Saturday, March 28, 2009

New Horizons

New Horizons is the first spacecraft ever to travel to the dwarf planet Pluto. The main objective of the mission is to observe the distant Pluto for the first time as well as photograph the five known moons Charon, Nix, Hydra, Kerberos, and Styx, and perhaps some moons not yet discovered.

New Horizons was launched on January 19, 2006 from Cape Canaveral, Florida. As it escaped Earth's gravity, it reached a speed of 36,360 miles per hour or 10.1 miles per second! Shortly after launch, it passed the moon, and by April 7, 2006, it had passed Mars's orbit. Soon after, in May 2006, New Horizons passed into the asteroid belt. It is true that there are many asteroids in the belt, but they are very far apart. The closest flyby of New Horizons to an asteroid was 132524 APL at about 60,000 miles on June 11-13, 2006, and the probe took this opportunity to test its instruments. However, at risk for damaging its instruments by looking at the Sun, New Horizons did not unveil its more powerful telescope, LORRI for this distant flyby.

New Horizons's pictures of the distant asteroid 132524 APL. The diameter of this asteroid is estimated at about 1.4 miles.

By late October, New Horizons had left the asteroid belt and was on target for its flyby of Jupiter. In early January 2007, New Horizons began its Jupiter encounter. As New Horizons passed Jupiter, it tracked and took photos of Jupiter's outer moon, Callirrhoe as practice for navigation. Also, New Horizons made observations of other of Jupiter's moons and edited their orbit information. On February 28, 2007, New Horizons passed its closest to Jupiter at about 1.3 million miles from the planet. On March 5, the Jupiter encounter came to an end.

On June 8, 2008, the probe passed Saturn's orbit. On March 18, 2011, New Horizons passed the next planet, Uranus. On December 2, 2011, New Horizons's distance from Pluto dropped below 983 million miles, the closest approach to Pluto ever by a spacecraft. This surpassed Voyager 2's record set in the 1980's, although this mission was not directed toward Pluto. In 2012, New Horizons began a series of simulations to test equipment for the Pluto encounter.

In 2013, analyses on the Pluto system assessed growing concerns that the many moons of the Pluto system suggested the presence of other extraneous debris - in other words, a possible threat to the spacecraft. The current planned trajectory takes New Horizons to a relatively safe area, namely near the orbit of Charon, Pluto's largest moon and its binary companion. The reason that this was theoretically safer was that Charon, being large, clears the debris in its orbit with its gravity, unlike its smaller satellite neighbors. However, just in case a closer look revealed danger, alternate flyby plans were devised. Later analyses concluded that the risk was not significant and that New Horizons could proceed as planned.

In July 2013, New Horizons was close enough to distinguish Pluto (bright spot at center) and its largest Moon Charon (dim spot just above and to the left of Pluto).

On its path towards Neptune, New Horizons approached the Neptune's L5 point in late 2013, allowing the spacecraft to take a few observations of recently discovered asteroids near that location. On October 25, 2013, New Horizons became less than 5 AU (astronomical units) from Pluto.

On August 25, 2014, the probe passed Neptune's orbit, at which point it was only about 2.5 AU from Pluto. On December 6, 2014, the spacecraft emerged from its final period of hibernation before the Pluto encounter. Over the next several weeks, the operations team checked the functioning of the system's instruments.

During the final approach, image resolution improved steadily. By early February, New Horizons was able to discern both Nix and Hydra, two of Pluto's smaller moons (though Styx and Kerberos are smaller still).

In the above image, Hydra is highlighted by a yellow diamond and Nix by an orange one. The lefthand image shows the Plutonian system and background stars, while the right has been processed to emphasize the Moons. The bright streak in each image is an artificial effect of the camera resulting from overexposure of Pluto.

On March 10, New Horizons passed its final symbolic milestone as it became less than 1 AU from the dwarf planet - closer than the Earth is to the Sun. New Horizons also realized its first color image of Pluto and Charon on April 14, 2015.

The following month, images from New Horizons became the best ever looks at Pluto. During June, the probe revealed reflective polar regions, dark spots and other tantalizing features of Pluto as well as on Charon, also capturing the massive difference in coloration between the two objects. Some of these features appear in the image below, taken between June 25 and June 27.

