What are sunspots? Did you know that sunspots occur as a part of a cycle? The dark spots are extremely hot, but they appear to be dark, because they are relatively cool (almost 2,000K cooler) compared to the regions of the photosphere surrounding them. The parts of the spots are called the umbra and the penumbra, just like the darker and lighter parts of the shadows cast during eclipses, and sunspots can be larger than the Earth! However, these spots are not shadows. They are brighter than the a full moon appears. The Sun rotates differentially, meaning that part of it rotates faster than the rest, and this sort of rotation seems to wrap its magnetic field up like a cord (according to the Babcock model). Every 11 years, this causes the magnetic poles of the Sun to flip, meaning that its north pole becomes its south pole, and its south pole becomes its north pole. The surface of the sun has a "magnetic carpet" of looped magnetic fields, and as the surface rotates, turbulence whips up the gas and giant loops of ionized gases get flung out, and sunspots are what we observe of these disturbances. This is why sunspots always come in pairs, and they are the north and south "poles" of these loops. Every cycle, there are anywhere from just a few to over 100 sunspots, and the Maunder butterfly diagram which portrays the pattern the sunspots make is aptly named. These magnetic disturbances can be measured using the Zeeman effect, where an atom being passed through a magnetic field is able to absorb multiple wavelengths of photons, each representing the strength of the magnetic field. The magnetic field of a sunspot can be a few thousand times that of Earth's, and the cool temperature is thought to be caused because the fields may slow down the gas, which in turn slows down local convection. The heat is then thought to be deflected around the sunspot since infrared images also detect greater heat surrounding these spots. Interestingly enough, since we were discussing ice ages in class today, there was an extremely low record of sunspot activity in the sixteen and seventeen hundreds which coincided with a "mini ice age" in the northern hemisphere, which may or may not spell causation, but it is still compelling. We still do not understand the Sun's magnetic cycle, and it is something I would like to study. Some other magnetic phenomena of the sun include: prominences, which are giant loops of ionized gas trapped inside a magnetic arch that go through the photosphere, chromosphere, and even the lower corona; solar flares, which are strong enough to affect Earth's magnetic field and even disrupt navigation systems; and coronal holes, which is when magnetic loops break and do not reconnect with the Sun. This solar magnetic activity also causes auroras on Earth. Many other stars have starspots, but they are too far away to see. Through spectroscopic observations, however, we are able to study them and their magnetic cycles. By the way, I got all of this information from Foundations of Astronomy by Michael A. Seeds.
"All cannot live on the piazza, but everyone may enjoy the sun." -Italian proverb
Friday, March 15, 2013
Nuclear Fusion
As we have learned, stars manufacture their energy through nuclear fusion. According to Modern Physics by Serway, Moses, and Moyer, nuclear fusion occurs "[w]hen two light nuclei combine to form a heavier nucleus," and is both a confirmation and consequence of special relativity. It is very thought-provoking that "the total rest mass of the products is less than that of the reactants" in a reaction, but this fact and the amount of energy released during nuclear fusion demonstrate Einstein's mass-energy equivalence. This has to do with the binding energy, and the amount depends on the atomic mass. The peak of the function of binding energy (pictured below) occurs at the mass number of iron, which means we cannot obtain elements "heavier" than iron through fusion.
Our textbook, An Introduction to Modern Astrophysics by Carroll and Ostlie, tells us that the likelihood of fusion depends on the kinetic energy of the collision. The peak from the graph below corresponds to the collision of two protons at the central temperature of the Sun and depends on a very narrow range of temperature of the gas and the charges and masses of the reactants involved. This makes fusion very difficult to achieve here on Earth...
Nevertheless, physicists are trying to find a way to convert water into deuterium and use it as fuel for controlled nuclear fusion. It seems impossible to even be able to originate nuclear fusion, let alone finding a way to control and then harness the staggering amount of energy it would create, and I don't like the fact that we would be using the most precious resource we have, but given our current energy crisis and the pursuit of cleaner, sustainable energy, it seems like at least a hopeful future candidate for a source of alternative fuel. On the plus side, the process isn't radioactive. Oh, and it isn't impossible. Click here to see how a 14-YEAR-OLD MADE A NUCLEAR FUSION REACTOR.
