Egyptian pyramid construction techniques

From Wikipedia, the free encyclopedia

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

The three main pyramids at Giza, together with subsidiary pyramids and the remains of other ancient structures

Egyptian pyramid construction techniques are the controversial subject of many hypotheses. These techniques seem to have developed over time; later pyramids were not constructed in the same way as earlier ones. Most of the construction hypotheses are based on the belief that huge stones were carved from quarries with copper chisels, and these blocks were then dragged and lifted into position. Disagreements chiefly concern the methods used to move and place the stones.

In addition to the many unresolved arguments about the construction techniques, there have been disagreements as to the kind of workforce used. The Greeks, many years after the event, believed that the pyramids were built by slave labour. Archaeologists now believe that the Great Pyramid of Giza (at least) was built by tens of thousands of skilled workers who camped near the pyramids and worked for a salary or as a form of tax payment (levy) until the construction was completed, pointing to workers' cemeteries discovered in 1990. For the Middle Kingdom pyramid of Amenemhat II, there is evidence from the annal stone of the king that foreigners from Canaan were employed.

A number of pseudoscientific theories have been put forth to explain how the pyramids were built.

Historical hypotheses

Writings of Herodotus and Diodorus Siculus

The unknowns of pyramid construction chiefly center on the question of how the blocks were moved up the superstructure. There is no known accurate historical or archaeological evidence that definitively resolves the question. Therefore, most discussion on construction methods involves functional possibilities that are supported by limited historical and archaeological evidence.

The first historical accounts of the construction of these monuments came centuries after the era of pyramid construction, by Herodotus in the 5th century BC and Diodorus Siculus in the 1st century BC. Herodotus's account claims that the Egyptians used a machine (now commonly referred to as the "Herodotus Machine"), stating:

A machine lifting a large stone column, by Leonardo da Vinci. Believed to be sketched based on Herodotus' description

This pyramid was made like stairs, which some call steps and others, tiers. When this, its first form, was completed, the workmen used short wooden logs as levers to raise the rest of the stones; they heaved up the blocks from the ground onto the first tier of steps; when the stone had been raised, it was set on another lever that stood on the first tier, and the lever again used to lift it from this tier to the next. It may be that there was a new lever on each tier of steps, or perhaps there was only one lever, quite portable, which they carried up to each tier in turn; I leave this uncertain, as both possibilities were mentioned. But this is certain, that the upper part of the pyramid was finished off first, then the next below it, and last of all the base and the lowest part.

Diodorus Siculus's account states:

And it's said the stone was transported a great distance from Arabia, and that the edifices were raised by means of earthen ramps, since machines for lifting had not yet been invented in those days; and most surprising it is, that although such large structures were raised in an area surrounded by sand, no trace remains of either ramps or the dressing of the stones, so that it seems not the result of the patient labor of men, but rather as if the whole complex were set down entire upon the surrounding sand by some god. Now Egyptians try to make a marvel of these things, alleging that the ramps were made of salt and natron and that, when the river was turned against them, it melted them clean away and obliterated their every trace without the use of human labor. But in truth, it most certainly was not done this way! Rather, the same multitude of workmen who raised the mounds returned the entire mass again to its original place; for they say that three hundred and sixty thousand men were constantly employed in the prosecution of their work, yet the entire edifice was hardly finished at the end of twenty years.

Diodorus Siculus's description of the shipment of the stone from Arabia is correct since the term "Arabia" in those days implied the land between the Nile and the Red Sea where the limestone blocks have been transported from quarries across the river Nile.

Materials

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Third through Fifth dynasties

During the earliest period, pyramids were constructed wholly of stone. Locally quarried limestone was the material of choice for the main body of these pyramids, while a higher quality of limestone quarried at Tura (near modern Cairo) was used for the outer casing. Granite, quarried near Aswan, was used to construct some architectural elements, including the portcullis (a type of gate) and the roofs and walls of the burial chamber. Occasionally, granite was used in the outer casing as well, such as in the Pyramid of Menkaure. In the early pyramids, the layers of stone (called courses) forming the pyramid body were laid sloping inwards; however, this configuration was found to be less stable than simply stacking the stones horizontally on top of each other. The Bent Pyramid at Dahshur seems to indicate acceptance of a new technique at a transition between these two building techniques. Its lower section is built of sloping courses while in its upper section the stones are laid horizontally.

Middle Kingdom and onward

During the Middle Kingdom, pyramid construction techniques changed again. Most pyramids built then were little more than mountains of mud-brick encased in a veneer of polished limestone. In several cases, later pyramids were built on top of natural hills to further reduce the volume of material needed in their construction. The materials and methods of construction used in the earliest pyramids have ensured their survival in a generally much better state of preservation than for the pyramid monuments of the later pharaohs.

Mortar

The stones forming the core of the pyramids were roughly cut, especially in the Great Pyramid. To fill the gaps, huge quantities of gypsum and rubble were needed. The filling has almost no binding properties, but it was necessary to stabilize the construction. To make the gypsum mortar, it had to be dehydrated by heating which requires large quantities of wood. According to Egyptologists, the findings of both the 1984 and 1995 David H. Koch Pyramids Radiocarbon Projects may suggest that Egypt had to strip its forest and scrap every bit of wood it had to build the pyramids of Giza and other even earlier 4th Dynasty pyramids. Carbon dating samples from core blocks and other materials revealed that dates from the 1984 study averaged 374 years earlier than currently accepted and the 1995 dating averaging 100–200 years. As suggested by team members, "We thought that it was unlikely that the pyramid builders consistently used centuries-old wood as fuel in preparing mortar. The 1984 results left us with too little data to conclude that the historical chronology of the Old Kingdom was wrong by nearly 400 years, but we considered this at least a possibility". Egyptologists propose that the old wood problem is responsible for the discrepancy, claiming the earlier dates were possibly derived from recycling large amounts of centuries-old wood and other earlier materials.

