Friday, May 27, 2016

Tropical Depression Two (2016)

Storm Active: May 27-

Around May 24, a broad area of scattered showers and thunderstorms began to develop in association with a low-pressure trough situated to the north of Hispaniola. Over the next few days, convection gradually became more concentrated near the low-pressure center as it deepened and moved generally toward the west-northwest. By the afternoon of May 27, the circulation had become well-defined with a curved band of strong thunderstorms just to the north and west of the center. This resulted in the designation of the system as Tropical Depression Two.

As of 5:00 pm EDT on May 27, 2016, Tropical Depression Two had sustained winds of 35 mph, a minimum pressure of 1009 mb, and was moving to the west-northwest at 13 mph. Tropical storm warnings are in effect for the coast of South Carolina.

Wednesday, May 18, 2016

Professor Quibb's Picks – 2016

My personal prediction for the 2015 North Atlantic Hurricane Season (written May 18, 2015) is as follows:

14 cyclones attaining tropical depression status*,
13 cyclones attaining tropical storm status*,
7 cyclones attaining hurricane status*, and
3 cyclones attaining major hurricane status.
*Note: Hurricane Alex formed on January 13, long before the official start of the season on June 1 and before I made these predictions.

This prediction calls for a nearly average Atlantic hurricane season, with predictions just barely exceeding historical averages in all categories.

The picture for the 2016 Atlantic hurricane season is unusually murky, due to several uncertainties regarding significant factors that influence tropical cyclone formation. First, the 2015-16 El Ninò event has continued to unfold, ranking in the top 3 historically in both intensity and duration. Positive sea surface temperature anomalies have persisted into May in the equatorial Pacific, indicating the continuation of the event. The chart below compares El Ninò events since 1950.

The 2015-16 event (black line) is probably most comparable to the 1997-98 event in its qualities, so if this trend were to repeat, the 2016 season would end with the ENSO in a negative phase. However, it has occurred that El Ninò events persist to the end of the second year, or that they become roughly neutral. A neutral ENSO (El Ninò Southern Oscillation) index, all else held equal, would lead to an average hurricane season, and a negative index to a more active season. The latest predictions indicate that neutral conditions will in fact prevail during the season's peak in September and October, but there is a great deal of uncertainty.

Second, the Atlantic Multi-Decadal Oscillation (AMO) (an empirically observed trend in tropical cyclone activity that has decades-long period) appears to be wrapping up the positive phase that led to busier hurricane seasons during the 2000's and early 2010's. However, this trend is harder to predict than the ENSO, and while some meteorological experts believe that it is now entering its negative phase, it is difficult to know for certain. The combination of these two factors yield an expectation of an average season, but with an unusually high probability of deviance from this prediction.

Finally, we examine a few more proximate factors to cyclone formation in the Atlantic. Current mean sea surface temperatures, as with all global temperatures, are anomalously high relative to historical data. However, temperatures in the Gulf of Mexico and along the U.S. Eastern seaboard are lower relative to average than the southern Caribbean and central tropical Atlantic. These latter areas may therefore be especially favorable to cyclonogenesis. Normally, preseason wind shear tendencies would also be relevant to my forecast, but due to the possible rapid changes in the ENSO index, these observations would have little predictive power.

My estimated risks for different parts of the Atlantic basin are as follows (with 1 indicating very low risk, 5 very high, and 3 average):

U.S. East Coast: 2
Neither the jet stream nor the negative anomaly in sea surface temperatures is as pronounced in this region as in 2015. Nevertheless, wind shear may still inhibit development in this region, leading to a lower risk of landfalls.

Yucatan Peninsula and Central America: 4
The southern Caribbean has some of the most anomalously warm temperatures in the Atlantic, and could fuel tropical cyclones that traverse it. After upper-level winds subside about midway through the season, there is potential for dangerous hurricanes to develop in this region.

Caribbean Islands: 3
The Caribbean Islands are at about average risk this year, with moderately warm temperatures and a diminishing El Ninò that will lead to a fair, but not exceedingly high likelihood of westward-tracking cyclones. Expect 2-3 tropical storms, at least one of which is of hurricane strength, to affect the islands.

Gulf of Mexico: 2
The Gulf remains rather safe this year, continuing the trend from the previous two seasons. Rather low temperatures will limit the potential for significantly damaging landfalls.

