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114-683: As early as 1839, the German mathematician and physicist Carl Friedrich Gauss postulated that an electrically conducting region of the atmosphere could account for observed variations of Earth's magnetic field. Sixty years later, Guglielmo Marconi received the first trans-Atlantic radio signal on December 12, 1901, in St. John's, Newfoundland (now in Canada ) using a 152.4 m (500 ft) kite-supported antenna for reception. The transmitting station in Poldhu , Cornwall, used

228-492: A heliometer from Fraunhofer . The scientific activity of Gauss, besides pure mathematics, can be roughly divided into three periods: astronomy was the main focus in the first two decades of the 19th century, geodesy in the third decade, and physics, mainly magnetism, in the fourth decade. Gauss made no secret of his aversion to giving academic lectures. But from the start of his academic career at Göttingen, he continuously gave lectures until 1854. He often complained about

342-423: A magnetometer in 1833 and – alongside Wilhelm Eduard Weber – the first electromagnetic telegraph in 1833. Gauss was the first to discover and study non-Euclidean geometry , coining the term as well. He further developed a fast Fourier transform some 160 years before John Tukey and James Cooley . Gauss refused to publish incomplete work and left several works to be edited posthumously . He believed that

456-664: A spark-gap transmitter to produce a signal with a frequency of approximately 500  kHz and a power of 100 times more than any radio signal previously produced. The message received was three dits, the Morse code for the letter S . To reach Newfoundland the signal would have to bounce off the ionosphere twice. Dr. Jack Belrose has contested this, however, based on theoretical and experimental work. However, Marconi did achieve transatlantic wireless communications in Glace Bay, Nova Scotia , one year later. In 1902, Oliver Heaviside proposed

570-411: A bit to absorption on frequencies above. However, during intense sporadic E events, the E s layer can reflect frequencies up to 50 MHz and higher. The vertical structure of the E layer is primarily determined by the competing effects of ionization and recombination. At night the E layer weakens because the primary source of ionization is no longer present. After sunset an increase in the height of

684-406: A butcher, bricklayer, gardener, and treasurer of a death-benefit fund. Gauss characterized his father as honourable and respected, but rough and dominating at home. He was experienced in writing and calculating, whereas his second wife Dorothea, Carl Friedrich's mother, was nearly illiterate. He had one elder brother from his father's first marriage. Gauss was a child prodigy in mathematics. When

798-554: A collection of short remarks about his results from the years 1796 until 1814, shows that many ideas for his mathematical magnum opus Disquisitiones Arithmeticae (1801) date from this time. Gauss graduated as a Doctor of Philosophy in 1799, not in Göttingen, as is sometimes stated, but at the Duke of Brunswick's special request from the University of Helmstedt, the only state university of

912-491: A considerable literary estate, too. Gauss referred to mathematics as "the queen of sciences" and arithmetics as "the queen of mathematics", and supposedly once espoused a belief in the necessity of immediately understanding Euler's identity as a benchmark pursuant to becoming a first-class mathematician. On certain occasions, Gauss claimed that the ideas of another scholar had already been in his possession previously. Thus his concept of priority as "the first to discover, not

1026-665: A critique of d'Alembert's work. He subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts clarified the concept of complex numbers considerably along the way. In the preface to the Disquisitiones , Gauss dates the beginning of his work on number theory to 1795. By studying the works of previous mathematicians like Fermat, Euler, Lagrange, and Legendre, he realized that these scholars had already found much of what he had discovered by himself. The Disquisitiones Arithmeticae , written since 1798 and published in 1801, consolidated number theory as

1140-431: A curious feature of his working style that he carried out calculations with a high degree of precision much more than required, and prepared tables with more decimal places than ever requested for practical purposes. Very likely, this method gave him a lot of material which he used in finding theorems in number theory. Gauss refused to publish work that he did not consider complete and above criticism. This perfectionism

1254-469: A decade. Therese then took over the household and cared for Gauss for the rest of his life; after her father's death, she married actor Constantin Staufenau. Her sister Wilhelmina married the orientalist Heinrich Ewald . Gauss's mother Dorothea lived in his house from 1817 until she died in 1839. The eldest son Joseph, while still a schoolboy, helped his father as an assistant during the survey campaign in

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1368-483: A discipline and covered both elementary and algebraic number theory . Therein he introduces the triple bar symbol ( ≡ ) for congruence and uses it for a clean presentation of modular arithmetic . It deals with the unique factorization theorem and primitive roots modulo n . In the main sections, Gauss presents the first two proofs of the law of quadratic reciprocity and develops the theories of binary and ternary quadratic forms . The Disquisitiones include

1482-689: A geomagnetic storm the F₂ layer will become unstable, fragment, and may even disappear completely. In the Northern and Southern polar regions of the Earth aurorae will be observable in the night sky. Lightning can cause ionospheric perturbations in the D-region in one of two ways. The first is through VLF (very low frequency) radio waves launched into the magnetosphere . These so-called "whistler" mode waves can interact with radiation belt particles and cause them to precipitate onto

1596-677: A habit in his later years, for example, the number of paths from his home to certain places in Göttingen, or the number of living days of persons; he congratulated Humboldt in December 1851 for having reached the same age as Isaac Newton at his death, calculated in days. Similar to his excellent knowledge of Latin he was also acquainted with modern languages. At the age of 62, he began to teach himself Russian , very likely to understand scientific writings from Russia, among them those of Lobachevsky on non-Euclidean geometry. Gauss read both classical and modern literature, and English and French works in

