Quanto Magazine

The last transit of Venus that you or I have been able to see..

The last transit of Venus in our lifetime, which took place on June 5th 2012. Surely not many of you could see it because the visibility was not too good. The most fortunate were the people of the north-eastern Oceania and Asia region, and of course in the Arctic Circle. Some of you perhaps saw it in June 2004. If not, I’m afraid you’re running out of life chances, as the next event will be late 2117, and since you’re reading these lines (then you’re at least the age of knowing how to read) the probability that you’ll live to that date is quite small.

To explain the magic of the event is but a simple geometric coincidence without major transcendence. It is but a small “eclipse” of the Sun, but instead of it being the Moon that causes the eclipse, in this case it is the tiny planet of Venus.

We must take advantage of every “rare” event to get the most juice possible. The most obvious application, which has been used since the eighteenth century is to use transits to measure the distance between the Sun and Earth (distance known as Astronomical Unit ), using trigonometry, or more precisely by measures of parallax . The first accurate determinations were made by Joseph La Lande which measured a distance of 153 ± 1 million km. A century later, Newcomb got the value of 149.59 ± 0.31 million km, which at the time was amazing accuracy.

The value currently considered (measured with radar and telemetry techniques from space probes) is known with a precision of a few meters. I want to highlight an educational initiative of ESO who used this technique during the transit of Venus in 2004, for thousands of schoolchildren and observers worldwide could combine their observations and thus regain the value of the Astronomical Unit. All the measures of the more than 1500 participants were combined to determine with GPS the moments in which the different phases of the transit happened. The value finally obtained was 149.6 ± 11.8 million km.

Other more “modern” applications for this last transit of 2012 are indirectly related to the search for extrasolar planets . It has attempted to determine the variation pattern of overall brightness of the sun during this mini-eclipse (which unlike 2004 the Sun happens being in an active period magnetically speaking), to try to learn how to apply it to detect planets in other Similar stars. Also study the atmosphere of Venus can serve both to better understand the meteorology of the planet, to have a reference of the atmospheres we might expect to see on extrasolar planets, in this case uninhabitable.

An example is the animation of the right where it is to study the twilight phenomenon or “twilight” that clearly is seen in the bottom of the disc of Venus, as the Sun passes behind. More information on the project page .

Finally, I leave here with a photo. (credit: JAXA, NASA), the transit of 2012 was taken with the instrument SOT on board the satellite HINODE .

The Physics of Quadcopters

A quadcopter is a type of multirotor helicopter, which is propelled by a set of four rotors. The have a simple mechanical design which means they are cheap to construct, and available for commercial use.

Quadcopters have seen a major increase in popularity in recent years. One of their most famous appearances in the public eye was their demonstration on TED in 2013.

The sheer maneuverability and control of these things is incredible! You’re probably wondering how these things work – so I’ll outline some of the physics behind them.

Well to start with the drone needs its key components: An electronic flight system, motors, a battery, and propellers. These are also intelligent machines and some quadcopters have miniature sensors in their on-board computer. These can detect changes in air pressure and air pressure, and provide obstacle avoidance and GPS location.

When the drone lifts off, the motors serve the purpose of rotating the blades at super-high speeds, approximately 30,000 RPM for high end drones.

Flight Dynamics

To calculate the Lift of the drone, the formula is:

L = 1/2 pV^2 x S x CL


p = Air Density (approx 1.2KG/m^3)
v = Velocity
S = Surface Area (of the blade)
CL = Coefficient of lift.

A quadcopter has four blades. When the quad lifts off, two of these blades will rotate clockwise, and the other two will rotate counterclockwise. T
he aim will be to generate enough force upward that is greater that the weight of the drone. Once the drone is in the air, it can then reduce the power, and when the upward force of the blades is equal to the weight of the drone, it will hover.

When the drone wants to move in a direction, the force increase in the two rotors at the opposite end of the direction that the drone wishes to move in. For example, if the drone wishes to move sideways right, the two rotors on the left side of the drone will increase in power.

If the drone wishes to turn in a direction, two diagonal rotors increase in power. For example, if the drone wishes to turn right (turn in a clockwise direction), the front left and back right motor will increase in power. This is because they are rotating counter clockwise.

There are also some commercial drones with more than four rotors, usually six or eight. The more rotors a drone has, the more stable it becomes. It also gives the pilot more accuracy and maneuverability when flying.


