Polytopes: Part II

This is the second part of a four part post. For the first part, see here.

The space in which all possible planes reside is the third dimension. In three dimensional space, one can define six directions: up, down, left, right, forward and back. The structures in this space are known as polyhedra (singular: polyhedron). Polyhedra are constructed with polygons, with a certain number of polygons at each vertex. Regular polyhedra have a certain number of identical regular polygons at each vertex. By extending the system from previously, these are denoted {p,q}.

One can construct the polygon {p,q} by having q regular polygons {p} at each vertex. As with polygons, situations with p≤2 are trivial, and q≤2 also has this property. This is due to the fact that any polyhedron {p,1} equals {p}, and any {p,2} is two overlapping polygons {p}, again trivially equaling {p}.

For any {p,q}, the total angular measure around each vertex is q internal angles of the regular polygon {p}. Since the internal angle of a regular polygon with p≥3 is (1-2/p)180º, the total angular measure around each vertex is (1-2/p)q180º. If this value is less than 360º, than the polyhedron curves inward, and eventually intersects itself, forming a finite polyhedron. Therefore, for a finite polyhedron,

(1-2/p)q180<360
(1-2/p)q<2
1-2/p<2/q
1<2/p+2/q
1/p+1/q>1/2

The final formula sets a limit on the number of possible finite regular polyhedra (assuming p,q>2), and only five pairs of {p,q}, where p and q are whole numbers, meet these requirements. These are called the Platonic Solids:

{3,3}
1/3+1/3=2/3>1/2
The above pair {3,3} defines a polyhedron with three triangles at each vertex. The resulting figure has four triangular faces, six lines, four vertices, and is known as the tetrahedron. It is pictured below.


{3,4}
1/3+1/4=7/12>1/2
The pair {3,4} defines another polyhedron with four triangles at each vertex. The resulting figure has eight triangular faces twelve lines, six vertices, and is known as the octahedron. It is pictured below.


{4,3}
1/4+1/3=7/12>1/2
The pair {4,3} defines another polyhedron with three squares at each vertex. The resulting figure has six square faces, twelve lines, eight vertices, and is known as the cube, or hexahedron. It is pictured below.


{3,5}
1/3+1/5=8/15>1/2
The pair {3,5} defines yet another polyhedron with five triangles at each vertex. The resulting figure has twenty triangular faces, thirty lines, twelve vertices, and is known as the icosahedron. It is pictured below.


{5,3}
1/5+1/3=8/15>1/2
The pair {5,3} defines a final polyhedron with three pentagons at each vertex. The resulting figure has twelve pentagonal faces, thirty lines, twenty vertices, and is known as the dodecahedron. It is pictured below.


The above polyhedrons are the only convex regular polyhedrons that are finite. They are finite due to the fact that they are tilings of a sphere, and therefore exist on the elliptic plane in elliptic geometry. Elliptic space is simply space with positive curvature. For more information on the three geometries, see here. In addition, convex regular polyhedra in elliptic 2-space or Euclidean 3-space (depending on point of view because elliptic and hyperbolic planes in n dimensions can only be accurately represented in n+1 dimensions) also have the property that the number of vertices plus the number of faces minus the number of sides always equals two. Also, for the above polyhedra, if the number of vertices and the number of faces are switched, the result is another polyhedron, known as the original polyhedron's dual. The dual of the tetrahedron is the tetrahedron, since both numbers are the same, and the other four come in dual pairs, with the cube's dual being the octahedron and the icosahedron's dual being the dodecahedron. This is equivalent to switching the numbers in {p,q} to {q,p}. As a result, if a polyhedron is finite, its dual will be also.

Another case to consider is when 1/p+1/q=1/2. In this case, the curvature is exactly zero, and the polyhedron tiles the Euclidean plane, and the polyhedron is known as a tiling. Since the flat Euclidean Universe is infinite, so are polyhedra that tile it. There are only three regular tilings of the Euclidean plane:

{3,6}

Polytopes: Part I

Polytopes, simply defined, are geometrical shapes. They can exist in any dimension, and are usually finite. For the purpose of the post, all polytopes are considered finite unless specifically designated to the contrary.

Starting in zero dimensions, there is no direction, and the only thing that can exist is an infinitesimal speck, called a point. In one dimension, an infinite number of distinct points exist. By connecting any two of these points, a line segment is formed.



A point and a line representing zero and one dimensions.

In abstract mathematical space the line must continue forever, representing the entire one-dimensional universe.

In two dimensions, there are an infinite number of lines possible, all crisscrossing what is called a plane. We are familiar with many shapes, called polygons, on this plane. One and two sided polygons aren't really possible with straight line segments, but one and two sided polygons with arcs of a circle as sides, called henagons and digons are technically possible. The simplest real polygon however, is the triangle. A triangle is the shape defined by the intersections of three distinct nonparallel line segments. A picture of a triangle is shown below.