On July 14, 2015, at precisely 11:49:57 UTC, the New Horizons spacecraft made its closest approach to Pluto at a distance of 7800 miles. During the flyby, all instruments were busy collecting data, so it was only hours afterward that the probe sent a signal to Earth confirming the flyby's success. After an additional 4.5 hour delay for the radio signal to travel, news of the mission's success reached Earth over 10 hours after closest approach. New Horizons took the following path through the Pluto system:

The above image shows the positions of Pluto and its moons when New Horizons made its closest approach (C/A). After closest approach, the probe briefly passed into the shadow of Pluto and then of Charon, allowing it to observe how sunlight interacted with the bodies' atmospheres.

The first image, taken before the Pluto flyby, shows the never-before-seen world in true color. The photograph, with its iconic heart-shaped feature, became one of the most famous images captured by a spacecraft. The second image shows the mostly gray coloration of Charon as well as its reddish polar cap.

The analysis of Pluto's atmosphere (primarily conducted as New Horizons passed behind the shadow of Pluto) indicated that it was dominated by nitrogen, and that this nitrogen escaped the relatively weak gravity at a rapid rate. This fact, added to the observed complexity of the surface's texture and composition from enhanced-color imagery, indicate that Pluto is geologically active.

In August, NASA identified the Kuiper belt object designated 2014 MU69 as the next target for New Horizons under the constraints of its diminished fuel supply.

Meanwhile, data from the Pluto encounter continued to produce astonishing discoveries. For example, the occultation images showed that Pluto's outer atmosphere is in fact a blueish color, an effect caused by the scattering of sunlight off of complex molecules. These molecules form through the interaction of nitrogen and methane with solar wind.

In addition, surface analysis revealed the presence of exposed water ice on Pluto, indicated in the image above. All of this analysis and more was completed before New Horizons even finished transmitting data from the Pluto encounter. Due to the limited power supply on the spacecraft as well as other factors, the images and instrument readings that were all collected within a few days took well over a year to transmit. It was only in late October 2016 that Earth finished receiving the stored data.

The above image shows New Horizons's path from Earth to Pluto.

By the summer of 2016, the data transfer from the spacecraft back to Earth was nearly complete.


Friday, March 20, 2009


MESSENGER (MErcury, Surface Space ENvironment, GEochemistry, and Ranging) is a spacecraft that had flybys of Earth, Venus, and its primary destination, Mercury. Named for Mercury being the messenger god, the main objective of MESSENGER was to observe Mercury, and also map the previously unmapped side of the planet (only 40% of the planet of Mercury was mapped in the mid 1970's by Mariner 10).

Due to the closeness of Mercury to the Sun, MESSENGER would be heading directly into the Sun's more powerful gravity and therefore would need to use a significant amount of fuel to eventually orbit Mercury. To save fuel, MESSENGER instead used gravity assists from Earth and Venus to help direct it on its path. The duration of the mission expanded because of this, but it would allow MESSENGER to practice flybys and test its equipment before reaching Mercury.

MESSENGER was launched on August 4, 2004 with a Delta II rocket from Cape Canaveral. After launch, it followed the Earth around its orbit and had a flyby of Earth on August 2, 2005, getting within 1,500 miles of Earth. Shortly after its flyby with Earth, it fired thrusters, this maneuver being called Deep Space Maneuver 1, or DSM-1. Using this thrust, MESSENGER switched orbits and fell into an elliptical orbit crossing that which is Venus's. While in this orbit, MESSENGER made two flybys of Venus. One on October 24, 2006 and one on June 5, 2007. In October 2007 MESSENGER made its second maneuver, DSM-2. This put MESSENGER directly on target for its first flyby of Mercury.

Track of MESSENGER from launch in 2004 to destination in 2011.

The space probe's first flyby of Mercury occurred on January 14, 2008. On this date, MESSENGER caught its first glimpse of the unseen side of Mercury. Below is MESSENGER's first image of this unknown surface. Two subsequent other flybys of Mercury happened on October 6, 2008 and September 29, 2009 to slow down MESSENGER's speed enough so that it could orbit Mercury.

The first image MESSENGER took during the first flyby of Mercury, showing its previously unknown side.

