Egyptian pyramids

 From Wikipedia, the free encyclopedia

A view of the Giza pyramid complex from the plateau to the south of the complex. From left to right, the three largest are: the Pyramid of Menkaure, the Pyramid of Khafre and the Great Pyramid of Giza. The three smaller pyramids in the foreground are subsidiary structures associated with Menkaure's pyramid.

Famous pyramids (cut-through with internal labyrinth layout).

The Egyptian pyramids are ancient masonry structures located in Egypt. Sources cite at least 118 identified "Egyptian" pyramids. Approximately 80 pyramids were built within the Kingdom of Kush, now located in the modern country of Sudan. Of those located in modern Egypt, most were built as tombs for the country's pharaohs and their consorts during the Old and Middle Kingdom periods.

The earliest known Egyptian pyramids are found at Saqqara, northwest of Memphis, although at least one step-pyramid-like structure has been found at Saqqara, dating to the First Dynasty: Mastaba 3808, which has been attributed to the reign of Pharaoh Anedjib, with inscriptions, and other archaeological remains of the period, suggesting there may have been others. The otherwise earliest among these is the Pyramid of Djoser built c. 2630–2610 BCE during the Third Dynasty. This pyramid and its surrounding complex are generally considered to be the world's oldest monumental structures constructed of dressed masonry.

The most famous Egyptian pyramids are those found at Giza, on the outskirts of Cairo. Several of the Giza pyramids are counted among the largest structures ever built. The Pyramid of Khufu is the largest Egyptian pyramid and the last of the Seven Wonders of the Ancient World still in existence, despite being the oldest by about 2,000 years.

Name

Unicode: 𓍋𓅓𓂋𓉴
Pyramid in hieroglyphs

The name for a pyramid in Egyptian is myr, written with the symbol 𓉴 (O24 in the Gardner Sign List). Myr is preceded by three other signs used as phonetics. The meaning of myr is unclear, as it only self-references the built object itself. By comparison, some similar architectural terms become compound words, such as the word for 'temple' (per-ka) becoming a compound of the words for 'house' and 'soul'. It has been speculated myr belongs to a class of words like djed and ankh, which refer to objects already in existence when the Egyptian language split off from afroasiatic.  A typical translation of myr is given as 'high place'. By graphical analysis, myr uses the same sign, O24, as benben. The benben is the mound of existence that arose out of the abyss, known as nun in the Egyptian creation myth. The relationship between myr and benben is further linked by the capstone architectural element of pyramids and obelisks, which was named benbenet, the feminine form of benben.

Sign O24 related terms

Hieroglyph	Sign	Egyptian	English
	O24	myr	Pyramid
	O24	benben	Primeval Mound
	O24	benbent	Pyramidon
	O24	Aaa	Pyramid tomb

Historical development

The Mastabat al-Fir’aun at Saqqara

Preceded by assumed earlier sites in the Eastern Sahara, tumuli with megalithic monuments developed as early as 4700 BCE in the Saharan region of Niger. Fekri Hassan (2002) indicates that the megalithic monuments in the Saharan region of Niger and the Eastern Sahara may have served as antecedents for the mastabas and pyramids of ancient Egypt. During Predynastic Egypt, tumuli were present at various locations (e.g., Naqada, Helwan).

From the time of the Early Dynastic Period (c. 3150–2686 BCE), Egyptians with sufficient means were buried in bench-like structures known as mastabas. At Saqqara, Mastaba 3808, dating from the latter part of the 1st Dynasty, was discovered to contain a large, independently built step-pyramid-like structure enclosed within the outer palace facade mastaba. Archaeological remains and inscriptions suggest there may have been other similar structures dating to this period.

The first historically documented Egyptian pyramid is attributed by Egyptologists to the 3rd Dynasty pharaoh Djoser. Although Egyptologists often credit his vizier Imhotep as its architect, the dynastic Egyptians themselves, contemporaneously or in numerous later dynastic writings about the character, did not credit him with either designing Djoser's pyramid or the invention of stone architecture. The Pyramid of Djoser was first built as a square mastaba-like structure, which as a rule were known to otherwise be rectangular, and was expanded several times by way of a series of accretion layers, to produce the stepped pyramid structure we see today. Egyptologists believe this design served as a gigantic stairway by which the soul of the deceased pharaoh could ascend to the heavens.

Though other pyramids were attempted in the 3rd Dynasty after Djoser, it was the 4th Dynasty, transitioning from the step pyramid to true pyramid shape, which gave rise to the great pyramids of Meidum, Dahshur, and Giza. The last pharaoh of the 4th Dynasty, Shepseskaf, did not build a pyramid and beginning in the 5th Dynasty; for various reasons, the massive scale and precision of construction decreased significantly leaving these later pyramids smaller, less well-built, and often hastily constructed. By the end of the 6th Dynasty, pyramid building had largely ended and it was not until the Middle Kingdom that large pyramids were built again, though instead of stone, mudbrick was the main construction material.

