Solutions to Homework 2
1. Write a paragraph each for comets and asteroids, describing their distribution in
the solar system, composition (what are they made of), range of sizes, and
anything that you think is relevant.
Comets: divided into short-period (inner solar system) and long-period (outer
solar system and beyond); thought to originate in Kuiper Belt (beyond Neptune)
and Oort Cloud (~50,000 AU); made of rock and ice, densities ~0.5 g/cc,
probably very fragile; typical sizes ~1 – 10 km, but some might consider Pluto
(1200 km) to be a comet; typical encounter speeds with Earth ~20-50 km/s; most
likely candidate for K-T impact.
Asteroids: mostly found in asteroid belt between Mars and Jupiter, but substantial
population in 1:1 resonance with Jupiter (Trojans) and a small fraction cross the
orbits of the inner planets; thought to be origin of most meteorites found on Earth;
mostly made of rock, bulk densities ~1-3 g/cc, possible rubblized structure in
some cases; typical sizes ~100 m – 500 km (Ceres); typical encounter speeds with
Earth ~10-20 km/s; some asteroids have satellites.
2. We believe that Tunguska-like events occur somewhere on the Earth’s surface
approximately once per millennium (1000 yr). Using this fact, estimate how
many times humankind have witnessed such an event in the past 12000 years.
Be sure to explain your calculation carefully, listing any assumptions that you
make. You will need to know that approximately 70% of the Earth’s surface is
covered by water. You will also need to make an estimate for the fraction of the
land that has been inhabited by humans over this time interval. How would such
events have been interpreted by ancient societies? If such an event had
occurred over the USA or the Soviet Union in the 1960s, how might it have been
interpreted?
One event per 1000 yr implies 12 events over 12000 yr, but ~70% of these would
occur over water. If we assume that over the past 12000 yr there was an average
of 108 humans on the planet but that they were concentrated in “city-units” of
typically 104 individuals, and that a typical explosion could be seen within 100
km of such a unit, then the fraction of land “monitored” by humans is ~ 108 
(100 km)2/(104  4R2) ~ 0.6, where R = 6400 km is the radius of the Earth.
This means that the total number of Tunguska events likely seen by civilization
over the past 12000 yr is roughly 12  (1 – 0.7)  0.6 ~ 2 (anything between 0.1
and 10 is perfectly reasonable). In ancient times, such events would likely have
been interpreted as acts of god, punishment for bad behavior, etc. In the 1960s,
such an event would likely have been interpreted as a preemptive nuclear strike
(although there would not have been any radioactive fallout).
3. In your reading, the method of radiometric dating for determining the age of a
rock was mentioned, such as measuring the U-235:Pb-207 and U-238:Pb-206
ratios in zircon spherules. Using the web, a textbook, or some other resource,
describe the method of carbon dating. Be sure to specify the parent isotope and
stable daughter product used in this method. Why is carbon dating not useful
for determining the age of most impact structures on Earth? What is carbon
dating typically used for instead?
14
C is an unstable carbon isotope that decays into 14N with a halflife of ~5600 yr.
Since carbon can be aspirated or ingested by living organisms, and assuming that
the ratio of radioactive to non-radioactive carbon isotopes in the Earth’s
atmosphere has remained roughly constant over time (14C is actually created by
cosmic rays striking 14N in the atmosphere), then the amount of 14C left in organic
material provides an estimate of when that material was created or was last living.
However, the relatively short halflife means that such materials can only be dated
back at most 60,000 yr or so (i.e. ~10 halflives). This means the radiocarbon
method is unsuitable for dating impact structures that for the most part are
millions, not thousands, of years old.