Finally, on March 17, 2011, MESSENGER fired the thrusters necessary to be ensnared by Mercury's gravity. It began orbiting the planet the next day, and began downloading scientific information on April 4. Many stunning images of the surface have been captured, in a higher resolution than ever before!

MESSENGER quickly completed several scientific inquires, thoroughly photographing the unknown side of Mercury, finding water in its atmosphere and possibly detecting a liquid core. Further experiments were conducted concerning the magnetosphere, as Mercury has the most volatile magnetosphere known in the Solar System.

After observing its cycle, MESSENGER determined that the magnetic field has a curious instability and asymmetry to it that is not found in another other known planetary body. The field is more concentrated to the north, resulting in very different geological formations in the north versus the south polar regions. The north polar region contains plains, with relative protection from erosion, (see below) while the south is open to a fierce bombardment of particles.

By observing the structure of craters on Mercury, MESSENGER also has indirectly determined the nature of Mercury's surface, as impacts of similar objects create different craters on different planetary bodies.

The north polar plains of Mercury in real-color (top), and false-color highlighting different rock types (bottom).

In November 2011, NASA extended the MESSENGER mission an entire year beyond its end date of March 17, 2012, in order to collect further data concerning the outer atmosphere and volcanic activity early in the planet's history. Also, this extension allowed MESSENGER to observe the effect on Mercury of the 2013 solar maximum, or a local peak in solar activity.

In addition, to facilitate more detailed observations, MESSENGER completed two orbital thrusts between April 16 and April 20, 2012 that shortened its orbital period from 11.6 hours to 8.0 hours. In its new position, the probe had more time at low altitudes, from where geologic and magnetic activity could be observed for longer periods at a time.

From its new vantage point, MESSENGER gathered enough data for scientists at NASA to make a groundbreaking announcement: there appears to be water (in the form of ice) on Mercury. Several pieces of evidence support this claim, including the fact that Mercury's near-zero axial tilt keeps many crater basins near the poles perpetually in shadow (and thus below freezing), and that chemical analyses suggested unusually high hydrogen concentrations in the same polar regions. However, perhaps the most compelling evidence is the reflectivity of these supposed ice deposits.

Areas highlighted in yellow illustrate high reflectivity, or albedo, of certain crater basins, precisely where the ice deposits would persist.

On December 30, 2012, MESSENGER captured an image of the last part of Mercury's previously unknown surface, mapping, for the first time, all 100% of the surface in daylight. This allowed the compilation of global mosaics, such as the Mercator projection below (obviously, just as with Earth maps of this type, the features near the poles are elongated).

MESSENGER's orbit continued to gradually tighten about the planet due to the influence of the Sun's gravity, and in April 2014 had its closest approach yet to the planet, dropping to an altitude of only 123.7 miles at its periapsis. In June 2014, MESSENGER made another altitude adjustment, raising its periapsis to extend the lifetime of its orbit about Mercury. Even with this adjustment, MESSENGER's orbit began to degrade over time. However, MESSENGER's decreasing altitude allowed more detailed and precise examination of Mercury than ever before. By August 3, 2014, on which date the probe had spent 10 years in space, the closest approach to Mercury had fallen below 62 miles (100 kilometers).

By September 12, the orbit had shrunk to a mere 15 miles to the surface at closest approach! This allowed high resolution images such as the one above, with only 6 meters per pixel. The smoothness of the landscape in the above image reflects past pyroclastic flow across the region. The spacecraft then performed the second of four orbital maneuvers to raise the periapsis and maintain orbit about Mercury. In October, the probe underwent another such maneuver. However, though MESSENGER's propellant was scheduled to run out in March 2015, scientists on the program's propulsion team devised a plan to use the helium which pressurized portions of the spacecraft as a makeshift fuel to extend the mission for a few additional weeks. The first application of this plan was executed successfully in January 2015, using a combination of the remaining propellant and pressurized helium to adjust MESSENGER's orbit.

Over the following months, the spacecraft took advantage of its low altitude to initiate a "hover campaign" in which the MESSENGER's magnetometer and neutron spectrometer would take observations at very low altitudes. The probe also completed its 4000th orbit of Mercury on March 27. In early April, another maneuver raised the periapsis of MESSENGER's orbit, before which it had sunk to a closest approach altitude of only 3.7 miles! After a few additional such maneuvers throughout April, the inexorable gravitational force of the Sun ultimately won out, causing the probe's anticipated crash into Mercury's surface on April 30, 2015. At this point, MESSENGER had spent well over ten years in space and four years in orbit of the Solar System's innermost planet, more than twice the original mission plan!