Most drones get very little airtime, with few reaching over 30 minutes. This is because as you increase the battery size, you also increase the weight on the drone. If we were to attach a battery powerful enough to last a few hours – it would be so heavy that the drone wouldn’t get off the ground!

To fuel the motors, most batteries are Lithium Polymer Batteries (LiPos). Never leave these batteries unattended hen charging as they have been known to explode!

For further information take a look at these articles

The different types of drone

This Is How Drones Work

Great Scientists: David J. Bohm

If you have already been browsing other articles in this magazine you may have come across a term as baffling as Bohmian. What is Bohmian? Well, so roughly speaking it is possible to say that Bohmian is anyone who sympathizes or is in any way related to the ideas of David Joseph Bohm, a scientist and philosopher. Probably most of you will not have heard of Bohm in life, although there is a certain professor of quantum physics who mentions him in his lectures in speaking of the experiment of the two slits. You can also find some mention about this character in some books of orthodox quantum physics or classic manuals to use. Even if you digress a little you can find in our library a quantum book written by him and containing an alleged dedication to his Spanish colleagues written in pencil on one of the first pages.

David Joseph Bohm was born in 1917 in Wilkes, Barre, Pennsylvania (United States). His first contacts with science came in his readings of science fiction, when he was still a child. No more information was available in that small mining town. David was fascinated by the forces of the universe and the great number of things that are beyond our understanding. He studied physics at the State College, continuing his training at CALTECH, where he did not last long, since he never fit into that atmosphere. The pace of constant problem solving and test-taking made him drop out of high school after the first semester to leave for the University of California at Berkeley. There he investigated during the Second World War the dispersion of nuclear particles under the supervision of J. Robert Oppenheimer. After finishing his doctorate in Berkeley (1943), he became an assistant professor at Princeton University in 1947.

For a physicist of the caliber of Richard Feynman, an area of ​​physics was only interesting if he could find a problem in it, and turned out to be a genius solving all sorts of problems. On the occasions when Bohm encountered a technical difficulty, he always distrusted the too abstract mathematical reasoning. As he pointed out, after all, in any mathematical development, there are things we assume without examining them too much, and the more complicated the mathematics employed, the easier it is for errors to remain. He preferred to proceed always intuitively, rather than logical or mechanical, and preferred to feel the answer and visualize it in his mind clearly before giving the pertinent mathematical steps. It is as if his method in solving problems is carried more by imagination and intuition than by pure logic.

For example, while still in the state of Pennsylvania, he had been trying to understand the functioning of the gyroscope, that puzzle that intrigues the children with their continuous swing. Usually, when we push an object so that its center of gravity falls out of equilibrium, it falls. A gyroscope, however, does not fall, its axis of rotation moves and undergoes precession. When most students of physics face the problem of the gyroscope, we will learn different formulas, such as the conservation of the moment, which lead to a rather incomplete explanation. But Bohm needed a direct perception of the intimate nature of this movement. One day, while walking in the field, he imagined his own person as a gyroscope, and through some kind of internal muscular movement he was able to understand the nature of the movement. In this way, and using his own body, he understood the functioning of the gyroscopes. The formulas and mathematics would come later, as a simple formal tool to explain their internalization.

This particular skill accompanies you throughout your professional life. One of his colleagues said, “Dave always comes to the right conclusions, but his math is terrible. I take his work home and find all kinds of mistakes and I have to spend all night looking for the right demonstration. , The result is the same as Dave directly visualized. ”

To give another example we will take the spin of an electron, a quantum concept far removed from the usual classic resemblance with a balloon circling over itself. This is a concept that begins to move away from common sense. Most physics students would be content to visualize the electron in terms of mathematical manipulations and equations, without explicit reference to anything physical. Bohm, however, found himself able to experience sensations with his body about the way the spin components are combined into something moving in a new direction.

When I was still studying in Berkeley, Bohm did quite new works with plasmas. He discovered that electrons ejected from atoms do not behave as individual particles in a high-temperature gas (known as plasma), but rather as part of an immense and organized whole (an idea that was deeply rooted). An enormous number of electrons would produce quite organized effects, as if an organic process were directing their collective behavior. Shortly after Bohm would say that these collective movements, now known as Bohm-diffusion, gave the impression that the sea of electrons was alive somehow. This was Bohm’s first major discovery in the field of physics, and he is directly engaged with the deep themes of the universe and the interconnections that would characterize his thinking and scientific work. Nevertheless, in his last years of life he elaborated a conception of the universe according to which it consists in the interconnection of all things, a notion to which he would give the name of “implicated order”.