A triangle. The line segments are called the sides, and each pair of sides marks an angle. The triangle is the interior region defined by three line segments, and including the line segments. The rest of the plane is known as the exterior.

For a triangle, if it is equilateral, or all the sides are the same length, it follows that the triangle is also equiangular, or that the angles all have the same measure. This only follows for the triangle and not any higher polygon. The triangle that is both equiangular and equilateral is called a regular triangle. It is given the symbol {3}. The angles of any triangle in regular space must add up to 180º, with a general Euclidean space polygon's angles add up to 180º(p-2) where p is the number of sides in the polygon.

Moving on, one can define a polygon with four sides, known as a quadrilateral. From the formula mentioned above, it is clear that the angles of a quadrilateral add up to 360º. With four sides, a polygon can be equilateral without be equiangular, the example being the rhombus, and can also be equiangular without being equilateral, the example being the rectangle. The two shapes are shown below, along with the quadrilateral that is both equilateral and equiangular, the square, which is denoted by {4}.



All of the above polygons mentioned are convex, meaning that none of their angles' measures exceed 180º. However, with polygons of four sides and up, concave polygons exist, and these can even intersect themselves. When a polygon intersects itself on the plane, it is known as a complex polygon.




The above is a concave quadrilateral, followed by a complex one.

In two dimensions, a polygon with any number of sides can exist and the regular convex pentagon {5}, regular convex hexagon {6}, regular convex heptagon {7}... etc. follow the quadrilateral. In general a regular convex polygon {p} has p points, or vertices, and p line segments, or sides. The total angular measure of a regular convex polygon is 180º(p-2), and each individual angle is 180(1-2/p), however, the sum of exterior angles of any regular convex polygon is always 360º.

A polygon with p sides can also be generated by taking the pth roots of one, and plotting them on the complex plane. Each vertices' coordinates is found with (cos(2πik/p),sin(2πik/p), where k is {0,1,2,3...p-1}. The pentagon found be connecting the 5th roots of one is shown below



Note that all five points lie on the unit circle, or the circle with radius one centered at the origin (the point (0,0)).

However, a regular polygon does not have to be convex. There are regular polygons which are complex. These are known as star polygons. Each star polygon with p sides has the same vertices as a regular convex polygon {p}, but the sides are connected in a different way. The simplest star polygon is known as the pentagram, and is shown below.



This particular star polygon is denoted {5/2} because the points are connected in such a way that every point is connected to the point two points away from it in a counterclockwise order. With five sides, {5/1} simply connects the points in order, simply resulting in {5}, {5/3} is a simple reflection of {5/2} and {5/4} is just the points connected in the opposite order, still resulting in {5}.

In the case of {6}, no continuous star polygon can exist, because {6/1} and {6/5} trivially equal {6}, and the remaining three, {6/2}, {6/3} and {6/4} are not continuous, meaning that despite what point of the hexagon one starts at, not all the points are accessed, and the polygon is made up of multiple parts, two for {6/2} (the Jewish star) and {6/4}, for which the parts are triangles, and three for {6/3}, the parts being lines.



The star polygon {6/2}. It is made up of two disconnected triangles, and is therefore not a true star polygon.

The number of distinct continuous star polygons, disregarding simple reflections and rotations, for all p up to 10 are

p=1-4: 0
p=5: 1; {5/2}
p=6: 0
p=7: 2 {7/2}, {7/3}
p=8: 1; {8/3}
p=9 2; {9/2}, {9/4}
p=10 1; {10/3}

All possible star polygons with p up to 16 are shown below (note p is replaced by n in the picture).



In the above picture, the red diagonal lines represent {p/1} which is simply {p}, {p/2}, {p/3} and so on and the blue diagonal lines show the common names for these star polygons. Note that the continuous star polygons are in boldface, while the discontinuous ones are not (click on the image to enlarge it).

In conclusion, a star polygon {p/x} is continuous when GCD(p,x)=1 (greatest common divisor of x and p). An equivalent statement is that p and x are relatively prime. GCD(p,x) is also the number of discontinuous elements required to include all the points. For example, since GCD(5,2)=1, {5/2} is a continuous star polygon, but GCD(6,2)=2 and two distinct pieces (in this case triangles) are required to include all the points with a {6/2} polygon. Additionally, if p-x=±1, then {p/x} is equivalent to {p} and is a special case of the more general star polygon, the convex polygon.

A final notable case of polygon is the star-shaped polygon, which is very different from a star polygon. For any star-shaped polygon, there exists one or more points on the interior of the polygon from which a line can be drawn to each vertex without intersecting any other part of the polygon. An example of a star polygon is shown below.



The top image shows how the polygon's interior contains at least one point from which all vertices are visible, or accessible with lines without obstruction. In addition, the set of points where this is true is named with the (quite odd) name, the kernel. The lower image shows the kernel of this particular star-shaped polygon. Complex polygons cannot be star-shaped, and, ironically, nor can star polygons. Convex polygons are all star-shaped, and their kernels coincide with the polygons themselves. Additionally, all triangles are star-shaped by definition, because they all are convex.