1710-548: A heart attack in Göttingen; and was interred in the Albani Cemetery there. Heinrich Ewald , Gauss's son-in-law, and Wolfgang Sartorius von Waltershausen , Gauss's close friend and biographer, gave eulogies at his funeral. Gauss was a successful investor and accumulated considerable wealth with stocks and securities, finally a value of more than 150 thousand Thaler; after his death, about 18 thousand Thaler were found hidden in his rooms. The day after Gauss's death his brain

1824-483: A high frequency (3–30 MHz) radio blackout that can persist for many hours after strong flares. During this time very low frequency (3–30 kHz) signals will be reflected by the D layer instead of the E layer, where the increased atmospheric density will usually increase the absorption of the wave and thus dampen it. As soon as the X-rays end, the sudden ionospheric disturbance (SID) or radio black-out steadily declines as

1938-493: A lectureship position in space physics at La Trobe University in 1968. She remained at La Trobe for the remainder of her career, though in 1974 and 1978/9 she had simultaneous positions at the US Air Force Geophysics Laboratory as part of her GPS research. Her initial work focused on the use of radio wave reflection to study irregularities in the ionosphere . Her work on radio transmission through

2052-469: A private scholar. He gave the second and third complete proofs of the fundamental theorem of algebra , made contributions to number theory , and developed the theories of binary and ternary quadratic forms. Gauss was instrumental in the identification of Ceres as a dwarf planet. His work on the motion of planetoids disturbed by large planets led to the introduction of the Gaussian gravitational constant and

2166-417: A radio wave reaches the ionosphere, the electric field in the wave forces the electrons in the ionosphere into oscillation at the same frequency as the radio wave. Some of the radio-frequency energy is given up to this resonant oscillation. The oscillating electrons will then either be lost to recombination or will re-radiate the original wave energy. Total refraction can occur when the collision frequency of

2280-479: A result of lightning activity. Their subsequent research has focused on the mechanism by which this process can occur. Due to the ability of ionized atmospheric gases to refract high frequency (HF, or shortwave ) radio waves, the ionosphere can reflect radio waves directed into the sky back toward the Earth. Radio waves directed at an angle into the sky can return to Earth beyond the horizon. This technique, called "skip" or " skywave " propagation, has been used since

2394-690: A scandal in public, Eugen suddenly left Göttingen under dramatic circumstances in September 1830 and emigrated via Bremen to the United States. He wasted the little money he had taken to start, after which his father refused further financial support. The youngest son Wilhelm wanted to qualify for agricultural administration, but had difficulties getting an appropriate education, and eventually emigrated as well. Only Gauss's youngest daughter Therese accompanied him in his last years of life. Collecting numerical data on very different things, useful or useless, became

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2508-415: A strong calculus as the sole tasks of astronomy. At university, he was accompanied by a staff of other lecturers in his disciplines, who completed the educational program; these included the mathematician Thibaut with his lectures, the physicist Mayer , known for his textbooks, his successor Weber since 1831, and in the observatory Harding , who took the main part of lectures in practical astronomy. When

2622-445: A university chair in Göttingen, "because he was always involved in some polemic." Gauss's life was overshadowed by severe problems in his family. When his first wife Johanna suddenly died shortly after the birth of their third child, he revealed the grief in a last letter to his dead wife in the style of an ancient threnody , the most personal surviving document of Gauss. The situation worsened when tuberculosis ultimately destroyed

2736-402: Is actually lower in the local summer months. This effect is known as the winter anomaly. The anomaly is always present in the northern hemisphere, but is usually absent in the southern hemisphere during periods of low solar activity. Within approximately ± 20 degrees of the magnetic equator , is the equatorial anomaly. It is the occurrence of a trough in the ionization in the F 2 layer at

2850-460: Is also common, sometimes to distances of 15,000 km (9,300 mi) or more. The F layer or region, also known as the Appleton–Barnett layer, extends from about 150 km (93 mi) to more than 500 km (310 mi) above the surface of Earth. It is the layer with the highest electron density, which implies signals penetrating this layer will escape into space. Electron production

2964-649: Is currently used to compensate for ionospheric effects in GPS . This model was developed at the US Air Force Geophysical Research Laboratory circa 1974 by John (Jack) Klobuchar . The Galileo navigation system uses the NeQuick model . GALILEO broadcasts 3 coefficients to compute the effective ionization level, which is then used by the NeQuick model to compute a range delay along the line-of-sight. The open system electrodynamic tether , which uses

3078-448: Is dominated by extreme ultraviolet (UV, 10–100 nm) radiation ionizing atomic oxygen. The F layer consists of one layer (F 2 ) at night, but during the day, a secondary peak (labelled F 1 ) often forms in the electron density profile. Because the F 2 layer remains by day and night, it is responsible for most skywave propagation of radio waves and long distance high frequency (HF, or shortwave ) radio communications. Above

3192-508: Is enough to absorb most (if not all) transpolar HF radio signal transmissions. Such events typically last less than 24 to 48 hours. The E layer is the middle layer, 90 to 150 km (56 to 93 mi) above the surface of the Earth. Ionization is due to soft X-ray (1–10 nm) and far ultraviolet (UV) solar radiation ionization of molecular oxygen (O 2 ). Normally, at oblique incidence, this layer can only reflect radio waves having frequencies lower than about 10 MHz and may contribute

3306-533: Is higher than the plasma frequency of the ionosphere, then the electrons cannot respond fast enough, and they are not able to re-radiate the signal. It is calculated as shown below: where N = electron density per m and f critical is in Hz. The Maximum Usable Frequency (MUF) is defined as the upper frequency limit that can be used for transmission between two points at a specified time. where α {\displaystyle \alpha } = angle of arrival ,

