Comet ISON Appears To Have Broken Up And Mostly Evaporated As It Travelled Around The Sun, But Something

Ask Ethan: Could The Fabric Of Spacetime Be Defective?

“The topic I’d like to suggest is high-energy relics, like domain walls, cosmic strings, monopoles, etc… it would be great to read more about what these defects really are, what their origin is, what properties they likely have, or, and this is probably the most exciting part for me, how we expect them to look like and interact with the ‘ordinary’ universe.”

So, you’d like to ruin the fabric of your space, would you? Similar to tying a knot in it, stitching it up with some poorly-run shenanigans, running a two-dimensional membrane through it (like a hole in a sponge), etc., it’s possible to put a topological defect in the fabric of space itself. This isn’t just a mathematical possibility, but a physical one: if you break a symmetry in just the right way, monopoles, strings, domain walls, or textures could be produced on a cosmic scale. These could show up in a variety of ways, from abundant new, massive particles to a network of large-scale structure defects in space to a particular set of fluctuations in the cosmic microwave background. Yet when it comes time to put up or shut up, the Universe offers no positive evidence of any of these defects. Save for one, that is: back in 1982, there was an observation of one (and only one) event consistent with a magnetic monopole. 35 years later, we still don’t know what it was.

It’s time to investigate the possibilities, no matter how outlandish they seem, on this week’s Ask Ethan!

Portraits of birds by Laila Jeffreys

Why Bennu?

Our OSIRIS-REx spacecraft will travel to a near-Earth asteroid, called Bennu, where it will collect a sample to bring back to Earth for study. 

But why was Bennu chosen as the target destination asteroid for OSIRIS-REx? The science team took into account three criteria: accessibility, size and composition.

Accessibility: We need an asteroid that we can easily travel to, retrieve a sample from and return to Earth, all within a few years time. The closest asteroids are called near-Earth objects and they travel within 1.3 Astronomical Units (AU) of the sun. For those of you who don’t think in astronomical units…one Astronomical Unit is approximately equal to the distance between the sun and the Earth: ~93 million miles.

For a mission like OSIRIS-REx, the most accessible asteroids are somewhere between 0.08 – 1.6 AU. But we also needed to make sure that those asteroids have a similar orbit to Earth. Bennu fit this criteria! Check!

Size: We need an asteroid the right size to perform two critical portions of the mission: operations close to the asteroid and the actual sample collection from the surface of the asteroid. Bennu is roughly spherical and has a rotation period of 4.3 hours, which is in our size criteria. Check!

Composition: Asteroids are categorized by their spectral properties. In the visible and infrared light minerals have unique signatures or colors, much like fingerprints. Scientists use these fingerprints to identify molecules, like organics. For primitive, carbon-rich asteroids like Bennu, materials are preserved from over 4.5 billion years ago! We’re talking about the start of the formation of our solar system here! These primitive materials could contain organic molecules that may be the precursors to life here on Earth, or elsewhere in our solar system.

Thanks to telescopic observations in the visible and the infrared, as well as in radar, Bennu is currently the best understood asteroid not yet visited by a spacecraft.

All of these things make Bennu a fascinating and accessible asteroid for the OSIRIS-REx mission.

Make sure to follow us on Tumblr for your regular dose of space: http://nasa.tumblr.com

We’re used to radiation being invisible. With a Geiger counter, it gets turned into audible clicks. What you see above, though, is radiation’s effects made visible in a cloud chamber. In the center hangs a chunk of radioactive uranium, spitting out alpha and beta particles. The chamber also has a reservoir of alcohol and a floor cooled to -40 degrees Celsius. This generates a supersaturated cloud of alcohol vapor. When the uranium spits out a particle, it zips through the vapor, colliding with atoms and ionizing them. Those now-charged ions serve as nuclei for the vapor, which condenses into droplets that reveal the path of the particle. The characteristics of the trails are distinct to the type of decay particle that created them. In fact, both the positron and muon were first discovered in cloud chambers! (Image credit: Cloudylabs, source)

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.)

Celebrating 17 Years of NASA’s ‘Little Earth Satellite That Could’

The satellite was little— the size of a small refrigerator; it was only supposed to last one year and constructed and operated on a shoestring budget — yet it persisted.

After 17 years of operation, more than 1,500 research papers generated and 180,000 images captured, one of NASA’s pathfinder Earth satellites for testing new satellite technologies and concepts comes to an end on March 30, 2017. The Earth Observing-1 (EO-1) satellite will be powered off on that date but will not enter Earth’s atmosphere until 2056. 

“The Earth Observing-1 satellite is like The Little Engine That Could,” said Betsy Middleton, project scientist for the satellite at NASA’s Goddard Space Flight Center in Greenbelt, Maryland. 

To celebrate the mission, we’re highlighting some of EO-1’s notable contributions to scientific research, spaceflight advancements and society. 

Scientists Learn More About Earth in Fine Detail

This animation shifts between an image showing flooding that occurred at the Arkansas and Mississippi rivers on January 12, 2016, captured by ALI and the rivers at normal levels on February 14, 2015 taken by the Operational Land Imager on Landsat 8. Credit: NASA’s Earth Observatory  

EO-1 carried the Advanced Land Imager that improved observations of forest cover, crops, coastal waters and small particles in the air known as aerosols. These improvements allowed researchers to identify smaller features on a local scale such as floods and landslides, which were especially useful for disaster support. 

On the night of Sept. 6, 2014, EO-1’s Hyperion observed the ongoing eruption at Holuhraun, Iceland as shown in the above image. Partially covered by clouds, this scene shows the extent of the lava flows that had been erupting.