Quarrying

There is good information concerning the location of the quarries, some of the tools used to cut stone in the quarries, transportation of the stone to the monument, leveling the foundation, and leveling the subsequent tiers of the developing superstructure. Workmen probably used copper chisels, drills, and saws to cut softer stone, such as most of the limestone. The harder stones, such as granite, granodiorite, syenite, and basalt, cannot be cut with copper tools alone; instead, they were worked with time-consuming methods like pounding with dolerite, drilling, and sawing with the aid of an abrasive, such as quartz sand. This occurred in a process known as sand abrasion. Blocks were transported by sledge likely lubricated by water.

Leveling the foundation may have been accomplished by use of water-filled trenches as suggested by Mark Lehner and I. E. S. Edwards or through the use of a crude square level and experienced surveyors.

Transport of stone blocks

A transport of a large statue on a sledge

One of the major problems faced by the early pyramid builders was the need to move huge quantities of stone. The Twelfth Dynasty tomb of Djehutihotep has an illustration of 172 men pulling an alabaster statue of him on a sledge. The statue is estimated to weigh 60 tons and Denys Stocks estimated that 45 workers would be required to start moving a 16,300 kg (35,900 lb; 16.3 t) lubricated block, or eight workers to move a 2,750 kg (6,060 lb; 2.75 t) block. Dick Parry has suggested a method for rolling the stones, using a cradle-like machine that had been excavated in various new kingdom temples. Four of those objects could be fitted around a block so it could be rolled easily. Experiments done by the Obayashi Corporation, with concrete blocks 0.8 metres (2 ft 7 in) square by 1.6 metres (5 ft 3 in) long and weighing 2.5 tonnes (2,500 kg; 5,500 lb), showed how 18 men could drag the block over a 1-in-4 incline ramp, at a rate of 18 metres per minute (1 ft/s). This idea was previously described by John Bush in 1977, and is mentioned in the Closing Remarks section of Parry's book. Vitruvius in De architectura described a similar method for moving irregular weights. It is still not known whether the Egyptians used this method but the experiments indicate it could have worked using stones of this size. Egyptologists generally accept this for the 2.5 ton blocks mostly used but do not agree over the methods used for the 15+ ton and several 70 to 80 ton blocks.

The diary of Merer, logbooks written more than 4,500 years ago by an Egyptian official and found in 2013 by a French archeology team under the direction of Pierre Tallet in a cave in Wadi al-Jarf, describes the transportation of limestone blocks from the quarries at Tura to Giza by boat.

Ramps

Example of a large straight ramp. This method was likely not used in pyramid construction according to current expert consensus (see text).

From left to right: Zig-zagging ramp (Uvo Hölscher), ramp using the incomplete part of the superstructure (Dieter Arnold), and a spiraling ramp supported by the superstructure (Mark Lehner)

Most Egyptologists acknowledge that ramps are the most tenable of the methods to raise the blocks, yet they acknowledge that it is an incomplete method that must be supplemented by another device. Archaeological evidence for the use of ramps has been found at the Great Pyramid of Giza and other pyramids. The method most accepted for assisting ramps is levering. The archaeological record gives evidence of only small ramps and inclined causeways, not something that could have been used to construct even a majority of the monument. To add to the uncertainty, there is considerable evidence demonstrating that non-standardized or ad hoc construction methods were used in pyramid construction.

Therefore, there are many proposed ramps and there is a considerable amount of discrepancy regarding what type of ramp was used to build the pyramids. One of the widely discredited ramping methods is the large straight ramp, and it is routinely discredited on functional grounds for its massive size, lack of archaeological evidence, huge labor cost, and other problems (Isler 2001: 213).

Other ramps serve to correct these problems of ramp size, yet either run into critiques of functionality and limited archaeological evidence. There are zig-zagging ramps, straight ramps using the incomplete part of the superstructure (Arnold 1991), spiraling ramps supported by the superstructure and spiraling ramps leaning on the monument as a large accretion are proposed. Mark Lehner speculated that a spiraling ramp, beginning in the stone quarry to the southeast and continuing around the exterior of the pyramid, may have been used. However, spiral ramps would have covered the building for decades fully and would not allow the regular layout of the exact pyramid square in equal distance to each base cornerstone as the only method to keep the exact geometric shape of the edges and side mantles. The stone blocks may have been drawn on sleds along the ramps lubricated by water.

As a recent study shows that the challenge was not just to account for the route the transported stones had to take but to account for the size and frequency of stones being moved—circa 1 ton being put in place every 2–3 minutes by human draw teams on a ramp of maximum 10%—to enable building of the Great Pyramid within 30 years. The special case of the pyramid extensions E2 and E3 in Meydum of ring-shaped extensions only 5 m wide around the previous building core shows the ramp system was effective and small in volume which applies for tangential ramps of 10 cubits or 5m width. The challenge was the geometry of the pyramid that provides shorter side lengths the higher the building grows, increases the necessity to turn maneuvers and allows less space for ramps leaning on the masonry. The slope change of the Bent Pyramid is probably the result of the discovery this can not be solved with steep gradients. Essential was also the effective organization of the building site by a module that allows the work division of the teams along plots and the transport causeways between ramp and workplace, including the return of pulling crew and sleds down, probably by a second ramp system. The problem of building the mantle at the ramp arrival point could be solved by bypass-systems.