This post is continued in Polytopes Part II.

Sources: Regular Polytopes by H.S.M. Coxeter, Google Images

Orbital Resonance

An orbital resonance is a gravitational phenomenon in which two bodies that are both orbiting around one parent body are in a specific pattern. For example, if for every orbit Planet Bill makes around a star, Planet Joe makes exactly three, the two bodies are in an orbital resonance (the notation for this is 1:3). It may seem unlikely that such a correspondence would occur in the physical universe and that this is a purely abstract notion, but this is not the case. In fact, gravity often pushes objects into these resonances when the objects are of comparable size, but ejects them if one is too much bigger than the other, as with Jupiter and an asteroid.

In fact, there are resonances in our Solar System! The simplest case is the one between Neptune and Pluto. For every two orbits Pluto makes, Neptune makes exactly three. This is why it is impossible for the two bodies every to collide, despite the fact that their orbits cross. This is the only stable resonance involving two planetary (although Pluto is a dwarf planet) bodies, but other objects outside of Neptune's orbit are in resonances with Neptune, known as Trans-Neptunian objects. Most of these are smaller than Pluto, but one larger one is known: the dwarf planet Eris. However, Eris is not known to be in an exact orbital resonance with Neptune.

Other Trans-Neptunian objects sometimes are in resonances with Neptune, the most common being (object:Neptune) 2:3, corresponding to Pluto and other bodies, 3:5, 4:7, 1:2, and other rarer ones such as 2:5, 3:4, 4:5, 1:4, 1:5, 1:3, 3:7, and 6:11. Some of the latter sometimes correspond to only one known object, and may be coincidental. Some are also unstable, and smaller objects can often be ejected from a resonance by a gravitational pull quite easily. The special case of a 1:1 orbital resonance is addressed in the post, Lagrangian Points.



A chart showing known objects, resonances, and distances of various objects beyond Neptune's orbit.

Resonances of a different kind impact asteroids in the main asteroid belt. Jupiter's gravitational pull has a great effect on the asteroid belt, and unlike Neptune's resonances, are much closer to each other. Therefore, rather than objects commonly existing at these resonances, repeated encounters with Jupiter ejects the asteroids onto another orbit. Due to this, there are gaps, called Kirkwood Gaps, that exist at the main orbital resonances with Jupiter, named after Daniel Kirkwood, who observed and explained the nature of the gaps in 1857. The population of asteroids in relation to the gaps is shown in the image below.



This image shows four main resonances and the effect they have on asteroid population. There are other weaker ones that cause a lower number of asteroids to keep stable orbits, but they are no nearly as drastic. Two examples are 7:3, shown at 2.71 AU, and 9:4, at 3.03 AU.

There are a few other major resonances in our Solar System, including a 1:2 between Saturn's moons Dione and Enceladus and the 3:4 one between Saturn's moons Hyperion and Titan. However, the most famous resonance in our Solar System is the only known Laplace resonance, or one involving more than two bodies.

It is the 1:2:4 resonance between three of the four Galilean moons of Jupiter. Io is the 1, Europa the 2, and Ganymede the 4. This remarkable property supposedly emerged with a gravitational encounter of Ganymede and another body, resulting with the instability of Ganymede's orbit. Due to the gravitational pull of Io and Europa coupled with that of Jupiter, Ganymede settled into its position in the resonance.

Outside of the Solar System, many other orbital resonances probably exist, between exosolar planets, and one known example is a pair of planets orbiting the star cataloged as GJ 876. They are comparable in size to Jupiter, and both orbit extremely close to their parent star, within what would be the orbit of Mercury, but have a 1:2 resonance. This type of resonance among major, Jupiter-sized planets is more significant than any other one known, and is unique, as far as we know.

There are also other types of resonances, such as secular resonance, which is the alignment of the precessions of two bodies. Precession is the cycle in which the axial tilt of a body orbits around back to its starting position. For the Earth, the axial tilt is 23.44º, and it takes approximately 41,000 years to run a full cycle. During this time, the axial tilt varies from 22º to 24º. Where the north pole points in space is known as the celestial north pole. Over time, this pole moves over the heavens, and the north star therefore changes. Currently, the closest major star is Polaris, but over the next tens of thousands years it will progressively move away and then back when the cycle is complete. As a result, Vega will also be the north pole star for a time in each cycle.

This and other types of resonance are common throughout the Solar System within seemingly complex and random gravitational interactions.

Another type of synchronization that may be called a resonance, is the tidal force of a satellite on a parent body. Over time, the pair of bodies tug on each other gravitationally, pushing the slightly heavier side of each inward so that the period of rotational of the smaller body is eventually exactly the same as its revolution period, and is tidally locked. Many moons in the Solar System exhibit this feature and the most well known is our own moon. We only see one face of it because it is tidally locked to the Earth. In addition to this however, there also is a gravitational tug from the moon that is slowly tidally locking the Earth and the Earth day is slowly getting longer, increasing by mere fractions of a second each year. In addition, the two moons of Mars, at least eight of Jupiter's moons, fifteen of Saturn's moons, five of Uranus, two of Neptune, and the Pluto-Charon system (both bodies are locked to each other). Also, there is one known extrasolar instance of mutual tidal locking in which a star and its giant planet are tidally locked to each other.

