Latest Posts by evisno - Page 3

The latest from Brock Davis - love his work! 

Portraits of birds by Laila Jeffreys

W. M. Keck Observatory telescopes logo. December 2, 2014

Image above: A dusty planetary system (left) is compared to another system with little dust in this artist’s conception. Image Credit: NASA/JPL-Caltech. Planet hunters received some good news recently. A...

Zeta Ophiuchus

A massive star plowing through the gas and dust floating in space. Zeta Oph is a bruiser, with 20 times the Sun’s mass. It’s an incredibly luminous star, blasting out light at a rate 80,000 times higher than the Sun! Even at its distance of 400 light years or so, it should be one of the brightest stars in the sky … yet it actually appears relatively dim to the eye.

Credit: NASA/Hubble

Star Clouds Toward the Southern Crown

Nearly every day Cassini sends back something amazing to sit and wonder at.

1) Saturn’s rings, 15 July 2014

2) Tethys / Saturn’s rings 14 July 2014

3) Disk of Saturn 14 July 2014

4) Prometheus / F Ring 13 July 2014

5) Pan in the Encke Gap 13 July 2014

All raw and unprocessed images from saturn.jpl.nasa.gov

Merging galaxies and droplets of starbirth

The Universe is filled with objects springing to life, evolving and dying explosive deaths. This new image from the NASA/ESA Hubble Space Telescope captures a snapshot of some of this cosmic movement. Embedded within the egg-shaped blue ring at the centre of the frame are two galaxies. These galaxies have been found to be merging into one and a “chain” of young stellar superclusters are seen winding around the galaxies’ nuclei.

At the centre of this image lie two elliptical galaxies, part of a galaxy cluster known as [HGO2008]SDSS J1531+3414, which have strayed into each other’s paths. While this region has been observed before, this new Hubble picture shows clearly for the first time that the pair are two separate objects. However, they will not be able to hold on to their separate identities much longer, as they are in the process of merging into one.

Finding two elliptical galaxies merging is rare, but it is even rarer to find a merger between ellipticals rich enough in gas to induce star formation. Galaxies in clusters are generally thought to have been deprived of their gaseous contents; a process that Hubble has recently seen in action. Yet, in this image, not only have two elliptical galaxies been caught merging but their newborn stellar population is also a rare breed.

The stellar infants — thought to be a result of the merger — are part of what is known as “beads on a string” star formation. This type of formation appears as a knotted rope of gaseous filaments with bright patches of new stars and the process stems from the same fundamental physics which causes rain to fall in droplets, rather than as a continuous column.

Nineteen compact clumps of young stars make up the length of this “string”, woven together with narrow filaments of hydrogen gas. The star formation spans 100,000 light years, which is about the size of our galaxy, the Milky Way. The strand is dwarfed, however, by the ancient, giant merging galaxies that it inhabits. They are about 330,000 light years across, nearly three times larger than our own galaxy. This is typical for galaxies at the centre of massive clusters, as they tend to be the largest galaxies in the Universe.

The electric blue arcs making up the spectacular egg-like shape framing these objects are a result of the galaxy cluster’s immense gravity. The gravity warps the space around it and creates bizarre patterns using light from more distant galaxies.

Image credit: NASA, ESA/Hubble and Grant Tremblay (European Southern Observatory)

Mathematics is full of wonderful but “relatively unknown” or “poorly used” theorems. By “relatively unknown” and “poorly used”, I mean they are presented and used in a much more limited scope than they could be, so the theorem may be well known, but only superficially, and its generality and scope can be very understated and underappreciated.

One of my favorites of such theorems was discovered around the 4th century by the Greek mathematician Pappus of Alexandria. It is known as Pappus’s Centroid Theorem.

Pappus’s theorem originally dealt with solids of revolution and their surface areas and volumes. This was all Pappus was able to figure out with the geometric proof methods available to him at the time. Later on, the theorem was rediscovered by Guldin, and studied by Leibniz, Cavalieri and Euler.

The theorem is usually split into two, one for areas and one for volumes. However, the basic principle that makes both work is exactly the same, though it is much more general in the case of areas on a plane, or volumes in space.