As we were saying some paragraphs ago, Bohm got a parking assistant professor at Princeton University in 1947. While teaching of quantum mechanics in the following years he wrote a book called Quantum Theory (1951), which is still a classic in the Field (this is the book that I talked about in the first paragraph). When he completed this work, Bohm was beginning to befriend Albert Einstein, who was also at Princeton at that time. Einstein apparently told Bohm that he had never seen the quantum theory presented as clearly as it appeared in Bohm’s new book, and the two scientists began to converse more assiduously. As their relationship grew closer they discovered that they had much in common in their basic conceptions of quantum theory, and together they deepened in the interpretations and metaphysical meaning of quantum theory. He would gladly give a large sum of money for being in one of these talks, with Bohm and Einstein putting the quantum to calving.

These discussions led Bohm to seriously consider the validity of the classical interpretation of quantum mechanics (Vienna circle). Encouraged by the confidence that his association with Einstein gave him, Bohm embarked on a great undertaking: revising the foundations of quantum theory, which led to his peculiar formulation of the same and eventually lead to a crusade for life In search of the knowledge and understanding that allowed him to describe all reality (a theory of everything).

By the same time Bohm brought out another important example of his peculiar way of being. Had some problems with American justice, since he had to appear before the Committee on Un – American Activities (1949) under the accusation baseless, he and some other lab partners radiation Berkeley sympathized the with communism. During World War II Oppenheimer referred to the FBI the names of supposedly philomarxist friends and acquaintances. Apparently, Bohm was one of them. As he passionately believed in freedom he refused to declare, for reasons of principle, what earned him the accusation of contempt of congress. After a trial he was acquitted. His Princeton students asked that he be reinstated in his post, and Einstein was said to want Bohm to become his personal assistant, but his contract, after an unfortunate incident, was not renewed, and would never again be taught in the States United. Einstein himself, who spent many years futilely searching for his own alternative theory of quantum mechanics, referred to Bohm as his “intellectual successor” saying that “if anyone can do it, that will be Bohm.”

Bohm moved to Brazil (1951), to the University of Sao Paulo, where he would be professor until 1955. The embassy requisitioned him the passport, with which he lost his nationality. There he worked on his second book, Causality and Chance in Modern Physics (1957), which is still used in some universities. From Brazil he went to the Technion Institute in Haifa, Israel, and then to the University of Bristol in England. There, he and a student in 1959 made another original contribution to quantum theory. He discovered with Yakir Aharonov what is now known as the Aharonov-Bohm effect. They showed that quantum mechanics predicts that the movement of charged particles is conditioned by the presence of magnetic fields, even when they do not penetrate where they are confined. Several experiments have confirmed this effect.

He was later acquitted of the contempt charge by allowing him to return to the United States, but it was too late for him, he settled permanently at Birkbeck College in London.

In the next thirty years the work of David Bohm focused on the foundations of quantum theory and the theory of relativity and its implications in various fields. As is often the case with advanced physicists, in later life he became interested in philosophical matters, holding interminable talks with Indian spiritual director J. Krishnamurti.

In this new stage of his life he elaborated a conception of the universe according to which it consists in the interconnection of all things, a notion to which it would give the name of “implicated order”. He wrote more books of physics ( Theory of Relativity in 1966), philosophy ( Wholeness and the Implicate Order in 1980), and even the nature of consciousness ( Science, Order and Creativity in 1987 with David Peat).

He died of a heart attack in 1992 when, in collaboration with other scientists, he was preparing a new volume on quantum mechanics. His friends and colleagues remind him of a man who was not only brilliant and audacious but also extraordinarily frank, polite and generous.