Overall, the 2016 season is expected to be around average, but there is an unusually low degree of confidence in this forecast due to expected shifts in climate throughout the year. Regardless, everyone should take sufficient preparedness measures, since dangerous storms can occur even in quiet seasons.


Saturday, May 14, 2016

Hurricane Names List – 2016

For the North Atlantic Basin, the list for naming tropical cyclones in 2016 is as follows:


This list is the same as that for the 2010 season, with the exception of Ian and Tobias, which replaced the retired names Igor and Tomas, respectively.

Saturday, April 16, 2016

ExoMars Mission

ExoMars, or Exobiology on Mars, is a mission jointly run by the European Space Agency (ESA) and the Russian Federal Space Agency (Roscosmos) to investigate possible traces of life on the planet Mars. The mission includes two launches: one in 2016 and one in 2018, with the first delivering an orbiter and a lander to Mars and the second the ExoMars rover.

The first launch took place on March 14, 2016 in Kazakhstan using a Russian-built launch vehicle. Both the Trace Gas Orbiter (TGO) and the Entry, Descent, and Landing Demonstrator Module (EDM) will arrive in the Martian system in October 2016.

The primary mission of the TGO, as the name suggests, is to refine our measurements of the scarcer components of the Martian atmosphere, including methane and water vapor. From an orbit about 250 miles above the surface of the red planet, the orbiter will obtain information orders of magnitude more accurate than any previous results. Methane in particular is generated by specific geological and organic processes. While the Trace Gas Orbiter would not be able to identify the cause of gaseous emissions by itself, it can pinpoint the sources geographically, aiding in the selection of the ExoMars rover landing site. The orbiter itself was constructed by the ESA while the Russian agency contributed several of its instruments.

Meanwhile the EDM lander (also called Schiaparelli after the Italian astronomer Giovanni Schiaparelli) will demonstrate crucial techniques for landing on the Martian surface shortly after the first spacecraft arrives at Mars. Weighing over 1300 pounds, the lander requires a controlled landing to reach the Martian surface safely, just like the Curiosity rover. The probe will use a heat shield and parachutes to slow its descent and a liquid propulsion braking system to control its final touchdown on Mars. In addition, the static lander will carry instruments to record the landing. Communicating through the TGO, the lander will transmit its data after the fact to guide future landings.

While Schiaparelli's main purpose is the demonstration of landing technologies, it also carries a science payload that will operate for roughly 2-8 (Earth) days after its arrival on the surface. The onboard instruments suite called DREAMS will provide standard weather measurements such as humidity, pressure, wind speed, and temperature. In addition, a camera onboard will capture images of the landing itself. Finally, the lander will measure atmospheric transparency and search for electric fields on the red planet's surface, the first such measurement of its kind.

The second launch will occur sometime in the latter half of 2018, carrying the European-built ExoMars rover and a surface platform on which it will land, contributed by Roscosmos. The spacecraft will arrive at Mars in early 2019 at a landing site chosen with help from the 2016 mission's data. The same technology demonstrated in the first landing will allow the second module to perform a soft touchdown on the surface of Mars. After landing, the surface platform will deploy ramps, off of which the rover will exit to begin its exploration of the surface.

The rover's mission will last at least six months. Its primary mission will be to search for organic substances on the Martian surface. Since the harsh conditions of the surface may have obliterated traces of chemicals, the ExoMars rover will have the ability to bore holes as deep as two meters to obtain better preserved samples. After collecting samples, the rover will transfer them to its onboard laboratory for chemical analysis. With its careful site selection and dedicated exobiology instruments, the ExoMars mission has perhaps the best opportunity yet of discovering definitive biosignatures on Mars. It also would accomplish the technological objective of honing the ability to make soft, precision landings on the red planet. Finally, the mission paves the way for the holy grail of Martian exploration: returning a sample from the red planet back to Earth. Sources:,

Saturday, March 26, 2016

The Projective Plane: An Algebraic Exploration II

This is the third post in a series discussing the projective plane. For the first, see here.

The previous post explained how certain types of polynomials, namely the homogeneous polynomials, define curves in the projective plane called projective varieties. This post will explicate the relation between projective varieties and affine varieties (typical curves in the plane) and indicate how projective varieties are in a way the extensions of affine varieties to include their points at infinity.

