Notes for Week 3.
The chapter on the Moon discusses its gross features (gross=big scale in this context),
gives some background on minerology and rock formation, talks about how ages for rocks
are derived, and then describes crater-forming processes. There is a lot of vocabulary,
and the words are those you can hear the astronauts using in films of the lunar landings.
Some amplification and comments:
The concept of radiative equilibrium temperature is very important. Here is the
basic derivation of this important quantity.
Consider a spherical planet a distance d from a star of luminosity L. The fraction of L that
this planet intercepts is
(cross-section of the planet)/(area of sphere of radius d) = πr2 /4πd2
Of the intercepted radiation, some fraction A will be reflected. For bodies in our solar
system, most of the solar radition is in the visible, so we consider an average value in the
visual: Avis.
A perfect blackbody planet at a uniform temperature T will radiate energy away at the rate:
LPlanet = 4πr2 σ T4 .
A basic property of blackbodies is that they emit radiation with the same efficiency that
they absorb it. The planet will radiate most of its energy in the infrared [a statement we
can check after finishing the calculation of T] so
fraction absorbed = 1-fraction reflected = 1-Avis
and
efficiency of re-radiation = (1-AIR)
This gives
(1-A vis ) (πr 2 /4πd 2 ) L = (1-A IR ) (4πr 2 ) ( T e q 4 )
as the fundamental equation for the radiative equilibrium temperature of a planet.
You should be able to answer these questions:
Why is it called "equilibrium" temperature? Hint: What would happen if the planet
were hotter than or colder than Teq ?
What would the equation look like if the Earth, instead of being round, were a cube
with one face perpendicular to the Sun-Earth line?
As we found with LC5, the above determines the averge value of T4 over the surface of
the planet; the local value may be quite different. Let's calculate the radiative equilibrium
temperature of a surface oriented at an angle θ to the line to the Sun:
perpendicular
θ
surface
to the Sun
For this situation:
(1-A vis ) (C/4πd 2 ) L = (1-A IR ) (A) ( T e q 4 )
where C = cross-sectional area in the beam of sunlight and A = surface area radiating.
We'll assume the surface is able to radiate only from the topside (as it is sitting on an
insulating layer of soil, for example). Then C/A = cos θ and
(
T e q 4 ) = [(1-A vis )/(1-A IR )] [(Lcos / 4πd 2 )]
The biggest factor making the summer warmer than the winter is this one: the Sun's
altitude* in the sky is greater at a given time of day in summer than in winter. Second
biggest factor is the length of the day - more hours of sunlight in the summer. The third
biggest factor is how much of the atmosphere the sunlight has to traverse as it heads for
the ground - more in the winter when the angle is lower, so more light is scattered out of
the path, so less reaches the ground. Why is the distance from the Earth to the Sun NOT
an important factor?
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The second big idea in this chapter is the description of how the melting and solidification
process works, and how this information can be used to deduce how deep and how hot a
rock was when it was melted.
Key terms:
eutectic temperature; eutectic composition
ternary phase diagram
Consider first a mixture of two minerals. Figure 3-12 illustrates how the solid, partially
melted, and melted phases for this substance depend on the temperature and the % of
each material present. Consider each of the following three possibilities:
(a) you have a sample of pure Pyroxene. As you heat the rock, it melts at what T?
(b) you have a sample of pure Plagioclase. As you heat the rock, it melts at what T?
(c) you have a mixture that is about 45% plagioclase, 55% pyroxene. It melts at what T?
* Altitude = angle from the horizon to the object, measured vertically. The zenith angle, θ
in our equation, is (90°-altitude) in degrees or (π/2 -altitude) in radians.
(d) you take a bunch of pure pyroxene rock and grind it up; you mix this with a similar
amount of pure pyroxene. As this is heated, what happens? What would happen if you
used mostly one and only a little of the other? A familiar case of 2 substances: Sprinkle
salt crystals onto an ice cube. What happens?
Consider next a mixture of 3 minerals - figure 3.13. If we start from liquid and let it solidify,
some portions will solidify first, leaving a changed composition of the liquid portion. The
path that the liquid portion will take in this diagram depends on the properties of the
minerals and on the pressure. Thus, the final rock will give evidence of the conditions
under which it solidified - small crystals for rapid solidification, big ones for slow
solidification. More of some components for a rock formed under higher pressure than
under lower pressure.
A familiar case of a mixture of three substances: Consider ice on a gravel road on a very
cold day, with added crystals of salt. What happens as it starts to melt?
Altogether, Moon rocks reveal:
(a) ongoing "processing" of lunar material by impacts.
(b) evidence for widespread (liquid) magma on the Moon soon after it formed.
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The third big idea is the use of radioactively unstable isotopes to date a rock.
For an unstable isotope U that decays forming (among other things) a product D:
dN U/dt = - λ t so N(t) = NoU e- λ t
while ND(t) = NoD + NoU (1-e- λ t)
If we know NoD then we can simply look at the ratio NU/ND = e- λ t /(1- e- λ t ) to determine t.
This might be the case for example if D is something that would not normally be part of the
rock (or the particular mineral in the rock) but that is trapped in it if it is produced there by
radioactive decay after the rock forms. Also, if D is one isotope of an element that has
other isotopes, then we can compare the ratio of the isotopes of D to what one finds where
D is present without any U. Another version is to look for variations within your sample in
the ratios U/D and D/D', where D' is the other isotope of D. This is the basis for the
analysis on page 54.
The analysis on page 54 assumes that the isotope ratio is constant while the ratios of Rb
to Sr depend on the exact conditions where the mineral grains formed. This is a good
assumption. If you start with ingredients with a uniform isotope ratio and form a variety of
minerals, the minerals (and resulting rocks) will have essentially the same isotope ratio.
Why? Well - two isotopes of one element have the same chemical properties. They differ
only in atomic mass. To separate isotopes is pretty hard; that was the main assignment
for the Ames Laboratory here during WWII.
What is meant by the "age" of a rock, measured this way? It is the time since the rock was
last melted. The oldest rocks on Earth are < 4 Gyr old, measured by this method; does
that mean the planet is <4 Gyr old?
A few things to think about in preparation for exam I:
We have covered
Chapter One: Topics related to Earth
Figuring out the arrangement and scale of the Solar System
Geometry of measuring planet orbits in AU
Techniques for determining the AU in km
Figuring out the origin of the Earth - processes that modify its surface.
Chapter Two: Topics related to the Sun
Temperature, blackbody radiation, Wien's law
The source of the Sun's energy
estimating the main sequence lifetime from L and mc2
Fusion and fission
The origin of the elements
Chapter Three: Topics related to the Moon
The history of the Moon's surface
The temperature of the Moon's surface
radiative equilibrium temperature
How rocks form
phase diagrams, ternary diagrams, eutectic melt
Finding the age of a rock
Where did we use:
conservation of energy (or mass-energy)?
geometry?
proportional reasoning?