Long after the end of Egypt's own pyramid-building period, a burst of pyramid-building occurred in what is present-day Sudan, after much of Egypt came under the rule of the Kingdom of Kush, which was then based at Napata. Napatan rule, known as the 25th Dynasty, lasted from 750 BCE to 664 BCE. The Meroitic period of Kushite history, when the kingdom was centered on Meroë, (approximately in the period between 300 BCE and 300 CE), experienced a full-blown pyramid-building revival, which saw about 180 Egyptian-inspired indigenous royal pyramid-tombs constructed in the vicinity of the kingdom's capital cities.

Al-Aziz Uthman (1171–1198), the second Ayyubid Sultan of Egypt, tried to destroy the Giza pyramid complex. He gave up after only damaging the Pyramid of Menkaure because the task proved too large.

Pyramid symbolism

Diagram of the interior structures of the Great Pyramid. The inner line indicates the pyramid's present profile, the outer line indicates the original profile.

The shape of Egyptian pyramids is thought to represent the primordial mound from which the Egyptians believed the earth was created. The shape of a pyramid is also thought to be representative of the descending rays of the sun, and most pyramids were faced with polished, highly reflective white limestone, in order to give them a brilliant appearance when viewed from a distance. Pyramids were often also named in ways that referred to solar luminescence. For example, the formal name of the Bent Pyramid at Dahshur was The Southern Shining Pyramid, and that of Senusret II at El Lahun was Senusret Shines.

While it is generally agreed that pyramids were burial monuments, there is continued disagreement on the particular theological principles that might have given rise to them. One suggestion is that they were designed as a type of "resurrection machine."

The Egyptians believed the dark area of the night sky around which the stars appear to revolve was the physical gateway into the heavens. One of the narrow shafts that extend from the main burial chamber through the entire body of the Great Pyramid points directly towards the center of this part of the sky. This suggests the pyramid may have been designed to serve as a means to magically launch the deceased pharaoh's soul directly into the abode of the gods.

All Egyptian pyramids were built on the west bank of the Nile, which, as the site of the setting sun, was associated with the realm of the dead in Egyptian mythology.

Number and location of pyramids

For a more comprehensive list, see List of Egyptian pyramids.

In 1842, Karl Richard Lepsius produced the first modern list of pyramids—now known as the Lepsius list of pyramids—in which he counted 67. A great many more have since been discovered. At least 118 Egyptian pyramids have been identified. The location of Pyramid 29 which Lepsius called the "Headless Pyramid", was lost for a second time when the structure was buried by desert sands after Lepsius's survey. It was found again only during an archaeological dig conducted in 2008.

Many pyramids are in a poor state of preservation or buried by desert sands. If visible at all, they may appear as little more than mounds of rubble. As a consequence, archaeologists are continuing to identify and study previously unknown pyramid structures.

The most recent pyramid to be discovered was that of Neith, a wife of Teti.

All of Egypt's pyramids, except the small Third Dynasty pyramid at Zawyet el-Maiyitin, are sited on the west bank of the Nile, and most are grouped together in a number of pyramid fields. The most important of these are listed geographically, from north to south, below.

Abu Rawash

Main article: Abu Rawash

The largely destroyed Pyramid of Djedefre

Abu Rawash is the site of Egypt's most northerly pyramid (other than the ruins of Lepsius pyramid number one), the mostly ruined Pyramid of Djedefre, son and successor of Khufu. Originally it was thought that this pyramid had never been completed, but the current archaeological consensus is that not only was it completed, but that it was originally about the same size as the Pyramid of Menkaure, which would have placed it among the half-dozen or so largest pyramids in Egypt.

Its location adjacent to a major crossroads made it an easy source of stone. Quarrying, which began in Roman times, has left little apart from about fifteen courses of stone superimposed upon the natural hillock that formed part of the pyramid's core. A small adjacent satellite pyramid is in a better state of preservation.

Giza

Main article: Giza pyramid complex

Map of the Giza pyramid complex

Aerial view of the Giza pyramid complex

The Giza Plateau is the location of the Pyramid of Khufu (also known as the "Great Pyramid" and the "Pyramid of Cheops"), the somewhat smaller Pyramid of Khafre (or Chephren), the relatively modest-sized Pyramid of Menkaure (or Mykerinus), along with a number of smaller satellite edifices known as "Queen's pyramids", and the Great Sphinx of Giza. Of the three, only Khafre's pyramid retains part of its original polished limestone casing, near its apex. This pyramid appears larger than the adjacent Khufu pyramid by virtue of its more elevated location, and the steeper angle of inclination of its construction—it is, in fact, smaller in both height and volume.

The Giza pyramid complex has been a popular tourist destination since antiquity and was popularized in Hellenistic times when the Great Pyramid was listed by Antipater of Sidon as one of the Seven Wonders of the Ancient World. Today it is the only one of those wonders still in existence.

Zawyet el-Aryan

See also: Zawyet el'Aryan

This site, halfway between Giza and Abusir, is the location for two unfinished Old Kingdom pyramids. The northern structure's owner is believed to be pharaoh Nebka, while the southern structure, known as the Layer Pyramid, may be attributable to the Third Dynasty pharaoh Khaba, a close successor of Sekhemkhet. If this attribution is correct, Khaba's short reign could explain the seemingly unfinished state of this step pyramid. Today it stands around 17 m (56 ft) high; had it been completed, it is likely to have exceeded 40 m (130 ft).