One other special type of tidal locking is the Mercury-Sun system. For every two revolutions of Mercury around the Sun, Mercury revolves thrice on its axis. This relation is known to be stable, but it is a unique case.

Resonances are mysterious connections in the heavens which are difficult to find and even more difficult to understand.

Sources: http://www.cfa.harvard.edu/mpec/K08/K08S05.html, http://upload.wikimedia.org/wikipedia/commons/1/1c/TheResonantTNO_90AU.svg, http://adsabs.harvard.edu/abs/1983Natur.301..201D, NASA, http://adsabs.harvard.edu/abs/1991Icar...94..399M

Lagrange Points

As of 2006, one condition specifying an object as a planet is that the object has "cleared its neighborhood". This refers to bodies large enough to gravitationally fling objects out of their orbits. Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune have all done that. Pluto and its Kuiper belt counterparts, known as Plutinos, often share orbits with one another, ending up in a jumble of asteroids. It is easy to see, by this definition, that Pluto (and other dwarf planets such as Eris and Ceres) is clearly not a planet. However, the gas giants have not completely cleared their orbits. Mysterious "Trojan" asteroids follow them around their orbits. But how can this be?

The answer requires physics. There are actually five points within a system of to bodies in which an object can orbit in stability, despite being within or very near a larger object's orbit. These are known as Lagrange points, named after Joseph Lagrange. The discovery of these points occurred while Lagrange was exploring the famous three-bodies problem in 1772 (calculating a stable system for three bodies with their respective masses, speeds, and positions at any given time). Despite the simplicity of a two body system, an additional object makes the result very complex, and many scientists devoted their time to researching this problem. Contrary to previous belief, there are actually five points at which stability is attained (rather than three, which was the result found by Leonhard Euler 20 years previously). All objects at these points have a 1:1 (read "one to one") orbital resonance with the smaller of the two main bodies, meaning that for every orbit that it makes, an object at any Lagrange point makes one too. They are described in the picture below.



The five Lagrange points are illustrated above, along with a description below. They are denoted by L1, L2, L3, L4, and L5, while the arrows denote the forces acting on the areas around these points.

L1
If a line is drawn between the two massive bodies, L1 will lie on that line, in between the two, at the point where the gravitational pulls of each cancel each other out. In the case of the Earth-Sun system, this point is much closer to Earth, due to the fact that the Sun is much more massive. In most cases, a body orbiting closer to the Sun than Earth, i.e. an object at L1, would orbit faster and have a smaller revolution period. However, the gravitational pull of the Earth keeps an object at L1 from speeding up, and the object is literally held at that point all the way around Earth's orbit. It therefore has the same orbital period as the Earth, at one Earth year.

L2
L2 lies on the same line as L1, but is outside the Earth's orbit rather than inside it. Here, the force from the Sun would usually have such an object orbit slower than the Earth, but the Earth's gravity adds pull to the object, speeding it up, and keeping its year equal to one Earth year. Essentially, both L1 and L2 are points at which the forces of the Earth and Sun add or subtract to from exactly one Earth year.

L3
L3, as with L1 and L2, again lies on the line extended from between the two massive bodies, here the Earth and the Sun. This time, the body is opposite the Earth on its orbit, lying outside the Earth's orbit, but also closer to the Sun than Earth. But how can this be? The above statement appears to be a contradiction, but is actually possible due to the pull that the Earth has on the Sun. Both bodies actually orbit around the barycenter of the Earth and the Sun, which, despite being well inside the Sun, is not at the center. Because of this, the Sun is always slightly farther from the Earth than it would be for a less massive object, such as a body at L3, and a position satisfying both statements is therefore possible. At L3, the forces of the Earth and the Sun again balance, and the body has the exact same orbital period as the Earth.

L4 and L5
L4 and L5 are grouped together because they are very similar and both have the same properties. They are the most notable of the five Lagrange Points. If the Sun and the Earth form two points of two equilateral triangles, then L4 and L5 are located at the third point of each triangle. L4 orbits 60º ahead of Earth, and L5 orbits 60º behind. Due to the fact that the points are equidistant from the Earth and the Sun, they receive as much pull from the Sun as the Earth does, and they are kept stable by the Earth's gravity. However, the points L4 and L5 are stable if and only if the ratio of the larger body to the smaller one exceeds 25. This is true for the system of the Sun and any planet, as well as the Earth-Moon system. Otherwise, an object orbiting at these points will be perturbed and its orbit changed.