From Bohm we have his alternative theory of quantum mechanics, which came to light more than forty years ago but has been ignored until recently. This theory, completely braided and absolutely different, also accounts for all known subatomic phenomena. In it chance plays no role and every material object always occupies a particular region of space. In addition, its laws form a unique set, applicable equally to all physical objects, even if we have to admit the nonlocality. In any case, the theory has not yet surpassed the relativistic test, which in these times would take it to the box of definitive oblivion. Meanwhile, a small group of Bohmians from around the world are facing the dominant majority, armed with hidden variables and pilot trajectories seeking to realize that dream that Einstein and Bohm wove in the afternoons of Princeton.

if you want to delve into the life and work of this scientist, take a look at https://en.wikipedia.org/wiki/David_Bohm Another valuable source of documents and links is the library of Birkbeck College, University of London, where Bohm was Professor of Theoretical Physics: http://www.bbk.ac.uk/lib/about/hours/bohm

Physics is Fun!

A major Physics magazine recently published recently a tribute to boredom, arguing that the only guarantee of success in Physics is the continuation of tedious and soporific work. To be a good physicist, the author seemed to say, one had to resign oneself to boredom and become a wet blanket.

It seems to me: if a science is not fun, exciting and challenging, this cannot be good. The great Archibald Wheeler, father of black holes and many other things and physical yet jovial and sparkling in his ninetieth year, often proposed as the first moral principle of physics the term “Do not do any calculation, without first knowing the outcome.” But it is necessary to state that there is another more fundamental still, a zero start, and Thermodynamics : “Learn to have fun with science”.

If we want to appreciate science, we must go further. After all, shoes or a can opener are useful objects, but they do not inspire deep feelings. To say that science is useful does not even give it justice. Because, José Antonio Marina says in his recent ‘Theory of creative intelligence’, “you can say many things about sciene: It is fun and solemn, loud, dazzling, opaque, terrible, mocking, enigmatic, discreet, overwhelming and even more”.

We must also defend an aesthetic conception of science, which brings it closer to art. The great Richard Feynman felt a deep emotion to the laws of nature, which he described as even religious, but added “few non-scientists can understand.” Though some, like the Portuguese poet Fernando Pesso to, for whom “the binomial of Newton is more beautiful than the Venus de Milo” . When I think of General Relativity in quantum theory or Newtonian Dynamics, I feel a similar emotion that I induce the music of Bach or Beethoven or pictures of Velázquez or El Bosco. Feynmann reminds me of Mozart, with his grace, his wit and penchant for gambling. Einstein was a builder of structures advancing security and poise of Bach. The serenity and balance of Faraday always remind me of Stravinsky.

Perhaps this example will clarify to us what the author was thinking about the article. The long years of study we have to follow scientists, with difficult calculations, so many laboratory sessions and learning complex techniques, correspond to those that musicians or painters have to follow. It is true that the hours of scales on the piano or repeatedly drawing hands and noses are annoying and heavy in themselves. But they become exciting and make sense, knowing that, behind them, will come the wonder of the finished work. That is why you have to see calculations and observations as stages to overcome a challenge and, as such, feel their fun.

For if in science you discover the grace of beauty, you also have the emotion of sport. It is a deeply human activity because it allows us to realize one of the most defining aspirations of our species: to accept challenges and overcome them. Athletes dedicate long efforts to surpass brands. A few are brilliant and they reach the newspapers, most of us are content to face our poor personal records, to keep the form, to travel a distance, to climb a mountain. But being small does not take away their emotion, because the brand itself is always the most important. The same impulse moves scientists to strive to understand a theory, to do the experiment, to be able to calculate something; No matter what others have already done, everyone has their challenges. That is why we must also learn to live science with the rejoicing of sport.

It is also a great adventure, collective and individual. Without it, human beings would not have arrived at what they are: after biological evolution, came a social, in which technology and science have been essential engines. Every personal work, no matter how modest it may seem to us, is a chapter of that great history, of the incredible feat of having come to elaborate such a marvelous description – in its pure sense: full of wonders – of the laws of nature, Thanks to the efforts of thousands and thousands of people. Without his work, the homo would be less sapiens today.

Physics must produce a great delight. You have to enjoy it, play with it, have fun with it. This helps you to understand it and make discoveries. If someone finds it boring, it is better to do something else.

We must be aware of our privilege and, therefore, the most important task that teachers have is to transmit that emotion. That is, the magic of science.

What is Ball Lightening?

Many of you have doubted if the “Ball Lightning” was serious or a joke. Those who think it was a joke should know that there is an international association dedicated to study and collect data on this curious phenomenon.