3420-411: Is sufficient to affect radio propagation . This portion of the atmosphere is partially ionized and contains a plasma which is referred to as the ionosphere. Ultraviolet (UV), X-ray and shorter wavelengths of solar radiation are ionizing, since photons at these frequencies contain sufficient energy to dislodge an electron from a neutral gas atom or molecule upon absorption. In this process

3534-405: Is the main reason for absorption of HF radio waves , particularly at 10 MHz and below, with progressively less absorption at higher frequencies. This effect peaks around noon and is reduced at night due to a decrease in the D layer's thickness; only a small part remains due to cosmic rays . A common example of the D layer in action is the disappearance of distant AM broadcast band stations in

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3648-501: Is useful in crossing international boundaries and covering large areas at low cost. Automated services still use shortwave radio frequencies, as do radio amateur hobbyists for private recreational contacts and to assist with emergency communications during natural disasters. Armed forces use shortwave so as to be independent of vulnerable infrastructure, including satellites, and the low latency of shortwave communications make it attractive to stock traders, where milliseconds count. When

3762-581: The Celestial police . One of their aims was the discovery of further planets. They assembled data on asteroids and comets as a basis for Gauss's research on their orbits, which he later published in his astronomical magnum opus Theoria motus corporum coelestium (1809). In November 1807, Gauss followed a call to the University of Göttingen , then an institution of the newly founded Kingdom of Westphalia under Jérôme Bonaparte , as full professor and director of

3876-582: The Committee on Space Research (COSPAR) and the International Union of Radio Science (URSI). The major data sources are the worldwide network of ionosondes , the powerful incoherent scatter radars (Jicamarca, Arecibo , Millstone Hill, Malvern, St Santin), the ISIS and Alouette topside sounders , and in situ instruments on several satellites and rockets. IRI is updated yearly. IRI is more accurate in describing

3990-494: The Gauss composition law for binary quadratic forms, as well as the enumeration of the number of representations of an integer as the sum of three squares. As an almost immediate corollary of his theorem on three squares , he proves the triangular case of the Fermat polygonal number theorem for n = 3. From several analytic results on class numbers that Gauss gives without proof towards

4104-487: The astronomical observatory , and kept the chair until his death in 1855. He was soon confronted with the demand for two thousand francs from the Westphalian government as a war contribution, which he could not afford to pay. Both Olbers and Laplace wanted to help him with the payment, but Gauss refused their assistance. Finally, an anonymous person from Frankfurt , later discovered to be Prince-primate Dalberg , paid

4218-467: The method of least squares , which he had discovered before Adrien-Marie Legendre published it. Gauss was in charge of the extensive geodetic survey of the Kingdom of Hanover together with an arc measurement project from 1820 to 1844; he was one of the founders of geophysics and formulated the fundamental principles of magnetism . Fruits of his practical work were the inventions of the heliotrope in 1821,

4332-546: The popularization of scientific matters. His only attempts at popularization were his works on the date of Easter (1800/1802) and the essay Erdmagnetismus und Magnetometer of 1836. Gauss published his papers and books exclusively in Latin or German . He wrote Latin in a classical style but used some customary modifications set by contemporary mathematicians. In his inaugural lecture at Göttingen University from 1808, Gauss claimed reliable observations and results attained only by

4446-448: The troposphere , extends from the surface to about 10 km (6 mi). Above that is the stratosphere , followed by the mesosphere. In the stratosphere incoming solar radiation creates the ozone layer . At heights of above 80 km (50 mi), in the thermosphere , the atmosphere is so thin that free electrons can exist for short periods of time before they are captured by a nearby positive ion . The number of these free electrons

4560-399: The 1920s to communicate at international or intercontinental distances. The returning radio waves can reflect off the Earth's surface into the sky again, allowing greater ranges to be achieved with multiple hops . This communication method is variable and unreliable, with reception over a given path depending on time of day or night, the seasons, weather, and the 11-year sunspot cycle . During

4674-443: The D layer, so there are many more neutral air molecules than ions. Medium frequency (MF) and lower high frequency (HF) radio waves are significantly attenuated within the D layer, as the passing radio waves cause electrons to move, which then collide with the neutral molecules, giving up their energy. Lower frequencies experience greater absorption because they move the electrons farther, leading to greater chance of collisions. This

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4788-441: The Duke was killed in the battle of Jena in 1806. The duchy was abolished in the following year, and Gauss's financial support stopped. When Gauss was calculating asteroid orbits in the first years of the century, he established contact with the astronomical community of Bremen and Lilienthal , especially Wilhelm Olbers , Karl Ludwig Harding , and Friedrich Wilhelm Bessel , as part of the informal group of astronomers known as

4902-686: The E layer maximum increases the range to which radio waves can travel by reflection from the layer. This region is also known as the Kennelly–Heaviside layer or simply the Heaviside layer. Its existence was predicted in 1902 independently and almost simultaneously by the American electrical engineer Arthur Edwin Kennelly (1861–1939) and the British physicist Oliver Heaviside (1850–1925). In 1924 its existence

5016-532: The F 1 layer. The F 2 layer persists by day and night and is the main region responsible for the refraction and reflection of radio waves. The D layer is the innermost layer, 48 to 90 km (30 to 56 mi) above the surface of the Earth. Ionization here is due to Lyman series -alpha hydrogen radiation at a wavelength of 121.6 nanometre (nm) ionizing nitric oxide (NO). In addition, solar flares can generate hard X-rays (wavelength < 1 nm ) that ionize N 2 and O 2 . Recombination rates are high in