The five Lagrange Points are not only in physics, but have been observed. Some spacecraft have used the point as a base for observations. The Earth-Sun L1 point is best suited for observations of the Sun, due to the fact that the view from this point will never be disrupted by the Earth or the Moon. One spacecraft that has used this point to its advantage is SOHO (Solar and Heliospheric Observatory) launched in 1995. The Earth-Sun L2 point is ideal for deep space observations, because its view will never be disrupted by the Earth, the Moon, or the Sun. One spacecraft planned to take advantage of the L2 point is the James Webb Space Telescope, which is planned to be launched in 2014 as the successor of the famous Hubble Space Telescope. Here, no light from the Earth or Sun will damage the equipment, and an unhindered view of the wonders of deep space will be available. L3 does not have any specific use, but some people once believed that another planet similar to Earth, called "Counter-Earth" existed at this point. Since this point was never visible from Earth, the possibility was real until more sophisticated technology disproved this hypothesis. L4 and L5 are possible sights of space colonization, and a base could be set up in a stable orbit in one of these points, if partial or total evacuation of Earth ever became necessary. However, this does not seem imminently likely.

In a perfect system with only three bodies, the points L1, L2, and L3 are technically stable, but in a multiple body system like our own Solar System, an object staying at any of these points over a long period of time is impossible. The gravitational effect from the other planets will eventually disrupt any of these orbits. However, L4 and L5 are a very different story, indeed. They are much broader than the other three points, and perturbations will not always disrupt their orbits. Rather, a perturbation at these two points will change the orbit of the point slightly, increasing the eccentricity, but still keeping the objects in the range of the points. Due to this, there is matter floating around in these areas for both the Sun-Earth system and the Earth-Moon system. However, in both cases, the debris is simply interplanetary dust, and does not form large objects. In the outer planets, however, there are larger objects. Within the orbits of Jupiter and Neptune, there are many asteroids occupying these areas. The are referred to as Trojan asteroids. Also, two moons of Saturn, Tethys and Dione, have two smaller satellites each orbiting in perfect synchronization with them.

The Lagrange points are important for many uses, many of which could be important to humanity's survival in the future.

Sources:
http://www.esa.int/esaSC/SEMM17XJD1E_index_0.html
http://en.wikipedia.org/wiki/Lagrangian_point
http://en.wikipedia.org/wiki/Three-body_problem

Rosetta

Rosetta was a space probe launched by the European Space Agency whose mission is to orbit the comet 67P/Churyumov-Gerasimenko (comets are named as a combination of their catalogue number and their discoverers) and study it to learn more about the Solar System's origins.

This space probe consisted of two parts: Rosetta, the main spacecraft, and Philae, the lander that landed on the comet.

Rosetta was originally planned for launched in 2003, to study a totally different comet. But the launch was delayed, and a new trajectory was conceived. Rosetta was launched by the European Space Agency on March 2, 2004, beginning its hugely complicated series of flybys in space. No rocket could propel the spacecraft directly to the comet, so Rosetta's trajectory had to be long and complex. The first encounter was an Earth flyby, about a year after launch on March 4, 2005, where Rosetta tested its instruments and received a gravitational pull which sent it on orbit that would take it close to Mars. On February 25, 2007, Rosetta made a successful flyby of Mars, and it was set towards the Earth once again. As Rosetta approached the Earth for the flyby of November 13, 2007, an astronomer mistook the spacecraft for a small asteroid, about 75 feet in diameter, and predicted that it would pass within only a few thousand miles of the Earth. This caused momentary panic, until its track was recognized as Rosetta's.

After its second Earth flyby, Rosetta flew by by an asteroid, 2867 Steins, but did not use many instruments on it, to save most of its memory for the comet encounter. The asteroid was only about 2.5 miles across and Rosetta approached within 480 miles of the body on September 5, 2008. Although not much information was gathered on the asteroid, its orbit was verified more accurately than ever before.

Rosetta then made its last flyby of Earth on November 13, 2009.

On July 10, 2010, the probe encountered the asteroid 21 Lutetia. At its closest approach Rosetta was only 1960 miles away, and a significant amount of data was gathered. The ongoing analysis of this data will determine the composition of the asteroid. In addition, the exact orbit characteristics, mass, and rotation have been obtained, as well as many photographs.

Rosetta's image of the asteroid 21 Lutetia at closest approach.

After the second asteroid encounter, Rosetta shut off its systems in deep space in order to save its energy for the comet encounter. This stage is known has "Deep-Space Hibernation" and it began in May 2011. In total, this stage lasted for almost three years and ended on January 20, 2014. On this date, Rosetta sent a signal indicating that the hibernation had been successful and the spacecraft's systems were functioning normally. By this time, the comet had passed its apogee and it was once again approaching the Sun, so that Rosetta was close enough to power itself on the star's energy.

Even then, though, both the spacecraft and the comet were traveling relatively slowly. Rosetta began observations in January of that year as it approached the comet. In May, Rosetta began a series of maneuvers to prepare to orbit the comet. At the same time, a halo began to develop around the comet as it approached the Sun. The following image from early May shows this halo, or coma. This coma forms from sublimating ice and the release of gas trapped inside the comet's nucleus. In late June 2014, Rosetta measured that about 2 glasses of water a second were released from the comet in the form of vapor. This rate increased as the comet approached the Sun.