How a nuclear fusion reactor works: (according to howstuffworks.com)
In order to achieve the high temperatures and pressures needed for nuclear fusion, reactors will either use magnetic confinement, which uses electromagnetism to heat and compress hydrogen plasma (which is the process France's International Thermonuclear Experimental Reactor (ITER) will be using) or inertial confinement, which uses lasers or ion beams to do the same thing. In order to get the hydrogen plasma, hydrogen gas streams will be heated by accelerators and then compressed by super magnets. Not many people at school know this, but I was an aviation electrician/mechanic in the Marine Corps, and one of my first licenses was my universal technician's certification to work with air conditioners. I'm telling you this because this nuclear reactor looks everything like an air conditioner to me:
See what I mean? The way that the reactor generates power is just like almost any other power plant, once the heat has been produced, by either process I mentioned before. The heat is used to turn water into steam which turns a generator. The simplicity of this design is beautiful.
I encourage you to research this on your own, and if you do, please comment below which method you favor. I like the magnetic confinement process, personally. The inertial confinement setup is much more complicated, and it takes 192 laser beams to work!
Also, click here to educate yourself about California's Renewable Energy Transmission Initiative (RETI)!
(P.S. happy belated birthday, Albert! 3.14)
Our textbook, An Introduction to Modern Astrophysics by Carroll and Ostlie, tells us that the likelihood of fusion depends on the kinetic energy of the collision. The peak from the graph below corresponds to the collision of two protons at the central temperature of the Sun and depends on a very narrow range of temperature of the gas and the charges and masses of the reactants involved. This makes fusion very difficult to achieve here on Earth...
Nevertheless, physicists are trying to find a way to convert water into deuterium and use it as fuel for controlled nuclear fusion. It seems impossible to even be able to originate nuclear fusion, let alone finding a way to control and then harness the staggering amount of energy it would create, and I don't like the fact that we would be using the most precious resource we have, but given our current energy crisis and the pursuit of cleaner, sustainable energy, it seems like at least a hopeful future candidate for a source of alternative fuel. On the plus side, the process isn't radioactive. Oh, and it isn't impossible. Click here to see how a 14-YEAR-OLD MADE A NUCLEAR FUSION REACTOR.
How a nuclear fusion reactor works: (according to howstuffworks.com)
In order to achieve the high temperatures and pressures needed for nuclear fusion, reactors will either use magnetic confinement, which uses electromagnetism to heat and compress hydrogen plasma (which is the process France's International Thermonuclear Experimental Reactor (ITER) will be using) or inertial confinement, which uses lasers or ion beams to do the same thing. In order to get the hydrogen plasma, hydrogen gas streams will be heated by accelerators and then compressed by super magnets. Not many people at school know this, but I was an aviation electrician/mechanic in the Marine Corps, and one of my first licenses was my universal technician's certification to work with air conditioners. I'm telling you this because this nuclear reactor looks everything like an air conditioner to me:
I encourage you to research this on your own, and if you do, please comment below which method you favor. I like the magnetic confinement process, personally. The inertial confinement setup is much more complicated, and it takes 192 laser beams to work!
Also, click here to educate yourself about California's Renewable Energy Transmission Initiative (RETI)!
(P.S. happy belated birthday, Albert! 3.14)
Class Summary
Very much of what we have discussed in class can be summed up in the Hertzsprung-Russell (H-R) diagram...
Some of the topics we discussed in class do not belong in the H-R diagram, but many do, and I will point them out along the way. We first discussed the celestial sphere, which is an inaccurate model, but useful nonetheless. It changes slowly enough that we can use it as a map and a calendar. (I find this an interesting, albeit rudimentary combination of space and time.) We also learned a bit of history; about other sets of orientations that astronomers use, such as the equatorial coordinates; and phenomena such as precession and the tilt of Earth's axis, which dictate which star is our "North star" and which season we are experiencing, and all of these concepts are fundamental to guide our observations.
However, measurements which appear on the H-R diagram such as the distances to and absolute magnitudes of stars cannot be obtained without celestial mechanics, which was the next chapter we covered. Kepler's laws of planetary motion, proper motion, Newton's laws, and stellar parallax gave us the tools we needed to make our first plots on the H-R diagram. Aristarchus was an astronomer in ancient Greece, and he was able to calculate the distance to the sun using geometry and solar parallax, but once we were able to understand the kinematics of our solar system, Christiaan Huygens and Giovanni Cassini were able to calculate the distance to the sun much more accurately using Venus, Mars, and stellar parallax. Since most stars are so far away, stellar parallax is extremely difficult to detect and was kind of a "missing link" for astronomers. For example, it provided proof that Earth moves and gave us a sort of measuring stick, but although its conceptual implications were vast, it was only marginally helpful to us mathematically. Even so, from the distance and apparent magnitude, the absolute magnitude of stars can be found, and the luminosity and size and mass may also be calculated using a bit of kinematics and studying binary systems as well.