First, we consider how projective varieties naturally give rise to normal affine plane curves. Consider the projective variety defined by the equation F(x:y:z) = 0, where F is a homogeneous polynomial. In the first post of this series, we saw that the plane z = 1 in three-dimensional space can represent the subset of the projective plane that corresponds to the normal affine plane (i.e., without the points at infinity). We repeat the image from the first post for convenience, where each line through the origin (a point of projective space) is represented by the point at which it intersects the plane.

Since we obtain the affine plane by setting z = 1, it seems reasonable that we should be able to "collapse" projective varieties algebraically by setting z = 1 in the equation F(x:y:z) = 0. This is indeed the case, since substituting z = 1 yields a polynomial in only two variables: f(x,y) = F(x,y,1). For example, if F(x:y:z) = x2y + 2yz2 - 5z3 (note that F is homogeneous and therefore defines a projective variety), then f(x,y) = F(x,y,1) = x2y + 2y*12 - 5*13 = x2y + 2y - 5. The projective variety F(x,y,z) = 0 therefore corresponds to an affine variety f(x,y), as desired.

There is also an algebraic process that does the reverse by taking a polynomial f(x,y) and producing a corresponding homogeneous polynomial in three variables, F(x:y:z). The process works as follows:
  1. Add up the powers of x and y in each term of f and let n be the greatest degree that appears
  2. Multiply each term by zn-k, where k is the degree of the term (this ensures that the resulting polynomial is homogeneous)
Let us take the polynomial f(x,y) = x2y + 2y - 5 from above and apply the new process. The degrees of the three terms are 3, 1, and 0, respectively (-5 = -5x0y0). The maximum of these is 3, so we multiply the first term by z3-3 = z0 = 1, the second by z3-1 = z2, and the third by z3-0 = z3. The resulting polynomial is F(x:y:z) = x2y + 2yz2 - 5z3, exactly the same as the original polynomial! It is easy to verify that these two processes are mutually inverse in general, except in certain special cases such as when F is only a function of z.

Now we may apply these algebraic tools to solve the problems introduced in the last post that cannot be solved visually. First, regarding the hyperbola, algebra confirms our intuition. To see this, take the equation xy - 1 = 0 and transform it into the corresponding projective variety. The result is easily calculated as xy - z2 = 0. The asymptotes x = 0 and y = 0 to this hyperbola (see image in previous post) are unchanged by the process since they only have one term, clearly of maximal degree.

Next, recall that the points at infinity in the projective plane are those for which z = 0 in the homogeneous coordinates (x:y:z). This can be seen in the above visualization, where only points of the form (a:b:1) belong to the affine subset of the projective plane. Now any (x:y:z) can be scaled to this form by multiplying each component by 1/z (remember, only the ratio of the coordinates matters), but only when z is nonzero. Therefore, we substitute z = 0 and solve the equations to see which points at infinity each curve intersects. For the hyperbola, this gives xy = 0, so x = 0 or y = 0. Therefore, the two points at infinity the hyperbola intersects are (0:1:0) and (1:0:0). Other coordinate triples satisfying xy = 0 such as (3:0:0) differ only by a scale factor from one of the two solutions above and therefore define the same point in the projective plane. It follows that (0:1:0) and (1:0:0) are the only solutions. But the asymptotes x = 0 and y = 0 hit exactly the same points, (0:1:0) and (1:0:0), respectively! This confirms our intuition: a hyperbola and its asymptotes really do intersect at infinity.

The cubic y - x3 = 0 has no asymptote, but clearly goes off to infinity in some manner. We may use our algebraic tools to investigate the function's behavior in the projective plane. The highest degree term is x3, of degree 3, so we must multiply the other term (namely the expression y, of degree 1) by z3-1 = z2. The projective variety corresponding to the cubic is therefore defined by the equation yz2 - x3 = 0. Substituting z = 0 yields x3 = 0, which has the single point (0:1:0) in the projective plane as a solution (since z is already set to 0). Note that even though the cubic goes to infinity in both the positive and negative directions, it meets only one point at infinity because opposite directions are identified (see the representation of the projective plane in a sphere in the first post). This indicates that the projective variety induced by the cubic meets that induced by the y-axis with equation x = 0 at infinity. Indeed, this makes some intuitive sense: as x becomes very large, it becomes insignificant relative to y = x3 and therefore the point (x,y) is "close" to the y-axis x = 0 (this can also be seen by zooming out a graph of the cubic - the graph eventually becomes nearly indistinguishable from the y-axis). We can also visualize the projective variety yz2 - x3 = 0 that extends the cubic on the sphere (see below).