On July 14, Rosetta was close enough to obtain images of the body of the comet, revealing an unusual two-lobed structure (above). Images continued to increase in resolution as Rosetta approached.

Through July and early August 2014, the spacecraft continued to use fuel to decrease its speed relative to 67P/Churyumov-Gerasimenko, since orbit around such a small body ultimately required Rosetta's relative speed to the comet to be only 1 m/s (a typical walking speed)! In addition, Rosetta took its first temperature and surface readings. On August 6, 2014, Rosetta became the first spacecraft to ever enter orbit around a comet.

Rosetta had moved within 60 miles of the comet by this time, and began to map the surface with high-resolution imagery (such as the image above) to identify possible landing sites. By the end of August, 5 possibilities for a landing site had been chosen. During September, Site J, which the white plus sign marks on the image below, was selected for landing. In early November, this site was renamed "Agilkia" after an island on the Nile river.

On the days leading up to landing, Rosetta altered its orbit to release trajectory. During the morning of November 12, separation of Rosetta and Philae was confirmed. Seven hours later, the lander successfully touched down at the Agilkia site, but its harpoons did not activate, and the lander bounced twice, ultimately landing about a kilometer away from its original intended site.

Philae took images, such as the above, showing the comet from 40 m above during its initial descent.

Upon its final descent, Philae unfortunately landed on its side, with most of its solar panels facing the ground or in shadow. On November 15, after using its battery for 57 hours, the lander entered hibernation. However, during its 57-hour period of activity, Philae successfully used all 10 of its scientific instruments. This included a drill, which chemically analyzed a sample from the comet's surface and relayed the data to Earth via Rosetta. Notably, Philae did detect the presence of organic molecules on the comet's surface.

Important observations continued to pile up from the Rosetta orbiter over the next several months. During December 2014, Rosetta measured the composition of water vapor released from the comet as it approached the Sun. This vapor had more deuterium (the isotope of hydrogen with one neutron) than the water found on Earth. The measurement suggested that comets such as 67P/Churyumov-Gerasimenko (theoretically originating in the Kuiper Belt near Pluto) may not have been the primary source of water for Earth's oceans, which were likely filled by impacting solar system bodies.

In February 2015, as the comet continued to approach the Sun and grow more active, Rosetta departed its near circular orbit and positioned itself for several close flybys to collect gas released from the comet. The first of these flybys took place on February 14 at a distance of less than 4 miles!

Rosetta took the above image of the two-lobed comet on March 14, 2015. It has been enhanced to indicate the increasing volume of dust and gas streaming from the comet as it approached perihelion.

The same month, Rosetta also completed the first ever detection of molecular nitrogen at a comet. This offered crucial information as to the comet's origin, because molecular nitrogen can only become trapped within a comet's ice at very low temperatures. Therefore, the measurement supported the theory that 67P/Churyumov-Gerasimenko originated in the Kuiper Belt. During the spring, researchers also used data from both Rosetta and Philae to conclude that the comet was not significantly magnetized, the first such measurement for a comet. This was significant because iron was a major component of protoplanetary dust and some models indicated that assembled objects would have been magnetized in the early Solar System. The result gives another way to test theories of the formation of the Solar System.

During June, communication with the lander Philae was sporadically established as its solar panels were illuminated. However, no more investigations were able to be made. Later that month, the ESA officially granted an extension to the Rosetta mission through mid-2016.

The above images show the higher comet activity as it approached perihelion. The comet's closest approach to the Sun did not take it inside Earth's orbit, but it reached a minimum distance of 1.28 AU on August 13, 2015.

The mission continued to yield new results after perihelion had passed. For instance, using data from July 2015, it was able to measure and investigate the size of the diamagnetic cavity surrounding the comet's nucleus. This cavity is a region where the gas streaming off of the comet deflects incoming charged particles from the solar wind. Assuming the comet itself is not magnetized (which Rosetta determined it wasn't), this region should be free of magnetic fields. However, it only becomes appreciable in size near perihelion, when more material is ejected from the comet.

The above chart shows the magnetic field measured by Rosetta during a period of a few hours on July 26, 2015. The blue highlighted region (in which the field is nearly 0) corresponds to when the spacecraft passed through the diamagnetic cavity along its orbit. Rosetta passed into the edges of the cavity many times, allowing for detailed measurements of its size.

Continued analysis of data from near perihelion in August indicated that certain organic substances were present on Rosetta, including glycine, the simplest of the amino acids. Though the discrepancy in isotope ratios discussed above indicates that comets of this exact type did not transport these compounds to Earth in its early history, it supports the general claim that glycine can originate from comets, a conclusion hinted at by earlier study but never unambiguously verified.