Ball lightening refers to “fireballs” which are tens of centimeters in diameter with a high luminosity. Their life usually ranges from a few seconds to a whole minute. When disappearing they can go on softly or with a loud bang (the name of the latter is “violent lightning ball” ). They are one of the few phenomena of nature that do not have a generally accepted scientific explanation. Of the many theories that try to define this phenomenon we can cite those defending antimatter meteors and others stating that they are just mere optical illusions .

To tell the truth, few people with scientific training have contemplated these phenomena. One of the latest experiences with this phenomenon recently occurred in an aircraft of the Air UK which was going through a storm. The crew was surprised by a fireball that appeared in the cockpit. The beauty of this phenomenon is such that the person who was a scientist who had the opportunity to see this did not hesitate to characterize this phenomenon as“the most beautiful thing in my life” when he saw through the hallway of his plane (only a scientist Could say something like that about a lightning bolt).

This phenomenon is close to being deciphered. In the words of the speaker: ” we assume that under the action of strong magnetic fields created at the foot of lightning, power lines are formed, called streamers (they have been able to be reproduced in laboratory ddp 10 million volts) Which act as the filaments of a light bulb. These streamers form a kind of tangled skein that gives the ball beam a stable structure that prevents it from bursting for a period of time.

The streamers constitute one millionth of the ball ray which indicates that it would only produce burns if they touched the filaments of its interior, that reach temperatures of about 16 thousand degrees Celsius. The rest of the matter, according to the theoretical model defended by these scientists, is air at room temperature.

If you seek more information magazine of the National Geographic Society speaks on the subject in an article.

Faster than light, nothing

An article and a commentary in the journal Nature describes an experiment by Wang, Kuzmich and Dogariu. The experiment consists in passing a pulse of light through a certain material – cesium vapor – specially prepared so that it has appropriate properties of emission and absorption of photons.

These articles received an unusual attention in the media, both in our country and in others, even saying that the experiment, which (supposedly) produced speeds higher than the light, “was a challenge to ideas Of Einstein.” Poor Einstein. He despised sensationalism, he must be turning in his grave.

And the comments in the media about the challenge on Einstein were exagerated, including saying the speeds could be 300 times the speed of light, when in fact it is a change of a few percent.

The commentary by Marangos, and even the article by Wang et al., Are written, it has to be said, in a somewhat obscure way and in a more sensationalist style than should be the norm for scientific work. It should be read with some care to make sure that what moves faster than light is called average group velocity, something quite different from the speed with which the photons of the light pulse. Careful analysis reveals that if this average speed is higher than the speed of light in vacuum,  c , this is due to a wrong definition because the average group velocity which is not the same as the average velocity of the photons.

This is evident if one thinks that all photons always travel at the speed c , which is known since 1905, and thanks precisely to Einstein that, in two fundamental articles established, first, that the speed of light is a universal constant, and second, that the light pulses consist of photons that travel at the speed precisely c. Proven results in literally millions of experiments.

The call rate of propagation of light in a medium, or the average group velocity, which can be denoted by v, is a useful measurement in certain approximations. Perhaps the best way to understand this for non-professionals is with a couple of examples.

e.g 1. If someone measures the distance in a straight line between Madrid (Spain) and Geneva (Switzerland), it is 1,000 kilometers. I have traveled this route many times by car, at an average speed of 100 kilometers per hour, and it has taken me 14 hours, not the 10 that would be expected.

The reason for this, of course, is that the road does not follow a straight line, and the actual route by car, is 1,400 km. The same thing happens to a photon when passing through a material medium. It collides with other atoms and its trajectory is no longer a straight line. So ts effective rate is lower: like my effective rate on the Madrid-Geneva is about 70 kilometers per hour.

However, it seems that the example of my trip to Geneva is not valid, because in them I always came out with an actual speed which is lower. How is it possible to understand that Wang and colleagues found a higher speed than c ? I’ve explained this in another example below.

Consider a set of 10 runners running an athletics track, all at 30 kilometers per hour. Suppose that the runners of the group begin to run separated from each other by a meter. Let us now call the speed of the group to the one with which the center of the group moves, center that is five meters behind the one that goes on head at the beginning of the race.

Now comes the trick. Each time the riders pass the finish line, a new rider is placed at the height of the head and start to run at 30 kilometers per hour.