Abusir

Main article: Abusir

The Pyramid of Sahure at Abusir, viewed from the pyramid's causeway

There are a total of fourteen pyramids at this site, which served as the main royal necropolis during the Fifth Dynasty. The quality of construction of the Abusir pyramids is inferior to those of the Fourth Dynasty—perhaps signaling a decrease in royal power or a less vibrant economy. They are smaller than their predecessors and are built of low-quality local limestone.

The three major pyramids are those of Niuserre, which is also the best-preserved, Neferirkare Kakai and Sahure. The site is also home to the incomplete Pyramid of Neferefre. Most of the major pyramids at Abusir were built using similar construction techniques, comprising a rubble core surrounded by steps of mudbricks with a limestone outer casing. The largest of these Fifth Dynasty pyramids, the Pyramid of Neferirkare Kakai, is believed to have been built originally as a step pyramid some 70 m (230 ft) high and then later transformed into a "true" pyramid by having its steps filled in with loose masonry.

Saqqara

Main article: Saqqara

The Pyramid of Djoser

Major pyramids located here include the Pyramid of Djoser—generally identified as the world's oldest substantial monumental structure to be built of dressed stone—the Pyramid of Userkaf, the Pyramid of Teti and the Pyramid of Merikare, dating to the First Intermediate Period of Egypt. Also at Saqqara is the Pyramid of Unas, which retains a pyramid causeway that is one of the best-preserved in Egypt. Together with the pyramid of Userkaf, this pyramid was the subject of one of the earliest known restoration attempts, conducted by Khaemweset, a son of Ramesses II. Saqqara is also the location of the incomplete step pyramid of Djoser's successor Sekhemkhet, known as the Buried Pyramid. Archaeologists believe that had this pyramid been completed, it would have been larger than Djoser's.

South of the main pyramid field at Saqqara is a second collection of later, smaller pyramids, including those of Pepi I, Djedkare Isesi, Merenre, Pepi II and Ibi. Most of these are in a poor state of preservation.

The Fourth Dynasty pharaoh Shepseskaf either did not share an interest in or have the capacity to undertake pyramid construction like his predecessors. His tomb, which is also sited at south Saqqara, was instead built as an unusually large mastaba and offering temple complex. It is commonly known as the Mastabat al-Fir’aun.

A previously unknown pyramid was discovered in north Saqqara in late 2008. Believed to be the tomb of Teti's mother, it currently stands approximately 5 m (16 ft) high, although the original height was closer to 14 m (46 ft).

Dahshur

Main article: Dahshur

Sneferu's Red Pyramid

This area is arguably the most important pyramid field in Egypt outside Giza and Saqqara, although until 1996 the site was inaccessible due to its location within a military base and was relatively unknown outside archaeological circles.

The southern Pyramid of Sneferu, commonly known as the Bent Pyramid, is believed to be the first Egyptian pyramid intended by its builders to be a "true" smooth-sided pyramid from the outset; the earlier pyramid at Meidum had smooth sides in its finished state, but it was conceived and built as a step pyramid, before having its steps filled in and concealed beneath a smooth outer casing of dressed stone. As a true smooth-sided structure, the Bent Pyramid was only a partial success—albeit a unique, visually imposing one; it is also the only major Egyptian pyramid to retain a significant proportion of its original smooth outer limestone casing intact. As such it serves as the best contemporary example of how the ancient Egyptians intended their pyramids to look. Several kilometres to the north of the Bent Pyramid is the last—and most successful—of the three pyramids constructed during the reign of Sneferu; the Red Pyramid is the world's first successfully completed smooth-sided pyramid. The structure is also the third-largest pyramid in Egypt, after the pyramids of Khufu and Khafra at Giza.

Also at Dahshur is one of two pyramids built by Amenemhat III, known as the Black Pyramid, as well as a number of small, mostly ruined subsidiary pyramids.

Mazghuna

Main article: Mazghuna

Located to the south of Dahshur, several mudbrick pyramids were built in this area in the late Middle Kingdom, perhaps for Amenemhat IV and Sobekneferu.

The Pyramid of Amenemhet I at Lisht

Lisht

Main article: Lisht

Two major pyramids are known to have been built at Lisht: those of Amenemhat I and his son, Senusret I. The latter is surrounded by the ruins of ten smaller subsidiary pyramids. One of these subsidiary pyramids is known to be that of Amenemhat's cousin, Khaba II. The site which is in the vicinity of the oasis of the Faiyum, midway between Dahshur and Meidum, and about 100 kilometres south of Cairo, is believed to be in the vicinity of the ancient city of Itjtawy (the precise location of which remains unknown), which served as the capital of Egypt during the Twelfth Dynasty.

Meidum

Main article: Meidum

The pyramid at Meidum

The pyramid at Meidum is one of three constructed during the reign of Sneferu, and is believed by some to have been started by that pharaoh's father and predecessor, Huni. However, that attribution is uncertain, as no record of Huni's name has been found at the site. It was constructed as a step pyramid and then later converted into the first "true" smooth-sided pyramid, when the steps were filled in and an outer casing added. The pyramid suffered several catastrophic collapses in ancient and medieval times. Medieval Arab writers described it as having seven steps, although today only the three uppermost of these remain, giving the structure its odd, tower-like appearance. The hill on which the pyramid is situated is not a natural landscape feature, it is the small mountain of debris created when the lower courses and outer casing of the pyramid gave way.

