NASA’s WISE Findings Poke Hole In Black Hole ‘doughnut’ Theory

The latest from Brock Davis - love his work! 

5 Vital Lessons Scientists Learn That Can Better Everyone’s Life

“4. Following your intuition will never get you as far as doing the math will. Coming up with a beautiful, powerful and compelling theory is the dream of many scientists worldwide, and has been for as long as there have been scientists. When Copernicus put forth his heliocentric model, it was attractive to many, but his circular orbits couldn’t explain the observations of the planets as well as Ptolemy’s epicycles – ugly as they were – did. Some 50 years later, Johannes Kepler built upon Copernicus’ idea and put forth his Mysterium Cosmographicum: a series of nested spheres whose ratios could explain the orbits of the planets. Except, the data didn’t fit right. When he did the math, the numbers didn’t add up.”

There are a lot of myths we have in our society about how the greatest of all scientific advances happened. We think about a lone genius, working outside the constraints of mainstream academia or mainstream thinking, working on something no one else works on. That hasn’t ever really been true, and yet there are actual lessons – valuable ones – to be learned from observing scientists throughout history. The greatest breakthroughs can only happen in the context of what’s already been discovered, and in that sense, our scientific knowledge base and our best new theories are a reflection of the very human endeavor of science. When Newton claimed he was standing on the shoulders of giants, it may have been his most brilliant realization of all, and it’s never been more true today.

Come learn these five vital lessons for yourself, and see if you can’t find some way to have them apply to your life!

Uneasiness in Observers of Unnatural Android Movements Explained

It has been decades in the making, but humanoid technology has certainly made significant advancements toward creation of androids - robots with human-like features and capabilities. While androids hold great promise for tangible benefits to the world, they may induce a mysterious and uneasy feeling in human observers. This phenomenon, called the “uncanny valley,” increases when the android’s appearance is almost humanlike but its movement is not fully natural or comparable to human movement. This has been a focus of study for many years; however, the neural mechanism underlying the detection of unnatural movements remains unclear.

The research is in Scientific Reports. (full open access)

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

We’re Way Below Average! Astronomers Say Milky Way Resides In A Great Cosmic Void

“If there weren’t a large cosmic void that our Milky Way resided in, this tension between different ways of measuring the Hubble expansion rate would pose a big problem. Either there would be a systematic error affecting one of the methods of measuring it, or the Universe’s dark energy properties could be changing with time. But right now, all signs are pointing to a simple cosmic explanation that would resolve it all: we’re simply a bit below average when it comes to density.”

When you think of the Universe on the largest scales, you likely think of galaxies grouped and clustered together in huge, massive collections, separated by enormous cosmic voids. But there’s another kind of cluster-and-void out there: a very large volume of space that has its own galaxies, clusters and voids, but is simply higher or lower in density than average. If our galaxy resided near the center of one such region, we’d measure the expansion rate of the Universe to be higher-or-lower than average when we used nearby techniques. But if we measured the global expansion rate, such as via baryon acoustic oscillations or the fluctuations in the cosmic microwave background, we’d actually arrive at the true, average rate.

We’ve been seeing an important discrepancy for years, and yet the cause might simply be that the Milky Way lives in a large cosmic void. The data supports it, too! Get the story today.

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.)

9 Squares 8

Top: David Stanfield, Skip Hursh, Jamie Muntean Middle: David Urbinati, Al Boardman, Jake Williams Bottom: Riccardo Albertini, Bee Grandinetti, Qais Sarhan

About the project

WATCH: Crystal Birth, a Beautiful Timelapse of Metallic Crystals Forming in Chemical Solutions [video]

Galaxies: Types and morphology

A galaxy is a gravitationally bound system of stars, stellar remnants, interstellar gas, dust, and dark matter. Galaxies range in size from dwarfs with just a few hundred million (108) stars to giants with one hundred trillion (1014) stars, each orbiting its galaxy’s center of mass.

Galaxies come in three main types: ellipticals, spirals, and irregulars. A slightly more extensive description of galaxy types based on their appearance is given by the Hubble sequence. 

Since the Hubble sequence is entirely based upon visual morphological type (shape), it may miss certain important characteristics of galaxies such as star formation rate in starburst galaxies and activity in the cores of active galaxies.