After 10 laps, all the runners that have been added go together in the head of the group, the stragglers have disappeared and the runner in the middle of the group goes five meters before what would have been half of the group if We would not have touched it; Therefore, the average speed of the group is greater than that of any of the runners.

Of course, the trick is that the runners who start the test are not the same as those who finished it. The average group velocity concept we have entered is correct if the group contains the same people at the beginning and the end; But ceases to be if we add and delete corridors.

The fact that such group speed exceeds 30 kilometers per hour does not imply that the riders could transmit the information faster than each one ran: if we give a witness to one of the runners, the witness will go at the speed of the runner . And that if we were lucky and chose one of the runners who survive to the end.

This is, in essence, also the mechanism of the photon experiment performed by Wang et al., And, as they themselves acknowledge in the references cited in their article, by quite a few other researchers before them. In the experiments performed by the predecessors, the leading photons excite atoms, which in turn produce photons, which are added to the head group; The photos that go in the queue are absorbed.

Although each photon, always moving at the speed  c ¸ the group velocity is greater than c . The merit of Wang, Kuzmich and Dogariu is to use a more subtle method to enhance / suppress photons, but neither here nor any one of them goes more speed than light in the vacuum. And if an experimenter were to say otherwise, it would be necessary to think that he had measured wrongly and asked for an independent repetition of his experiment: as Belmonte said, what can not be can not be, and besides is impossible. The photons that pass through a medium traveling, all the while, at the speed c , as, on the other hand, it is clear in the articles themselves of Nature if you read them carefully.

The definition of group speed that authors use is inappropriate. Wang, Kuzmich, and Dogariu point him out at the end of his article, though, astutely, they maintain their definition because in that way an experiment that is simply curious seems to have fundamental consequences.

I have no doubt that it is highly unlikely that the work will succeed in “leading to profound implications for the issue of signal propagation,” as Wang and Adláters suggest, or in putting Einstein’s ideas at bay, as the Media, but there is no doubt that it has tended to get publicity for the authors. This does not mean that it has no value; Above all, technical. We must recognize the remarkable ability of Wang and colleagues: it is not easy to construct an experiment that carries out the multiplication and annihilation of photons.

Chaos Theory Revolution


At the end of the nineteenth century the great majority of physicists thought that Newtonian mechanics and Maxwell’s electromagnetism could explain all the phenomena of nature. It was the deterministic image of the physical world. The classical scheme postulated that if we could determine the positions and velocities at a certain moment of all the particles of the universe then we could calculate anything from Newton’s equation. The interactions would be given by electric and magnetic fields and by Newtonian gravity. Mathematics would do the rest.

The famous Lorenz attractor represents complexity in a dynamic system.

The arrival of Einsteinian (special and general) relativity changed our view of space and time inherited from Newton, although it was the quantum theory that definitively collapsed the deterministic vision of the universe.

At the same time, and very slowly, some results were emerging that would eventually become another revolution, another radical change in the way we see the world. One of the first to note that the classic scheme could not work was the great French mathematician Henri Poincaré (1854-1912).He did not need to build a new physical theory. It remained within the framework of the classical equations and observed that for systems formed by 3 bodies in gravitational interaction Newton’s equations are not sufficient It is not that we are not smart enough to find closed solutions, but no matter how clever they may be the mathematicians of the future, they will never find them. There is simply no finite algorithm.

His research on the qualitative behavior of dynamical systems opened a new path in the mathematical study of physics. The new topological and geometric tools would reinforce the old analytical techniques. Poincaré was one of the first to obtain information on equations without the need to solve them.

His work has developed throughout the twentieth century and is still far from complete. Great mathematicians like Birkhoff, Kolmogorov, Moser, Smale, Arnold … have made important contributions. And many more. It would be impossible here to point them all out. But what is this revolution about which I speak? What does it mean for our world view?

This is a general scheme applicable to disciplines as diverse as planetary mechanics, fluid dynamics, electromagnetism, biology or even economics. The common denominator: differential equations. The result: the complexity in the behavior of the variables we consider. The objective: to understand qualitatively, without solving the equations, some aspects of the system. The problem: Numerical methods fail when it comes to long-term predictions.

This is commonly referred to as chaos theory. But what do we mean when we talk about chaos? Ordinarily it is understood that something is chaotic when there is confusion or disorder. This is not the chaos of which I speak. When it is said that the problem of n bodies is chaotic, it does not mean that it is disordered, after all the dynamic evolution is given by very orderly differential equations.