Hawara

Main article: Hawara

The Pyramid of Amenemhet III at Hawara

Amenemhat III was the last powerful ruler of the Twelfth Dynasty, and the pyramid he built at Hawara, near the Faiyum, is believed to post-date the so-called "Black Pyramid" built by the same ruler at Dahshur. It is the Hawara pyramid that is believed to have been Amenemhet's final resting place.

El Lahun

Main article: El Lahun

The Pyramid of Senusret II. The pyramid's natural limestone core is clearly visible as the yellow stratum at its base.

The Pyramid of Senusret II at El Lahun is the southernmost royal-tomb pyramid structure in Egypt. Its builders reduced the amount of work necessary to construct it by using as its foundation and core a 12-meter-high natural limestone hill.

El-Kurru

Main article: El-Kurru

Piye's pyramid at El-Kurru

Piye, the king of Kush who became the first ruler of the Twenty-fifth Dynasty, built a pyramid at El-Kurru. He was the first Egyptian pharaoh to be buried in a pyramid in centuries.

Nuri

Main article: Nuri

Taharqa's pyramid at Nuri

Taharqa, a Kushite ruler of the Twenty-fifth Dynasty, built his pyramid at Nuri. It was the largest in the area (North Sudan).

Construction dates and heights

The following table lays out the chronology of the construction of most of the major pyramids mentioned here. Each pyramid is identified through the pharaoh who ordered it built, his approximate reign, and its location.

Pyramid (Pharaoh)	Reign	Field	Height
Pyramid of Djoser (Djoser)	c. 2670 BCE	Saqqara	62 meters (203 feet)
Red Pyramid (Sneferu)	c. 2612–2589 BCE	Dahshur	104 meters (341 feet)
Meidum Pyramid (Sneferu)	c. 2612–2589 BCE	Meidum	65 meters (213 feet) (ruined) Would have been 91.65 meters (301 feet) or 175 Egyptian Royal cubits.
Great Pyramid of Giza (Khufu)	c. 2589–2566 BCE	Giza	146.7 meters (481 feet) or 280 Egyptian Royal cubits
Pyramid of Djedefre (Djedefre)	c. 2566–2558 BCE	Abu Rawash	60 meters (197 feet)
Pyramid of Khafre (Khafre)	c. 2558–2532 BCE	Giza	136.4 meters (448 feet) Originally: 143.5 m (471 ft) or 274 Egyptian Royal cubits
Pyramid of Menkaure (Menkaure)	c. 2532–2504 BCE	Giza	65 meters (213 feet) or 125 Egyptian Royal cubits
Pyramid of Userkaf (Userkaf)	c. 2494–2487 BCE	Saqqara	48 meters (161 feet)
Pyramid of Sahure (Sahure)	c. 2487–2477 BCE	Abusir	47 meters (155 feet)
Pyramid of Neferirkare (Neferirkare Kakai)	c. 2477–2467 BCE	Abusir	72.8 meters (239 feet)
Pyramid of Nyuserre (Nyuserre Ini)	c. 2416–2392 BCE	Abusir	51.68 m (169.6 feet) or 99 Egyptian Royal cubits
Pyramid of Amenemhat I (Amenemhat I)	c. 1991–1962 BCE	Lisht	55 meters (181 feet)
Pyramid of Senusret I (Senusret I)	c. 1971–1926 BCE	Lisht	61.25 meters (201 feet)
Pyramid of Senusret II (Senusret II)	c. 1897–1878 BCE	el-Lahun	48.65 m (159.6 ft; 93 Egyptian Royal cubits) or 47.6 m (156 ft; 91 Egyptian Royal cubits)
Black Pyramid (Amenemhat III)	c. 1860–1814 BCE	Dahshur	75 meters (246 feet)
Pyramid of Khendjer (Khendjer)	c. 1764–1759 BCE	Saqqara	about 37 metres (121 ft), now completely ruined
Pyramid of Piye (Piye)	c. 721 BCE	El-Kurru	20 meters (66 feet) or 30 meters (99 feet)
Pyramid of Taharqa (Taharqa)	c. 664 BCE	Nuri	40 meters (132 feet) or 50 meters (164 feet)

Construction techniques

Drawing showing transportation of a colossus. The water poured in the path of the sledge, long dismissed by Egyptologists as ritual, but now confirmed as feasible, served to increase the stiffness of the sand, and likely reduced by 50% the force needed to move the statue.

Main article: Egyptian pyramid construction techniques

Further information: Diary of Merer

Constructing the pyramids involved moving huge quantities of stone. While most blocks came from nearby quarries, special stones were transported on great barges from distant locations, for instance white limestone from Tura and granite from Aswan.