EO-1’s other key instrument Hyperion provided an even greater level of detail in measuring the chemical constituents of Earth’s surface— akin to going from a black and white television of the 1940s to the high-definition color televisions of today. Hyperion’s level of sophistication doesn’t just show that plants are present, but can actually differentiate between corn, sorghum and many other species and ecosystems. Scientists and forest managers used these data, for instance, to explore remote terrain or to take stock of smoke and other chemical constituents during volcanic eruptions, and how they change through time.  

Crowdsourced Satellite Images of Disasters   

EO-1 was one of the first satellites to capture the scene after the World Trade Center attacks (pictured above) and the flooding in New Orleans after Hurricane Katrina. EO-1 also observed the toxic sludge in western Hungary in October 2010 and a large methane leak in southern California in October 2015. All of these scenes, which EO-1 provided quick, high-quality satellite imagery of the event, were covered in major news outlets. All of these scenes were also captured because of user requests. EO-1 had the capability of being user-driven, meaning the public could submit a request to the team for where they wanted the satellite to gather data along its fixed orbits. 

This image shows toxic sludge (red-orange streak) running west from an aluminum oxide plant in western Hungary after a wall broke allowing the sludge to spill from the factory on October 4, 2010. This image was taken by EO-1’s Advanced Land Imager on October 9, 2010. Credit: NASA’s Earth Observatory

 Artificial Intelligence Enables More Efficient Satellite Collaboration

This image of volcanic activity on Antarctica’s Mount Erebus on May 7, 2004 was taken by EO-1’s Advanced Land Imager after sensing thermal emissions from the volcano. The satellite gave itself new orders to take another image several hours later. Credit: Earth Observatory

EO-1 was among the first satellites to be programmed with a form of artificial intelligence software, allowing the satellite to make decisions based on the data it collects. For instance, if a scientist commanded EO-1 to take a picture of an erupting volcano, the software could decide to automatically take a follow-up image the next time it passed overhead. The Autonomous Sciencecraft Experiment software was developed by NASA’s Jet Propulsion Laboratory in Pasadena, California, and was uploaded to EO-1 three years after it launched. 

This image of Nassau Bahamas was taken by EO-1’s Advanced Land Imager on Oct 8, 2016, shortly after Hurricane Matthew hit. European, Japanese, Canadian, and Italian Space Agency members of the international coalition Committee on Earth Observation Satellites used their respective satellites to take images over the Caribbean islands and the U.S. Southeast coastline during Hurricane Matthew. Images were used to make flood maps in response to requests from disaster management agencies in Haiti, Dominican Republic, St. Martin, Bahamas, and the U.S. Federal Emergency Management Agency.

The artificial intelligence software also allows a group of satellites and ground sensors to communicate and coordinate with one another with no manual prompting. Called a “sensor web”, if a satellite viewed an interesting scene, it could alert other satellites on the network to collect data during their passes over the same area. Together, they more quickly observe and downlink data from the scene than waiting for human orders. NASA’s SensorWeb software reduces the wait time for data from weeks to days or hours, which is especially helpful for emergency responders. 

Laying the Foundation for ‘Formation Flying’

This animation shows the Rodeo-Chediski fire on July 7, 2002, that were taken one minute apart by Landsat 7 (burned areas in red) and EO-1 (burned areas in purple). This precision formation flying allowed EO-1 to directly compare the data and performance from its land imager and the Landsat 7 ETM+. EO-1’s most important technology goal was to test ALI for future Landsat satellites, which was accomplished on Landsat 8. Credit: NASA’s Goddard Space Flight Center

EO-1 was a pioneer in precision “formation flying” that kept it orbiting Earth exactly one minute behind the Landsat 7 satellite, already in orbit. Before EO-1, no satellite had flown that close to another satellite in the same orbit. EO-1 used formation flying to do a side-by-side comparison of its onboard ALI with Landsat 7’s operational imager to compare the products from the two imagers. Today, many satellites that measure different characteristics of Earth, including the five satellites in NASA’s A Train, are positioned within seconds to minutes of one another to make observations on the surface near-simultaneously.

For more information on EO-1’s major accomplishments, visit: https://www.nasa.gov/feature/goddard/2017/celebrating-17-years-of-nasa-s-little-earth-satellite-that-could

Make sure to follow us on Tumblr for your regular dose of space: http://nasa.tumblr.com/.

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!

If it is just us, seems like an awful waste of space.

Carl Sagan (from Contact)

‘아렌델’은 빅뱅 이후 첫 10억년 이내에 존재했던 별입니다. 우리 지구에 닿기까지 129억년이나 걸린 것이지요!

우리 태양보다 적어도 50배 크고, 몇백만 배 밝아요. 평소에는 지구에서 볼 수 없지만, 우리와 ‘아렌델’ 사이에 있는 은하단이 렌즈 역할을 했습니다!

A View into the Past

Our Hubble Space Telescope just found the farthest individual star ever seen to date!

Nicknamed “Earendel” (“morning star” in Old English), this star existed within the first billion years after the universe’s birth in the big bang. Earendel is so far away from Earth that its light has taken 12.9 billion years to reach us, far eclipsing the previous single-star record holder whose light took 9 billion years to reach us.

Though Earendel is at least 50 times the mass of our Sun and millions of times as bright, we’d normally be unable to see it from Earth. However, the mass of a huge galaxy cluster between us and Earendel has created a powerful natural magnifying glass. Astronomers expect that the star will be highly magnified for years.

Earendel will be observed by NASA’s James Webb Space Telescope. Webb's high sensitivity to infrared light is needed to learn more about this star, because its light is stretched to longer infrared wavelengths due to the universe's expansion.