Posts tagged: order of magnitude
There will be an annular solar eclipse taking place this evening (5/20/2012) viewable from the western parts of the US and eastern Asia.
This is what happens when the Moon blocks some of the light from the Sun by passing in front of it from our vantage point on Earth. It is an annular eclipse and not a total eclipse because the Moon won’t completely cover up the Sun in the sky as it does during a total solar eclipse. Solar eclipses happen in these two varieties because the distance from the Earth to the Moon varies during its orbit around Earth. Therefore, the apparent size of the Moon looks different depending on how close the Moon is to Earth, and will only cover up as much of the Sun as it can. In fact, the Moon happens to be near its furthest point now.
It is interesting to note how close the Moon comes to completely covering up the Sun, and how this depends on the geometry of the situation.
During a total solar eclipse, the Moon will cover up the Sun almost exactly without much overlap. This happens because the apparent size of the Moon as viewed from Earth is nearly the same as the apparent size of the Sun as viewed from Earth.
This can be measured by comparing the ratio of the Sun’s radius and its distance to Earth to the ratio of the Moon’s radius and its distance to Earth. According to my own calculations they only differ by factor of about 3% of the Moon ratio. This is justified since the the Moon happens to be about 400 times smaller than the Sun, but is also about 400 times closer. Bigger differences in these ratios would imply that the Moon looks smaller or larger than the Sun during an eclipse. This closeness is equivalent to the claim that the two right triangles drawn in this diagram are close to being similar.
Is there any physical reason for why these ratios are the way they are? It seems plausible that the Moon could have had a different size, and orbited a little closer or further away from Earth thereby preventing such a ‘perfect’ total eclipse from happening. The configurations we witness almost seem like some coincidence!
So what? Is there any value to this special orientation during an eclipse?
Actually, the perfect total eclipses we are lucky enough to experience are valuable opportunities for astronomical observation.
In 1919, Arthur Eddington observed a total solar eclipse and was able to experimentally verify the phenomenon of gravitational lensing—one of the theoretical predictions of general relativity, which involved the bending of distant star light due to the Sun’s gravity. In addition, these perfect total eclipses also allow for other observations of solar phenomenon.
If it were not for this ecliptic coincidence and things were any different, then how much more difficult would it have been for scientists to learn about these other astrophysical phenomenon?
Here is another way to justify how humans are “central” in this universe.
Consider the range in size of what is physically relevant in our universe. This range spans about 60 orders of magnitude (1060) from the size of the visible universe at about 13.7 billion light years (~1028 cm) all the way down to the Planck length, which is the smallest scale in which notions of size and distance essentially breakdown according to standard theories, at ~10-33 cm.
As humans, at about 102 cm in size, we happen to exist near the middle of this range. This really is no coincidence, nor is it some kind of trickery with units of measurement.
It can be argued that if humans were any smaller our brains wouldn’t have room to develop their complexity allowing for our intelligence, and if we were any bigger our brains would lose their practical efficiency which depends on the brain’s ability to interact with itself. In fact, this must be true for all intelligent life.
So if you ever happen to feel down about your size relative to the rest of the universe, know that you are much more significant than any other structure in this universe exactly the way you are. Even David Deutsch once wrote: “The size of the universe is no more depressing than the size of a cow.”
However, it does seem rather peculiar that we humans exist and are making these observations of size at the point in time when the Universe has expanded to its current size, which just so happens to place us in the middle of this size range. Couldn’t this have been any different?
On the other end of things, one may even be willing to pass the size scale set by the Plank length as being mere coincidence, since the fundamental constants which determine its value could have possibly taken on different values themselves making things very different.
If for some reason the existential risks mentioned in the last post made you feel insignificant in anyway, let the following serve as consolation and reason to feel significant (and quite literally “central”) to the universe as human beings existing in this period of time.
From a temporal perspective of the evolution of the universe, we live close to a point in time where the universe is switching over from slowing down its expansion to speeding up its expansion. This is important from an observational standpoint, because as the expansion continues distant galaxies and regions of the universe will be disappearing from sight leaving the Milky Way and some of its closest neighbors in isolation. After a sufficient amount of time, light will no longer be able to reach us in ways that permit practical observation of these distant regions. This also includes light that gives evidence of the “big bang” via the cosmic background radiation. Thus, new astronomers in the future may not even be able to observe other galaxies to convince them that they live in a universe with many others. It is also somewhat peculiar that we exist in a period of time where the expansion is neither too weak nor too strong to even detect in the first place. In fact, this is essentially the only time period in the universe’s evolution where a detection of this expansion and the existence of dark energy could be possible! [Read more about this here]
Moving from this larger cosmic scale to our Solar System, there are still more ways we are living in a central moment in time. Within an order of magnitude, we exist halfway through the main life of our Solar System, which began about 4.5 billion years ago and will end in 5-6 billion years when the Sun turns into a red giant and ultimately a white dwarf.