In 2013, papyri, named Diary of Merer, were discovered at an ancient Egyptian harbor at the Red Sea coast. They are logbooks written over 4,500 years ago by an official with the title inspector, who documented the transport of white limestone from the Tura quarries, along the Nile River, to the Great Pyramid of Giza, the tomb of the Pharaoh Khufu.

It is possible that quarried blocks were then transported to the construction site by wooden sleds, with sand in front of the sled wetted to reduce friction. Droplets of water created bridges between the grains of sand, helping them stick together. Workers cut the stones close to the construction site, as indicated by the numerous finds of cutting tools. The finished blocks were placed on the pre-prepared foundations. The foundations were levelled using a rough square level, water trenches and experienced surveyors.

Biopolitics

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Biopolitics

Biopolitics is a concept introduced by the French philosopher Michel Foucault in the mid-20th century. At its core, biopolitics explores how governmental power operates through the management and regulation of a population's bodies and lives.

This interdisciplinary field scrutinizes the mechanisms through which political authorities and institutions exercise control over populations which goes beyond conventional forms of governance. This encompasses areas such as the regulation of health, reproduction, sexuality, and other aspects of biological existence. The governmental power of biopolitics is exerted through practices such as surveillance, healthcare policies, population control measures, gender-based laws, and the implementation of biometric identification systems.

Foucault's thesis claims that contemporary power structures are increasingly preoccupied with the administration of life itself, rather than solely focusing on individual behaviors or actions. Accordingly, biopolitics entails the governance of populations as biological entities, with an emphasis on optimizing their health, productivity, and reproductive capacities in manners conducive to broader political and economic objectives. In its essence, biopolitics investigates how political power intersects with biological life, shaping the bodies, behaviors, and well-being of populations through diverse strategies and controls.

Notions of biopolitics

Previous notions of the concept can be traced back to the Middle Ages in John of Salisbury's work Policraticus, in which the term body politic was coined and used. The term biopolitics was first used by Rudolf Kjellén, a political scientist who also coined the term geopolitics, in his 1905 two-volume work The Great Powers. Kjellén used the term in the context of his aim to study "the civil war between social groups" (comprising the state) from a biological perspective, and thus named his putative discipline "biopolitics". In Kjellén's organicist view, the state was a quasi-biological organism, a "super-individual creature." The Nazis also subsequently used the term in the context of their racial policy, with Hans Reiter using it in a 1934 speech to refer to their concept of nation and state based on racial supremacy.

In contemporary US political science studies, usage of the term is mostly divided between a poststructuralist group using the meaning assigned by Foucault (denoting social and political power over life) and another group that uses it to denote studies relating biology and political science. In the work of Foucault, biopolitics refers to the style of government that regulates populations through "biopower" (the application and impact of political power on all aspects of human life).

Morley Roberts, in his 1938 book Bio-politics argued that a correct model for world politics is "a loose association of cell and protozoa colonies". Robert E. Kuttner used the term to refer to his particular brand of "scientific racism," as he called it, which he worked out with noted antisemite Eustace Mullins, with whom Kuttner co-founded the Institute for Biopolitics in the late 1950s, and also with Glayde Whitney, a behavioral geneticist. Most of his opponents label his model as antisemitic. Kuttner and Mullins were inspired by Morley Roberts, who was in turn inspired by Arthur Keith, or both were inspired by each other and either co-wrote together (or with the Institute of Biopolitics) Biopolitics of Organic Materialism dedicated to Roberts and reprinted some of his works.

In the work of Michael Hardt and Antonio Negri, biopolitics is framed in terms of anti-capitalist insurrection using life and the body as weapons; examples include flight from power and, "in its most tragic and revolting form", suicide terrorism, conceptualized as the opposite of biopower, which is seen as the practice of sovereignty in biopolitical conditions.

According to Professor Agni Vlavianos Arvanitis, biopolitics is a conceptual and operative framework for societal development, promoting bios (Greek for "life") as the central theme in every human endeavor, be it policy, education, art, government, science or technology. This concept uses bios as a term referring to all forms of life on our planet, including their genetic and geographic variation.

Alternative usages

One usage concerns the interplay and interdisciplinary studies relating biology and political science, primarily the study of the relationship between biology and political behavior. Most of these works agree on three fundamental aspects. First, the object of investigation is primarily political behavior, which—and this is the underlying assumption—is caused in a substantial way by objectively demonstrable biological factors. For example, the relationship of biology and political orientation, but also biological correlates of partisanship and voting behavior. (See also sociobiology.) Note here Ernst Haeckel's famous proposition that "[p]olitics is applied biology."

Another common usage is per a political spectrum that reflects and or advocates various positions towards regarding the biotech revolution.

A less common one sometimes surfaces in the green politics of bioregionalism.

In the colonial setting

Biopolitics, read as a variation of Foucault's Biopower, has proven to be a substantive concept in the field of postcolonial studies. Foucault's term refers to the intersection between power (political, economic, judicial etc.) and the individual's bodily autonomy. According to postcolonial theorists, present within the colonial setting are various mechanisms of power that consolidate the political authority of the colonizer; Biopolitics is thus the means by which a colonising force utilises political power to regulate and control the bodily autonomy of the colonized subject, who are oppressed and subaltern. Edward Said, in his work Orientalism, analysed the means by which colonial powers rationalised their relationship with the colonized societies they inhabited through discursive means, and how these discourses continue to influence modern-day depictions of the Orient. Franz Fanon applied a psychoanalytic frame to his theories of subjectivity, arguing that the subjectivity of the colonized is in constant dialogue with the oppressive political power of the colonizer, a mirroring of the Oedipal father-son dynamic. While not using the term himself, Fanon's work has been cited as a major development in the conceptualisation of biopolitics in the colonial setting.