The invention/discovery of calculus eventually brought to the table much more general methods that made Pappus’s theorem somewhat limited in comparison.

Still, the theorem is based on a very clever idea that is quite satisfying both conceptually and intuitively, and brilliant in its simplicity. By knowing this key idea one can greatly simplify some problems involving volumes and areas, so the theorem can still be useful, especially when associated with methods from calculus.

Virtually all of the literature that mentions the theorem focus on solids of revolution only, as if the theorem was merely a curiosity, and they never really address why centroids would play a role. This is a big shame. The theorem is much more general, useful and conceptually interesting, and that’s why I’m writing this long post about it. But first, a few words on centroids.

Centroids

Centroids are the purely geometric analogues of centers of mass. They are a single point, not necessarily lying inside a shape, that defines the weighted “average position” of a shape in space. Naturally, the center of mass of any physical object with uniform density coincides with its geometric centroid, since the mass is uniformly distributed along the object’s volume.

Centers of mass are useful because they give you a point of balance: you can balance any object by any point directly under its center of mass (under constant vertical gravity). This comes from the fact the torques acting on the shape exactly cancel out in this configuration, so the object does not tilt either way.

Centroids exist for objects with any dimension you wish. A scattering of points has a centroid that is just their average position. In 1 dimension we have a line segment, whose centroid is always at its center. In 2, 3 or more dimensions, things get a bit trickier. We can always find the centroid by integrating all over the shape and weighting each point by that point’s position. This is generally a complicated enough problem by itself, but for simpler shapes this can be trivial.

Centroids, as well as centers of mass, also have the nice property of being additive: the union of two shapes has a centroid that is the weighted average of both centroids. So if you can decompose a shape into simpler ones, you can find its centroid without any hassle.

Pappus’s Centroid Theorems (for surfaces of revolution)

So, let’s imagine we have ourselves some generic planar curve, which we’ll call the generator. This can be a line segment or some arc of a circle, like in the main animation of this post, above. Anything will do, as long as it lies on a plane.

If we rotate the generator around an axis lying on its plane, and which does not cross the generator, it will “sweep” an area in space. Pappus’s theorem then states:

The surface area swept by a generator curve is equal to the length of the generator c multiplied by the length of the path traced by the geometric centroid of the generator L. That is: A = Lc.

In other words, if you have a generating curve with length c, and its centroid is at a distance a from the axis of rotation, then after a complete turn the generator will have swept an area A = 2πac, where L = 2πa is just the circumference of the circle traced by the centroid. This is what the first animation in this post is showing.

Note that in that animation we are ignoring the circular parts on top and bottom of the cylinder and cone, for simplicity. But this isn’t really a limitation of the theorem at all. If we add line segments for generating those regions we get a new generator with a different centroid, and the theorem still holds. Here’s what that setup looks like:

Try doing the math! It’s nice to see how things work out in the end.

The second theorem is exactly the same statement, but it deals with volumes. So, instead of a planar curve, we now have a closed planar shape with an area A.

If we now rotate it around an axis, the shape will sweep a volume in space. The volume is then just area of the generating shape times the length of the path traced by its centroid, that is, V = LA = 2πRA, where R is the distance the centroid is from the axis of rotation.

Why it works

Now, why does this work? What’s the big deal about the centroid? Here’s an informal, intuitive way to understand it.

You may have noticed that I drew the surfaces in the previous animations in a peculiar translucent way. This was done intentionally, not just to give the surfaces a physically “real” feel, but to illustrate the reason why the theorem works. (It also looks cooler!)

Here are the closed cylinder and cone after they were generated by a rotating curve:

Notice how the center of the top and bottom of the cylinder and the cone are a darker, denser color? This results from the fact that the generator is sweeping more “densely” in those regions. Farther out from the center of rotation, the surface is lighter, indicating a lower “density”.

To better convey this idea, let’s consider a line segment and a curve on a plane.

Below, we see equally spaced line segments of the same length placed along a red curve, perpendicular to it, in two different ways. Under the segments, we see the light blue area that would be swept by the line segment as it moved along the red curve in this way.