Another breakthrough in being able to take measurements was the invention of quantum mechanics. Understanding blackbody radiation enabled us to add color and temperature to our H-R diagram! Kirchhoff's laws helped us to understand the temperature a little better, and they also helped us to understand the density of gases in and around stars. Stellar spectra gave us information about the composition, surface composition, temperature, and size of the stars, and even their radial velocities! Spectral series, absorption and emission features, the photoelectric effect, and equations from Maxwell, Boltzmann, and Saha gave us even more insight into the stars, and this information is represented on the H-R diagram by stellar classification (OBAFGKMLTY).
Radiative transport helped us to learn more about the energies, differential temperatures, optical depths, random walk paths, and opacities of stars, which actually gave us information about the stars' ionizations, cores, radiative and convective zones, and overall structures. Using these energies along with densities, competing pressures, and sizes of stars and conservation of mass, we were able to understand the composition even more intricately. Understanding how stars generate their power through fusion opened the door to understanding their life cycles, and their impressive temperature dependences are evident on the H-R diagram.
Finally, returning to the Virial theorem which we were introduced to in the first half of the quarter, we were able to explain how and why stars form. Using the initial mass function, the circle of life of stars can be traced on the H-R diagram. All true stars (those which generate their fuel through hydrogen fusion) begin their zero-age on the main sequence. Smaller stars "burn" their fuel at conservative rates, and it burns so uniformly and completely that they never develop a hydrogen shell and never become giants. Since they do lose some mass, they slide down the main sequence a little, and only leave once there is no longer any hydrogen fusion in their cores. Since they are not massive enough to ignite the helium they spent their lifetimes creating, they simply die unremarkable deaths. Medium mass stars and stars with greater mass experience much more eventful lives, and the crazier the shorter. These stars move all over the H-R diagram! Once the hydrogen in their cores is exhausted, the stars condense and may leave the main sequence. At this point, less massive stars experience a helium flash, and more massive stars become more degenerate. Then the triple alpha process takes over, which is the fusion of helium, and the stars move up and to the right on the H-R diagram. If the stars are massive enough, they will begin fusing carbon. The most massive stars will fuse all the way up to iron. Due to the Virial theorem, there will be thermal pulsing as the stars expand and contract, each time throwing off mass and energy. All the while, these stars are migrating around the H-R diagram. This is the general lifecycle for these types of stars, and depending on their masses, they will either become white dwarfs and settle down at the lower left corner of the H-R diagram, neutron stars, or black holes. Companion stars transfer their lost mass to each other and even go through multiple novas!
Using the initial mass function and the H-R diagram, we can get a good idea of the measurements we can't take directly or even indirectly. (Stars tend to be guilty by association.)
Of course, we also learned about telescopes and interferometry, exoplanets, and even global warming. The next step is to take what we've learned and apply it...on the final! (And in life.) Good luck, everyone, and don't forget to do your part to minimize your carbon foot print!
Some of the topics we discussed in class do not belong in the H-R diagram, but many do, and I will point them out along the way. We first discussed the celestial sphere, which is an inaccurate model, but useful nonetheless. It changes slowly enough that we can use it as a map and a calendar. (I find this an interesting, albeit rudimentary combination of space and time.) We also learned a bit of history; about other sets of orientations that astronomers use, such as the equatorial coordinates; and phenomena such as precession and the tilt of Earth's axis, which dictate which star is our "North star" and which season we are experiencing, and all of these concepts are fundamental to guide our observations.
However, measurements which appear on the H-R diagram such as the distances to and absolute magnitudes of stars cannot be obtained without celestial mechanics, which was the next chapter we covered. Kepler's laws of planetary motion, proper motion, Newton's laws, and stellar parallax gave us the tools we needed to make our first plots on the H-R diagram. Aristarchus was an astronomer in ancient Greece, and he was able to calculate the distance to the sun using geometry and solar parallax, but once we were able to understand the kinematics of our solar system, Christiaan Huygens and Giovanni Cassini were able to calculate the distance to the sun much more accurately using Venus, Mars, and stellar parallax. Since most stars are so far away, stellar parallax is extremely difficult to detect and was kind of a "missing link" for astronomers. For example, it provided proof that Earth moves and gave us a sort of measuring stick, but although its conceptual implications were vast, it was only marginally helpful to us mathematically. Even so, from the distance and apparent magnitude, the absolute magnitude of stars can be found, and the luminosity and size and mass may also be calculated using a bit of kinematics and studying binary systems as well.