5130-455: The F 2 layer daytime ion production is higher in the summer, as expected, since the Sun shines more directly on the Earth. However, there are seasonal changes in the molecular-to-atomic ratio of the neutral atmosphere that cause the summer ion loss rate to be even higher. The result is that the increase in the summertime loss overwhelms the increase in summertime production, and total F 2 ionization

5244-483: The F layer, the number of oxygen ions decreases and lighter ions such as hydrogen and helium become dominant. This region above the F layer peak and below the plasmasphere is called the topside ionosphere. From 1972 to 1975 NASA launched the AEROS and AEROS B satellites to study the F region. An ionospheric model is a mathematical description of the ionosphere as a function of location, altitude, day of year, phase of

5358-490: The French language. Gauss was "in front of the new development" with documented research since 1799, his wealth of new ideas, and his rigour of demonstration. Whereas previous mathematicians like Leonhard Euler let the readers take part in their reasoning for new ideas, including certain erroneous deviations from the correct path, Gauss however introduced a new style of direct and complete explanation that did not attempt to show

5472-450: The Sun at any one time. Sunspot active regions are the source of increased coronal heating and accompanying increases in EUV and X-ray irradiance, particularly during episodic magnetic eruptions that include solar flares that increase ionization on the sunlit side of the Earth and solar energetic particle events that can increase ionization in the polar regions. Thus the degree of ionization in

5586-475: The act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again. The posthumous papers, his scientific diary , and short glosses in his own textbooks show that he worked to a great extent in an empirical way. He was a lifelong busy and enthusiastic calculator, who made his calculations with extraordinary rapidity, mostly without precise controlling, but checked

5700-580: The act of learning, not possession of knowledge, provided the greatest enjoyment. Gauss confessed to disliking teaching, but some of his students became influential mathematicians, such as Richard Dedekind and Bernhard Riemann . Gauss was born on 30 April 1777 in Brunswick in the Duchy of Brunswick-Wolfenbüttel (now in the German state of Lower Saxony ). His family was of relatively low social status. His father Gebhard Dietrich Gauss (1744–1808) worked variously as

5814-441: The angle of the wave relative to the horizon , and sin is the sine function. The cutoff frequency is the frequency below which a radio wave fails to penetrate a layer of the ionosphere at the incidence angle required for transmission between two specified points by refraction from the layer. There are a number of models used to understand the effects of the ionosphere on global navigation satellite systems. The Klobuchar model

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5928-449: The atmosphere near the magnetic poles increasing the ionization of the D and E layers. PCA's typically last anywhere from about an hour to several days, with an average of around 24 to 36 hours. Coronal mass ejections can also release energetic protons that enhance D-region absorption in the polar regions. Geomagnetic storms and ionospheric storms are temporary and intense disturbances of the Earth's magnetosphere and ionosphere. During

6042-625: The best-paid professors of the university. When Gauss was asked for help by his colleague and friend Friedrich Wilhelm Bessel in 1810, who was in trouble at Königsberg University because of his lack of an academic title, Gauss provided a doctorate honoris causa for Bessel from the Philosophy Faculty of Göttingen in March 1811. Gauss gave another recommendation for an honorary degree for Sophie Germain but only shortly before her death, so she never received it. He also gave successful support to

6156-517: The birth of Louis, who himself died a few months later. Gauss chose the first names of his children in honour of Giuseppe Piazzi , Wilhelm Olbers, and Karl Ludwig Harding, the discoverers of the first asteroids. On 4 August 1810, Gauss married Wilhelmine (Minna) Waldeck, a friend of his first wife, with whom he had three more children: Eugen (later Eugene) (1811–1896), Wilhelm (later William) (1813–1879), and Therese (1816–1864). Minna Gauss died on 12 September 1831 after being seriously ill for more than

6270-701: The burdens of teaching, feeling that it was a waste of his time. On the other hand, he occasionally described some students as talented. Most of his lectures dealt with astronomy, geodesy, and applied mathematics , and only three lectures on subjects of pure mathematics. Some of Gauss's students went on to become renowned mathematicians, physicists, and astronomers: Moritz Cantor , Dedekind , Dirksen , Encke , Gould , Heine , Klinkerfues , Kupffer , Listing , Möbius , Nicolai , Riemann , Ritter , Schering , Scherk , Schumacher , von Staudt , Stern , Ursin ; as geoscientists Sartorius von Waltershausen , and Wappäus . Gauss did not write any textbook and disliked

6384-490: The contemporary school of Naturphilosophie . Gauss had an "aristocratic and through and through conservative nature", with little respect for people's intelligence and morals, following the motto " mundus vult decipi ". He disliked Napoleon and his system, and all kinds of violence and revolution caused horror to him. Thus he condemned the methods of the Revolutions of 1848 , though he agreed with some of their aims, such as

6498-411: The daytime. During solar proton events , ionization can reach unusually high levels in the D-region over high and polar latitudes. Such very rare events are known as Polar Cap Absorption (or PCA) events, because the increased ionization significantly enhances the absorption of radio signals passing through the region. In fact, absorption levels can increase by many tens of dB during intense events, which

6612-532: The discovery of HF radio propagation via the ionosphere in 1923. In 1925, observations during a solar eclipse in New York by Dr. Alfred N. Goldsmith and his team demonstrated the influence of sunlight on radio wave propagation, revealing that short waves became weak or inaudible while long waves steadied during the eclipse, thus contributing to the understanding of the ionosphere's role in radio transmission. In 1926, Scottish physicist Robert Watson-Watt introduced