Moreover, we also exist about halfway through the period of time on Earth which permits the survival of life like us. In the past, until about half a billion years ago, the Sun was too faint and the Earth lacked an oxygen-rich atmosphere. In the next half a billion years, the Sun’s luminosity will be increasing causing temperatures on Earth to rise high enough to where the Earth’s oceans will evaporate away and the atmosphere will escape into outer space. This stage will have some drastic consequence’s for Earth in the future, but goes to show that we are currently in the most suitable period for life on Earth.
Relative to the more recent evolution of humans on Earth, there is one more way we are in a pivotal moment in time. We are at the end of an exponential expansion in human population. In the past century, the human population quadrupled, and doubled twice in just the last 100 years. This is the first time this has ever happened in history, and will likely be the last since there are strong doubts that Earth could even handle such a large capacity.
Whether you want to regard these as mere coincidences, or formulate some kind of anthropic arguments as justification, know that this all really happens to be the case. So while we continue to Tumbl, perhaps we can let this increase our awareness and inspire us to live accordingly.
Most of the ideas mentioned here were brought to my own awareness during a course offered at UCSC, called Cosmology and Culture, taught by Joel Primack and Nancy Abrams. This course has resulted in their informative book, The View from the Center of the Universe, which explores these ideas amongst others more thoroughly.
Compiled below is a selection of estimated dates for some events given certain assumptions in the evolution of Earth, the Solar System, and the Universe. Most events are of an astronomical and cosmological nature though some are geological. A more complete list from which the ones included here were taken can be found on Wikipedia.
- In 10,000 years - The end of humanity, according to Brandon Carter’s Doomsday argument, which assumes that half of the humans who will ever have lived have already been born.[3]
- In 50,000 years - Niagara Falls erodes away the remaining 20 miles to Lake Erie and ceases to exist.[6]
- In 500,000 years - By this time Earth will have likely been impacted by a meteorite of roughly 1 km in diameter.[9]
- In 1 million years - Highest estimated time until the red supergiant star Betelgeuse explodes in a supernova. The explosion is expected to be easily visible in daylight.[10][11]
- In 50 million years - The Californian coast begins to be subducted into the Aleutian Trench[15]
Africa will have collided with Eurasia, closing the Mediterranean Basin and creating a mountain range similar to the Himalayas.[16]
- In ~240 million years - From its present position, the Solar System will have completed one full orbit of the Galactic center.[18]
- In 250 million years - All the continents on Earth fuse into a possible new supercontinent.[19][20]
- In 1 billion years - The Sun’s luminosity increases by 10%, causing Earth’s surface temperatures to reach an average of 47°C and the oceans to boil away.[22]
- In 1.5 billion years - The Sun’s circumstellar habitable zone moves outwards as its increased luminosity causes carbon dioxide to increase in Mars’s atmosphere, raising its surface temperature to levels akin to Earth during the ice age.[23]
- In 5.4 billion years - The Sun becomes a red giant.[29]Mercury, Venus and possibly Earth are destroyed.[30]
- In 7 billion years - Collision between the Milky Way and Andromeda galaxies.[32]
- In 14.4 billion years - Sun becomes a black dwarf as its luminosity falls below three trillionths its current level, while its temperature falls to 2239 K, making it invisible to human eyes.[36]
- In 20 billion years - The end of the Universe in the Big Rip scenario.[37] Observations of galaxy cluster speeds by the Chandra X-Ray Observatory suggest that this will not occur.[38]
- In 100 billion years - The Universe’s expansion causes all evidence of the Big Bang to disappear beyond the practical observational limit, rendering cosmology impossible.[41]
- In 1012 (1 trillion) years - Low estimate for the time until star formation ends in galaxies as galaxies are depleted of the gas clouds they need to form stars.[43],
- In 2×1012 (2 trillion) years - All galaxies outside the Local Supercluster are no longer detectable in any way, assuming that dark energy continues to make the Universe expand at an accelerating rate.[44]
- In 1015 (1 quadrillion) years - Estimated time until stellar close encounters detach all planets in the Solar System from their orbits.[43]
By this time, the Sun will have cooled to five degrees above absolute zero.[47]
- In 3×1043 years - Estimated time for all nucleons in the observable Universe to decay, if the proton half-life takes the largest possible value, 1041 years,[43] assuming that the Big Bang was inflationary and that the same process that made baryons predominate over anti-baryons in the early Universe makes protons decay.[53] By this time, if protons do decay, the Black Hole Era, in which black holes are the only remaining celestial objects, begins.[46][43]
