would have occurred were YU55 to have struck the ocean and raised a large tsunami. . . . Objects like YU55 are hard to detect because they are so faint and move so fast. However, humanity’s ability to scan the sky to detect, catalog, and analyze such objects has increased notably in recent years.”

And we were still safe!

But there
have
been meteoric impacts during the Earth’s long history. Let’s revisit two such instances (without a time machine), and do a little mathematics
in the process. About 65 million years ago such an encounter occurred (Alvarez et al. 1980), and it may well have caused the demise of the dinosaurs. The site where the explosive encounter is believed to have taken place is called the Chicxulub crater; it is an ancient impact crater buried underneath the Yucatán Peninsula in Mexico. Dust from the impact was lofted into the upper atmosphere all around the globe, where it lingered for at least several months. Eventually it settled back to the surface of the earth, having done a superb job of blocking sunlight and thus devastating plant and animal life. On the dark and cold Earth that temporarily resulted, many life-forms became extinct. Available evidence suggests that about 20% of the asteroid’s mass ended up as dust settling out of the upper atmosphere. This dust amounted to an average of about 0.02 gm/cm
²
on the surface of the Earth. If this ancient asteroid had a density of about 2 gm/cm
³
(about the density of moon rock),
how large was it
?

The radius of the Earth is about 4000 miles, or 6400 km. This is 6.4 × 10
⁸
cm, so the area of the Earth’s surface is about 4 × 3 × 40 × 10
¹⁶
≈ 5 × 10
¹⁸
cm
²
. Each cm
²
contained, according to hypothesis, 0.02 gm of asteroid dust. The total mass that settled out was therefore about 10
¹⁷
gm, and the mass of the asteroid was about five times this, or 5 × 10
¹⁷
gm. Since the shape was almost certainly irregular, let’s replace it with the cube of the same mass; the size of the cube will differ little from the average dimension of the asteroid. A cube of side
L
with this mass and supposed density 2 gm cm
⁻³
means that 2
L
³
= 5 × 10
¹⁷
, or
L
= (0.25 × 10
¹⁸
)
^1/3
cm ≈ 6 km. This suggests that the asteroid was, to the nearest order of magnitude, about 10 km in size. This is not unreasonable for an asteroid (though the dinosaurs might well have disagreed). The impact is estimated to have released 4 × 10
²³
joules of energy, equivalent to 10
⁸
megatons of TNT (trinitrotoluene) on impact. Comparing this with equivalent TNT yield of the atomic bombs dropped on Hiroshima (12–15 kilotons) and Nagasaki (20–22 kilotons) in August 1945, one can quite see why such an impact can wreak so much devastation.

Far more recently, between 20,000 and 50,000 years ago, the Earth had another visitor. A much smaller object with diameter between 40 m and 50 m in size hurtled toward Flagstaff, Arizona, before there was a Flagstaff, or even an Arizona. Its speed is estimated to have been about 72,000 km/hr (20,000 m/s). Made mostly of iron, its density was probably about 8000 kg/m
³
(
question:
what is this in gm/cm
³
, the units used above?). A cube of side 45 m made of this material would have a mass
M
≈ 8000 × (45)
³
≈ 7 × 10
⁸
kg, or about 700,000 metric tons. Some studies suggest the object was smaller, approximately a sphere of diameter 40 m, with volume about 260,000 metric tons. Either way it is pretty large; the battleship
Iowa
by comparison displaces about 50,000 metric tons. The kinetic energy on impact for the smaller mass is 1/2 × mass × (speed)
²
≈ 5 × 10
¹⁶
joules, roughly equivalent to the energy released by a thermonuclear bomb (20 tons of TNT).

Suppose that the object called 1996 AJ1(comparable in size with 2005 YU55) had indeed struck the Earth . . . what would have been the “impact,” and how does it compare with that of the Winslow meteorite? It was estimated to be about ten times the size of that one, so the volume must be roughly a thousand times greater. However, it is believed to be composed of rock and ice with an average density of about 3000 kg m
^-3
, considerably less than the chunk of iron that created the hole in Arizona. The speed at its closest approach was about 16,000m/s, or 80% of the estimate for the Winslow object. Then the kinetic energy would have been about 10
³
×
× (0.8)
²
≈ 240 times larger; say between two and three hundred times larger. Since the Earth’s surface is 70% water, there is a good chance it would have hit the ocean, creating huge tsunami waves. These would of course have been extremely destructive, as we know from earthquake-initiated events around the globe. Had it hit a large city such as New York or Washington, D.C., the devastation would have been widespread. To quote one source [
42
]:

So what is the probability of a big chunk of rock hitting a city? It is of course extremely difficult to assess this risk, but we can obtain an upper bound (of sorts) on the probability that
if an asteroid is heading directly for the Earth
(so
that impact somewhere is inevitable), it will hit a major population center. Note that this is
very
different from the probability that an asteroid will impact the Earth. According to one online source [
43
] (using data from January 2007) ranking the largest cities by land area, that ranked first is the New York Metropolitan area with about 9000 km
²
(I am rounding these area and population figures to the nearest thousand and million respectively). With a population of about 18 million according to this source, the population density was about 2000 people/km
²
. The 100th city in this ranking is Jeddah in Saudi Arabia at about 1000 km
²
(this is a slight overestimate); the population in January 2007 was about 3 million, so the corresponding population density was approximately 3000 people/km
²
. The cities in places 50 and 51 are respectively Delhi, India, and Denver, USA, with land areas about 1300 km
²
, though the contrast between them could not be more striking. Delhi had a population of around14 million at that time; Denver’s was nearly 2 million, with a much lower population density of course. The corresponding population densities are about 10,000 and 2000 people/km
²
. We can get a “handle” on the average population density from these figures using the Goldilocks principle, noting that it will likely be between 1000 and 10,000 people/km
²
. This gives about 3000/km
²
. We noted in
Chapter 18
that by 2007 more than half the world’s population was living in cities. Taking a world population of 7 billion (declared by the United Nations as reached October 31, 2011), we estimate that the combined area of all cities is

.

How does this compare with the cross-sectional area of the Earth? This is what the asteroid will “see” as it heads toward us, and with a radius of about 6400 km, this area is π ×(6400)
²
≈ 10
⁸
km
²
. The ratio of the combined metropolitan area to that of the Earth’s cross section is about 10
⁻²
, so it’s not a tiny figure, relatively speaking. So on the basis of a
very
crude calculation, it appears that
if
the asteroid is heading directly for our planet, so an impact somewhere is inevitable, there’s about a 1% chance that a city will be hit. But it may not be yours! The probability of that occurring is much lower, of course. We know that about half the world’s population reside in cities. City populations in general tend to be larger than 10
⁵
and smaller than 10
⁷
(excluding really large cities, of course), so we’ll apply the Goldilocks principle one more time to obtain our guesstimate for an average population: one million. Dividing three billion
by this number, we have three thousand such “average” cities. We have noted above that, given an impending direct collision, there’s about a 1% chance that a city will be hit. Dividing 10
^-2
by three thousand indicates that the chances of your city (or mine) being hit by an asteroid heading straight for Earth may be somewhere between 10
^-5
and 10
^-6
. Does that make you feel a little less concerned? And although the surface of the Earth is about 70% covered by water, this has no bearing on our calculation; it is just the relative area occupied by city dwellers that is important here. Of course, if the impacting body is large enough, then everyone is doomed whether it hits an ocean or not.