MESSENGER's images and data led to a enormous variety of crucial discoveries in addition to those mentioned above. For example, the probe discovered that Mercury is in fact shrinking over time (albeit very slowly) as its core cools and compresses.

The diagonal ridge in the above image was formed when Mercury's crust buckled in on itself.

The MESSENGER mission was innovative, efficient, and prolific in its results. The probe employed more inner solar system gravity assists than any prior mission, continued to operate months after its supply of propellant was exhausted, and generated many terabytes of scientific data along with over a quarter-million images as the first ever Mercury orbiter. MESSENGER revolutionized our understanding of the innermost planet.

For more info, see the MESSENGER main page.

Saturday, March 14, 2009

The Planck Constant and Its Applications

The Planck constant is the basis of a fundamental natural law. The Planck constant was derived from a relation that Max Planck devised. This equation is


The E stands for energy and the v stands for the frequency of the electromagnetic wave. The quantity h, is the Planck constant, which is the proportion constant between the values of E and v (for every E, there are h v's). For example, consider a star. The star gives off more light and is brighter if it has a lot of energy. And since energy creates heat, hotter stars give off more light (i.e. a higher frequency of the electromagnetic wave).

Using the Planck constant, the speed of light in a vacuum (186,383 miles per second), and the gravitational constant, the Planck length can be determined. According to the theory, and supported by all our current knowledge of physics, the Planck length is the smallest length anything can be. This may not seem possible, but there is a length that is simply indivisible. A good analogy for this is a pixel on a computer screen. To our eyes, the many dots blend into a full image, but at subatomic (well perhaps sub-subatomic) levels, there is actually a smallest length. No one knows what kind of matter (if it exists) could be at this primitive level. The Planck length is exactly 1.61625281*10^35 meters (0.0000000000000000000000000000000000161625281 meters). A theory of the Big Bang violates this claim, because if the Universe began from an infinitesimal point it would go through a period, however short, where the dimensions of the Universe would be smaller than the smallest possible length. There are two possible ways to excuse this possible exception. One is that the Universe never got that small, and a Big Bounce occurred. A Big Bounce occurs when a Universe contracts into a very small region and then re-expands into another, separate Universe. Therefore, the Universe never reached a size below the Planck length before billowing out into another Universe. The other possibility is that the Universe did go through a Big Bang, but the time in which the Universe would have been smaller than the Planck length is shorter than (you guessed it) the Planck time. The Planck time is another unit indirectly derived from the Planck constant. The Planck time can directly be calculated after the Planck length is known, because the Planck time is the amount of time it takes for light (the fastest thing in the Universe. Actually, there is a theory that there is a particle that travels faster than the speed of light, see the Tachyon) to travel over one Planck length. The actual value of one unit of Planck time is 5.3912427*10^-44 seconds (the actual value is 0.00000000000000000000000000000000000000000053912427 seconds).

There are Planck values for all units. The rest of the Planck values are: the Planck mass, the Planck charge, the Planck temperature, the Planck area, the Planck volume, the Planck momentum, the Planck energy, the Planck force, the Planck power, the Planck density, the Planck angular frequency, the Planck pressure, the Planck current, the Planck voltage, and the Planck impedance. Since I cannot express all of these units at length, I will discuss a few special cases. Some Planck units are the smallest possible of that unit, such as the Planck Length, while some are the maximum, such as the Planck temperature. The Planck temperature is the highest possible temperature. The lower bound on temperature is absolute zero, or -459.67 degrees Fahrenheit. It may seem odd that the lowest possible temperature is so high (on a larger scale) compared to the highest possible temperature, which is over 400,000,000,000,000,000,000,000,000,000,000 (400 nonillion) Fahrenheit. The Planck mass is also a special example. The Planck mass is around a milligram, and some particles, such as the electron neutrino and the electron antineutrino weigh much less than the Planck mass. However, when energy and mass are added together, the energy-mass of the electron neutrino, and for this matter anything, exceeds the Planck mass. The Planck area and volume are easily calculable from the Planck length, for obvious reasons. Another Planck unit, the Planck Density, is derived using the Planck Length and the Planck Mass (since mass within a certain volume is density). This density is 10^96 kilograms per square centimeter or 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 kilograms per square centimeter. To put this into perspective, imagine the Milky Way. At this density, it would take the mass of over a trillion Milky Way's (each with 300 billion stars) to fill the space of a single atom. This is even denser than black hole singularities (see here and here for more info about Black Holes).