Data from the probe indicated that the comet "breathed out" primordial oxygen near perihelion; that is, heating up the body of 67P released gases that had been trapped there for billions of years. While other mechanisms had been proposed for the molecular oxygen found around the comet, the Rosetta mission verified that a majority must have appeared via this process.

On September 5, 2016, as Rosetta began its final approach toward the comet, it spotted the lander Philae from a distance of 2.7 km after its precise location had not been known for two years! The image below has an astonishing resolution of 5 cm/pixel, and shows the lander in the bottom right, wedged in a crack. This explains why Philae was not able to operate under solar power after its landing.

Late on September 29, 2016, Rosetta's thrusters fired for the last time, sending it on a collision course with the comet. The spacecraft continued to transmit valuable data and images during its descent, taking advantage of its only opportunity to collect information from such a low altitude above the comet. The last image transmitted from Rosetta, shown below, was taken at an altitude of only about 20 meters, just before impact. The final impact occurred on September 30, ending communication with the probe.

The 12-year Rosetta mission provided invaluable knowledge about the structure, composition, and evolution of the comet 67P, and through it a better understanding of the formation of our Solar System billions of years ago. Comets are among the most pristine of time capsules for investigating the beginning of the Solar System as well as possible sources of water and other important molecules on Earth. The Rosetta mission will continue to shape our conception of our origin for years to come.

Sources: (Photos from ESA) ESA-Rosetta, http://en.wikipedia.org/wiki/Rosetta_(spacecraft),http://www.bbc.com/news/science-environment-27498534, http://www.astronomy.com/news/2014/07/the-twofold-comet-comet-67pchuryumov-gerasimenko, http://www.esa.int/Our_Activities/Space_Science/Rosetta/Rosetta_arrives_at_comet_destination, http://www.esa.int/var/esa/storage/images/esa_multimedia/images/2014/09/philae_s_primary_landing_site/14819792-1-eng-GB/Philae_s_primary_landing_site.png, http://www.esa.int/Our_Activities/Space_Science/Rosetta/Rosetta_s_comet_contains_ingredients_for_lifehttp://www.esa.int/Our_Activities/Space_Science/Rosetta/Rosetta_and_Philae_find_comet_not_magnetised, http://www.esa.int/Our_Activities/Space_Science/Rosetta/Rosetta_finds_magnetic_field-free_bubble_at_comet,http://www.esa.int/Our_Activities/Space_Science/Rosetta/Rosetta_s_comet_contains_ingredients_for_life, http://www.esa.int/Our_Activities/Space_Science/Rosetta/Mission_complete_Rosetta_s_journey_ends_in_daring_descent_to_comet

Tornadoes

This is the second part of a two part post. For the first part see here.

Simple thunderstorms are simply clouds developing at a frontal boundary, and these are the most common, but other, more complicated types also exist.

Although some thunderstorms from exclusively on the frontal boundary, many spawn other thunderstorms miles, or even tens of miles away whenever there is warm, humid air present. This is because thunderstorms cause downdrafts, or cool air to flow down from upper altitudes. When this air hits the ground, it spreads out and flows parallel to the ground. This moving air can help to create new thunderstorms in two different ways. Either the cold air pushes the warm air up, and as we already know, a warm updraft causes thunderstorms, or, a cool wind gust can encounter another one, and they push each other up, creating additional thunderstorms.

In many cases, thunderstorms are composed of individual units, called cells. These cells form and die in minutes, but as one dies, its downdrafts produces another cell, and on. The progression of these cells tend to move because cells form in humid air at the boundary of air masses, and as the boundary moves, the area of formation moves. However, there are cells that form in a more independent way then this.

The most powerful type of cell is known (rather appropriately) as the supercell. Like a normal cell, the supercell has an updraft fueling its development: a mesocyclone. A mesocyclone is a rotating column of air that feeds the storm with the moisture and rotation it needs to be powerful. A mesocylone is typically between 1 and 5 miles in diameter and can extend up to 10 miles in to the sky. It forms when strong high altitude winds and weak ground winds blow past each other, creating a tube of air. An updraft lifts this tube, and it eventually breaks into two tubes, one spinning clockwise, the other counterclockwise. Due to the Coriolis Effect, the clockwise column is neutralized in the northern hemisphere and the counterclockwise column is strengthened, while this is reversed in the southern hemisphere.

Once the mesocyclone has formed, if there is a frontal boundary present, a colossal anvil shaped cumulonimbus cloud is formed, often stretching over 60,000 feet into the air! Warm middle layer winds feed the cyclone, while upper level winds spark fierce downdrafts. In addition, the supercell creates such a temperature difference in the air, that its downdrafts create their very own mini-frontal boundary. All this, coupled with the powerful mesocyclone at the center, brings heavy rain or hail for hours, and a supercell can travel over 300 miles before dying out.

As if this wasn't enough, sudden huge downdrafts called microbursts can also form during the duration of a supercell. Often, this downdraft pushes down another coil of air spinning parallel to the ground, again forming two columns. The counterclockwise swirl near the ground is expanded by updrafts to form one huge rotating air mass, from the ground way into the clouds. This is the tornado.