- In 1065 years - Assuming that protons do not decay, estimated time for rigid objects like rocks to rearrange their atoms and molecules via quantum tunneling. On this timescale all matter is liquid.[49]
- In 1.7×10106 years - Estimated time until a supermassive black hole with a mass of 20 trillion solar masses decays by the Hawking process.[54] This marks the end of the Black Hole Era. Beyond this time, if protons do decay, the Universe enters the Dark Era, in which all physical objects have decayed to subatomic particles, gradually winding down to their final energy state.[46][43]
- In
years - Estimated time for a Boltzmann brain to appear in the vacuum via a spontaneous entropy decrease.[55] - In
years - Estimated time for random quantum fluctuations to generate a new Big Bang, according to Caroll and Chen.[56] - In
years - Scale of an estimated Poincaré recurrence time for the quantum state of a hypothetical box containing an isolated black hole of stellar mass.[57] This time assumes a statistical model subject to Poincaré recurrence. A much simplified way of thinking about this time is that in a model in which history repeats itself arbitrarily many times due to properties of statistical mechanics, this is the time scale when it will first be somewhat similar (for a reasonable choice of “similar”) to its current state again.
Here is something to think about that was only casually mentioned in passing in the recent video that was posted.
The sunlight you may or may not have experienced today finally managed to reach you after a ~100,000 year long journey since it was originally created at the Sun’s core!
Since the speed of light is finite, about 300,000,000 meters/second (or about 671,000,000 miles/hour), it takes time for it to travel from one point in space to another.
Given that the distance from Earth to the Sun is about 150,000,000,000 meters (about 93,000,000 miles) it takes about 8 minutes for light to reach us!
But this is just the time it takes light to reach us from the surface of the sun.
The light coming from the surface of the Sun is itself created as a by-product of nuclear fusion occurring deep in the Sun’s core.
Once light is created at at the Sun’s core it begins its journey to the surface of the Sun some 700,000,000 meters (430,000 miles) away from the core.
One might assume that this light takes the shortest path and heads straight to the surface, which would only take a couple seconds of travel time.
However, this is not the case because there is all kinds of star stuff that gets in the way.
An actual photon may only travel a mere fraction of a centimeter (anywhere between .01 and .3 centimeters depending on how close it is to the surface) before it makes a collision with other matter thereby diverting its path to some other random direction.
Photons continue moving in these seemingly random trajectories, bumping into other particles along the way, and don’t actually reach the surface until about 100,000 years later (give or take an order of magnitude)!
This kind of behavior characterizing the photons motion is modeled by something called a random walk, and is illustrated in a few different instances in the animations above.
Random walks have widespread applications through out the sciences and mathematics. The idea of random walks are even used in some computer algorithms to allow for more efficient solutions to some problems.
One particular application of personal interest, and a rather abstract generalization of the idea, is the quantum random walk, in which the superposition principle of quantum mechanics is used to put the trajectory into a combination of multiple possible trajectories to assist quantum computers in solving problems. The workings of Grover’s search algorithm can be thought of in this way. This isn’t the only instance that relates quantum mechanics to the workings of the Sun (see here).
Anyway, next time you are out in the relentless light of the Sun you may wonder what was going on some 100,000 years ago when that light first originated in the Sun, or maybe even where you’ll be 100,000 years from now when the light being created in the Sun at this moment finally reaches Earth.
(GIFs created from this Java app)
There is a useful quantum algorithm for quantum computers discovered by Lov Grover which pertains to the problem of searching large unstructured databases. Think of this problem as having to search through a large list or table of entries (like names and phone numbers) trying to find one particular entry. Like the factoring problem, the best a classical computer can do in searching a large unstructured database for a particular item is to proceed by brute force and exhaust all the possibilities. Grover’s search algorithm, though it does not provide an exponential speedup as in the case of factoring large integers, offers a quadratic speedup in searching such databases. However, this still has practical benefits.