Another breakthrough in being able to take measurements was the invention of quantum mechanics. Understanding blackbody radiation enabled us to add color and temperature to our H-R diagram! Kirchhoff's laws helped us to understand the temperature a little better, and they also helped us to understand the density of gases in and around stars. Stellar spectra gave us information about the composition, surface composition, temperature, and size of the stars, and even their radial velocities! Spectral series, absorption and emission features, the photoelectric effect, and equations from Maxwell, Boltzmann, and Saha gave us even more insight into the stars, and this information is represented on the H-R diagram by stellar classification (OBAFGKMLTY).
Radiative transport helped us to learn more about the energies, differential temperatures, optical depths, random walk paths, and opacities of stars, which actually gave us information about the stars' ionizations, cores, radiative and convective zones, and overall structures. Using these energies along with densities, competing pressures, and sizes of stars and conservation of mass, we were able to understand the composition even more intricately. Understanding how stars generate their power through fusion opened the door to understanding their life cycles, and their impressive temperature dependences are evident on the H-R diagram.
Finally, returning to the Virial theorem which we were introduced to in the first half of the quarter, we were able to explain how and why stars form. Using the initial mass function, the circle of life of stars can be traced on the H-R diagram. All true stars (those which generate their fuel through hydrogen fusion) begin their zero-age on the main sequence. Smaller stars "burn" their fuel at conservative rates, and it burns so uniformly and completely that they never develop a hydrogen shell and never become giants. Since they do lose some mass, they slide down the main sequence a little, and only leave once there is no longer any hydrogen fusion in their cores. Since they are not massive enough to ignite the helium they spent their lifetimes creating, they simply die unremarkable deaths. Medium mass stars and stars with greater mass experience much more eventful lives, and the crazier the shorter. These stars move all over the H-R diagram! Once the hydrogen in their cores is exhausted, the stars condense and may leave the main sequence. At this point, less massive stars experience a helium flash, and more massive stars become more degenerate. Then the triple alpha process takes over, which is the fusion of helium, and the stars move up and to the right on the H-R diagram. If the stars are massive enough, they will begin fusing carbon. The most massive stars will fuse all the way up to iron. Due to the Virial theorem, there will be thermal pulsing as the stars expand and contract, each time throwing off mass and energy. All the while, these stars are migrating around the H-R diagram. This is the general lifecycle for these types of stars, and depending on their masses, they will either become white dwarfs and settle down at the lower left corner of the H-R diagram, neutron stars, or black holes. Companion stars transfer their lost mass to each other and even go through multiple novas!
Using the initial mass function and the H-R diagram, we can get a good idea of the measurements we can't take directly or even indirectly. (Stars tend to be guilty by association.)
Of course, we also learned about telescopes and interferometry, exoplanets, and even global warming. The next step is to take what we've learned and apply it...on the final! (And in life.) Good luck, everyone, and don't forget to do your part to minimize your carbon foot print!
"We are from another generation of stars before the sun." -Professor Siana
10% of my final grade from Astronomy 1B. :)
(I don't know if I need to cite this, but I used my Astronomy 1A and 1B notes, homework, and test answers for some of this post.)
Thursday, March 14, 2013
The Ending Has Been Written
In case you were wondering about how it's all going to end, here's your "confirmation."
(See Entropy Statistics.)
(See Entropy Statistics.)
Saturday, March 9, 2013
My Sundial and My Compass
When I was in Qatar six or seven years ago, I bought this brass sundial/compass for my grandpa, because he was a Lieutenant Junior Grade in the Navy and drove a pt boat (pictured below) during World War II. He survived the war (mostly because it ended just as he arrived in Manila), but sadly, the reason I have this sundial is because he passed away a couple years ago, and my grandma returned it to me. Then she joined him in heaven only weeks after he passed. I'm very glad that I got to know my grandparents, and that I got to see them both and say goodbye before they left this earth. They were my moral compass when I was growing up, and I hope to pass on what I learned from them to my children and grandchildren. (I have so many pictures of both of my grandparents that I wanted to post, but I thought these two of my grandpa and his pt would be the most relevant for this post.)
selvbiografi
By the way, I wrote a book a few years ago.
I thought I'd post some excerpts from the book after writing about the harmony of our solar system...
The Conductor
He steps onto the floor unnoticed,
Pauses for a moment, then
With humble dignity he takes
His place on the great stage.
The orchestra is silent now,
Though moments gone they were bedlam,
And now so void of movement that
You hear him turn the page.
As though to his fingers attached
Are notes and treble clefs and staffs
The players keep their eyes on them
As every song is played.