This image shows the cubic curve in the affine plane as well as its projection (via lines through the origin, the center of the sphere) onto the surface of the sphere. It differs slightly from our earlier sphere representation since the plane is below and not above the sphere, but this makes little difference. At the bottom of the sphere, the origin of the plane touches the sphere (which is a point on the curve). At first, the path veers away from the y-axis (the grid line from top to bottom through the origin), but notice how when the curve approaches the equator of the sphere (infinity), it comes back to hover above the y-axis. Images like these help to interpret the results of our algebraic manipulations.

The projective plane has very elegant geometric properties (every two lines in the plane intersect in exactly one point, for example) and gives us a sturdy mathematical grounding for the slippery concept of behavior "at infinity." Generalizations of this concept are crucial in the study of polynomial curves and their corresponding equations.


Saturday, March 5, 2016

The Projective Plane: An Algebraic Exploration I

This is the second post in a series on projective space. For the first, see here.

The idea of "adding points at infinity" to the plane introduces new behavior to the study of the intersections of lines and curves. Since there are different points of infinity for each direction in the affine plane (as discussed in the last post) and parallel lines intersect at infinity, it is reasonable to suppose that certain lines and curves also intersect at infinity (see below).

For example, the hyperbola above is given by the equation xy = 1. Away from the origin, the two branches of this curve approach the x- and y-axes, defined by the equations y = 0 and x = 0, respectively. Since the distance between the curve and these lines (called asymptotes of the curve) approaches 0 far from the origin, it makes sense to suppose that the hyperbola intersects with these asymptotes at infinity. On the other hand, for other curves that clearly "go off to infinity" like the cubic curve y = x3 shown below, there is no asymptote. What point at infinity, if any, does the cubic intersect?

Answering this question is difficult in our as yet fuzzy picture of the structure of the projective plane. However, the algebraic definition of the projective plane provides the tools necessary for solving this and many other related problems. Introducing this machinery is the purpose of this post.

Lines, parabolas, cubics, hyperbolas and many other curves in the plane may be expressed in the following algebraic form: f(x,y) = 0, where f is a polynomial function of x and y. This means that it is a sum of terms of the form a*xiyj where i and j are nonnegative integers and a is a constant coefficient. For example, the equation for the hyperbola written above may be written xy - 1 = 0 and the equation for the cubic y - x3 = 0. Any curve defined by an equation of this form is known as an affine variety.

The previous post introduced the projective plane as the set of lines through the origin in three-dimensional space. It then illustrated two different ways in which certain representatives may be chosen from the lines to get a "picture" of the projective plane in three-dimensional space. We show that for equations of certain forms, it does not matter which representative we choose from a given line through the origin. First, let P = (x,y,z) be a point distinct from the origin in three-dimensional space (that is, at least one of x, y, and z is nonzero). Then since any two distinct points determine a line, P determines a unique line L through the origin. The point (ax,ay,az) is then on L for any constant a and every point on L is of this form. In other words, only the ratio of the coordinates to one another is required to determine on which line through the origin a given point lies. This fact can more easily be seen in two dimensions, as in the figure below.

The line above has equation y = 2/3*x. It passes through the origin and has slope 2/3, so any point (x,y) for which y/x = 2/3 is on the line (as demonstrated by the construction of a suitable triangle, as above). With this fact in mind, we introduce the concept of homogeneous coordinates. Homogeneous coordinates (x:y:z), where at least one is nonzero, define a point of the projective plane, with the understanding that only the ratio of x, y, and z matters. Thus (1:2:3) = (3:6:9), for example. With these identifications in mind, every point in the projective plane may be assigned homogeneous coordinates (though in many equivalent ways).

Next we consider projective varieties, i.e. certain types of curves in the projective plane. As before, they are defined as the set of points satisfying a certain polynomial equation, but in three variables instead of two: F(x,y,z) = 0. However, in light of the equivalence between points with different x-y-z coordinates, we must consider only polynomial equations that have the same points of the projective plane as solutions for any coordinate representation of the given points. These are called homogeneous polynomials. A polynomial is homogeneous if each one of its terms, or monomials, is of the same degree, meaning that the sum of the exponents in each term are the same. For example, F(x,y,z) = x2yz3 is trivially homogeneous of degree 6 because it has only one term and the sum of its powers are 2 + 1 + 3 = 6. F(x,y,z) = xy2 + z3 is homogeneous of degree 3 because the sum of the exponents of the xy2 term is 1 + 2 = 3 and is obviously also 3 for the second term, z3. The crucial property of homogeneous polynomials is that if F(x,y,z) = 0, then F(ax,ay,az) = 0 for any constant a:

The crucial fact used in the proof (click to enlarge) is that the exponents of each term (the p's, q's, and r's) must always add up to the same degree n. The an term can then be factored out, confirming that F(x,y,z) = 0 always implies F(ax,ay,az) = 0. This means that for any point in the projective plane, a homogeneous polynomial that is zero on one representative is zero on all. Conversely, if F(ax,ay,az) = 0, then the same proof (using 1/a) shows that F(x,y,z) = 0 so long as a is not zero. All this manipulation distills down to the following crucial statement: it is meaningful to say that a homogeneous polynomial is zero at a point in the projective plane since any representative gives the same result. We can thus denote projective varieties by the equation F(x:y:z) = 0 in homogeneous coordinates.

It follows that a homogeneous polynomial in three coordinates has a solution set of points (a curve) in the projective plane. These solution sets are the projective varieties. The next post (coming soon) continues to fill in the algebraic picture of the projective plane and relates affine varieties to projective ones, ultimately answering the questions posed at the beginning of this post.


Saturday, February 20, 2016

The Detection of Gravitational Waves

For an introduction to gravitational waves, see here.

Before 2016, a nobel prize had already been rewarded for an observation that was consistent with, and seemed to confirm, the existence of gravitational waves. In 1974, Russell Hulse and Joesph Taylor discovered a very compact binary system of objects at a distance of 21,000 light years, consisting of two neutron stars orbiting one another. One of the bodies was also a pulsar, meaning that the radiation beams emitted from its poles periodically point toward Earth as it rotates. Since the rotation rate of a neutron star changes only very slowly over time, pulsars are fairly precise clocks. However, Hulse and Taylor detected that the pulses did not reach Earth precisely on time, but varied slightly from the expected arrival time. They were sometimes sooner, sometimes later in a regular pattern, indicating that the pulsar in question was in fact part of a binary system.

The above diagram depicts the binary system consisting of pulsar B1913+16 and its companion, another neutron star. No radiation from the companion has been observed on Earth, indicating that its poles oriented away from us. However, its presence can be inferred from the fact that the pulsar moves farther and closer to Earth in a short, regular period, indicating an orbit. The difference in arrival times is about 3 seconds, indicating that the orbit is about 3 light-seconds across. Further, the orbital period is 7.75 hours.

This discovery provided an excellent opportunity to confirm the predictions of general relativity: such a compact system with rapidly orbiting masses would radiate fairly large quantities of gravitational radiation. However, direct detection was well beyond 1970's technology. Instead, Taylor observed the pulsar system over a number of decades, and found the following:

Since the discovery of the pulsar, its orbital period had been decreasing very slowly, though steadily and measurably, by about 35 seconds over a timespan of 30 years. This is very little relative to the total period of 7.75 hours, but the data matched the predictions of general relativity almost precisely: as energy was lost to gravitational waves, the neutron stars gradually spiral inward toward one other as their orbits becoming shorter and shorter. This remarkable confirmation of a prediction of relativity won Hulse and Taylor the Noble Prize in physics in 1993.

And there the matter sat. Though detectors grew more and more advanced, no direct detections of gravitational waves were made for over 20 years. This all changed in 2015.

On September 14, 2015, at 09:50:45 UTC, shortly after LIGO (the Laser Interferometer Gravitational-Wave Observatory) resumed activity following an upgrade, the two detectors in Washington State and Louisiana picked up a transient gravitational wave signal, the first ever observed by humankind. The announcement of the discovery was made several months later, on February 11, 2016.

The above image shows the signals recorded at Hanford, Washington (left) and Livingston, Louisiana (right). The signals are also superimposed on the right to demonstrate their similarity. The horizontal axis is time, measured relative to 09:50:45 UTC on that day. The reader may notice that the event was distinguishable from the surrounding noise in the detector for only about 0.05 seconds (the third row charts the residual noise after the theoretical waveform in the second row is subtracted out). The final row shows the rapid increase in gravitational wave amplitude during the event and the subsequent silence. The vertical dimension in the first several rows is the relative strain on the detectors, or the amount by which the different arms of LIGO were stretched or compressed by the ripples in spacetime. The scale for these axes measures strain by parts in 10-21. This corresponds to extraordinarily minute changes in length: the 4 kilometer arms of the LIGO detector changed by only about 10-18 meters, only about one thousandth the diameter of a proton!

