Download HOMEWORK 4

ATM S 211 – Climate and Climate Change
HOMEWORK 4: Ancient atmospheres and climate. (DUE MONDAY MARCH 1)
Qu. 1) (a) Explain where Earth’s atmosphere and ocean came from (in no more than 2 to
4 sentences)? [6 pts]
(b) Which gases are thought to have made up the Hadean atmosphere, after the
Earth’s core had formed? [6 pts]
Qu. 2) The age of the Solar System (and hence the Earth) can be determined from the
decay of radioisotopes within meteorites. The technique involves looking at how many
stable “daughter” atoms have been produced from the decay of a radioactive “parent”
atom. An example of a parent is 87Rb (a radioisotope of rubidium), which radioactively
decays to the daughter atom, 87Sr (a stable isotope of strontium). The following graph
shows some values from three meteorite samples.
The following equation is used to calculate the age of the samples and hence the age of
the Solar System:
 87 Sr 
 87 Sr 
t  87 Rb 
=  86 
+  86 
 86 
 Sr  TODAY  Sr START τ  Sr  TODAY
 87 Sr 
is just the intercept on the vertical axis (close to 0.7), which
The quantity  86 
 Sr START
corresponds to the ratio of strontium isotopes that were originally present in the
meteorites when the Solar System formed. The number of 86Sr atoms has remained the
same in the sample (because 86Sr is a stable isotope), but the number of 87Sr atoms has
 87 Sr 
accumulated from the decay of radioactive 87Rb. The quantities  86 
and
 Sr  TODAY
1
ATM S 211 – Climate and Climate Change
 87 Rb 
are the values on the y (vertical) and x (horizontal) axes, respectively, for
 86 
 Sr  TODAY
each data point. The slope of a line connecting the values for the three samples is
equivalent to t/τ, where t is the total time that has elapsed since the meteorites were
formed. The slope is found to be 0.06449. The quantity τ is given by τ = T/0.693 where T
is the half-life of 87Rb.
(a) What are isotopes? [2 pts]
(b) What is meant by the term “half-life”? [2 pts]
(c) Use the above equation, and the values given, to calculate the age of the Solar
System. (You will also need the half-life, T, of 87Rb, which is 49 billion years).
[4 pts]
(d) In another planetary system around another star, intelligent extraterrestrials living
on one of the planets find that the slope of the same graph using data from
meteorites within their planetary system is larger. Is their “solar system” older or
younger than ours? Provide a brief explanation. [3 pts]
Qu. 3) Former Vice-President of the U.S.A., Dan Quayle, said, “Mars is essentially in the
same orbit. Mars is somewhat the same distance from the Sun, which is very important.
We have seen pictures where there are canals, we believe, and water. If there is water that
means there is oxygen. If there is oxygen, then we can breathe.” (August, 1989)
Describe three technical mistakes in the above. (You will not receive credit for saying
that it was a mistake to have someone with Dan Quayle’s intellect as Vice-President).
[3 pts]
Qu. 4) How do carbon isotopes in sedimentary carbonate rocks tell us about rates of
production of atmospheric O2? (If stuck, see Chapter 11 of the textbook). [5 pts]
Qu. 5) Why is the planet Mars red? The simplistic answer is because Mars has redcolored iron oxides on its surface. The deeper answer is that Mars has lost hydrogen
atoms and molecular hydrogen to space. Explain how losing hydrogen to space would
have turned Mars red. [5 pts]
Qu. 6) Why is methane thought to have been an important greenhouse gas in the Earth’s
atmosphere during the Archean Eon? [3 pts]
2
ATM S 211 – Climate and Climate Change
Qu. 7) In the Milankovitch theory of ice ages, changes of Earth’s orbital parameters (such
as eccentricity, obliquity, and precession of perihelion) cause snow and ice to extend
much farther from the polar regions than they do today. What orbital configuration favors
glaciation? Why? (If stuck, check Chapter 14 of the textbook). [6 pts]
Qu. 8) The orbit of the Earth is not a circle but an ellipse. An ellipse is defined as the
locus of all points such that the sum of the distances to two fixed points, called the foci, is
a constant. We can easily show that this constant is equal to 2a, where a is called the
semimajor axis of the ellipse. (See Box Figure 14-1 of the textbook (reproduced below)
and accompanying discussion in Box 14-1.) The eccentricity e of the ellipse is defined
such that the distance from one focus to the midpoint of the figure is ae. An ellipse with
e = 0 is a circle, i.e., the Sun would be at the center of a circular orbit.
Box Figure 14-1 from textbook
(a) Kepler’s first law (see Box 14-1) states that the planets move around the Sun in
elliptical orbits with the Sun at one focus. Earth’s present orbit has a semimajor axis of 1
AU (Astronomical Unit) and an eccentricity of 0.017. The point of closest approach to
the Sun is the perihelion; the point farthest away is the aphelion. By considering Box Fig
14-1, how much closer is Earth to the Sun at perihelion than at aphelion? Express your
answer in astronomical units. [4 pts]
(b) The Milankovitch theory of the ice ages holds that the most important forcing factor
is the difference in solar heating at high latitudes when Northern Hemisphere summer
occurs at perihelion as opposed to aphelion. Using the inverse square law (Chapter 3),
find the solar flux at perihelion and at aphelion. Recall that the solar flux at 1 AU is 1370
W/m2. How much higher is the solar flux at perihelion than at aphelion today? Express
3
ATM S 211 – Climate and Climate Change
your answer as a percentage. How much warmer is the effective radiating temperature of
Earth (Chapter 3)? [9 pts]
(c) The eccentricity of Earth’s orbit varies with time as a consequence of gravitational
perturbations caused by the other planets. Repeat part (b) for e at its maximum value of
0.06. [6 pts]
4