In this first case, the segments are placed with one edge on the curve. You can see that when the red curve bends, the spacing between the segments is not constant, since they are not always parallel to each other. This is what results in the different density in the “sweeping” of the surface’s area. Multiplying the length of those line segments by the length of the red curve will NOT give you the total area of the blue strip in this case, because the segments are not placed along the curve centered at their geometric centroids.

What this means is that the differences in areas being swept by the segment as the curve bends don’t cancel out, that is, the bits with more density don’t make up for the ones with a lower density, canceling out the effect of the bend.

However, in this second case, we place the segment so that its centroid is along the curve. In this case, the area is correctly given by Pappus’s theorem, because whenever there’s a bend in the curve one side of the segment is sweeping in a lower density and the other is sweeping at a higher density in a such a way that they both exactly cancel out.

This happens because that “density” is inversely proportional to the “speed” of each point of the line segment as it is moving along the red curve. But this “speed” is directly proportional to the distance to the point of rotation, which lies along the red curve.

Therefore, things only cancel out when you use the centroid as the anchor/pivot on the curve, which allows us to assume a constant density throughout the entire path, which in turn means there’s a constant area/volume being swept per unit of length traversed. The theorem follows directly from this result.

The same argument works in 3D for volumes.

Generalizations and caveats

As mentioned, the theorem is much more general than solids and surfaces of revolution, and in fact works for a lot of tracing curves (open or closed) and generators, as long as certain conditions are met. These are described in detail in the article linked at the end.

First, the tracing curve, the one where the generator moves along (always colored red in these illustrations), needs to be sufficiently smooth, otherwise you can’t properly define the movement of the generator along its extent. Secondly, the generator must be two dimensional, always lying on a plane perpendicular to the tracing curve. Third, in order for the theorem to remain simple, the curve and the plane must intersect at the centroid of the generator.

This means that, in two dimensions, the generator can only be a line segment (or pieces of it), as in the previous images. In this case, both the “2D volume” (the area) and “2D area” (the lateral curves that bound the area, traced by the ends of the segment) can be properly described by the theorem. If the tracing curve has sharp enough bends or crosses itself, then different regions may be covered more than once. This is something that needs to be accounted for via other means.

The immediate extension of this 2D case to 3D is perfectly valid as well, where the line segment gets replaced by a circle as a generator. This gives us a cylindrical tube of constant radius along the tracing curve in space, no longer confined to a plane. Both the lateral surface area of the tube and its volume can be computed directly by the theorem. You can even have knots as the curve!

In 3D, we could also have other shapes instead of a circle as the generator. In this case there are complications, mostly because now the orientation of the shape matters. Say, for example, that we have a square-shaped generator. As it moves along a curve it can also twist around, as seen in the animation below:

In both cases, the volume is exactly the same, and is given by Pappus’ theorem. This can be understood as a generalization of Cavalieri’s principle along the red curve, which also only works because we’re using the centroid as our anchor.

However, the surface areas in this case are NOT the same: the surface area of the twisted version is larger.

But if the tracing curve is planar (2D) itself and the generator does not rotate relative to the curve (that is, it remains “upright” all along the path), like in the case of surfaces of revolution, then the theorem works fine for areas in 3D.

So the theorem holds nicely for “2D volumes” (planar areas) and 3D volumes, but usually breaks down for surface areas in 3D. The theorem only holds for surface areas in 3D in a particular orientation of the generator along the curve (see reference).

In all valid cases, however, the centroid is the only point where you get the direct statement of the theorem as mentioned before.

Further generalizations

Since the theorem holds for 2D and 3D volumes, we can do a lot more with it. So far, we only considered a generator that is constant along the curve, which is why we have the direct expression for the volume. We actually don’t need this restriction, but then we have to use calculus.

For instance, in 3D, given a tracing curve parametrized by 0 ≤ s ≤ L, and a generator as a shape of area A(s), which varies along the curve in such a way that the centroid is always in the curve, then we can compute the total volume simply by evaluating the integral:

V = ∫0LA(s) ds

Which is basically a line integral along the scalar field given by A(s).