The theoretical wave form above was a simulation of the event that generated the gravitational waves: the final in-spiraling and ultimate merging of two black holes. The increasing frequency and amplitude of the signals corresponds to the final moments of the collapsing system as the two black holes orbit faster and faster and tighter and tighter around one another before finally combining. Further, the signals at the two detectors were separated by 6.9 ms, smaller than the light travel time between the sites of 10 ms. The delay between the arrival times allows the direction of the source to be identified.

This image shows the region in the sky from which the signals likely originated. The colors indicate the confidence that the source lay within the indicated region: purple is the 90% confidence region and yellow the 50% confidence region. The uncertainty arises from the fact that there were two detectors, and not the three required for a full triangulation.

In addition to the location of the source, the analysis of the waveform yields more. The distance of the system was roughly 1.2 billion light-years, meaning that the merger that we are just now observing occurred over a billion years ago. The two black holes had respective masses of about 36 and 29 solar masses, while the final black hole after the merger weighed in at 62 solar masses. This corresponds to a loss of about 3 solar masses, which was all converted into energy released as gravitational waves as the holes merged. The magnitude of this cataclysm can scarcely be overstated: at its peak, the rate of energy release was an estimated 3.6x1049 W, greater than the radiation emitted from all stars in the observable universe combined!

In addition to being a resounding confirmation of general relativity, the observation was the first truly direct detection of black holes: the fact that such massive objects came within hundreds of kilometers of one another indicates that they had extremely high densities, densities only possible in black holes. But while significant, cosmologists were already nearly certain that both gravitational waves and black holes existed. However, this discovery marks the opening of a brand new field of astronomy. Gravitational waves, which pass unimpeded through nearly anything over nearly any distance, allow us to "hear" cosmic events that we could not have detected before. In theory, these waves could allow us to observe the earliest stages of the universe, before it became transparent to electromagnetic radiation. In 2016, 100 years after Einstein predicted gravitational waves, we took the first step towards seeing the universe in a new way.


Friday, February 12, 2016

The Projective Plane: A Visual Introduction

The statement "any two distinct lines intersect in a point" is almost true in normal plane geometry. The exception, of course, is the case of two parallel lines. However, from real experience we know from the rules of perspective that two parallel lines "converge" very far away, even if we know that they in fact maintain the same distance apart.

From this, we naturally comes the intuition that "parallel lines intersect at infinity." Certainly this tidies up our intersection statement because it provides a way for even parallel lines to intersect. But what does "at infinity" mean? Is there really a "point" there? The notion of projective space makes these ideas explicit and rigorous.

We focus on the (real) projective plane, the extension of the normal plane to include these "points at infinity" where parallel lines intersect. The set of points in the projective plane is defined, somewhat enigmatically, as "the set of lines through the origin in three-dimensional space." Defining each point to be a line in a different space seems extremely confusing at first, but there are multiple ways to visualize this concept.

The first method of visualization illustrates how the projective plane is related to the ordinary plane (sometimes called the affine plane). Consider three-dimensional space with ordinary coordinates x,y, and z as shown. The plane labeled z = 1 contains all points for which the z coordinate is 1, namely all those of the form (x,y,1). Clearly this plane is just like the ordinary two-dimensional plane (under the correspondence (x,y,1) → (x,y)), only embedded in three dimensions, like a flat sheet of paper in our world (but infinite). The dotted line shown that passes through the origin intersects the plane at the particular point (a,b,1). Remembering that the projective plane is meant to be an extension of the affine plane, we identify the dotted line with the point where it intersects the plane. Clearly, for each point in the z = 1 plane, there exists exactly one line through the origin and the given point. This shows how the ordinary plane is a subset of projective space (the set of lines through the origin)!

However, not every line through the origin intersects the plane z = 1. For instance, the x-axis, the y-axis (both shown), and any other line in the plane of these two axes only contain points for which the z coordinate is 0 and can never intersect the plane z = 1 (to be clear, the three-dimensional space considered here does not have its own points at infinity!). Therefore, these lines cannot correspond to points on the ordinary plane. These special lines, in fact, are the points at infinity in the projective plane.