Their sheets of music are useless
As he commands the tubas play
Yet picks out delicate notes and
Then sends them up the harpists' way.
Though he has only two hands and
One baton vice their multitude,
Each player knows exactly what
The conductor conveys.
And even in the audience,
And though his back is turned to you,
You feel his passion, though you can
Deny it, as you may.
(As a side note for coincidence's sake, Molchanov cast Jupiter as the role of conductor of this orchestra, and as a Sagittarius, Jupiter is my ruling planet. Also, this poem is found on page 23 of my book, and I was born on the 23rd of November. However, those are only coincidences. I wrote The Conductor for my brother, who shares a birthday with Galilei Galileo, and is therefore an Aquarius and ruled by Uranus.)
Wonder
The purest form of innocence
Explores the world wide-eyed
Bare-bottomed little child carefree
She has nothing to hide
Each new day brings discovery
The sky is limitless
What keeps it from cascading down?
Becomes a noble quest
If the answer cannot be found
A fable takes its place
While drifting clouds do fill her courts
Each presenting his case
Those Who Seek Still Find
I've looked for wisdom everywhere
To the ends of the Earth
Although my toes touched seawater
I continued my search
Tallest mountain overtaken
And oldest tree studied
I held my ear up to the wind
To hear who set it free
I've found something strong that was made
From something delicate
A rose, though fragile, produces
A most powerful scent
The terrible engulfing fire
Is honest though it burns
And many things in life are fair
Gravity works, Earth turns
My hunt led me to Orion
And I whispered to him
How close am I in my pursuit?
Can you see truth's footprints?
He met my query with silence
As I had known he would
A storyteller in the square
Did me a deal more good
"A thief approached a sleeping town
As did the dawning sun
The stranger forfeited the race
And so the morning won..."
"Sometimes," he told me, "we must learn
"Not just how far to go
"But also when we should give up
"It's important to know."
I still seek out wisdom today
But don't travel as far
It is not required of those
With a discerning heart
Solarium
How does the delicate clear glass
Support the universe,
And not the rigid beams of brass
Like seams, when pressured, burst?
A woman now knows the answer
To this simple question
But long ago a child entered
The great Solarium
"Who left you diamonds on the roof?"
She asked constellations
They had no voice, but spoke of truth
Found in Solariums
One day a sparrow hit a wall,
Wanting inside the room
The girl cried as she watched it fall
Death by Solarium
A kiss was sheltered from the wind
But not protected from
Two wounded wet eyes looking in
The bright Solarium
Beyond a secret garden hedge
Down an old statue comes
Not everything can be witnessed
Inside Solariums
Green plants do grow rather nicely
When someone sings to them
A nursery quite becoming
Of a Solarium
How many scholars studied there,
Where curious minds run?
A healthy knowledge fills the air
In the Solarium
And mirrored in the crystal plane
Captive her reflection
Old soul tonight with moon will wane
Goodbye Solarium
Musical Spheres
Fellow classmate and blogger The Mad Physi gave me a beautiful gift this quarter: a book called QUADRIVIUM.
This book is actually a collection of four ancient books, partly written during the time of Pythagorus, and have been studied by Cassiodonus, Philolaus, Timaeus, Archytus, Plato, Aristotle, Eudemus, Euclid, Cicero, Philo the Jew, Nichomachus, St. Clement of Alexandria, St. Origen, Plotinus, Dionysius the Areopagite, Bede, Alcuin, Al-Khwarizmi, Al-Kindi, Eriugena, Gerbert d'Aurillac, the Bretheren of Purity, Fulbert, Ibn Stina (Avicenna), Hugo of St. Victor, Bernardus Silvestris, Bernard of Clairvaux, Hildegard of Bingen, Alanus ab Insulis, Joachim of Fiore, Ibn Arabi, Grosseteste ("the great English scientist"), Roger Bacon, Thomas Aquinas, Dante, and Kepler. (Talk about name dropping.)
In Book IV, harmonics, scales, chord progression and more are discussed in the most fascinating way, with amazing visual representations of harmony through the use of a harmonograph (speaking of which, I am going to build one, and you can, too!) There is a section in this book entitled The Music of the Spheres, and the spheres it discusses are the planets of our solar system. I've heard of a more general concept like this before, more of a symphony of the entire universe, and I would like to share what the book had to say about "planets playing in tune" and some more recent developments on the subject...