Michel Foucault

See also: Biopower

French philosopher and social theorist Michel Foucault first discussed his thoughts on biopolitics in his lecture series "Society Must Be Defended" given at the Collège de France from 1975 to 1976. Foucault's concept of biopolitics is largely derived from his own notion of biopower, and the extension of state power over both the physical and political bodies of a population. While only mentioned briefly in his "Society Must Be Defended" lectures, the conceptualisation of biopolitics developed by Foucault has become prominent in social science and the humanities.

Foucault described biopolitics as "a new technology of power...[that] exists at a different level, on a different scale, and [that] has a different bearing area, and makes use of very different instruments." More than a disciplinary mechanism, Foucault's biopolitics acts as a control apparatus exerted over a population as a whole or, as Foucault stated, "a global mass." In the years that followed, Foucault continued to develop his notions of the biopolitical in his "The Birth of Biopolitics" and "The Courage of Truth" lectures.

Foucault gave numerous examples of biopolitical control when he first mentioned the concept in 1976. These examples include "ratio of births to deaths, the rate of reproduction, the fertility of a population, and so on." He contrasted this method of social control with political power in the Middle Ages. Whereas in the Middle Ages pandemics made death a permanent and perpetual part of life, this was then shifted around the end of the 18th century with the introduction of milieu into the biological sciences. Foucault then gives different contrasts to the then physical sciences in which the industrialisation of the population was coming to the fore through the concept of work, where Foucault then argues power starts to become a target for this milieu by the 17th century. The development of vaccines and medicines dealing with public hygiene allowed death to be held (and/or withheld) from certain populations. This was the introduction of "more subtle, more rational mechanisms: insurance, individual and collective savings, safety measures, and so on."

Elliptic orbit

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Elliptic_orbit

Animation of Orbit by eccentricity
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Two bodies with similar mass orbiting around a common barycenter with elliptic orbits.

Two bodies with unequal mass orbiting around a common barycenter with circular orbits.

Two bodies with highly unequal mass orbiting a common barycenter with circular orbits.

An elliptical orbit is depicted in the top-right quadrant of this diagram, where the gravitational potential well of the central mass shows potential energy, and the kinetic energy of the orbital speed is shown in red. The height of the kinetic energy decreases as the orbiting body's speed decreases and distance increases according to Kepler's laws.

In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1 (thus excluding the circular orbit). In a wider sense, it is a Kepler orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

In a gravitational two-body problem with negative energy, both bodies follow similar elliptic orbits with the same orbital period around their common barycenter. Also the relative position of one body with respect to the other follows an elliptic orbit.

Examples of elliptic orbits include Hohmann transfer orbits, Molniya orbits, and tundra orbits.

Velocity

Under standard assumptions, no other forces acting except two spherically symmetrical bodies m₁ and m₂, the orbital speed (${\displaystyle v}$) of one body traveling along an elliptic orbit can be computed from the vis-viva equation as:

${\displaystyle v=\sqrt{\mu \left(\frac{2}{r}-\frac{1}{a}\right)}}$

where:

• ${\displaystyle \mu }$ is the standard gravitational parameter, G(m₁+m₂), often expressed as GM when one body is much larger than the other.
• ${\displaystyle r}$ is the distance between the orbiting body and center of mass.
• ${\displaystyle a}$ is the length of the semi-major axis.

The velocity equation for a hyperbolic trajectory has either + ${\displaystyle \frac{1}{a}}$, or it is the same with the convention that in that case a is negative.

Orbital period

Under standard assumptions the orbital period (${\displaystyle T}$) of a body travelling along an elliptic orbit can be computed as:

${\displaystyle T=2\pi \sqrt{\frac{a^3}{\mu }}}$

where:

• ${\displaystyle \mu }$ is the standard gravitational parameter.
• ${\displaystyle a}$ is the length of the semi-major axis.

Conclusions:

• The orbital period is equal to that for a circular orbit with the orbital radius equal to the semi-major axis (${\displaystyle a}$),
• For a given semi-major axis the orbital period does not depend on the eccentricity (See also: Kepler's third law).

Energy

Under standard assumptions, the specific orbital energy (${\displaystyle ϵ}$) of an elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form:

${\displaystyle \frac{v^2}{2}-\frac{\mu }{r}=-\frac{\mu }{2a}=ϵ<0}$

where:

• ${\displaystyle v}$ is the orbital speed of the orbiting body,
• ${\displaystyle r}$ is the distance of the orbiting body from the central body,
• ${\displaystyle a}$ is the length of the semi-major axis,
• ${\displaystyle \mu }$ is the standard gravitational parameter.