Levering

Levering methods are considered to be the most tenable solution to complement ramping methods, partially due to Herodotus's description; and partially to the shadoof, a lever-enabled irrigation device first depicted in Egypt during the New Kingdom and found concomitantly with the Old Kingdom in Mesopotamia. In Lehner's (1997: 222) point of view, levers should be employed to lift the top 3% of the material of the superstructure. It is important to note that the top 4% of this material comprises 1⁄3 of the total height of the monument. In other words, in Lehner's view, levers should be employed to lift a small amount of material and a great deal of vertical height of the monument.

In the milieu of levering methods, there are those that lift the block incrementally, as in repeatedly prying up alternating sides of the block and inserting wooden or stone shims to gradually move the stone up one course; and there are other methods that use a larger lever to move the block up one course in one lifting procedure. Since the discussion of construction techniques to lift the blocks attempts to resolve a gap in the archaeological and historical record with a plausible functional explanation, the following examples by Isler, Keable, and Hussey-Pailos list experimentally tested methods. Isler's method (1985, 1987) is an incremental method and, in the Nova experiment (1992), used wooden shims or cribbing. Isler was able to lift a block up one tier in approximately one hour and 30 minutes. Peter Hodges's and Julian Keable's method is similar to Isler's method and instead used small manufactured concrete blocks as shims, wooden pallets, and a pit where their experimental tests were performed. Keable was able to perform his method in approximately 2 minutes. Scott Hussey-Pailos's (2005) method uses a simple levering device to lift a block up a course in one movement. This method was tested with materials of less strength than historical analogs (tested with materials weaker than those available in ancient Egypt), a factor of safety of 2, and lifted a 2500-pound block up one course in under a minute. This method is presented as a levering device to work complementary with Mark Lehner's idea of a combined ramp and levering techniques.

Harbors

Egyptians used the now-disappeared branch of the Nile to transport the tons of construction materials. A 2012 study led by geographer Hader Sheisha at Aix-Marseille University proposed that the former waterscapes and higher river levels around 4,500 years ago facilitated the construction of the Giza Pyramid Complex. The Nile's present waterways have receded too far from the pyramid sites to be of use.

A new study published in May 2024 mapped an extinct branch of the Nile, Ahramat Branch, which once flowed near Egypt's Great Pyramid and other Giza monuments. Using satellite imaging and sediment core analysis, researchers found the 64 kilometres (40 mi) waterway was crucial for transporting materials and labor for pyramid construction. The branch which was about 0.5 kilometres (0.31 mi) wide with a depth of at least 25 metres (82 ft) disappeared likely due to drought and desertification.

Pyramid building experiments

Yoshimura

In 1978, Nippon TV funded the pyramid building project conceived by archaeologist Sajuki Yoshimura. It was originally planned as a 1 to 5 scale model of the Great Pyramid. Because of the limited budget, the size had to be drastically reduced when the price of limestone rose as the project gained publicity. A concrete foundation had to be poured as the selected site offered no bedrock basis. With the help of two cranes and a forklift, the pyramid was built to reach a height of 11 metres (36 ft), with a 15 metres (49 ft) base. The structure was ultimately dismantled and hauled away.

Nova

In 1992, Egyptologist Mark Lehner and stonemason Roger Hopkins conducted a three-week pyramid-building experiment for a Nova television episode. They built a pyramid 6 metres (20 ft) high by 9 metres (30 ft) wide, consisting of a total of 162 cubic metres (5,700 cu ft), or about 405 tons. It was made out of 186 stones weighing an average of 2.2 tons each. Twelve quarrymen carved 186 stones in 22 days, and the structure was erected using 44 men. They used iron hammers, chisels and levers (this is a modern shortcut, as the ancient Egyptians were limited to using copper and later bronze and wood). But Lehner and Hopkins did experiments with copper tools, noting that they were adequate for the job in hand, provided that additional manpower was available to constantly resharpen the ancient tools. They estimated they would have needed around 20 extra men for this maintenance. Another shortcut taken was the use of a front-end loader or fork lift truck, but modern machinery was not used to finish the construction. They used levers to lift the capstone to a height of 20 feet (6.1 m). Four or five men were able to use levers on stones less than one ton to flip them over and transport them by rolling, but larger stones had to be towed. Lehner and Hopkins found that by putting the stones on wooden sledges and sliding the sledges on wooden tracks, they were able to tow a two-ton stone with 12 to 20 men. The wood for these sledges and tracks would have to have been imported from Lebanon at great cost since there was little, if any, wood in ancient Egypt. While the builders failed to duplicate the precise jointing created by the ancient Egyptians, Hopkins was confident that this could have been achieved with more practice.

Great Pyramid

Further information: Diary of Merer

Some research suggests other estimates to the accepted workforce size. For instance, physicist Kurt Mendelssohn calculated that the workforce may have been 50,000 men at most, while Ludwig Borchardt and Louis Croon placed the number at 36,000. According to Miroslav Verner, a workforce of no more than 30,000 was needed in the Great Pyramid's construction. Evidence suggests that around 5,000 were permanent workers on salaries with the balance working three- or four-month shifts in lieu of taxes while receiving subsistence "wages" of ten loaves of bread and a jug of beer per day. Zahi Hawass believes that the majority of workers may have been volunteers. Most archaeologists agree that only about 4,000 of the total workforce were labourers who quarried the stone, hauled blocks to the pyramid, and set the blocks in place. The vast majority of the workforce provided support services such as scribes, toolmakers, and other backup services. The tombs of supervisors contain inscriptions regarding the organisation of the workforce. There were two crews of approximately 2,000 workers sub-divided into named gangs of 1,000. The gangs were divided into five phyles of 200 which were in turn split into groups of around 20 workers grouped according to their skills, with each group having their own project leader and a specific task.