6726-643: The duchy. Johann Friedrich Pfaff assessed his doctoral thesis, and Gauss got the degree in absentia without further oral examination. The Duke then granted him the cost of living as a private scholar in Brunswick. Gauss subsequently refused calls from the Russian Academy of Sciences in St. Peterburg and Landshut University . Later, the Duke promised him the foundation of an observatory in Brunswick in 1804. Architect Peter Joseph Krahe made preliminary designs, but one of Napoleon's wars cancelled those plans:

6840-423: The electrons in the D-region recombine rapidly and propagation gradually returns to pre-flare conditions over minutes to hours depending on the solar flare strength and frequency. Associated with solar flares is a release of high-energy protons. These particles can hit the Earth within 15 minutes to 2 hours of the solar flare. The protons spiral around and down the magnetic field lines of the Earth and penetrate into

6954-510: The elementary teachers noticed his intellectual abilities, they brought him to the attention of the Duke of Brunswick who sent him to the local Collegium Carolinum , which he attended from 1792 to 1795 with Eberhard August Wilhelm von Zimmermann as one of his teachers. Thereafter the Duke granted him the resources for studies of mathematics, sciences, and classical languages at the University of Göttingen until 1798. His professor in mathematics

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7068-412: The end of the fifth section, it appears that Gauss already knew the class number formula in 1801. In the last section, Gauss gives proof for the constructibility of a regular heptadecagon (17-sided polygon) with straightedge and compass by reducing this geometrical problem to an algebraic one. He shows that a regular polygon is constructible if the number of its sides is either a power of 2 or

7182-430: The equator and crests at about 17 degrees in magnetic latitude. The Earth's magnetic field lines are horizontal at the magnetic equator. Solar heating and tidal oscillations in the lower ionosphere move plasma up and across the magnetic field lines. This sets up a sheet of electric current in the E region which, with the horizontal magnetic field, forces ionization up into the F layer, concentrating at ± 20 degrees from

7296-602: The existence of the Kennelly–Heaviside layer of the ionosphere which bears his name. Heaviside's proposal included means by which radio signals are transmitted around the Earth's curvature. Also in 1902, Arthur Edwin Kennelly discovered some of the ionosphere's radio-electrical properties. In 1912, the U.S. Congress imposed the Radio Act of 1912 on amateur radio operators , limiting their operations to frequencies above 1.5 MHz (wavelength 200 meters or smaller). The government thought those frequencies were useless. This led to

7410-415: The first half of the 20th century it was widely used for transoceanic telephone and telegraph service, and business and diplomatic communication. Due to its relative unreliability, shortwave radio communication has been mostly abandoned by the telecommunications industry, though it remains important for high-latitude communication where satellite-based radio communication is not possible. Shortwave broadcasting

7524-666: The first investigations, due to mislabelling, with that of the physician Conrad Heinrich Fuchs , who died in Göttingen a few months after Gauss. A further investigation showed no remarkable anomalies in the brains of both persons. Thus, all investigations on Gauss's brain until 1998, except the first ones of Rudolf and Hermann Wagner, actually refer to the brain of Fuchs. Gauss married Johanna Osthoff on 9 October 1805 in St. Catherine's church in Brunswick. They had two sons and one daughter: Joseph (1806–1873), Wilhelmina (1808–1840), and Louis (1809–1810). Johanna died on 11 October 1809, one month after

7638-456: The first radio modification of the ionosphere; HAARP ran a series of experiments in 2017 using the eponymous Luxembourg Effect . Edward V. Appleton was awarded a Nobel Prize in 1947 for his confirmation in 1927 of the existence of the ionosphere. Lloyd Berkner first measured the height and density of the ionosphere. This permitted the first complete theory of short-wave radio propagation. Maurice V. Wilkes and J. A. Ratcliffe researched

7752-440: The first to publish" differed from that of his scientific contemporaries. In contrast to his perfectionism in presenting mathematical ideas, he was criticized for a negligent way of quoting. He justified himself with a very special view of correct quoting: if he gave references, then only in a quite complete way, with respect to the previous authors of importance, which no one should ignore; but quoting in this way needed knowledge of

7866-521: The health of his second wife Minna over 13 years; both his daughters later suffered from the same disease. Gauss himself gave only slight hints of his distress: in a letter to Bessel dated December 1831 he described himself as "the victim of the worst domestic sufferings". By reason of his wife's illness, both younger sons were educated for some years in Celle , far from Göttingen. The military career of his elder son Joseph ended after more than two decades with

7980-460: The history of science and more time than he wished to spend. Soon after Gauss's death, his friend Sartorius published the first biography (1856), written in a rather enthusiastic style. Sartorius saw him as a serene and forward-striving man with childlike modesty, but also of "iron character" with an unshakeable strength of mind. Apart from his closer circle, others regarded him as reserved and unapproachable "like an Olympian sitting enthroned on

8094-463: The idea of a unified Germany. As far as the political system is concerned, he had a low estimation of the constitutional system; he criticized parliamentarians of his time for a lack of knowledge and logical errors. Some Gauss biographers have speculated on his religious beliefs. He sometimes said "God arithmetizes" and "I succeeded – not on account of my hard efforts, but by the grace of the Lord." Gauss

8208-493: The interactions of the ions and electrons with the neutral atmosphere and sunlight, or it may be a statistical description based on a large number of observations or a combination of physics and observations. One of the most widely used models is the International Reference Ionosphere (IRI), which is based on data and specifies the four parameters just mentioned. The IRI is an international project sponsored by