Conclusions:

• For a given semi-major axis the specific orbital energy is independent of the eccentricity.

Using the virial theorem to find:

• the time-average of the specific potential energy is equal to −2ε
  • the time-average of r⁻¹ is a⁻¹
• the time-average of the specific kinetic energy is equal to ε

Energy in terms of semi major axis

It can be helpful to know the energy in terms of the semi major axis (and the involved masses). The total energy of the orbit is given by

${\displaystyle E=-G\frac{Mm}{2a}}$,

where a is the semi major axis.

Derivation

Since gravity is a central force, the angular momentum is constant:

${\displaystyle \dot{\mathbf{L}}=\mathbf{r}\times \mathbf{F}=\mathbf{r}\times F(r)\hat{\mathbf{r}}=0}$

At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore:

${\displaystyle L=rp=rmv}$.

The total energy of the orbit is given by

${\displaystyle E=\frac{1}{2}m{v}^2-G\frac{Mm}{r}}$.

Substituting for v, the equation becomes

${\displaystyle E=\frac{1}{2}\frac{L^2}{m{r}^2}-G\frac{Mm}{r}}$.

This is true for r being the closest / furthest distance so two simultaneous equations are made, which when solved for E:

${\displaystyle E=-G\frac{Mm}{r_1+r_2}}$

Since $r_1=a+aϵ$ and ${\displaystyle r_2=a-aϵ}$, where epsilon is the eccentricity of the orbit, the stated result is reached.

Flight path angle

The flight path angle is the angle between the orbiting body's velocity vector (equal to the vector tangent to the instantaneous orbit) and the local horizontal. Under standard assumptions of the conservation of angular momentum the flight path angle ${\displaystyle \varphi }$ satisfies the equation:

${\displaystyle h=rv\cos \varphi }$

where:

• ${\displaystyle h}$ is the specific relative angular momentum of the orbit,
• ${\displaystyle v}$ is the orbital speed of the orbiting body,
• ${\displaystyle r}$ is the radial distance of the orbiting body from the central body,
• ${\displaystyle \varphi }$ is the flight path angle

${\displaystyle \psi }$ is the angle between the orbital velocity vector and the semi-major axis. ${\displaystyle \nu }$ is the local true anomaly. ${\displaystyle \varphi =\nu +\frac{\pi }{2}-\psi }$, therefore,

${\displaystyle \cos \varphi =\sin (\psi -\nu )=\sin \psi \cos \nu -\cos \psi \sin \nu =\frac{1+e\cos \nu }{\sqrt{1+e^2+2e\cos \nu }}}$

${\displaystyle \tan \varphi =\frac{e\sin \nu }{1+e\cos \nu }}$

where ${\displaystyle e}$ is the eccentricity.

The angular momentum is related to the vector cross product of position and velocity, which is proportional to the sine of the angle between these two vectors. Here ${\displaystyle \varphi }$ is defined as the angle which differs by 90 degrees from this, so the cosine appears in place of the sine.

Equation of motion

Main article: orbit equation

From initial position and velocity

An orbit equation defines the path of an orbiting body ${\displaystyle m_2}$ around central body ${\displaystyle m_1}$ relative to ${\displaystyle m_1}$, without specifying position as a function of time. If the eccentricity is less than 1 then the equation of motion describes an elliptical orbit. Because Kepler's equation ${\displaystyle M=E-e\sin E}$ has no general closed-form solution for the Eccentric anomaly (E) in terms of the Mean anomaly (M), equations of motion as a function of time also have no closed-form solution (although numerical solutions exist for both).

However, closed-form time-independent path equations of an elliptic orbit with respect to a central body can be determined from just an initial position (${\displaystyle \mathbf{r}}$) and velocity (${\displaystyle \mathbf{v}}$).

For this case it is convenient to use the following assumptions which differ somewhat from the standard assumptions above:

1. The central body's position is at the origin and is the primary focus (${\displaystyle \mathbf{F}\mathbf{1}}$) of the ellipse (alternatively, the center of mass may be used instead if the orbiting body has a significant mass)
2. The central body's mass (m1) is known
3. The orbiting body's initial position(${\displaystyle \mathbf{r}}$) and velocity(${\displaystyle \mathbf{v}}$) are known
4. The ellipse lies within the XY-plane

The fourth assumption can be made without loss of generality because any three points (or vectors) must lie within a common plane. Under these assumptions the second focus (sometimes called the "empty" focus) must also lie within the XY-plane: ${\displaystyle \mathbf{F}\mathbf{2}=\left(f_x,f_y\right)}$ .

Using vectors

The general equation of an ellipse under these assumptions using vectors is:

${\displaystyle |\mathbf{F}\mathbf{2}-\mathbf{p}|+|\mathbf{p}|=2a\mid z=0}$

where:

• ${\displaystyle a}$ is the length of the semi-major axis.
• ${\displaystyle \mathbf{F}\mathbf{2}=\left(f_x,f_y\right)}$ is the second ("empty") focus.
• ${\displaystyle \mathbf{p}=\left(x,y\right)}$ is any (x,y) value satisfying the equation.