A construction management study carried out by the firm Daniel, Mann, Johnson, & Mendenhall in association with Mark Lehner, and other Egyptologists, estimates that the total project required an average workforce of 14,567 people and a peak workforce of 40,000. Without the use of pulleys, wheels, or iron tools, they used critical path analysis to suggest the Great Pyramid was completed from start to finish in approximately 10 years. Their study estimates the number of blocks used in construction was between 2 and 2.8 million (an average of 2.4 million), but settles on a reduced finished total of 2 million after subtracting the estimated volume of the hollow spaces of the chambers and galleries. Most sources agree on this number of blocks somewhere above 2.3 million. Their calculations suggest the workforce could have sustained a rate of 180 blocks per hour (3 blocks/minute) with ten-hour workdays for putting each individual block in place. They derived these estimates from modern third-world construction projects that did not use modern machinery, but conclude it is still unknown exactly how the Great Pyramid was built. As Dr. Craig Smith of the team points out:

The logistics of construction at the Giza site are staggering when you think that the ancient Egyptians had no pulleys, no wheels, and no iron tools. Yet, the dimensions of the pyramid are extremely accurate and the site was leveled within a fraction of an inch over the entire 13.1-acre base. This is comparable to the accuracy possible with modern construction methods and laser leveling. That's astounding. With their 'rudimentary tools', the pyramid builders of ancient Egypt were about as accurate as we are today with 20th-century technology.

Average core blocks of the Great Pyramid weigh about 1.5 tons each, and the granite blocks used to roof the burial chambers are estimated to weigh up to 80 tons each.

The entire Giza Plateau is believed to have been constructed over the reign of five pharaohs in less than a hundred years, which generally includes: the Great Pyramid, Khafre and Menkaure's pyramids, the Great Sphinx, the Sphinx, and Valley Temples, 35 boat pits cut out of solid bedrock, and several causeways, as well as paving nearly the entire plateau with large stones. This does not include Khafre's brother Djedefre's northern pyramid at Abu Rawash, which would have also been built during this time frame of 100 years. In the hundred years prior to Giza—beginning with Djoser, who ruled from 2687 to 2667 BC, and amongst dozens of other temples, smaller pyramids, and general construction projects—four other massive pyramids were built: the Step pyramid of Saqqara (believed to be the first Egyptian pyramid), the pyramid of Meidum, the Bent Pyramid, and the Red Pyramid. Also during this period (between 2686 and 2498 BC) the Sadd el-Kafara dam, which used an estimated 100,000 cubic meters of rock and rubble, was built.

In October 2018, a team of archaeologists from the Institut Français d'Archéologie Orientale and University of Liverpool announced the discovery of the remains of a 4,500-year-old ramp contraption at Hatnub, excavated since 2012. This method, which aided in lifting the heavy alabaster stones up from their quarries, may have been used to build Egypt's Great Pyramid as well. Yannis Gourdon, co-director of the joint mission at Hatnub, said:

This system is composed of a central ramp flanked by two staircases with numerous post holes, using a sled which carried a stone block and was attached with ropes to these wooden posts, ancient Egyptians were able to pull up the alabaster blocks out of the quarry on very steep slopes of 20 percent or more ... As this system dates back at least to Khufu's reign, that means that during the time of Khufu, ancient Egyptians knew how to move huge blocks of stone using very steep slopes. Therefore, they could have used it for the construction [of] his pyramid.

Internal ramp hypothesis

Main article: Jean-Pierre Houdin

Houdin's father was an architect who, in 1999, thought of a construction method that, it seemed to him, made more sense than any existing method proposed for building pyramids. To develop this hypothesis, Jean-Pierre Houdin, also an architect, gave up his job and set about drawing the first fully functional CAD architectural model of the Great Pyramid of Giza. His scheme involves using a regular external ramp to build the first 30% of the pyramid, with an "internal ramp" taking stones up beyond that height. The stones of the external ramp are re-cycled into the upper stories, thus explaining the otherwise puzzling lack of evidence for ramps.

After four years working alone, Houdin was joined by a team of engineers from the French 3D software company Dassault Systèmes, who used the most modern computer-aided design technology available to further refine and test the hypothesis, making it (according to Houdin) the only one proven to be a viable technique. Houdin published his theory in the books Khufu: The Secrets Behind the Building of the Great Pyramid in 2006 and The Secret of the Great Pyramid, co-written in 2008 with Egyptologist Bob Brier.

In Houdin's method, each ramp inside the pyramid ended at an open space, a notch temporarily left open in the edge of the construction. This 10-square-meter clear space housed a crane that lifted and rotated each 2.5-ton block, to ready it for eight men to drag up the next internal ramp. There is a notch of sorts in one of the right places, and in 2008 Houdin's co-author Bob Brier, with a National Geographic film crew, entered a previously unremarked chamber that could be the start of one of these internal ramps. In 1986 a member of the French team (see below) saw a desert fox at this notch, rather as if it had ascended internally.

Houdin's thesis remains unproven and in 2007, Egyptologist David Jeffreys from the University College London described the internal spiral hypothesis as "far-fetched and horribly complicated", while Oxford University's John Baines, declared he was "suspicious of any theory that seeks to explain only how the Great Pyramid was built".

Houdin has another hypothesis developed from his architectural model, one that could finally explain the internal "Grand Gallery" chamber that otherwise appears to have little purpose. He believes the gallery acted as a trolley chute/guide for counterbalance weights. It enabled the raising of the five 60-ton granite beams that roof the King's Chamber. Houdin and Brier and the Dassault team are already credited with proving for the first time that cracks in beams appeared during construction, were examined and tested at the time and declared relatively harmless.