As you may know, Kepler studied the motion of the planets. During his study, he wrote Harmoniae Mundi, or "harmony of the world," in which he compared the planets' angular velocities with harmonics. He set about calculating these harmonies based on the association previously given the seven known "planets" (including Earth's moon) with the seven musical notes. Below is a photo from the book showing the comparison between the ancient system and Kepler's interpretation.
The pentagons inscribed within the
(at the time this book was written) known orbits of Mercury and Venus and by Earth and Mars (pictured at right) are used in the book to prove the harmony of the worlds. Book II is all about geometry, and it was highly regarded during the time these books were written and studied; therefore, any explanation using geometry was not only a good argument, but it was actually regarded as being sacred.
The illustration below depicts two models which show how the planets were believed to orbit and sadly reveal a source of discord in the ancient sheet music...
Heavenly Harmony and Earthly Harmonics, a paper from the Quarterly Journal of the Royal Astronomical Society, tells us that the ancient Greeks, including Plato and Aristotle, believed in the "harmony of the spheres," and that the planets played music as they swept through space. Throughout the Middle Ages, this belief persisted, with the romanticized piety of notion that this music was the planets themselves praising God. In 1596, Kepler wrote Mysterium Cosmographicum, within a couple years after Shakespeare wrote about the "harmony of heaven" in The Merchant of Venice. In Mysterium, Kepler used simple geometry to provide a logical explanation for this accepted harmony. I assume this is the same geometry as I showed you earlier with the pentagons. However, although most of his calculations were astonishingly accurate, they were based on the assumptions that there were only six planets, as you saw in the inaccurate illustration of the motion of the planets.
Later work by less famous Molchanov cast Jupiter as the "conductor" of this orchestra, and included the entire solar system. His "commensurability" concept which uses linear equations of the planets' orbits and their frequencies to explain their resonance is much more accurate, and much more rigorous. Here are the linear equations of the frequencies, where m denotes frequency:
.
If you found this interesting, be sure to check out what my fellow classmate wrote about galactic geometry on her blog, The Mad Physi, and read what my other classmate wrote about the physics behind music in his blog, Up and Atom! Also, here is a video of a harmonograph in action!
In Book IV, harmonics, scales, chord progression and more are discussed in the most fascinating way, with amazing visual representations of harmony through the use of a harmonograph (speaking of which, I am going to build one, and you can, too!) There is a section in this book entitled The Music of the Spheres, and the spheres it discusses are the planets of our solar system. I've heard of a more general concept like this before, more of a symphony of the entire universe, and I would like to share what the book had to say about "planets playing in tune" and some more recent developments on the subject...
As you may know, Kepler studied the motion of the planets. During his study, he wrote Harmoniae Mundi, or "harmony of the world," in which he compared the planets' angular velocities with harmonics. He set about calculating these harmonies based on the association previously given the seven known "planets" (including Earth's moon) with the seven musical notes. Below is a photo from the book showing the comparison between the ancient system and Kepler's interpretation.
The pentagons inscribed within the
(at the time this book was written) known orbits of Mercury and Venus and by Earth and Mars (pictured at right) are used in the book to prove the harmony of the worlds. Book II is all about geometry, and it was highly regarded during the time these books were written and studied; therefore, any explanation using geometry was not only a good argument, but it was actually regarded as being sacred.
The illustration below depicts two models which show how the planets were believed to orbit and sadly reveal a source of discord in the ancient sheet music...
Heavenly Harmony and Earthly Harmonics, a paper from the Quarterly Journal of the Royal Astronomical Society, tells us that the ancient Greeks, including Plato and Aristotle, believed in the "harmony of the spheres," and that the planets played music as they swept through space. Throughout the Middle Ages, this belief persisted, with the romanticized piety of notion that this music was the planets themselves praising God. In 1596, Kepler wrote Mysterium Cosmographicum, within a couple years after Shakespeare wrote about the "harmony of heaven" in The Merchant of Venice. In Mysterium, Kepler used simple geometry to provide a logical explanation for this accepted harmony. I assume this is the same geometry as I showed you earlier with the pentagons. However, although most of his calculations were astonishingly accurate, they were based on the assumptions that there were only six planets, as you saw in the inaccurate illustration of the motion of the planets.
Later work by less famous Molchanov cast Jupiter as the "conductor" of this orchestra, and included the entire solar system. His "commensurability" concept which uses linear equations of the planets' orbits and their frequencies to explain their resonance is much more accurate, and much more rigorous. Here are the linear equations of the frequencies, where m denotes frequency:
In every case, he is off in his calculations by less than a percent, and his second equation gets replaced by
.