The semi-major axis length (a) can be calculated as:

${\displaystyle a=\frac{\mu |\mathbf{r}|}{2\mu -|\mathbf{r}|{\mathbf{v}}^2}}$

where ${\displaystyle \mu =G{m}_1}$ is the standard gravitational parameter.

The empty focus (${\displaystyle \mathbf{F}\mathbf{2}=\left(f_x,f_y\right)}$) can be found by first determining the Eccentricity vector:

${\displaystyle \mathbf{e}=\frac{\mathbf{r}}{|\mathbf{r}|}-\frac{\mathbf{v}\times \mathbf{h}}{\mu }}$

Where ${\displaystyle \mathbf{h}}$ is the specific angular momentum of the orbiting body:

${\displaystyle \mathbf{h}=\mathbf{r}\times \mathbf{v}}$

Then

${\displaystyle \mathbf{F}\mathbf{2}=-2a\mathbf{e}}$

Using XY Coordinates

This can be done in cartesian coordinates using the following procedure:

The general equation of an ellipse under the assumptions above is:

${\displaystyle \sqrt{{\left(f_x-x\right)}^2+{\left(f_y-y\right)}^2}+\sqrt{x^2+y^2}=2a\mid z=0}$

Given:

${\displaystyle r_x,r_y}$ the initial position coordinates

${\displaystyle v_x,v_y}$ the initial velocity coordinates

and

${\displaystyle \mu =G{m}_1}$ the gravitational parameter

Then:

${\displaystyle h=r_x{v}_y-r_y{v}_x}$ specific angular momentum

${\displaystyle r=\sqrt{r_x^2+r_y^2}}$ initial distance from F1 (at the origin)

${\displaystyle a=\frac{\mu r}{2\mu -r\left(v_x^2+v_y^2\right)}}$ the semi-major axis length

${\displaystyle e_x=\frac{r_x}{r}-\frac{h{v}_y}{\mu }}$ the Eccentricity vector coordinates

${\displaystyle e_y=\frac{r_y}{r}+\frac{h{v}_x}{\mu }}$

Finally, the empty focus coordinates

${\displaystyle f_x=-2a{e}_x}$

${\displaystyle f_y=-2a{e}_y}$

Now the result values fx, fy and a can be applied to the general ellipse equation above.

Orbital parameters

The state of an orbiting body at any given time is defined by the orbiting body's position and velocity with respect to the central body, which can be represented by the three-dimensional Cartesian coordinates (position of the orbiting body represented by x, y, and z) and the similar Cartesian components of the orbiting body's velocity. This set of six variables, together with time, are called the orbital state vectors. Given the masses of the two bodies they determine the full orbit. The two most general cases with these 6 degrees of freedom are the elliptic and the hyperbolic orbit. Special cases with fewer degrees of freedom are the circular and parabolic orbit.

Because at least six variables are absolutely required to completely represent an elliptic orbit with this set of parameters, then six variables are required to represent an orbit with any set of parameters. Another set of six parameters that are commonly used are the orbital elements.

Solar System

In the Solar System, planets, asteroids, most comets, and some pieces of space debris have approximately elliptical orbits around the Sun. Strictly speaking, both bodies revolve around the same focus of the ellipse, the one closer to the more massive body, but when one body is significantly more massive, such as the sun in relation to the earth, the focus may be contained within the larger massing body, and thus the smaller is said to revolve around it. The following chart of the perihelion and aphelion of the planets, dwarf planets, and Halley's Comet demonstrates the variation of the eccentricity of their elliptical orbits. For similar distances from the sun, wider bars denote greater eccentricity. Note the almost-zero eccentricity of Earth and Venus compared to the enormous eccentricity of Halley's Comet and Eris.

Distances of selected bodies of the Solar System from the Sun. The left and right edges of each bar correspond to the perihelion and aphelion of the body, respectively, hence long bars denote high orbital eccentricity. The radius of the Sun is 0.7 million km, and the radius of Jupiter (the largest planet) is 0.07 million km, both too small to resolve on this image.

Radial elliptic trajectory

A radial trajectory can be a double line segment, which is a degenerate ellipse with semi-minor axis = 0 and eccentricity = 1. Although the eccentricity is 1, this is not a parabolic orbit. Most properties and formulas of elliptic orbits apply. However, the orbit cannot be closed. It is an open orbit corresponding to the part of the degenerate ellipse from the moment the bodies touch each other and move away from each other until they touch each other again. In the case of point masses one full orbit is possible, starting and ending with a singularity. The velocities at the start and end are infinite in opposite directions and the potential energy is equal to minus infinity.

The radial elliptic trajectory is the solution of a two-body problem with at some instant zero speed, as in the case of dropping an object (neglecting air resistance).

See also: Free fall § Inverse-square law gravitational field

History

The Babylonians were the first to realize that the Sun's motion along the ecliptic was not uniform, though they were unaware of why this was; it is today known that this is due to the Earth moving in an elliptic orbit around the Sun, with the Earth moving faster when it is nearer to the Sun at perihelion and moving slower when it is farther away at aphelion.

In the 17th century, Johannes Kepler discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun at one focus, and described this in his first law of planetary motion. Later, Isaac Newton explained this as a corollary of his law of universal gravitation.