Tornadoes are immensely powerful and are actually, contrary as it is to common belief, invisible. Only when debris is picked up by the tornado does it assume a color and this depends totally upon the terrain. Tornadoes range from virtually harmless to incredibly deadly and can last from seconds to hours. Also, multiple tornadoes can come from one mesocyclone and often orbit around each other within it, assuming the form of a funnel cloud, which appears to be one tornado and can be largely hidden among clouds until it hits.



A funnel cloud hidden in the atmosphere. There is a vague suggestion of a rotation in the cloud structure, but there is no obvious tornado, like there is above.

Although a majority of tornadoes are spawned from supercells, some do not. It is possible for a weak tornado to develop from a regular thunderstorm, but this is uncommon, and a tornado in this situation is usually so weak that it is virtually undetectable. In some cases, tornadoes also from over water, forming what is known as a waterspout. As with land tornadoes, the waterspout assumes the color of water, and in most cases, appears blue. Some waterspouts have been known to develop mysteriously in fair weather, but most, known as tornadic waterspouts, form in thunderstorms. Hurricanes can be a source of tornadoes and waterspouts, when circulations form in their thunderstorm bands, and some hurricanes have caused over 100 tornadoes! Three Atlantic hurricanes have attained this: Hurricane Beulah in 1967, with about 115, Hurricane Frances in 2004, with 103, and Hurricane Ivan in 2004, with 117. Of Frances's tornadoes, 6 were considered significant, or with winds exceeding 110 mph, and Ivan had 19 significant tornadoes.



A waterspout off Florida's coast.

Also, dust devils, or swirling gusts of wind, can form in a dry environment, usually a desert. One last type of rotating mesocyclonic structure is a "fire whirl" or "steam devil". If there is a significantly heated environment, as in a fire, or heated water, the updrafts caused by the heated air can be strong enough to start a mesocyclone. The latter types of rotating mesocyclonic structures are not associated with thunderstorms, but are related since they are similar to tornadoes. They are usually weak in comparison to a real tornado.



A dust devil that formed in Nevada. Note the fair weather conditions.

Until recently, most areas used the Fujita scale to measure tornado strength. However, the U.S. thought the categories and their respective damages educated guesses at best, and the scale was changed into what is now known as the Enhanced Fujita Scale.



A comparison of the original Fujita scale introduced in 1971, and the new Enhanced Fujita Scale, which was first used in 2007. Although the Fujita scale covers a higher range of wind speeds, the Enhanced version gives more accurate damage information, and considers that anything under 65 mph is not a tornado at all, but perhaps a spinning funnel cloud that doesn't reach the ground.

The most powerful tornado recorded had peak winds around 300 mph, and wreaked havoc in Oklahoma City in 1999. This tornado also caused the second most damage of any single one, with adjusted damages of $2 billion. However, a tornado in 1896 has the greatest adjusted damage value, at a stunning $2.9 billion. In terms of width, one tornado had a damage path width of 2.5 miles in 2004. In terms of path length, the infamous Tri-State Tornado is the record holder, traveling 219 miles over three states, Missouri, Illinois, and Indiana. It also holds the record for the deadliest U.S. tornado with 695 deaths recorded. The huge F5 tornado wreaked havoc in 1925. In the world, the deadliest tornado probably belongs to the Bangladesh 1989 tornado which took an astounding 1300 lives. Finally, the largest tornado outbreak occurred in 1974, on April 3 and 4 of that year. An extremely powerful system spawned an amazing 148 tornadoes recorded and probably more weak ones going unrecorded. Among these, 23 were F4 and 6 were F5! The number of major tornadoes was simply unheard of, and this outbreak is still, after over 35 years, the largest one in history.



All of the tracks of the tornadoes in the Super Outbreak of 1974. They are numbered in order of development, from 1 to 148.

There have been many exceptional tornado oddities, most involving debris and people being picked up and placed back down unharmed. There have been many claims of this type and some have been disputed. For more information, see here.

Tornadoes are among the most violent weather events on the Earth, and they are the most deadly, there being no accurate way to predict them. Although tornado watches and warnings are put into effect when there is a risk or a sighting of a tornado, these can only be given a few minutes in advance. Climatology wise, tornadoes occur in various parts of the world. As well as the U.S. southern Canada and northern Mexico, much of Europe, South Africa, areas of Argentina, Australia, and Eastern Asia also experience tornadoes regularly.

Currently, Earth is the only celestial body on which tornadoes have been detected. There are rotating systems on other planets, such as Jupiter (the Great Red Spot and various other systems), and many of these may feature mesocyclones, but no real evidence for extraterrestial tornadoes has been collected.

Sources: The Weather Book by Jack Williams, http://www.srh.noaa.gov/mfl/?n=waterspouts, http://www.nssl.noaa.gov/edu/safety/tornadoguide.html, National Hurricane Center