8322-462: The ionosphere and decrease the ionization. Sydney Chapman proposed that the region below the ionosphere be called neutrosphere (the neutral atmosphere ). At night the F layer is the only layer of significant ionization present, while the ionization in the E and D layers is extremely low. During the day, the D and E layers become much more heavily ionized, as does the F layer, which develops an additional, weaker region of ionisation known as

8436-421: The ionosphere follows both a diurnal (time of day) cycle and the 11-year solar cycle . There is also a seasonal dependence in ionization degree since the local winter hemisphere is tipped away from the Sun, thus there is less received solar radiation. Radiation received also varies with geographical location (polar, auroral zones, mid-latitudes , and equatorial regions). There are also mechanisms that disturb

8550-404: The ionosphere is less than the radio frequency, and if the electron density in the ionosphere is great enough. A qualitative understanding of how an electromagnetic wave propagates through the ionosphere can be obtained by recalling geometric optics . Since the ionosphere is a plasma, it can be shown that the refractive index is less than unity. Hence, the electromagnetic "ray" is bent away from

8664-598: The ionosphere led to some of her best-known work in communications between ground and satellites. In the early US Air force's development of GPS (then called Navstar), she was the only Australian involved in the design. Her collaborations with Australian Antarctic Division and the Co-operative Research Centre for Satellite Systems led to her having a significant role in the development of Australia's FedSat satellite (active 2002-2007 ). Essex-Cohen became hospitalised for mesothelioma in December 2002. After

8778-444: The ionosphere, adding ionization to the D-region. These disturbances are called "lightning-induced electron precipitation " (LEP) events. Additional ionization can also occur from direct heating/ionization as a result of huge motions of charge in lightning strikes. These events are called early/fast. In 1925, C. T. R. Wilson proposed a mechanism by which electrical discharge from lightning storms could propagate upwards from clouds to

8892-464: The ionosphere, is being researched. The space tether uses plasma contactors and the ionosphere as parts of a circuit to extract energy from the Earth's magnetic field by electromagnetic induction . Carl Friedrich Gauss This is an accepted version of this page Johann Carl Friedrich Gauss (German: Gauß [kaʁl ˈfʁiːdʁɪç ˈɡaʊs] ; Latin : Carolus Fridericus Gauss ; 30 April 1777 – 23 February 1855)

9006-473: The ionosphere. On July 26, 1963, the first operational geosynchronous satellite Syncom 2 was launched. On board radio beacons on this satellite (and its successors) enabled – for the first time – the measurement of total electron content (TEC) variation along a radio beam from geostationary orbit to an earth receiver. (The rotation of the plane of polarization directly measures TEC along the path.) Australian geophysicist Elizabeth Essex-Cohen from 1969 onwards

9120-646: The ionosphere. Around the same time, Robert Watson-Watt, working at the Radio Research Station in Slough, UK, suggested that the ionospheric sporadic E layer (E s ) appeared to be enhanced as a result of lightning but that more work was needed. In 2005, C. Davis and C. Johnson, working at the Rutherford Appleton Laboratory in Oxfordshire, UK, demonstrated that the E s layer was indeed enhanced as

9234-462: The light electron obtains a high velocity so that the temperature of the created electronic gas is much higher (of the order of thousand K) than the one of ions and neutrals. The reverse process to ionization is recombination , in which a free electron is "captured" by a positive ion. Recombination occurs spontaneously, and causes the emission of a photon carrying away the energy produced upon recombination. As gas density increases at lower altitudes,

9348-425: The magnetic dip equator, where the geomagnetic field is horizontal, this electric field results in an enhanced eastward current flow within ± 3 degrees of the magnetic equator, known as the equatorial electrojet . When the Sun is active, strong solar flares can occur that hit the sunlit side of Earth with hard X-rays. The X-rays penetrate to the D-region, releasing electrons that rapidly increase absorption, causing

9462-417: The magnetic equator. This phenomenon is known as the equatorial fountain . The worldwide solar-driven wind results in the so-called Sq (solar quiet) current system in the E region of the Earth's ionosphere ( ionospheric dynamo region ) (100–130 km (60–80 mi) altitude). Resulting from this current is an electrostatic field directed west–east (dawn–dusk) in the equatorial day side of the ionosphere. At

9576-590: The mathematician Gotthold Eisenstein in Berlin. Gauss was loyal to the House of Hanover . After King William IV died in 1837, the new Hanoverian King Ernest Augustus annulled the 1833 constitution. Seven professors, later known as the " Göttingen Seven ", protested against this, among them his friend and collaborator Wilhelm Weber and Gauss's son-in-law Heinrich Ewald. All of them were dismissed, and three of them were expelled, but Ewald and Weber could stay in Göttingen. Gauss

9690-423: The normal rather than toward the normal as would be indicated when the refractive index is greater than unity. It can also be shown that the refractive index of a plasma, and hence the ionosphere, is frequency-dependent, see Dispersion (optics) . The critical frequency is the limiting frequency at or below which a radio wave is reflected by an ionospheric layer at vertical incidence . If the transmitted frequency

9804-590: The observatory was completed, Gauss took his living accommodation in the western wing of the new observatory and Harding in the eastern one. They had once been on friendly terms, but over time they became alienated, possibly – as some biographers presume – because Gauss had wished the equal-ranked Harding to be no more than his assistant or observer. Gauss used the new meridian circles nearly exclusively, and kept them away from Harding, except for some very seldom joint observations. Brendel subdivides Gauss's astronomic activity chronologically into seven periods, of which