Antiscience

From Wikipedia, the free encyclopedia

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

Antiscience is a set of attitudes that involve a rejection of science and the scientific method. People holding antiscientific views do not accept science as an objective method that can generate universal knowledge. Antiscience commonly manifests through rejection of scientific ideas such as climate change and evolution. It also includes pseudoscience, methods that claim to be scientific but reject the scientific method. Antiscience leads to belief in conspiracy theories and alternative medicine. Lack of trust in science has been linked to the promotion of political extremism and distrust in medical treatments.

History

In the early days of the scientific revolution, scientists such as Robert Boyle (1627–1691) found themselves in conflict with those such as Thomas Hobbes (1588–1679), who were skeptical of whether science was a satisfactory way to obtain genuine knowledge about the world.

Hobbes' stance is regarded by Ian Shapiro as an antiscience position:

In his Six Lessons to the Professors of Mathematics,...[published in 1656, Hobbes] distinguished 'demonstrable' fields, as 'those the construction of the subject whereof is in the power of the artist himself,' from 'indemonstrable' ones 'where the causes are to seek for.' We can only know the causes of what we make. So geometry is demonstrable, because 'the lines and figures from which we reason are drawn and described by ourselves' and 'civil philosophy is demonstrable, because we make the commonwealth ourselves.' But we can only speculate about the natural world, because 'we know not the construction, but seek it from the effects.'

In his book Reductionism: Analysis and the Fullness of Reality, published in 2000, Richard H. Jones wrote that Hobbes "put forth the idea of the significance of the nonrational in human behaviour". Jones goes on to group Hobbes with others he classes as "antireductionists" and "individualists", including Wilhelm Dilthey (1833–1911), Karl Marx (1818–1883), Jeremy Bentham (1748–1832) and J S Mill (1806–1873), later adding Karl Popper (1902–1994), John Rawls (1921–2002), and E. O. Wilson (1929–2021) to the list.

Jean-Jacques Rousseau, in his Discourse on the Arts and Sciences (1750), claimed that science can lead to immorality. "Rousseau argues that the progression of the sciences and arts has caused the corruption of virtue and morality" and his "critique of science has much to teach us about the dangers involved in our political commitment to scientific progress, and about the ways in which the future happiness of mankind might be secured". Nevertheless, Rousseau does not state in his Discourses that sciences are necessarily bad, and states that figures like René Descartes, Francis Bacon, and Isaac Newton should be held in high regard. In the conclusion to the Discourses, he says that these (aforementioned) can cultivate sciences to great benefit, and that morality's corruption is mostly because of society's bad influence on scientists.

William Blake (1757–1827) reacted strongly in his paintings and writings against the work of Isaac Newton (1642–1727), and is seen as being perhaps the earliest (and almost certainly the most prominent and enduring) example of what is seen by historians as the aesthetic or Romantic antiscience response. For example, in his 1795 poem "Auguries of Innocence", Blake describes the beautiful and natural robin redbreast imprisoned by what one might interpret as the materialistic cage of Newtonian mathematics and science. Blake's painting of Newton depicts the scientist "as a misguided hero whose gaze was directed only at sterile geometrical diagrams drawn on the ground". Blake thought that "Newton, Bacon, and Locke with their emphasis on reason were nothing more than 'the three great teachers of atheism, or Satan's Doctrine'...the picture progresses from exuberance and colour on the left, to sterility and blackness on the right. In Blake's view Newton brings not light, but night". In a 1940 poem, W.H. Auden summarises Blake's anti-scientific views by saying that he "[broke] off relations in a curse, with the Newtonian Universe".

One recent biographer of Newton considers him more as a renaissance alchemist, natural philosopher, and magician rather than a true representative of scientific Enlightenment, as popularized by Voltaire (1694–1778) and other Newtonians.

Antiscience issues are seen as a fundamental consideration in the historical transition from "pre-science" or "protoscience" such as that evident in alchemy. Many disciplines that pre-date the widespread adoption and acceptance of the scientific method, such as geometry and astronomy, are not seen as anti-science. However, some of the orthodoxies within those disciplines that predate a scientific approach (such as those orthodoxies repudiated by the discoveries of Galileo (1564–1642)) are seen as being a product of an anti-scientific stance.

Friedrich Nietzsche in The Gay Science (1882) questions scientific dogmatism:

"[...] in Science, convictions have no rights of citizenship, as is said with good reason. Only when they decide to descend to the modesty of a hypothesis, of a provisional experimental point of view, of a regulative fiction, maybe they be granted admission and even a certain value within the realm of knowledge – though always with the restriction that they remain under police supervision, under the police of mistrust. But does this not mean, more precisely considered, that a conviction may obtain admission to science only when it ceases to be a conviction? Would not the discipline of the scientific spirit begin with this, no longer to permit oneself any convictions? Probably that is how it is. But one must still ask whether it is not the case that, in order that this discipline could begin, a conviction must have been there already, and even such a commanding and unconditional one that it sacrificed all other convictions for its own sake. It is clear that Science too rests on a faith; there is no Science 'without presuppositions.' The question whether truth is needed must not only have been affirmed in advance, but affirmed to the extent that the principle, the faith, the conviction is expressed: 'nothing is needed more than truth, and in relation to it, everything else has only second-rate value".

