Section 11 (for Hunter's Sections) and Prelim 3 Review

V. Hunter Adams
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from IPython.display import Latex
import matplotlib.pyplot as plt
%matplotlib inline
import numpy
from matplotlib import animation, rc
from IPython.core.display import HTML 
from IPython.display import HTML
from ipywidgets import *
from scipy.integrate import odeint
from IPython.display import Image

Life Cycle of the Sun

Birth

A star begins as an enormous cloud of gas and dust (light-years in diameter). This gas and dust (mostly hydrogen, with some helium and a little bit of other stuff) collapses under its own gravity. As the volume occupied by this gas decreases, its temperature increases. Eventually, the temperature in the core gets hot enough for the hydrogen to ignite. When this happens, a star is born.

The Proton-Proton Chain

There is a sequence of reactions that yield Helium-4 (2 protons, 2 neutrons) from four protons. First, two protons collide and, before they can separate, one of the protons decays into a neutron. The result is deuterium (1 proton, 1 neutron).

After this, a proton collides with the deuterium and sticks, yielding Helium-3 (2 protons, 1 neutron).

Finally, two of these Helium-3's collide, resulting in Helium-4 (2 protons, 2 neutrons), and two protons being ejected. The net result is Helium-4 from 4 protons. As we learned, the mass of Helium-4 is less than the mass of four protons. The "lost" mass is converted directly into energy, according to $E=mc^2$.

Hydrostatic Equilibrium

For most of their lifetimes, stars are relatively simple objects that can be characterized by two things: mass and age. Mass is important because the mass of a star determines its internal pressure. The higher the pressure, the greater the tempreature and the faster the rate of nuclear fusion, making the star brighter. When the star is stable, the pressure at any depth must balance the force of gravity from the gas above it, and the total energy generated in the core must equal the total energy radiated at the surface. If these conditions are met, the star is in hydrostatic equilibrium.

For a star in hydrostatic equilibrium, a compression leads to an increase in pressure in the core, which increases the temperature, which increases the rate of fusion, which inflates the star. An expansion decreases the pressure, which decreases the temperature, which slows fusion, which allows gravity to press the star back together. These competing forces oppose one another.

The Main Sequence

As we just discussed, for a star to be in hydrostatic equilibrium then there is a relationship between its mass and its luminosity means that if you change one of these variables, the other changes predictably. Thus, hydrogen-burning stars in hydrostatic equilibrium will only be able to have specific values for mass and luminosity. These values define the main sequence.

The Giants

The sun will leave the main sequence in about 5 billion years.

As hydrogen in the core burns, it produces inert helium that accumulates in the core. Thus, the hydrogen/helium mixture in the core becomes denser and denser, which increases the pressure, and thus increases the rate of hydrogen fusion. The result is that the Sun's luminosity is slowly increasing. This process starts out slow.

At some point, however, the core becomes so depleted of hydrogen that energy production falls despite the increasing pressure. This causes the density of the core to increase even more. Since the core can no longer use fusion to fight back against gravity, the only option is to compress until the internal pressure is high enough to hold up the weight of the star. The increased pressure in the outer core causes hydrogen burining to increase in that area, causing the star's luminosity to increase.

Ultimately, we end up with an inert helium core surrounded by a hydrogen burining shell. When the core gets depleted of hydrogen, the star becomes a Red Giant. The shell causes more and more helium to accumulate in the core, increasing the pressure, increasing the rate of hydrogen burining, increasing the amount of helium in the core. It's a positive feedback loop.

No star remains a red giant for long. It is consuming fuel at an alarming rate, and will continue to do so until something stops it.

Helium Burning

Eventually something gives. Internal pressure can no longer support the weight of the star, and the helium core begins to fuse into Carbon. The enormous energy output from the Sun will cause it to balloon out as far as Venus' orbit.

End of Life

The Sun is now a carbon-oxygen core, surrounded by a helium-burining shell and an outer hydrogen-burning shell. The key fact, however, is that the Sun must consume helium much more quickly than it consumes hydrogen. A process similar to that which happened previously happens again. The star is tearing through its fuel, but it is never able to obtain the temperature and pressure required to start fusing carbon. The degenerate core grows more and more dense as the Sun expands out to the orbit of Jupiter.

The star then coughs itself to death. The helium-burning shell catches up to the hydrogen burning shell. All of the helium gets exhausted, but then the remaining hydrogen burns some more into existance, which ignites again. This process repeats some number of times, until the outer layers separate entirely from the white-dwarf core.

Greenhouse Warming

We've beaten the hell out of this topic. Can you all explain how it works? Can you talk about the role that it plays in Venus' atmosphere?

The Earth is not 0F because of the Greenhouse Effect. What is the wavelength of incident light from the the Sun? Wien's Law:

\begin{align} \lambda_{max} &= \frac{2898}{T}\\ &= \frac{2898}{5778}\\ &= 0.6 \mu m = 600 nm \end{align}

What is the wavelength of light re-emitted by the Earth?

\begin{align} \lambda_{max} &= \frac{2898}{T}\\ &= \frac{2898}{255}\\ &= 11.36 \mu m = 11,360 nm \end{align}

That's a much longer wavelength. Remember what we learned about absorption and emission spectra? Specific molecules will absorb and re-emit light at very specific wavelengths, and it will allow light that isn't at those wavelengths to pass unimpeded. Which direction does the light get re-emitted? All directions.