The above visualization illustrates the connection between the projective plane and the affine plane. It also indicates that there are many points at infinity, one for each line through the origin "lying flat" in the xy-plane. However, it fails to indicate how points at infinity are truly the intersections of parallel lines. For this, we use another visualization that chooses different representative points.

Using a sphere (or a hemisphere, to be more precise) to represent the projective plane is just as legitimate as using a plane: all that matters is that there is one point for each line through the origin. It does not matter which points we choose.

In fact, nearly every person is intimately familiar with this representation of projective space! Imagine that it is a clear night and you go out to look at the stars. You catch sight of the familiar constellation Orion, the hunter. The stars marking Orion's shoulders are Betelgeuse and Bellatrix, which we perceive to be neighboring stars that connect to form the figure of Orion. In fact, however, Betelgeuse is between two and three times as distant as Bellatrix. When we look up at the sky, we do not perceive the true three-dimensional space but points of light etched into the inner surface of the celestial sphere passing overhead. Stars in similar directions, regardless of their distances, are projected onto nearby points. This is why the result of treating all points along a line through the origin as equivalent is known as projective space.

It is clear, however, that every line through the origin intersects the sphere at exactly two points, while there can only be one representative for a point of projective space. Thus, by convention we consider only intersections with the upper hemisphere (just as in our example of the night sky - one cannot see stars looking downward!). This leaves only the "horizontal" lines intersecting the equator of the sphere twice. For these, we choose the points of intersection for positive y-values (the area colored dark green above) and finally the x-axis is represented by the dark red point of positive x. The projective plane is therefore the union of the yellow upper hemisphere, the dark green semicircle, and the dark red point. The latter two parts are the points at infinity.

The above image shows how the affine plane (and our first visualization) relate to our second visualization of the projective plane as (part of) a sphere. Lines through the origin (O) and a point in the upper hemisphere intersect the plane to form a one-to-one correspondence. As we would expect, points at infinity correspond to lines through the sphere's equator that are parallel to the plane and are therefore not part of our original affine plane.

Finally, the sphere illustrates how the projective plane solves the motivating problem of parallel lines than began this post.

Two parallel lines in the plane correspond to precisely the same lines in our first visualization, which indeed embeds a "copy" of the affine plane in three-dimensional space. When these parallel line are transferred to the sphere in the same manner that the point was above (remember: each transferred point represents a line through the origin and "transferring" a point is merely choosing a different representative), the figure above is the result. However, it is evident that the resulting arcs on the sphere intersect at the equator (green circle) and we know the equator contains the points at infinity! Though there appear to be two intersections, recall that points diametrically opposite from one another are on the same line through the center, so that these points are identified as one in the projective plane. We have our desired result: two parallel lines intersect in exactly one point.

The next post provides an algebraic description of the projective plane and explores more of its properties.

Sources: Algebraic Curves: An Introduction to Algebraic Geometry by William Fulton,,,,

Friday, January 22, 2016

Applications of Ion Propulsion

This is the second part of a two-part post. The first post, describing the function of ion propulsion engines, may be found here.

Ion thrusters have many advantages over other forms of propulsion. In comparison with traditional chemical propellent, they are roughly 10 times more efficient, lightening the load of space-traveling craft and saving massive amounts of fuel for launch. This efficiency originates in part from the higher exhaust velocity of the xenon propellant particles, which are ejected from the spacecraft at speeds of 20-50 km/s! In addition, the electric power required to run ion engines is relatively small, on the order of a few kilowatts. In comparison, a typical microwave oven consumes 1.1 kW, and the power consumption is significantly less than that of a typical automobile. Solar panels can meet this power demand during flight, allowing ion thrusters to create smooth and continuous acceleration over the entire duration of a mission.

However, this efficiency comes at a price: ion propulsion produces very small thrusts. The first model of ion engine actually used in spaceflight, the NSTAR engine, produced a thrust of around 90 millinewtons. This is the same force that your hand would experience from a single piece of paper on Earth by gravity, a barely detectable force! However, this minute force can operate continuously, adding up to a significant acceleration over time in the frictionless environment of space. In comparison, space probes which operate on chemical propellant may exert thrusts in the hundreds or thousands of newtons (the equivalent of a couple hundred pounds at Earth's surface), but only for very short times.

The small thrust of ion engines pales even more in comparison to that of rockets that launch payloads from Earth's surface. To escape Earth's gravity, such rockets must exert a force greater than the force of gravity on the often huge rockets. For example, the Saturn V rocket (shown above) that launched humans to the Moon generated an astounding 34,500,000 newtons of thrust at launch. For this reason, ion engines cannot be used to launch spacecraft.