In brief terms, Planck's relation and the application of the Planck constant to create the various Planck units was an advance in science that helped us to understand the fundamentals of Quantum Physics and the origin of the Universe.

Monday, March 9, 2009

Is the Universe all that there is?

Many times has the history of our Universe appeared on this blog. However, what exists outside our Universe, and what is the true meaning of infinity? Many theories exist. Some think that our Universe is the only Universe and that it is infinite, while others support the idea of a Multiverse. A Multiverse is the infinite set of Universes. The Multiverse is defined in many different ways.

The first theory, that the Universe is infinite, is clearly impossible if the Big Bang is to be believed. Personally, I find it hard to believe that an infinite Universe could come from a Universe that, at one point, definitely had finite space. Also, due to the Planck constant, infinitely small isn't possible either.

As for the actual 'definition' of Multiverse, there is none. Some believe that all Universes are connected to ours, except we can't see them. I believe I can safely eliminate this possibility by using the same argument as before with the fact the infinite space cannot rise from finite space.

Of course, some say that there aren't any other Universes, and our Universe is all that there is, finite or infinite. If for a moment, this theory was to be believed, questions would arise. Such as "If our Universe is the only Universe, than where did the matter now in this Universe originate?" Again, short of saying the matter was merely floating around in a vacuum, the only explanation is that this Universe must have been preceded by another. Therefore the Big Bang is the Big Bounce. In my previous post "Black Holes and Universe Budding" I described the third main theory of the Multiverse. This theory is that, through black holes and Big Bounces, all current Universes (including our own) are descended from some "ancestor Universe". This theory is based on the possibility that the very physical laws of a Universe are random. For example, Universe A with 3 spacial dimensions, like ours, spawns a Universe B through a Big Bounce, which has 5 spacial dimensions. Still others, take the Darwinian approach and say that Universes have there own set of cosmological 'genes', which determine the physical characteristics of that Universe. In this theory, some of these 'genes' are passed on to the next Universe. This theory may seem unlikely, but little is known about anything on a Multiuniversial scale.

Another theory is called "bubble universes". This theory is also, like the previous theory, based on the fact that all current Universes are descended form an "ancestor universe". However, instead of black holes and Big Bounces, a "patch" of space expands into a Universe, independent of its parent. Let's call the ancestor Universe A. If this theory is to be believed, all Universes are 'inside' Universe A. Our Universe could be a sphere floating in the sky of another Universe, but to us our Universe is huge. Size is perspective and probably not constant to all Universes.

The final theory is more radical than all the others. Some think that there are infinite Universes in the Multiverse which suggests the true existence of infinity in space. This single idea turns many away from this theory, but it gets even better. The infinite Universes are determined by possibilities of other Universes. Let me explain. For example, a random person named Bob, is purchasing an ice cream cone. He cannot choose between vanilla and chocolate ice cream. Finally, in our Universe, he decides vanilla. However, another Universe is created at this precise moment, and in that Universe, he buys chocolate. In yet another Universe, he may realize that he has no money in his wallet, or possibly he chooses chocolate, but there is none left. As you can see, many possible Universes can exist just because of Bob's simple decision. The idea of the theory is that somehow, a Universe is created for each and every decision or possible outcome in any situation in any Universe. If we accept this fact, we can clearly see that an infinite set of Universes can result from all the situations in the past 13 billion years. In fact, even Bob's simple decision can lead to an infinite set of Universes because anything could happen at that exact moment, however improbable (e.g. a meteor hitting the ice cream shop).