So the atmosphere, which has C02, methane, etc., allows short-wavelength light from the Sun to pass unimpeded. However, it absorbs and re-emits light at longer wavelengths. Because that light is re-emitted in all directions, some is sent back down to Earth, warming it more.

Orbital Resonances

From Kepler's Third Law, we know that the period of an orbiting body is related to its semi-major axis. For particular semi-major axis, the orbital periods will have simple integer relationships (maybe one object orbits once everytime another orbits twice, or maybe one orbits twice in the time it takes for the other to orbit 3 times, you get the point). What this means is that there are occasional moments when these bodies get close to each other. What does that allow for?

It allows for them to tug on one another. The two bodies can exchange energy, much like pushing a child on a swingset. What are some repurcussions of this in our own solar system?

Venus Atmosphere

Much of what I will cover for the Venutian atmosphere will be a review of previous Greenhouse material. This isn't a bad thing. This particular topic is very important for this class, so revisiting it isn't a bad idea. Also, it's our first section back after Spring Break. I don't want to cover really heavy material straight away.

Something to keep in mind is that we only came to understand the Greenhouse Effect (how it works and its importance here on Earth) by studying Venus. This particular topic is an excellent example of the importance of space exploration for understanding our own planet.

Venus is an absolute hellhole. The atmosphere on Venus is thick. Really thick. The atmospheric pressure at the surface of Venus is 90 bar. That's 90 times more atmospheric pressure than on the surface of the Earth, and approximately equivalent to the pressure that one feels 1 km underwater (the world record SCUBA depth is 332 meters). The temperature at the surface is 740K, sufficient to melt lead.

Venus is an excellent example of the fragility of the complex sequence of events that leads to habitable conditions on a planet. In many regards, Earth and Venus are twin sisters. They're approximately the same age, approximately the same mass, and they're located at approximately the same distance from the Sun (Venus is of course a bit closer). Earth became a life-sustaining paradise, and Venus became an absolute hellhole that rains sulfuric acid and has temperatures hot enough to melt lead. What went wrong?

Initial Conditions

When they were born, volcanic processes spewed H2O, CO2, SO2, N2, etc. into both the Venutian and Earth atmospheres.

Venus Divergence

Venus, being a little bit closer to the Sun, is slightly warmer than the Earth. As a result, more H2O remains in the atmosphere as water vapor instead of condensing into liquid water on the surface. H2O, as we've learned, is a very efficient greenhouse gas. Lots of water vapor in the atmosphere leads to increased heat, which evaporates any liquid water that may have condensed, which leads to more heat, etc.

Without liquid water on the surface, CO2 is not sequestered out of the atmosphere and contributes to greenhouse gassing as well. As a result, the planet cooks.

Earth Divergence

Earth, a bit cooler, has liquid water condense onto its surface. This liquid water forms oceans, which dissolves CO2 out of the atmosphere to form carbonate rocks. This reduces the greenhouse warming of the Earth, resulting in an atmosphere of mostly N2.

Why the Asymmetry in Oxygen?

This is a review question. Oxygen is a byproduct of biological processes. We see an excess of oxygen in the Earth's atmosphere because life put it there.

Where is all of Venus' Water Now?

The story of Venus' evolution that we just went over requires that Earth and Venus started with approximately the same amount of water. Where did all of that water go?

In the upper reaches of the atmosphere, water molecules can be dissociated. This means that they are broken into their constitudent pieces (O and H and H) by UV radiation. Do you guys remember what H2 is called? Answer: Deuterium (the same that is the critical step in the proton-proton chain).

On Venus, we see a much higher concentration of Deuterium than on Earth. What could this mean?

While, on Earth, much of the water remains in liquid form on the surface; Venus has all of its H2O in the atmosphere. This H2O is broken into O2 and Deuterium. Deuterium is heavier than normal hydrogen, and therefore has a higher escape velocity (it's harder for Deuterium to escape the atmosphere than it is for ordinary Hydrogen to escape). It makes sense then that the remaining Hydrogen in the Venutian atmosphere is enriched in Deuterium.

Furthermore, spacecraft have observed Hydrogen and Oxygen ions escaping from the planet into the solar wind. The flux of Hydrogen is twice that of oxygen, which is precisely what you'd expect if the source of these ions is dissociated water.

How do gas molecules escape a planet?

Temperature and kinetic energy have a direct relationship, given by:

\begin{align} \left<KE\right> &= \frac{3}{2}k_BT\\ \left<\frac{1}{2}mv^2\right> &= \frac{3}{2}k_BT \end{align}

Where $m$ is the mass of the particle in kg, $v$ is the velocity of the particle in m/s, $k_B$ is the Boltzmann Constant ($1.38 \times 10^{-23}$ J/K), and $T$ is the temperature in Kelvin. As temperature goes up, the kinetic energy goes up at the same rate.