This means we can use Pappus theorem to find the volume of all sorts of crazy shapes along a curve in space, which is quite nifty. Think of tentacles, bent pyramids and crazy helices, like this one:

What we have here are five equal equilateral triangles positioned on the vertices of a regular pentagon. Their respective centroids lie on their centers, and since all of them have the same area the overall centroid (the blue dot) is exactly in the middle of the pentagonal shape, which is true no matter how you rotate the pentagon or the individual triangles.

This means we can generate a solid along the red curve by sweeping these triangles while everything rotates in any crazy way we want (like the overall pentagon and each individual triangle separately, as in the animation), and Pappus’s theorem will give us the volume of this shape just the same. (But not the area!)

If this doesn’t convince you this theorem is awesome and underappreciated, I don’t know what will.

So there you go. A nifty theorem that doesn’t get enough love and appreciation.

Reference

For a great, detailed and proper generalization of the theorem (which apparently took centuries to get enough attention of someone) see:

A. W. Goodman and Gary Goodman, The American Mathematical Monthly, Vol. 76, No. 4 (Apr., 1969), pp. 355-366. (You can read it for free online, you just need an account.)

A Glowing Pool Of Light

"NGC 3132 is a striking example of a planetary nebula. This expanding cloud of gas, surrounding a dying star, is known to amateur astronomers in the southern hemisphere as the "Eight-Burst" or the "Southern Ring" Nebula.

The name “planetary nebula” refers only to the round shape that many of these objects show when examined through a small visual telescope. In reality, these nebulae have little or nothing to do with planets, but are instead huge shells of gas ejected by stars as they near the ends of their lifetimes. NGC 3132 is nearly half a light year in diameter, and at a distance of about 2000 light years is one of the nearer known planetary nebulae. The gases are expanding away from the central star at a speed of 9 miles per second.

This image, captured by NASA’s Hubble Space Telescope, clearly shows two stars near the center of the nebula, a bright white one, and an adjacent, fainter companion to its upper right. (A third, unrelated star lies near the edge of the nebula.) The faint partner is actually the star that has ejected the nebula. This star is now smaller than our own Sun, but extremely hot. The flood of ultraviolet radiation from its surface makes the surrounding gases glow through fluorescence. The brighter star is in an earlier stage of stellar evolution, but in the future it will probably eject its own planetary nebula”

Credit: The Hubble Heritage Team

Radar Observations of Asteroid 2014 HQ124

Radar data of asteroid 2014 HQ124 taken over for hours on June 8, 2014, when the asteroid was between 864.000 miles (1.39 million kilometers) and 902.00 miles (1,45 million kilometers) from Earth. The data reveals asteroid 2014 HQ124 to be an elongated, irregular object that is at least 1200 feet (370 meters) wide on it long axis. The radar was obtained using NASA’s 70 meters Goldstone antenna, the same antenna used for communicating with spacecraft in deep space. The Goldstone radar team paired with the Arecibo Observatory (Goldstone sending radar, Arecibo receiving) for the first five frames of this movie in order to collect higher quality data resulting in shaper images. The other frames were made by both sending and receiving with antennas at the Goldstone complex.

Credit: NASA/JPL

New Discoveries about Star Formation in the Flame Nebula

Stars are often born in clusters, in giant clouds of gas and dust. Astronomers have studied two star clusters using NASA’s Chandra X-ray Observatory and infrared telescopes and the results show that the simplest ideas for the birth of these clusters cannot work.

This composite image shows one of the clusters, NGC 2024, which is found in the center of the so-called Flame Nebula about 1,400 light years from Earth. In this image, X-rays from Chandra are seen as purple, while infrared data from NASA’s Spitzer Space Telescope are colored red, green, and blue.

A study of NGC 2024 and the Orion Nebula Cluster, another region where many stars are forming, suggest that the stars on the outskirts of these clusters are older than those in the central regions. This is different from what the simplest idea of star formation predicts, where stars are born first in the center of a collapsing cloud of gas and dust when the density is large enough.

Credit: NASA/Spitzer/Chandra

Windswept by Charles Sowers

Though we cannot physically hold wind or see its swirling forms around us, we can definitely feel it.