Consider a database consisting of N entries that needs to be searched through. It can be shown that the best known classical algorithms, on average, can find a given entry in N/2 computational steps. On the other hand, Grover’s algorithm allows this to be accomplished in √N steps on a quantum computer.
As an illustration, imagine some future quantum computer that is performing Grover’s search algorithm with a processing speed capable of performing 108 computational steps per second—a realistic processing speed for classical computers. Suppose it is engaged in a particularly arduous search through a space of N = 1030 entries looking for a single unique entry. How long will it take? Well, it would require √N = 1015 computational steps, which at a rate of 108 steps per second, would take 107 seconds, or about 4 months. Though this may seem like a long time, let us calculate how long it would take a classical computer to search through the same database with the same processing speed of 108 computational steps per second. It would take N/2 = 5×1029 steps and thus require 5 × 1021 seconds, or very nearly the age of the universe (13.7 billion years)! That is the size of the practical benefit of the quantum search algorithm over the classical. In general, the larger the search is, the larger the gain the quantum computer has over the classical.
The theoretical implications of these algorithms are even more mind-boggling, because of the way quantum computers work. In the example of Grover’s search algorithm, the quantum computer is not just doing 1015 computational steps in parallel, which can be mimicked classically by making all the computers on Earth work on nothing but this problem. The quantum computer is doing 1030 possible computations in parallel every time a computational step is invoked. If all the silicon in the whole of our planet were made into microchips, and they were all set performing different computations, there would still be fewer distinct computations going on in that giant parallel computer than there would be in a single quantum computer performing Grover’s algorithm. That is a measure of the complexity, structure, and the process that exists in ordinary matter just beyond our perception since quantum processes are, of course, going on all the time everywhere. In a quantum computer, some small part of nature’s complexity is put to good use!
The quantum computing model is inherently probabilistic. When an algorithm is successfully run on a quantum computer, the output corresponding to the answer to some problem may not always be correct. Instead, we demand that the algorithm gives the correct solution with a probability greater than 1/2. If this is the case, then simply repeating the algorithm multiple times will eventually yield an answer that will be correct with high probability. This is a just a consequence of the Chernoff bound in probability theory. Chris Lomont gives an example of this in The Hidden Subgroup Problem: Review and Open Problems:
Thus the majority is wrong very rarely. For example, we will make most algorithms succeed with probability 3/4 … Although it sounds like a lot, taking 400 repetitions of the algorithm causes our error to drop below 10-20, at which point it is more likely our computer fails than the algorithm fails. And since the algorithms we are considering are usually exponentially faster than classical ones, there is still a net gain in performance. If we do 1000 runs, our error drops below 10-55, at which point it is probably more likely you’ll get hit by lightning while reading this sentence than the algorithm itself will fail.
Simulating quantum systems using classical computers is hard to do because of the size and complexities of quantum systems. To describe a single particle in a classical system, you would need 6 different parameters: one for each of the 3 directions of motion and another 3 for the velocities of the particle in each of these directions. So for a system consisting on N particles, one would need 6N different parameters to completely specify the state space of the system.
On the other hand, due to the superposition principle and the linearity of quantum mechanics, to describe the simplest quantum system consisting of N quantum particles (or qubits), 2N parameters are needed to describe the state space completely. The thing is, for large enough values of N (greater than 4), 2N > 6N. Moreover, 2N grows exponentially faster than 6N as N increases. In essence, this is one of the main reasons why quantum computers would be able to outperform their classical counterparts.
Here is an example illustrating how large and how quickly exponential behavior like this can grow.
Suppose you have a chess board and a lot of coins. There are 64 spaces in total on the board. Start in one corner and place 2 coins on that square. Move to the next square and stack 22=4 coins on it. Then stack 23=8 coins on the third square and continue in this manner stacking 2N coins on the Nth square. The last square will have 264 = 18,446,744,073,709,551,616 coins stacked on top of each other.
How high do you think this stack would be?
It would stretch further than the distance from Earth to the Moon (400,000 kilometers), further than than the distance from the Earth to the Sun (150 million kilometers), and would be able to reach the next closest star Proxima Centauri (about 4.2 light-years or 3.97×1013 kilometers)!