Also, not only do these equations work, there are also accurate equations for the planets' moons. Computer systems are currently being programmed and tested in order to better understand these resonances. These models have not found fault with the numerology of Molchanov, meaning that there really is a harmony to the solar system. The frequency ratios of arbitrary solar systems in these computer models 2, 9/4 to 7/3, and 5/2 are favored, and these are representative of our solar system.
However, there is also discord predicted in the symphony of all solar systems. Just as in a poorly engineered bridge, improper resonances lead to the destruction of weak celestial instruments: inadequately small masses and or playing out of "key" (with the wrong frequencies), according to the computer models. Smaller masses with unstable orbits will either be "swallowed up or smashed." (I imagine the cacophony of someone banging, disharmoniously, on the low keys of a piano at this moment.) Hopefully, this already happened in our solar system, or it won't affect us while we are living in this beautiful evolution of harmony.
If you found this interesting, be sure to check out what my fellow classmate wrote about galactic geometry on her blog, The Mad Physi, and read what my other classmate wrote about the physics behind music in his blog, Up and Atom! Also, here is a video of a harmonograph in action!
Sunday, March 3, 2013
Fiat Lux
Why can't we see past the sun's photosphere?
We learned in class that optical depth depends on the opacity of a gas, and that opacity is determined by energy, essentially, whether it be from bound-bound transitions of elements in atoms, bound-free or free-free absorption, or from electron scattering. Furthermore, we learned about thermodynamic equilibrium, specifically, local thermodynamic equilibrium (LTE) and the "random walk" that photons take on their journey to freedom. If the path of the average length of one of these little strolls is much shorter than the distance over which the temperature changes significantly, the photons experience LTE, meaning they will be quite comfortable and needn't dress in massless layers or bother bringing a massless coat. In fact, they needn't wear anything at all, for they won't even be seen. They will be traveling under the cloak of opacity. This occurs in the interior of stars where the temperature is so high that photons can get into their birthday suits without worrying about peeping tom's engaging in voyeurism due to the fact that all of the atoms will be ionized, and there will be no wavelengths which permit transparency from a bound-bound transition, for example. (As of now, scientists have not been able to disprove the fact that photons living inside stars have formed nudist colonies, and that they simply don't want to be seen.)
But at the photosphere, these jaybirds exchange their safe LTE, which only depends on single parameter temperature, for a more risqué set of conditions, which includes many more degrees of freedom and greatly increases their chances of being seen. According to Harvard's "Breakdown of Local Thermodynamic Equilibrium" by George W. Collins, II, Maxwell-Boltzmann statistics and the Saha-Boltzmann ionization-excitation are sufficient to describe the distribution of energy levels, and that these levels will be constant under LTE. However, Collins states that in the upper atmosphere of stars, the density is not enough to support LTE, and here we must use different equations to lure our little photon friends from their primitive way of life and get them to realize their full potential in the form of useful light. Okay, so I'm paraphrasing. However, Collins does discuss some of the "phenomena which produce departure from Local Thermodynamic Equilibrium": the principle of detailed balancing, interlocking, and collisional ionization and photoionization.
In detailed balancing, every process, such as absorption and emission, must be balanced, and the jumps in energy levels must also balanced. In the upper atmospheres of stars, the rate of energy emitted per wavelength through atomic collisions is not able to be represented by the Planck function, and since this is not possible under thermal equilibrium, the fact that these collisions are able to occur are evidence of a breakdown in LTE. The example Collins gives is when transition from the first to the third energy level is more favorable than a transition to the second energy level in a hypothetical atom which only has three energy levels, and the fact that the transitions back down to the second and first energy levels will not be balanced.
Interlocking is when two different absorption lines have the same upper energy level. To be honest, I barely understood detailed balancing, and interlocking is even more confusing to me. Basically, the same breakdown of LTE occurs when there is an imbalance of the cyclical processes as there was in detailed balancing, only in addition, strong lines which are formed under non-LTE conditions appear weaker because they share the highest energy level with weaker lines from inside the star. The example given was that red, singly ionized Calcium lines appear abnormally weak due to the photons given off by their interlocked lines. Also, to summarize both detailed balancing and interlocking, when there are more reactions among photons than there are of particles, there will be a breakdown of LTE. It makes sense that this would happen when there are comparatively less ions, since that means there will be less particles zipping about.