In order to help visualize wind-currents, artist Charles Sowers created a kinetic installation consisting of 612 aluminum weather vanes called “Windswept” (2011). These were then meticulously placed on the side of the Randall Museum in San Francisco. Through this installation, we are able to see the patterns in the wind; where the currents go, how they turn, and sometimes how wind can abruptly change direction. This gives us a visual representation of the natural, invisible, force which moves around us, and sometimes with enough force, pushes and pulls us.

As the artist states: “Our ordinary experience of wind is as a solitary sample point of a very large invisible phenomenon. Windswept is a kind of large sensor array that samples the wind at its point of interaction with the Randall Museum building and reveals the complexity and structure of that interaction.”

This sort of installation creates a better understanding, and appreciation, of the wind. It is not just one large gust; a single wave can be made up of smaller currents, going in their own directions from the main flow. A dialogue begins to form between the building and the wind, the weather vanes acting as translators.

-Anna Paluch

death of a star by a supernova explosion,

and the birth of a black hole

New supernova likely arose from massive Wolf-Rayet star

They’ve been identified as possible causes for supernovae for a while, but until now, there was a lack of evidence linking massive Wolf-Rayet stars to these star explosions. A new study was able to find a “likely” link between this star type and a supernova called SN 2013cu, however.

“When the supernova exploded, it flash ionized its immediate surroundings, giving the astronomers a direct glimpse of the progenitor star’s chemistry. This opportunity lasts only for a day before the supernovablast wave sweeps the ionization away. So it’s crucial to rapidly respond to a young supernova discovery to get the flash spectrum in the nick of time,” the Carnegie Institution for Science wrote in a statement.

“The observations found evidence of composition and shape that aligns with that of a nitrogen-rich Wolf-Rayet star. What’s more, the progenitor star likely experienced an increased loss of mass shortly before the explosion, which is consistent with model predictions for Wolf-Rayet explosions.”

The star type is known for lacking hydrogen (in comparison to other stars) — which makes it easy to identify spectrally — and being large (upwards of 20 times more massive than our Sun), hot and breezy, with fierce stellar winds that can reach more than 1,000 kilometres per second. This particular supernova was spotted by the Palomar 48-inch telescope in California, and the “likely progenitor” was found about 15 hours after the explosion.

Researchers also noted that the new technique, called “flash spectroscopy”, allows them to look at stars over a range of about 100 megaparsecs or more than 325 million light years — about five times further than what previous observations with the Hubble Space Telescope revealed.

Image credit: ESO

Pitch Black: Cosmic Clumps Cast the Darkest Shadows

Astronomers have found cosmic clumps so dark, dense and dusty that they throw the deepest shadows ever recorded. Infrared observations from NASA’s Spitzer Space Telescope of these blackest-of-black regions paradoxically light the way to understanding how the brightest stars form.

The clumps represent the darkest portions of a huge, cosmic cloud of gas and dust located about 16,000 light-years away. A new study takes advantage of the shadows cast by these clumps to measure the cloud’s structure and mass.

Continue Reading

N44C nebula

On the middle left of the image is a source of its artistic likeness, a network of nebulous filaments surrounding the Wolf-Rayet star. This type of rare star is characterized by an exceptionally vigorous “wind” of charged particles. The shock of the wind colliding with the surrounding gas causes the gas to glow.

The Wolf-Rayet star is part of N44C, a nebula of glowing hydrogen gas surrounding young stars in the Large Magellanic Cloud. Visible from the Southern Hemisphere, the Large Magellanic Cloud is a small companion galaxy to the Milky Way.

What makes N44C peculiar is the temperature of the star that illuminates it. The most massive stars — those that are 10 to 50 times more massive than the Sun — have maximum temperatures of 30,000 to 50,000 degrees Celsius (54,000 to 90,000 degrees Fahrenheit). The temperature of this star is about 75,000 degrees Celsius (135,000 degrees Fahrenheit). This unusually high temperature may be due to a neutron star or black hole that occasionally produces X-rays but is now inactive.