9918-411: The original languages. His favorite English author was Walter Scott , his favorite German Jean Paul . Gauss liked singing and went to concerts. He was a busy newspaper reader; in his last years, he used to visit an academic press salon of the university every noon. Gauss did not care much for philosophy, and mocked the "splitting hairs of the so-called metaphysicians", by which he meant proponents of

10032-467: The problem by accepting offers from Berlin in 1810 and 1825 to become a full member of the Prussian Academy without burdening lecturing duties, as well as from Leipzig University in 1810 and from Vienna University in 1842, perhaps because of the family's difficult situation. Gauss's salary was raised from 1000 Reichsthaler in 1810 to 2400 Reichsthaler in 1824, and in his later years he was one of

10146-425: The product of a power of 2 and any number of distinct Fermat primes . In the same section, he gives a result on the number of solutions of certain cubic polynomials with coefficients in finite fields , which amounts to counting integral points on an elliptic curve . An unfinished eighth chapter was found among left papers only after his death, consisting of work done during 1797–1799. One of Gauss's first results

10260-575: The railroad system in the US for some months. Eugen left Göttingen in September 1830 and emigrated to the United States, where he joined the army for five years. He then worked for the American Fur Company in the Midwest. Later, he moved to Missouri and became a successful businessman. Wilhelm married a niece of the astronomer Bessel ; he then moved to Missouri, started as a farmer and became wealthy in

10374-437: The rank of a poorly paid first lieutenant , although he had acquired a considerable knowledge of geodesy. He needed financial support from his father even after he was married. The second son Eugen shared a good measure of his father's talent in computation and languages, but had a vivacious and sometimes rebellious character. He wanted to study philology, whereas Gauss wanted him to become a lawyer. Having run up debts and caused

10488-435: The reader the author's train of thought. Gauss was the first to restore that rigor of demonstration which we admire in the ancients and which had been forced unduly into the background by the exclusive interest of the preceding period in new developments. But for himself, he propagated a quite different ideal, given in a letter to Farkas Bolyai as follows: It is not knowledge, but the act of learning, not possession but

10602-406: The recombination process prevails, since the gas molecules and ions are closer together. The balance between these two processes determines the quantity of ionization present. Ionization depends primarily on the Sun and its Extreme Ultraviolet (EUV) and X-ray irradiance which varies strongly with solar activity . The more magnetically active the Sun is, the more sunspot active regions there are on

10716-411: The results by masterly estimation. Nevertheless, his calculations were not always free from mistakes. He coped with the enormous workload by using skillful tools. Gauss used a lot of mathematical tables , examined their exactness, and constructed new tables on various matters for personal use. He developed new tools for effective calculation, for example the Gaussian elimination . It has been taken as

10830-599: The shoe business in St. Louis in later years. Eugene and William have numerous descendants in America, but the Gauss descendants left in Germany all derive from Joseph, as the daughters had no children. In the first two decades of the 19th century, Gauss was the only important mathematician in Germany, comparable to the leading French ones; his Disquisitiones Arithmeticae was the first mathematical book from Germany to be translated into

10944-567: The sum. Gauss took on the directorate of the 60-year-old observatory, founded in 1748 by Prince-elector George II and built on a converted fortification tower, with usable, but partly out-of-date instruments. The construction of a new observatory had been approved by Prince-elector George III in principle since 1802, and the Westphalian government continued the planning, but Gauss could not move to his new place of work until September 1816. He got new up-to-date instruments, including two meridian circles from Repsold and Reichenbach , and

11058-558: The summer of 1821. After a short time at university, in 1824 Joseph joined the Hanoverian army and assisted in surveying again in 1829. In the 1830s he was responsible for the enlargement of the survey network to the western parts of the kingdom. With his geodetical qualifications, he left the service and engaged in the construction of the railway network as director of the Royal Hanoverian State Railways . In 1836 he studied

11172-421: The summit of science". His close contemporaries agreed that Gauss was a man of difficult character. He often refused to accept compliments. His visitors were occasionally irritated by his grumpy behaviour, but a short time later his mood could change, and he would become a charming, open-minded host. Gauss abominated polemic natures; together with his colleague Hausmann he opposed to a call for Justus Liebig on

11286-407: The sunspot cycle and geomagnetic activity. Geophysically, the state of the ionospheric plasma may be described by four parameters: electron density, electron and ion temperature and, since several species of ions are present, ionic composition . Radio propagation depends uniquely on electron density. Models are usually expressed as computer programs. The model may be based on basic physics of

11400-532: The term ionosphere in a letter published only in 1969 in Nature : We have in quite recent years seen the universal adoption of the term 'stratosphere'..and..the companion term 'troposphere'... The term 'ionosphere', for the region in which the main characteristic is large scale ionisation with considerable mean free paths, appears appropriate as an addition to this series. In the early 1930s, test transmissions of Radio Luxembourg inadvertently provided evidence of

11514-478: The topic of radio propagation of very long radio waves in the ionosphere. Vitaly Ginzburg has developed a theory of electromagnetic wave propagation in plasmas such as the ionosphere. In 1962, the Canadian satellite Alouette 1 was launched to study the ionosphere. Following its success were Alouette 2 in 1965 and the two ISIS satellites in 1969 and 1971, further AEROS-A and -B in 1972 and 1975, all for measuring

11628-537: The variation of the electron density from bottom of the ionosphere to the altitude of maximum density than in describing the total electron content (TEC). Since 1999 this model is "International Standard" for the terrestrial ionosphere (standard TS16457). Ionograms allow deducing, via computation, the true shape of the different layers. Nonhomogeneous structure of the electron / ion - plasma produces rough echo traces, seen predominantly at night and at higher latitudes, and during disturbed conditions. At mid-latitudes,