Ellipticals

The Hubble classification system rates elliptical galaxies on the basis of their ellipticity, ranging from E0, being nearly spherical, up to E7, which is highly elongated. These galaxies have an ellipsoidal profile, giving them an elliptical appearance regardless of the viewing angle. Their appearance shows little structure and they typically have relatively little interstellar matter. Consequently, these galaxies also have a low portion of open clusters and a reduced rate of new star formation. Instead they are dominated by generally older, more evolved stars that are orbiting the common center of gravity in random directions.

Spirals

Spiral galaxies resemble spiraling pinwheels. Though the stars and other visible material contained in such a galaxy lie mostly on a plane, the majority of mass in spiral galaxies exists in a roughly spherical halo of dark matter that extends beyond the visible component, as demonstrated by the universal rotation curve concept.

Spiral galaxies consist of a rotating disk of stars and interstellar medium, along with a central bulge of generally older stars. Extending outward from the bulge are relatively bright arms. In the Hubble classification scheme, spiral galaxies are listed as type S, followed by a letter (a, b, or c) that indicates the degree of tightness of the spiral arms and the size of the central bulge.

Barred spiral galaxy

A majority of spiral galaxies, including our own Milky Way galaxy, have a linear, bar-shaped band of stars that extends outward to either side of the core, then merges into the spiral arm structure. In the Hubble classification scheme, these are designated by an SB, followed by a lower-case letter (a, b or c) that indicates the form of the spiral arms (in the same manner as the categorization of normal spiral galaxies). 

Ring galaxy

A ring galaxy is a galaxy with a circle-like appearance. Hoag’s Object, discovered by Art Hoag in 1950, is an example of a ring galaxy. The ring contains many massive, relatively young blue stars, which are extremely bright. The central region contains relatively little luminous matter. Some astronomers believe that ring galaxies are formed when a smaller galaxy passes through the center of a larger galaxy. Because most of a galaxy consists of empty space, this “collision” rarely results in any actual collisions between stars.

Lenticular galaxy

A lenticular galaxy (denoted S0) is a type of galaxy intermediate between an elliptical (denoted E) and a spiral galaxy in galaxy morphological classification schemes. They contain large-scale discs but they do not have large-scale spiral arms. Lenticular galaxies are disc galaxies that have used up or lost most of their interstellar matter and therefore have very little ongoing star formation. They may, however, retain significant dust in their disks.

Irregular galaxy

An irregular galaxy is a galaxy that does not have a distinct regular shape, unlike a spiral or an elliptical galaxy. Irregular galaxies do not fall into any of the regular classes of the Hubble sequence, and they are often chaotic in appearance, with neither a nuclear bulge nor any trace of spiral arm structure.

Dwarf galaxy

Despite the prominence of large elliptical and spiral galaxies, most galaxies in the Universe are dwarf galaxies. These galaxies are relatively small when compared with other galactic formations, being about one hundredth the size of the Milky Way, containing only a few billion stars. Ultra-compact dwarf galaxies have recently been discovered that are only 100 parsecs across.

Interacting

Interactions between galaxies are relatively frequent, and they can play an important role in galactic evolution. Near misses between galaxies result in warping distortions due to tidal interactions, and may cause some exchange of gas and dust. Collisions occur when two galaxies pass directly through each other and have sufficient relative momentum not to merge.

Starburst

Stars are created within galaxies from a reserve of cold gas that forms into giant molecular clouds. Some galaxies have been observed to form stars at an exceptional rate, which is known as a starburst. If they continue to do so, then they would consume their reserve of gas in a time span less than the lifespan of the galaxy. Hence starburst activity usually lasts for only about ten million years, a relatively brief period in the history of a galaxy.

Active galaxy

A portion of the observable galaxies are classified as active galaxies if the galaxy contains an active galactic nucleus (AGN). A significant portion of the total energy output from the galaxy is emitted by the active galactic nucleus, instead of the stars, dust and interstellar medium of the galaxy.

The standard model for an active galactic nucleus is based upon an accretion disc that forms around a supermassive black hole (SMBH) at the core region of the galaxy. The radiation from an active galactic nucleus results from the gravitational energy of matter as it falls toward the black hole from the disc. In about 10% of these galaxies, a diametrically opposed pair of energetic jets ejects particles from the galaxy core at velocities close to the speed of light. The mechanism for producing these jets is not well understood.

The main known types are: Seyfert galaxies, quasars, Blazars, LINERS and Radio galaxy.

source

images: NASA/ESA, Hubble (via wikipedia)

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)