In the uppermost atmospheres of stars, there is still ionization occurring, through collisions and photoionization, although at a greatly reduced rate than that inside the star. And this is very interesting, but not all particles depart from LTE at the same temperature! But this makes sense. Remember that local thermodynamic equilibrium depends on the mean free path, and in a gas, different particles have different cross sections and number densities, therefore different mean free paths. Since electrons are the last particles to remain in thermodynamic equilibrium, when comparing collisions and photoionization, only at high temperatures and densities can there be more reactions among particles than among photons, so it only makes sense that at the cooler, less dense outer atmospheres of stars can photoionization dominate over collisional ionization. In conclusion, this is additional evidence of departure from LTE in the upper atmosphere of most stars.
Although scientists haven't been able to prove that photons aren't little nudists, I'm going to be bold and conjecture that they are not. I pretty much just added that bit to see if anyone was actually reading this.
We learned in class that optical depth depends on the opacity of a gas, and that opacity is determined by energy, essentially, whether it be from bound-bound transitions of elements in atoms, bound-free or free-free absorption, or from electron scattering. Furthermore, we learned about thermodynamic equilibrium, specifically, local thermodynamic equilibrium (LTE) and the "random walk" that photons take on their journey to freedom. If the path of the average length of one of these little strolls is much shorter than the distance over which the temperature changes significantly, the photons experience LTE, meaning they will be quite comfortable and needn't dress in massless layers or bother bringing a massless coat. In fact, they needn't wear anything at all, for they won't even be seen. They will be traveling under the cloak of opacity. This occurs in the interior of stars where the temperature is so high that photons can get into their birthday suits without worrying about peeping tom's engaging in voyeurism due to the fact that all of the atoms will be ionized, and there will be no wavelengths which permit transparency from a bound-bound transition, for example. (As of now, scientists have not been able to disprove the fact that photons living inside stars have formed nudist colonies, and that they simply don't want to be seen.)
But at the photosphere, these jaybirds exchange their safe LTE, which only depends on single parameter temperature, for a more risqué set of conditions, which includes many more degrees of freedom and greatly increases their chances of being seen. According to Harvard's "Breakdown of Local Thermodynamic Equilibrium" by George W. Collins, II, Maxwell-Boltzmann statistics and the Saha-Boltzmann ionization-excitation are sufficient to describe the distribution of energy levels, and that these levels will be constant under LTE. However, Collins states that in the upper atmosphere of stars, the density is not enough to support LTE, and here we must use different equations to lure our little photon friends from their primitive way of life and get them to realize their full potential in the form of useful light. Okay, so I'm paraphrasing. However, Collins does discuss some of the "phenomena which produce departure from Local Thermodynamic Equilibrium": the principle of detailed balancing, interlocking, and collisional ionization and photoionization.
In detailed balancing, every process, such as absorption and emission, must be balanced, and the jumps in energy levels must also balanced. In the upper atmospheres of stars, the rate of energy emitted per wavelength through atomic collisions is not able to be represented by the Planck function, and since this is not possible under thermal equilibrium, the fact that these collisions are able to occur are evidence of a breakdown in LTE. The example Collins gives is when transition from the first to the third energy level is more favorable than a transition to the second energy level in a hypothetical atom which only has three energy levels, and the fact that the transitions back down to the second and first energy levels will not be balanced.
Interlocking is when two different absorption lines have the same upper energy level. To be honest, I barely understood detailed balancing, and interlocking is even more confusing to me. Basically, the same breakdown of LTE occurs when there is an imbalance of the cyclical processes as there was in detailed balancing, only in addition, strong lines which are formed under non-LTE conditions appear weaker because they share the highest energy level with weaker lines from inside the star. The example given was that red, singly ionized Calcium lines appear abnormally weak due to the photons given off by their interlocked lines. Also, to summarize both detailed balancing and interlocking, when there are more reactions among photons than there are of particles, there will be a breakdown of LTE. It makes sense that this would happen when there are comparatively less ions, since that means there will be less particles zipping about.
In the uppermost atmospheres of stars, there is still ionization occurring, through collisions and photoionization, although at a greatly reduced rate than that inside the star. And this is very interesting, but not all particles depart from LTE at the same temperature! But this makes sense. Remember that local thermodynamic equilibrium depends on the mean free path, and in a gas, different particles have different cross sections and number densities, therefore different mean free paths. Since electrons are the last particles to remain in thermodynamic equilibrium, when comparing collisions and photoionization, only at high temperatures and densities can there be more reactions among particles than among photons, so it only makes sense that at the cooler, less dense outer atmospheres of stars can photoionization dominate over collisional ionization. In conclusion, this is additional evidence of departure from LTE in the upper atmosphere of most stars.
Although scientists haven't been able to prove that photons aren't little nudists, I'm going to be bold and conjecture that they are not. I pretty much just added that bit to see if anyone was actually reading this.
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