An interesting variation of this theory is the possibility that the alternate Universes are "already there". In the infinite set of Universes, there is another Universe that is exactly the same, save that one detail. Let me return to my Bob ice cream example. When Bob purchases the ice cream, our Universe comes "in contact" with our Universe (note that the closeness isn't in physical terms. It is actually close in a sense that the Universes are very closely related when the same situation occurred in both of them, with differing outcomes).

Personally, I think that the reality is a combination of the theories mentioned above. My theory is there was one Universe in the beginning and that the Universes branch every time two possible outcomes come about. However, there is a catch, which I stumbled upon, but picked myself up shortly afterward. Bob gets vanilla one day, and chocolate on another branch of the Universes. However, in both Universes, he gets vanilla again the next day. Do the branches combine for this situation? However, the fact that he chose differently yesterday will change the situation of the next day very slightly, and therefore they will still be distinct. As for whether black holes "bud off" and become new Universes, I think that they do. I think this holds with my theory. Let us step back for a moment. According to my theory, all Universes branch at the time that a situation comes about, such as Bob's. Each of these outcomes has its own "path" on the tree of time. When a Universe buds off from a parent, both Universes are still on the same path, because they are connected physically rather than by an ancestry further back toward the "root" of the "tree".

Not much absolute information is known on this topic and for more info, see here and here.

Sunday, March 1, 2009

Other Types of Primes

To discover very high primes, e.g. some over 10,000 digits, one cannot merely look at random numbers. For there are some functions that commonly produce primes, although no function actually is guaranteed to produce primes. Some common types of primes are factorial primes, primorial primes, repunit primes and Mersenne primes. For all the lists of primes mentioned on this post, please visit the Prime Pages.

A factorial prime is a prime that is one more or one less than a factorial. A factorial, represented n!, is the product of all the counting numbers below and including n. For example, 3!=3*2*1=6, and 5!=5*4*3*2*1=120. Therefore, a factorial prime is one of the form n!+1 or n!-1. The first few factorial primes are 2 (1!+1), 3 (2!+1), 5 (3!-1), 7 (3!+1), 23 (4!-1), 719 (6!-1), 5039 (7!-1), 39916801 (11!+1), 479001599 (12!-1), and 87178291199 (14!-1). Currently, the highest factorial prime known is 34790!-1 which has 142,891 digits!

The next type of prime, the primorial prime, is a prime that is one more or one less than a primorial. Similar to a factorial a primorial (n#) is the product of all primes up to and possibly including n. For example, 11#=2*3*5*7*11=2310. The first few primorial primes are 3 (2#+1), 5 (3#-1), 7 (3#+1), 29 (5#-1), 31 (5#+1), 211 (7#+1), 2309 (11#-1), 2311 (11#+1), 30029 (13#-1), 200560490131 (31#+1), 304250263527209 (41#-1). The highest known primorial prime is 392113#+1, which has 169,966 digits.

The top eight largest primes known are all Mersenne primes. All Mersenne primes are of the form 2^n-1. It has been proven that 2^n-1 can only be prime if n is prime. The first few Mersenne primes are 3 (2^2-1), 7 (2^3-1), 31 (2^5-1), 127 (2^7-1), 8191 (2^13-1), 131071 (2^17-1), 524287 (2^19-1), 2147483647 (2^31-1) and 2305843009213693951 (2^61-1). In total, there are only 46 Mersenne primes in existence. The highest Mersenne prime known is 2^43112609-1, which has 12,978,189 digits. For all the Mersenne primes and their english names (e.g. 2,147,483,647= two billion, one hundred forty seven million, four hundred eighty three thousand, six hundred forty seven) see here. For the highest Mersenne prime in existence, the first -illion in the name (million, billion, trillion, quadrillion) is quattuormilliamilliatrecensexviginmilliaunsexagintillion (wow)!

The final type I will discuss here, the repunit prime, is, on my opinion, the most peculiar type of prime. A repunit is a number consisting of only ones (e.g. 111 or 11111111). Only a handful of this are repunit primes. A repunit is written is Rn with n ones (R3=111) The first few repunits that are prime (as a trivial case, 1 is excluded) are R2=11, R19=1111111111111111111, R23=11111111111111111111111, and R317=11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111. The highest repunit prime is only R1031, but there are many probable primes that are in the process of being proved prime.

For the 5000 highest primes, visit here.

There are many other types of primes, only a few of which I described here.