The idea of electric rock propulsion dates to the 1930's and the first test of ion engines in space came during the 1960's. However, no operational mission utilized ion thrusters until NASA's Deep Space 1 probe, launched in 1998. This spacecraft performed a flyby of the asteroid 9989 Braille and the comet 19P/Borrelly. To meet the acceleration requirements of the mission extension to comet Borrelly, Deep Space 1 changed its velocity by 4.3 km/s using less than 74 kilograms of xenon. The Hayabusa probe, launched by the Japan Aerospace Exploration Agency (JAXA) in 2003, was another important demonstration of ion thruster technology. This probe used ion thrusters on its mission to land on the near-Earth asteroid 25143 Itokawa, collect samples, and return them to Earth for analysis. In 2010, the probe successfully returned the sample to Earth, completing the first ever asteroid sample return mission. Hayabusa 2, another asteroid sample return mission, launched in 2014 with similar objectives.

An even greater demonstration of ion propulsion technology began in 2007 with the launch of the Dawn spacecraft. This spacecraft carried three ion engines, operating alternately throughout the mission. Thrusting frequently throughout its eight-year journey, Dawn followed a spiral outward from the Earth, past Mars, into orbit of the asteroid Vesta, and subsequently into orbit of the asteroid Ceres.

The above image shows Dawn's outward spiral as well as the intervals during which one of its ion engines was in operation. With the exception of the gravitational assist at Mars, Dawn thrusted almost continuously, moving outward under its own power. Its total velocity change exceeded 10 km/s, the greatest yet for a spacecraft under its own power. When it successfully entered orbit around Ceres in April 2015, Dawn became the first spacecraft to orbit two different extraterrestrial targets. This feat would not have been possible using traditional methods of chemical propulsion.

Meanwhile, advances continued in ion thruster technology. NASA's Evolutionary Xenon Thruster (NEXT) performed a 48000 hour test in a vacuum chamber lasting from 2004 to 2009 to demonstrate its successful operation. Using power at a higher rate but also providing somewhat greater thrust, the NEXT engine (shown above) is at least 30% more efficient than its predecessors and can operate for a longer time. Continued development of ion propulsion technology promises to provide the foundation necessary for more ambitious interplanetary space missions.


Wednesday, January 13, 2016

Hurricane Alex (2016)

Storm Active: January 13-15

On January 7, a low pressure system situated along a front northeast of the Bahamas began deepening, producing a large area of strong winds over the western Atlantic. Though strong upper-level winds and cool ocean temperatures precluded immediate development into a tropical or subtropical cyclone, the National Hurricane Center began to monitor the disturbance. The low moved eastward, remaining frontal in nature, but strengthened even more, producing maximum winds to hurricane force on January 10. Over the next few days, shower activity increased modestly near the system's center as it took a southeast heading into the far eastern Atlantic. By January 12, bands of shallow convection surrounded a well-defined center. The next day, despite marginal sea surface temperatures, thunderstorm activity increased near the center. Due to the relatively shallow convection and gale force winds associated with the system, it was classified Subtropical Storm Alex that afternoon. Alex was only the fourth known tropical or subtropical system to form in the north Atlantic basin in January.

By the time of its formation, Alex had turned toward the northeast and was headed in the direction of the Azores. Meanwhile, despite marginal sea surface temperatures, convection continued to deepen and Alex developed a well-defined eye feature. At the same time, the upper-level low situated over Alex moved away, allowing the cyclone to transition to a tropical cyclone. Since the eyewall now had hurricane force winds, Alex was upgraded to a hurricane during the morning of January 14. It became the first hurricane in January since 1955, and the first to form during the month since 1938. By the afternoon, the outer bands began to affect the Azores Islands. The same evening, it reached its peak intensity of 85 mph winds and a central pressure of 981 mb. The next morning, the center of circulation passed among the central Azores, bringing hurricane-force winds to the region as it sped northward. Meanwhile, the eyewall disintegrated and the convective structure became lopsided as Alex began extratropical transition and weakened slightly. The system became extratropical that afternoon.

The above image shows Alex at peak intensity less than a day before it passed over the Azores. The hurricane developed a remarkable eye feature highly unusual for an off-season storm.

The track of Alex includes several days during which the system was an extratropical system producing winds near hurricane force.