11742-421: The years since 1820 are taken as a "period of lower astronomical activity". The new, well-equipped observatory did not work as effectively as other ones; Gauss's astronomical research had the character of a one-man enterprise without a long-time observation program, and the university established a place for an assistant only after Harding died in 1834. Nevertheless, Gauss twice refused the opportunity to solve

11856-630: Was Abraham Gotthelf Kästner , whom Gauss called "the leading mathematician among poets, and the leading poet among mathematicians" because of his epigrams . Astronomy was taught by Karl Felix Seyffer , with whom Gauss stayed in correspondence after graduation; Olbers and Gauss mocked him in their correspondence. On the other hand, he thought highly of Georg Christoph Lichtenberg , his teacher of physics, and of Christian Gottlob Heyne , whose lectures in classics Gauss attended with pleasure. Fellow students of this time were Johann Friedrich Benzenberg , Farkas Bolyai , and Heinrich Wilhelm Brandes . He

11970-576: Was a German mathematician , astronomer , geodesist , and physicist who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and professor of astronomy from 1807 until his death in 1855. He is widely considered one of the greatest mathematicians ever. While studying at the University of Göttingen , he propounded several mathematical theorems . Gauss completed his masterpieces Disquisitiones Arithmeticae and Theoria motus corporum coelestium as

12084-781: Was a member of the Lutheran church , like most of the population in northern Germany. It seems that he did not believe all dogmas or understand the Holy Bible quite literally. Sartorius mentioned Gauss's religious tolerance , and estimated his "insatiable thirst for truth" and his sense of justice as motivated by religious convictions. In his doctoral thesis from 1799, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root . Mathematicians including Jean le Rond d'Alembert had produced false proofs before him, and Gauss's dissertation contains

12198-689: Was deeply affected by this quarrel but saw no possibility to help them. Gauss took part in academic administration: three times he was elected as dean of the Faculty of Philosophy. Being entrusted with the widow's pension fund of the university, he dealt with actuarial science and wrote a report on the strategy for stabilizing the benefits. He was appointed director of the Royal Academy of Sciences in Göttingen for nine years. Gauss remained mentally active into his old age, even while suffering from gout and general unhappiness. On 23 February 1855, he died of

12312-960: Was detected by Edward V. Appleton and Miles Barnett . The E s layer ( sporadic E-layer) is characterized by small, thin clouds of intense ionization, which can support reflection of radio waves, frequently up to 50 MHz and rarely up to 450 MHz. Sporadic-E events may last for just a few minutes to many hours. Sporadic E propagation makes VHF-operating by radio amateurs very exciting when long-distance propagation paths that are generally unreachable "open up" to two-way communication. There are multiple causes of sporadic-E that are still being pursued by researchers. This propagation occurs every day during June and July in northern hemisphere mid-latitudes when high signal levels are often reached. The skip distances are generally around 1,640 km (1,020 mi). Distances for one hop propagation can be anywhere from 900 to 2,500 km (560 to 1,550 mi). Multi-hop propagation over 3,500 km (2,200 mi)

12426-579: Was educated at Grafton High . She subsequently completed a PhD in Physics at Australia's University of New England , investigating ionospheric irregularities under Frank Hibberd, graduating in 1966. Essex-Cohen was the fourth woman in Australia to receive a PhD in physics. After graduating her PhD, Elizabeth Essex-Cohen worked at University of the West Indies and James Cook University before taking up

12540-517: Was in keeping with the motto of his personal seal Pauca sed Matura ("Few, but Ripe"). Many colleagues encouraged him to publicize new ideas and sometimes rebuked him if he hesitated too long, in their opinion. Gauss defended himself, claiming that the initial discovery of ideas was easy, but preparing a presentable elaboration was a demanding matter for him, for either lack of time or "serenity of mind". Nevertheless, he published many short communications of urgent content in various journals, but left

12654-477: Was likely a self-taught student in mathematics since he independently rediscovered several theorems. He solved a geometrical problem that had occupied mathematicians since the Ancient Greeks , when he determined in 1796 which regular polygons can be constructed by compass and straightedge . This discovery ultimately led Gauss to choose mathematics instead of philology as a career. Gauss's mathematical diary,

12768-503: Was removed, preserved, and studied by Rudolf Wagner , who found its mass to be slightly above average, at 1,492 grams (3.29 lb). Wagner's son Hermann , a geographer, estimated the cerebral area to be 219,588 square millimetres (340.362 sq in) in his doctoral thesis. In 2013, a neurobiologist at the Max Planck Institute for Biophysical Chemistry in Göttingen discovered that Gauss's brain had been mixed up soon after

12882-449: Was the empirically found conjecture of 1792 – the later called prime number theorem – giving an estimation of the number of prime numbers by using the integral logarithm . Elizabeth Essex-Cohen Elizabeth Essex-Cohen (1940-2004) was an Australian physicist who worked in global positioning satellite physics and was among the first women in Australia to be awarded a PhD in physics. Elizabeth Annette Essex-Cohen, née Essex,

12996-506: Was using this technique to monitor the atmosphere above Australia and Antarctica. The ionosphere is a shell of electrons and electrically charged atoms and molecules that surrounds the Earth, stretching from a height of about 50 km (30 mi) to more than 1,000 km (600 mi). It exists primarily due to ultraviolet radiation from the Sun . The lowest part of the Earth's atmosphere ,

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