N44C is part of a larger complex that includes young, hot, massive stars, nebulae, and a “superbubble” blown out by multiple supernova explosions. Part of the superbubble is seen in red at the very bottom left of the Hubble image.

Credit: NASA/JPL/Hubble

NASA’s WISE findings poke hole in black hole ‘doughnut’ theory

A survey of more than 170,000 supermassive black holes, using NASA’s Wide-field Infrared Survey Explorer (WISE), has astronomers reexamining a decades-old theory about the varying appearances of these interstellar objects.

The unified theory of active, supermassive black holes, first developed in the late 1970s, was created to explain why black holes, though similar in nature, can look completely different. Some appear to be shrouded in dust, while others are exposed and easy to see.

The unified model answers this question by proposing that every black hole is surrounded by a dusty, doughnut-shaped structure called a torus. Depending on how these “doughnuts” are oriented in space, the black holes will take on various appearances. For example, if the doughnut is positioned so that we see it edge-on, the black hole is hidden from view. If the doughnut is observed from above or below, face-on, the black hole is clearly visible.

However, the new WISE results do not corroborate this theory. The researchers found evidence that something other than a doughnut structure may, in some circumstances, determine whether a black hole is visible or hidden. The team has not yet determined what this may be, but the results suggest the unified, or doughnut, model does not have all the answers.

Every galaxy has a massive black hole at its heart. The new study focuses on the “feeding” ones, called active, supermassive black holes, or active galactic nuclei. These black holes gorge on surrounding gas material that fuels their growth.

With the aid of computers, scientists were able to pick out more than 170,000 active supermassive black holes from the WISE data. They then measured the clustering of the galaxies containing both hidden and exposed black holes — the degree to which the objects clump together across the sky.

If the unified model was true, and the hidden black holes are simply blocked from view by doughnuts in the edge-on configuration, then researchers would expect them to cluster in the same way as the exposed ones. According to theory, since the doughnut structures would take on random orientations, the black holes should also be distributed randomly. It is like tossing a bunch of glazed doughnuts in the air — roughly the same percentage of doughnuts always will be positioned in the edge-on and face-on positions, regardless of whether they are tightly clumped or spread far apart.

But WISE found something totally unexpected. The results showed the galaxies with hidden black holes are more clumped together than those of the exposed black holes. If these findings are confirmed, scientists will have to adjust the unified model and come up with new ways to explain why some black holes appear hidden.

Image credit: NASA/JPL-Caltech

All the times science fiction became fact

I don’t usually go for these really-big-ads-disguised-as-infographics (Really? Sci-fi ink & toner?), but this one was too cool to pass up.

Unfortunately, no hoverboards yet. But we’ve still got 15 months before time runs out on that one:

Bonus: Why are some science fiction authors so good at predicting the future? Check out this episode of It’s Okay To Be Smart where I talk all about that:

(via io9)

Anticrepuscular Rays Over Florida

(no word to describe this feeling)

eat mor chikin

Comet ISON appears to have broken up and mostly evaporated as it travelled around the Sun, but something has made it around. It will be seen how much and in what condition. (Source of images) UPDATE: It is now confirmed that the comet is gone. Rest in pieces, ISON!

In engineering we talk a lot about tools. Some people have a favorite collection of software, some a metaphorical belt filled with tips, tricks, and techniques, and others a literal box or lab bench filled with instruments. In my experience, a good engineer not only maintains all three, but seeks...

I made my 22 mo old daughter an LED light suit costume. Looks hilarious!

> This might just be the coolest baby costume in the entire world.

> Her voice and movement remind me of Boo from Monsters Inc.

> You should probably let her run loose in all the dark areas of your neighborhood on Halloween night.

> Sweet, I can’t wait to show my 421 month old girlfriend this!

mydraco font pack #5

p.1: funny and useful fonts to have! download

p.2: beautiful and pretty fonts that I use a lot! download

Window phone concept

Is it a window, is it a phone? No. Actually, it’s Window phone and this is the part where you are saying “Whaaat? What are you talking about?”. The phone is actually a concept with extraordinary features. Surely if this could be actually put into production, it would set a new standard for the term “cool”.