There are many problems out there that require computers to use resources that grow exponentially like in this example. It has been difficult to say whether or not classical computers could even compute fast enough to make them practical problem solvers in this case, but we do have reasons to believe that quantum computers may be able to help out in this regard.
There exists a supermassive black hole located in the Perseus galaxy cluster 250 million light years from Earth that is emitting sound waves with a frequency of about 3x10^(-14) Hz. This corresponds to one wave oscillation every 10 million years, with a wavelength of about 1 light year. Musically speaking, this note would be close to a B flat 57 octaves below middle C!
Despite being located so far from us, the sound waves coming from the black hole manage to travel hundreds of thousands of light years from their source. Moreover, the lower limit of human hearing is around 20 Hz making the black hole’s sound some million billion times lower than what we can hear.
These sound waves could solve the mystery of why trillions of stars are not forming in this gaseous region, since the energies involved in these sound waves may be responsible for keeping the regions around the black hole hot enough to prevent star formation. If so, this black hole’s cosmic drone would remain constant for about 2.5 billion years!
The animation below zooms into the cluster revealing the black hole (central white spot) with its bipolar jets that are creating the two cavities, which are each 50 thousand light years wide. The sound waves then propagate into space from the edges of these cavities.
Among the many ramifications of quantum computation for apparently distant fields of study are its implications for the notion of mathematical proof. Performing any computation that provides a definite output is tantamount to proving that the observed output is one of the possible results of the given computation. Since we can describe the computer’s operations mathematically, we can always translate such a computation into the proof of some mathematical theorem. This was the case classically too, but in the absence of interference effects it is always possible to keep a record of the steps of the computation, and thereby produce (and check the correctness of) a proof that satisfies the classical definition - as “a sequence of propositions each of which is either an axiom or follows from earlier propositions in the sequence by the given rules of inference”. Now we are forced to leave that definition behind. Henceforward, a proof must be regarded as a process — the computation itself — for we must accept that in future, quantum computers will prove theorems by methods that neither a human brain nor any other arbiter will ever be able to check step-by-step, since if the ‘sequence of propositions’ corresponding to such a proof were printed out, the paper would fill the observable universe many times over.
- Deutsch, Ekert, and Lupacchini
“Machines, Logic and Quantum Physics”
Humans on Earth have been transmitting radio waves for over 100 years. The furthest these waves could have traveled away from Earth by now is about 100 light years. Thus, in communicating with extraterrestrial civilizations we have only managed to cover a region 200 light years in diameter around the Earth. Relative to the size of our Milky Way galaxy, which stretches about 100,000 light years in diameter, this is like comparing the area covered by Los Angeles to the surface area of the Earth! Since transmitted radio waves have not had the power to get through Earth’s ionosphere until about the 50s this distance is even smaller still. Nevertheless, after about 1 light year radio waves would probably be indistinguishable from the background radiation anyway. Perhaps this provides some justification for the Fermi paradox.
You may have to view this image (taken from here) in higher resolution just to see the blue dot depicting the extent of radio broadcasts.
The pigeonhole principle is a counting argument which states that if N items are to be placed into M boxes (with M less than N), then at least one of those boxes will contain more than one item. So for example, if there are 3 pigeons but only 2 pigeonholes, and each pigeon has to be placed into one of these pigeonholes, then one of the pigeonholes must contain at least 2 pigeons. Here is a surprising application of this principle applied to people and the numbers of hairs on their heads.
youngandpretty:
The earth has over 6,000,000,000 people in it. It’s safe to assume that each person has less than 1,000,000 hairs on their head. Now let’s say we sort all of these 6,000,000,000+ people into groups based on the number of hairs on their head. Each person is put into one group based on hair number ranging from 0 - 999,999. There are 1,000,000 groups and over 6,000,000,000 people. No matter how the distribution happens, one group must have at least 6,000 people in it. Therefore, there are at least 6,000 people in the world that have the exact same number of hairs. I wonder who they could be.
Working on the same principle, we can arrive at one very important conclusion about the people of Chicago. Chicago has over 1,000,000 people. Again, each person has to have less than 1,000,000 hairs. Once again, we distribute the people into groups. When we do this distribution, there must be at least one group that will have at least two people in it. Therefore, there are (at least) two people in Chicago with the exact same number of hairs. I’m pretty sure those two people have to be soul mates.
- Jason
There are some more interesting applications here.