The term "scientism", originating in science studies, was adopted and is used by sociologists and philosophers of science to describe the views, beliefs and behavior of strong supporters of applying ostensibly scientific concepts beyond its traditional disciplines. Specifically, scientism promotes science as the best or only objective means to determine normative and epistemological values. The term scientism is generally used critically, implying a cosmetic application of science in unwarranted situations considered not amenable to application of the scientific method or similar scientific standards. The word is commonly used in a pejorative sense, applying to individuals who seem to be treating science in a similar way to a religion. The term reductionism is occasionally used in a similarly pejorative way (as a more subtle attack on scientists). However, some scientists feel comfortable being labelled as reductionists, while agreeing that there might be conceptual and philosophical shortcomings of reductionism.

However, non-reductionist (see Emergentism) views of science have been formulated in varied forms in several scientific fields like statistical physics, chaos theory, complexity theory, cybernetics, systems theory, systems biology, ecology, information theory, etc. Such fields tend to assume that strong interactions between units produce new phenomena in "higher" levels that cannot be accounted for solely by reductionism. For example, it is not valuable (or currently possible) to describe a chess game or gene networks using quantum mechanics. The emergentist view of science ("More is Different", in the words of 1977 Nobel-laureate physicist Philip W. Anderson) has been inspired in its methodology by the European social sciences (Durkheim, Marx) which tend to reject methodological individualism.

Political

Elyse Amend and Darin Barney argue that while antiscience can be a descriptive label, it is often used as a rhetorical one, being effectively used to discredit ones' political opponents and thus charges of antiscience are not necessarily warranted.

Secular

Further information: Populism

Left-wing

Further information: Left-wing populism

One expression of antiscience is the "denial of universality and... legitimisation of alternatives", and that the results of scientific findings do not always represent any underlying reality, but can merely reflect the ideology of dominant groups within society. Alan Sokal states that this view associates science with the political right and is seen as a belief system that is conservative and conformist, that suppresses innovation, that resists change and that acts dictatorially. This includes the view, for example, that science has a "bourgeois and/or Eurocentric and/or masculinist world-view".

The anti-nuclear movement, often associated with the left, has been criticized for overstating the negative effects of nuclear power, and understating the environmental costs of non-nuclear sources that can be prevented through nuclear energy. Opposition to genetically modified organisms (GMOs) has also been associated with the left.

Right-wing

Further information: Right-wing populism

The origin of antiscience thinking may be traced back to the reaction of Romanticism to the Enlightenment-this movement is often referred to as the 'Counter-Enlightenment'. Romanticism emphasizes that intuition, passion and organic links to Nature are primal values and that rational thinking is merely a product of human life. There are many modern examples of conservative antiscience polemics. Primary among the latter are the polemics about evolutionary biology cosmology, historical geology, and origin of life research being taught in high schools, and environmental issues related to global warming and energy crisis.

Characteristics of antiscience associated with the right include the appeal to conspiracy theories to explain why scientists believe what they believe, in an attempt to undermine the confidence or power usually associated to science (e.g., in global warming conspiracy theories).

In modern times, it has been argued that right-wing politics carries an anti-science tendency. While some have suggested that this is innate to either rightists or their beliefs, others have argued it is a "quirk" of a historical and political context in which scientific findings happened to challenge or appeared to challenge the worldviews of rightists rather than leftists.

Religious

Main article: Relationship between religion and science

In this context, antiscience may be considered dependent on religious, moral and cultural arguments. For this kind of religious antiscience philosophy, science is an anti-spiritual and materialistic force that undermines traditional values, ethnic identity and accumulated historical wisdom in favor of reason and cosmopolitanism. In particular, the traditional and ethnic values emphasized are similar to those of white supremacist Christian Identity theology, but similar right-wing views have been developed by radically conservative sects of Islam, Judaism, Hinduism, and Buddhism. New religious movements such as New Age thinking also criticize the scientific worldview as favouring a reductionist, atheist, or materialist philosophy.

A frequent basis of antiscientific sentiment is religious theism with literal interpretations of sacred text. Here, scientific theories that conflict with what is considered divinely-inspired knowledge are regarded as flawed. Over the centuries religious institutions have been hesitant to embrace such ideas as heliocentrism and planetary motion because they contradicted the dominant interpretation of various passages of scripture. More recently the body of creation theologies known collectively as creationism, including the teleological theory of intelligent design, have been promoted by religious theists in response to the process of evolution by natural selection.

To the extent that attempts to overcome antiscience sentiments have failed, some argue that a different approach to science advocacy is needed. One such approach says that it is important to develop a more accurate understanding of those who deny science (avoiding stereotyping them as backward and uneducated) and also to attempt outreach via those who share cultural values with target audiences, such as scientists who also hold religious beliefs.

Areas

There is a cult of ignorance in the United States, and there has always been. The strain of anti-intellectualism has been a constant thread winding its way through our political and cultural life, nurtured by the false notion that democracy means that "my ignorance is just as good as your knowledge".

Isaac Asimov, "A Cult of Ignorance", Newsweek, 21 January 1980

Historically, antiscience first arose as a reaction against scientific materialism. The 18th century Enlightenment had ushered in "the ideal of a unified system of all the sciences", but there were those fearful of this notion, who "felt that constrictions of reason and science, of a single all-embracing system... were in some way constricting, an obstacle to their vision of the world, chains on their imagination or feeling". Antiscience then is a rejection of "the scientific model [or paradigm]... with its strong implication that only that which was quantifiable, or at any rate, measurable... was real". In this sense, it comprises a "critical attack upon the total claim of the new scientific method to dominate the entire field of human knowledge". However, scientific positivism (logical positivism) does not deny the reality of non-measurable phenomena, only that those phenomena should not be adequate to scientific investigation. Moreover, positivism, as a philosophical basis for the scientific method, is not consensual or even dominant in the scientific community (see philosophy of science).