We must, therefore, give another definition of chaotic system. The ingredients that are generally considered essential for chaos are: hypersensitivity in the initial conditions (that is, if we calculate the evolution from similar but not identical conditions will give us very different results), existence of complicated structures or “strange “(They are not worth spheres, cylinders or things like that) and freedom. Especially freedom (well, the technical term is topological transitivity but it seems clear to me the other).

When I speak of freedom I mean that there are no restricted regions, that if we wait long enough the system will evolve from one region to another. Freedom also refers to the “geometry” of solutions, whose complexity grows with dimension, and which is not classified at all (there are indeed theorems that in certain cases there is no classification algorithm).

We must visualize this graph as a body of revolution rotating it around the vertical axis of the left. It represents the opposite of the previous example: the ordered behavior of a dynamic system. The surfaces obtained by rotating the figure in the indicated direction contain magnetic fields. As you can see, they are bulls, which implies bounded magnetic orbits.

But there are many more fascinating and complex things that are not included within what is called chaos. Instead of talking about chaos, it is more revealing to talk about the different creatures that can populate a system: bifurcations, stability or instability (of many types), attractors or repulsors, knots, borders, symmetries, conserved “magnitudes”, ergodicity … and many More, some of which we know, and others we can not even imagine.

In short, it is an unlimited and complex universe of which we still know little. The chaos is just one among many interesting things. I simply encourage all those who have drawn attention to the topic to browse some books. You will find will not disappoint your imagination.

The next great era of awakening of the human intellect may very well produce a method of understanding the qualitative content of equations.” Richard P. Feynman.

Further reading:

R. Abraham, JE Marsden: Foundations of Mechanics. Benjamin, Reading Mass (1992).
S. Wiggins: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990).
KR Meyer, GR Hall: Introduction to Hamiltonian Dynamical Systems and the N-body problem. Springer, New York (1992).

How Many Planets Are There? (3/3)

It seems that the answer to this question is rather elusive. About a week ago we said that after the fall of Pluto from the planetary pantheon, the answer was 8. Then we saw some more can be found planets orbiting strange and inhospitable objects, such as neutron stars, but we only found a dozen by this method .

But as I said, knowing the number of stars in the universe (a number) accounts simply do not exist. Indeed, we are already seeing more and more planets, which convinces us that are not so strange.

And how can we discover these extra solar planets? Because it’s one of the most active branches in current astrophysics, detection methods have multiplied in recent years. The following image, published by M. Perryman in 2000, shows one of the most commonly used methods:

The method of radial velocities is based on the oscillatory motion. This wobble of the star will be more noticeable the larger the mass of the planet in relation to the mass of the star, so this method is more efficient to detect large planets, several Jupiter masses. It is called radial velocity because thanks to the Doppler effect, one can measure when the star moves away or approaches us, measuring the frequency shift of a spectral line. That is, when the star approaches us (because the planet is pulling it toward us in its movement) the spectral line will run a little to the blue, while if the star is moving away, the line will run until the Red. By measuring the frequency shift and having an estimate of the mass of the star, one can limit the size of the planet and some orbital parameters. This all sounds we known, because the method is similar to how stars were discovered pulsars .

The other method mentioned is the method of transits, which measure the changes in brightness in a star as the planet orbits it passes in front of it (relative to us). These small “eclipses” produce variations of brightness tiny but measurable. As in the previous case, the high – mass planets tend to obscure more than smaller planets star, so the first will be easier to detect.

Today they have reached more than 750 planets detected by making use of all these methods, and the census can go at every day in the catalog of the Extrasolar Planets Encyclopaedia .

There is currently a great competition in terms observatories and instruments capable of detecting the highest number of extrasolar planets. Examples are space missions COROT  (the European Space Agency ) and Kepler (from NASA ) using the transit method. On Earth, we have as an example the project SuperWASP (transits) in which participates actively IAC , and the instrument HARPS (radial velocities) of the  European Southern Observatory  (ESO), which runs on the 3.6m telescope at La Silla Observatory in Chile.