Considering the Consequences

Given this information, let's say we have a mixture of Nitrogen and Oxygen (like in our atmosphere) that is all at the same temperature. Which particles are moving more quickly?

Because all of the gas is at the same temperature, all of the gas has the same kinetic energy. But! Look at the equation for kinetic energy:

\begin{align} KE &= \frac{1}{2}mv^2 \end{align}

In order to keep the kinetic energy the same value, heavier particles (bigger $m$) must be moving more slowly (smaller $v$). Since Nitrogen has less mass than oxygen, it will be moving more quickly.

Some practice

What is the average speed of Hydrogen in the Sun's corona, which has a temperature in excess of 1,000,000K?

With this temperature, we can solve for the kinetic energy of the gas in the corona:

\begin{align} KE &= \frac{3}{2}k_BT\\ &= \frac{3}{2}\left(1.38 \times 10^{-23}\right)\left(1000000\right)\\ &= 2.07 \times 10^{-17} \end{align}

With the Kinetic Energy, we can solve for the velocity. Note that the mass of a Hydrogen (proton) is $1.67\times 10^{-27}$ kg.

\begin{align} KE &= \frac{1}{2}mv^2\\ v &= \sqrt{\frac{2KE}{m}} \end{align}

Substituting:

\begin{align} v &= \sqrt{\frac{2\left(2.07\times 10^{-17}\right)}{1.67\times 10^{-27}}}\\ &= 157450\text{ $m/s^2$} \end{align}

If the particle were heavier, what would happen to the velocity?

Escape Velocity

Recall that the equation for the escape velocity from a planet is given by:

\begin{align} v_{esc} &= \sqrt{\frac{2GM}{r}} \end{align}

Question: How could you derive this?

Answer: Conservation of energy.

Reflecting on the Above

As the mass of a planet increases, what do you expect to observe in the rate of thermal escape of the atmosphere?

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Clouds on Venus

The clouds on Venus form at altitudes between 20 and 80 km. It is at these altitudes that UV radiation can break H20, CO2, and SO2 molecules into the consitutent pieces, and those pieces ca re-form into H2SO4, which combines with water vapor to form sulfuric acid.

The lower altitudes are clear because the temperatures are so intense. Sulfuric acid droplets vaporize and break back into H20 and SO3, making the air clear.

Venutian Geology

How do we see through the Venutian clouds to study its surface? Answer: RADAR.

Given its size, Venus almost certainly has a differentiated, metal-rich core. It does not, however, have a magnetic field. Why might this be?

A magnetic field requires three things:

  1. A layer of electrically conducting fluid
  2. Convection in that layer
  3. Rotation

Moons!

Titan

  1. Moon of Saturn
  2. Rich in organics (methane/ethane clouds, rain, and lakes)
  3. Subsurface liquid water ocean
  4. Wide range of river valley morphologies
  5. Thick atmosphere (high atmospheric pressure)
  6. RADAR imaging used for studying the surface

Enceladus

  1. Moon of Saturn
  2. Liquid water plumes
  3. Cassini found sodium and potassium salts in the plume (salty ocean underneath)
  4. Global ocean underneath ice crust that interfaces with rocky core

Europa

  1. Moon of Jupiter
  2. Liquid water ocean underneath thin ice crust
  3. 2x more liquid water than Earth
  4. Internal heat generated by tidal flexing
  5. Icy surface covered in ridges, chaos, bands, and some impacts
  6. Stable (been in about the same situation for 4 billion years)

Io

  1. Moon of Jupiter
  2. Active Volcanoes due to internal tidal heating
  3. Approximately the size of Earth's Moon
  4. Lakes of molten sulfur
  5. Massive lava flows

Ganymede

  1. Moon of Jupiter
  2. Large-scale tectonics
  3. Magnetic Field
  4. Largest moon in the solar system

Europa, Io, and Ganymede are in a 1:2:4 orbital resonance.

Callisto

  1. Moon of Jupiter
  2. Battered Surface
  3. No tidal heating
  4. Very old surface terrain of water, CO2, silicates, and organics

Martian Surface Processes

How do we know that water flowed on the surface of Mars in the past?

A lot of evidence lives in and around the large craters on Mars. We see the crater rims eroded by gullies that appear to have come from running water. We see inflow channels into the craters, and outflow channels out of the craters.

Additionally, we see water valleys between lakes, and patterns on the floors of craters that are characteristic of sediment deposition.

We also see layering in rocks, indentations, and spherical shapes suggesting formation from sediment in water.

What evidence is there to suggest that liquid water occasionally still flows on the Martian surface?

We see Recurring Slope Linea (RSL). These are features that return each summer, linear patterns striping down crater walls. These are likely from melting ground ice which finds an outlet and runs downhill. Where do you think it's most likely to find RSL's?.

What are the biggest differences between the southern and northern hemispheres of Mars?

The northern hemisphere is younger (less cratered) and lower (less elevation). This is likely due to a very large impact in the past, which left a crater the size of the northern hemisphere of Mars.

What other processes have been observed on Mars?

We've seen dust devils, global and regional dust storms, annual polar caps of CO2, permanent polar caps of water ice, polar hoods of CO2.

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