Recent developments and discussions around antiscience attitudes reveal how deeply intertwined these beliefs are with social, political, and psychological factors. A study published by Ohio State News on July 11, 2022, identified four primary bases that underpin antiscience beliefs: doubts about the credibility of scientific sources, identification with groups holding antiscience attitudes, conflicts between scientific messages and personal beliefs, and discrepancies between the presentation of scientific messages and individuals’ thinking styles. These factors are exacerbated in the current political climate, where ideology significantly influences people's acceptance of science, particularly on topics that have become politically polarized, such as vaccines and climate change. The politicization of science poses a significant challenge to public health and safety, particularly in managing global crises like the COVID-19 pandemic.

The following quotes explore this aspect of four major areas of antiscience: philosophy, sociology, ecology and political.

Philosophy

Philosophical objections against science are often objections about the role of reductionism. For example, in the field of psychology, "both reductionists and antireductionists accept that... non-molecular explanations may not be improved, corrected or grounded in molecular ones". Further, "epistemological antireductionism holds that, given our finite mental capacities, we would not be able to grasp the ultimate physical explanation of many complex phenomena even if we knew the laws governing their ultimate constituents". Some see antiscience as "common...in academic settings...many people see that there are problems in demarcation between science, scientism, and pseudoscience resulting in an antiscience stance. Some argue that nothing can be known for sure".

Many philosophers are "divided as to whether reduction should be a central strategy for understanding the world". However, many agree that "there are, nevertheless, reasons why we want science to discover properties and explanations other than reductive physical ones". Such issues stem "from an antireductionist worry that there is no absolute conception of reality, that is, a characterization of reality such as... science claims to provide".

Sociology

Sociologist Thomas Gieryn refers to "some sociologists who might appear to be antiscience". Some "philosophers and antiscience types", he contends, may have presented "unreal images of science that threaten the believability of scientific knowledge", or appear to have gone "too far in their antiscience deconstructions". The question often lies in how much scientists conform to the standard ideal of "communalism, universalism, disinterestedness, originality, and... skepticism". "scientists don't always conform... scientists do get passionate about pet theories; they do rely on reputation in judging a scientist's work; they do pursue fame and gain via research". Thus, they may show inherent biases in their work. "[Many] scientists are not as rational and logical as the legend would have them, nor are they as illogical or irrational as some relativists might say".

Ecology and health sphere

Within the ecological and health spheres, Levins identifies a conflict "not between science and antiscience, but rather between different pathways for science and technology; between a commodified science-for-profit and a gentle science for humane goals; between the sciences of the smallest parts and the sciences of dynamic wholes... [he] offers proposals for a more holistic, integral approach to understanding and addressing environmental issues". These beliefs are also common within the scientific community, with for example, scientists being prominent in environmental campaigns warning of environmental dangers such as ozone depletion and the greenhouse effect. It can also be argued that this version of antiscience comes close to that found in the medical sphere, where patients and practitioners may choose to reject science and adopt a pseudoscientific approach to health problems. This can be both a practical and a conceptual shift and has attracted strong criticism: "therapeutic touch, a healing technique based upon the laying-on of hands, has found wide acceptance in the nursing profession despite its lack of scientific plausibility. Its acceptance is indicative of a broad antiscientific trend in nursing".

Glazer also criticises the therapists and patients, "for abandoning the biological underpinnings of nursing and for misreading philosophy in the service of an antiscientific world-view". In contrast, Brian Martin criticized Gross and Levitt by saying that "[their] basic approach is to attack constructivists for not being positivists," and that science is "presented as a unitary object, usually identified with scientific knowledge. It is portrayed as neutral and objective. Second, science is claimed to be under attack by 'antiscience' which is composed essentially of ideologues who are threats to the neutrality and objectivity that are fundamental to science. Third, a highly selective attack is made on the arguments of 'antiscience'". Such people allegedly then "routinely equate critique of scientific knowledge with hostility to science, a jump that is logically unsupportable and empirically dubious". Having then "constructed two artificial entities, a unitary 'science' and a unitary 'academic left', each reduced to epistemological essences, Gross and Levitt proceed to attack. They pick out figures in each of several areas – science studies, postmodernism, feminism, environmentalism, AIDS activism – and criticise their critiques of science".

The writings of Young serve to illustrate more antiscientific views: "The strength of the antiscience movement and of alternative technology is that their advocates have managed to retain Utopian vision while still trying to create concrete instances of it". "The real social, ideological and economic forces shaping science...[have] been opposed to the point of suppression in many quarters. Most scientists hate it and label it 'antiscience'. But it is urgently needed, because it makes science self-conscious and hopefully self-critical and accountable with respect to the forces which shape research priorities, criteria, goals".

Genetically modified foods also bring about antiscience sentiment. The general public has recently become more aware of the dangers of a poor diet, as there have been numerous studies that show that the two are inextricably linked. Anti-science dictates that science is untrustworthy, because it is never complete and always being revised, which would be a probable cause for the fear that the general public has of genetically modified foods despite scientific reassurance that such foods are safe.

Antivaccinationists rely on whatever comes to hand presenting some of their arguments as if scientific; however, a strain of antiscience is part of their approach.

Political

Political scientist Tom Nichols, from Harvard Extension School and the U.S. Naval War College, points out that skepticism towards scientific expertise has increasingly become a symbol of political identity, especially within conservative circles. This skepticism is not just a result of misinformation but also reflects a broader cultural shift towards diminishing trust in experts and authoritative sources. This trend challenges the traditional neutrality of science, positioning scientific beliefs and facts within the contentious arena of political ideology.