Finally, I cite the recent press release from the ESO, very relevant as to the question at hand:

Our new observations with HARPS mean that about 40% of all red dwarf stars have a super-Earth orbiting in their habitable zone, a zone that allows the existence of liquid water on the surface of the planet.” says Xavier Bonfils (IPAG, Observatoire des Sciences of the Universe of Grenoble, France), who leads the team. “Because red dwarfs are so common – there are about 160 billion in the Milky Way – this leads us to the conclusion that there are tens of billions of such planets in our galaxy alone.

There is still much to learn and to say about this fascinating topic so much interest. But what is certain is that in the coming years will go knowing more about planets, the composition and conditions of their atmospheres, and if they meet the requirements to support life.

How many planets are there? (2/3)

We spoke a few days ago what one immediately thinks for “planet” when you hear that word. Mercury, Venus … etc, until a few years ago, it was certainly no more than ten. In recent decades there have been dozens of space missions to different planets, comets and satellites. The results of these explorations have largely been available to the general public that have seen detailed images of Mars sent by the probes Viking and Mars Pathfinder.

But even though the exploration of the solar system still has a long way to go, the idea of worlds (and life) around other stars is surely back to the dawn of human imagination. Interestingly the first two planets detected are not found orbiting a quiet star (such as our beloved Sun), but in a strange place
where no one would have thought to find such a thing: the pulsar PSR 1257 +12 that is in the Virgo constellation about 2,000 light years from us. It came to fame in 1992 for harboring two planets of a size about twice that of our Earth. The discoverers of such a finding were the Polish Alexander Wolszczan and Canadian Dale Frail.

The pulsars (see the artistic recreation of one on the right) are tiny neutron stars that spin at breakneck speeds (specifically PSR B1257 + 12 has a diameter of about 30 km and a rotation period of 6 milliseconds, or that is, rotates about 160 times per second!). A curiosity is that both the discovery of this pulsar and observations that revealed the presence of these exoplanets were made at the Arecibo radio telescope in Puerto Rico. Antenna we have all seen in the movie “Contact” based on the novel by Carl Sagan. Interestingly this telescope today provides data project SETI @ home that seeks to detect radio emissions by extraterrestrial civilizations.

The method to detect these planets around pulsars is quite simple to understand. Pulsars have a period of extremely regular rotation, so the radio broadcast we received with a radio telescope has a fairly well-defined period. One example is the following video showing an auditory representation of electromagnetic pulses that are detected in some famous pulsars:

Now imagine a small planet revolving around one of these pulsars. This pulls gravity on the planet in the same way that the planet pulls on the pulsar, or in other words, both bodies will revolve around the center of mass system common. In the case of system PSR B1257 + 12 and the largest of its planets (called B as the second in distance) the center of mass is about 700 km from the pulsar (remember that its diameter is about 30 km). To calculate this figure just we need to know that the planet B has an estimated mass 4.3 times the mass of Earth and is orbiting at about 0.36 astronomical units pulsar).

From our reference system on Earth, the pulsar will have a very small oscillation around its center of mass, which will add to the intrinsic rotation of the neutron star. Thus the periodic pulse emitted will be modulated in turn by this oscillation another almost imperceptible.

The residue left after subtracting the period of the pulsar can determine the axis of the orbit of the object that produces the disturbance (the supposed planet), as well as having an upper limit on its mass.

Of course all this is complicated when instead of having a single planet we have a planetary system with several components. Each planet then exerts a little tug on the pulsar and slightly modulate its intrinsic pulse. So it becomes even more complicated demodulation. This is the case of PSR B1257 + 12 in which 4 planets have been detected in different sizes and orbital distances.

The advantage of this method is that it is quite accurate because the pulsar signals can be digitized with enough time to be picked up by radio telescopes resolution. In addition to being the natural clocks so precise pulsar is easy to detect tiny anomalies in the period due to relatively small planets, even smaller than Earth. No other method to discover planets as small.

The disadvantage is that there are not many known pulsars which severely limits the potential planets to discover. In fact, today, less than a dozen extra solar planets have been discovered using this technique. In addition, another point that takes some of the interest the matter is that all these planets would not harbor life almost certainly. Pulsars are neutron stars that remain as residue after the recent explosion of a massive star as a supernova . Therefore any planet in the area would have to be formed at some time after the explosion and would be subjected to intense high – energy radiation that would prevent the formation of life.

I guess I still need to answer the question. We’ll save that for next week.

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