The COVID-19 pandemic, for example, conflicting responses to public health measures and vaccine acceptance have highlighted the extent to which science has been politicized. Such polarization suggests that for some, rejecting scientific consensus or public health guidance serves as an expression of political allegiance or skepticism towards perceived authority figures.

This politicization of science complicates efforts to address public health crises and undermines the broader social contract that underpins scientific research and its application for the public good. The challenge lies not only in combating misinformation but also in bridging ideological divides that affect public trust in science. Strategies to counteract antiscience attitudes may need to encompass more than just presenting factual information; they might also need to engage with the underlying social and psychological factors that contribute to these attitudes, fostering dialogue that acknowledges different viewpoints and seeks common ground.

Antiscience media

Major antiscience media include portals Natural News, Global Revolution TV, TruthWiki.org, TheAntiMedia.org and GoodGopher. Antiscience views have also been supported on social media by organizations known to support fake news such as the web brigades.

Kepler's laws of planetary motion

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion

Illustration of Kepler's laws with two planetary orbits.

1. The orbits are ellipses, with foci F₁ and F₂ for Planet 1, and F₁ and F₃ for Planet 2. The Sun is at F₁.
2. The shaded areas A₁ and A₂ are equal, and are swept out in equal times by Planet 1's orbit.
3. The ratio of Planet 1's orbit time to Planet 2's is $(a_1/a_2{)}^{3/2}$.

In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The three laws state that:

1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3. The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.

The elliptical orbits of planets were indicated by calculations of the orbit of Mars. From this, Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits. The second law helps to establish that when a planet is closer to the Sun, it travels faster. The third law expresses that the farther a planet is from the Sun, the slower its orbital speed, and vice versa.

Isaac Newton showed in 1687 that relationships like Kepler's would apply in the Solar System as a consequence of his own laws of motion and law of universal gravitation.

A more precise historical approach is found in Astronomia nova and Epitome Astronomiae Copernicanae.

Comparison to Copernicus

Johannes Kepler's laws improved the model of Copernicus. According to Copernicus:

1. The planetary orbit is a circle with epicycles.
2. The Sun is approximately at the center of the orbit.
3. The speed of the planet in the main orbit is constant.

Despite being correct in saying that the planets revolved around the Sun, Copernicus was incorrect in defining their orbits. Introducing physical explanations for movement in space beyond just geometry, Kepler correctly defined the orbit of planets as follows:

1. The planetary orbit is not a circle with epicycles, but an ellipse.
2. The Sun is not at the center but at a focal point of the elliptical orbit.
3. Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the area speed (closely linked historically with the concept of angular momentum) is constant.

The eccentricity of the orbit of the Earth makes the time from the March equinox to the September equinox, around 186 days, unequal to the time from the September equinox to the March equinox, around 179 days. A diameter would cut the orbit into equal parts, but the plane through the Sun parallel to the equator of the Earth cuts the orbit into two parts with areas in a 186 to 179 ratio, so the eccentricity of the orbit of the Earth is approximately

${\displaystyle e\approx \frac{\pi }{4}\frac{186-179}{186+179}\approx 0.015,}$

which is close to the correct value (0.016710218). The accuracy of this calculation requires that the two dates chosen be along the elliptical orbit's minor axis and that the midpoints of each half be along the major axis. As the two dates chosen here are equinoxes, this will be correct when perihelion, the date the Earth is closest to the Sun, falls on a solstice. The current perihelion, near January 4, is fairly close to the solstice of December 21 or 22.

Nomenclature

It took nearly two centuries for the current formulation of Kepler's work to take on its settled form. Voltaire's Eléments de la philosophie de Newton (Elements of Newton's Philosophy) of 1738 was the first publication to use the terminology of "laws". The Biographical Encyclopedia of Astronomers in its article on Kepler (p. 620) states that the terminology of scientific laws for these discoveries was current at least from the time of Joseph de Lalande. It was the exposition of Robert Small, in An account of the astronomical discoveries of Kepler (1814) that made up the set of three laws, by adding in the third. Small also claimed, against the history, that these were empirical laws, based on inductive reasoning.

Further, the current usage of "Kepler's Second Law" is something of a misnomer. Kepler had two versions, related in a qualitative sense: the "distance law" and the "area law". The "area law" is what became the Second Law in the set of three; but Kepler did himself not privilege it in that way.

History

Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of Tycho Brahe. Kepler's third law was published in 1619. Kepler had believed in the Copernican model of the Solar System, which called for circular orbits, but he could not reconcile Brahe's highly precise observations with a circular fit to Mars' orbit – Mars coincidentally having the highest eccentricity of all planets except Mercury. His first law reflected this discovery.

In 1621, Kepler noted that his third law applies to the four brightest moons of Jupiter. Godefroy Wendelin also made this observation in 1643. The second law, in the "area law" form, was contested by Nicolaus Mercator in a book from 1664, but by 1670 his Philosophical Transactions were in its favour. As the century proceeded it became more widely accepted. The reception in Germany changed noticeably between 1688, the year in which Newton's Principia was published and was taken to be basically Copernican, and 1690, by which time work of Gottfried Leibniz on Kepler had been published.

Newton was credited with understanding that the second law is not special to the inverse square law of gravitation, being a consequence just of the radial nature of that law, whereas the other laws do depend on the inverse square form of the attraction. Carl Runge and Wilhelm Lenz much later identified a symmetry principle in the phase space of planetary motion (the orthogonal group O(4) acting) which accounts for the first and third laws in the case of Newtonian gravitation, as conservation of angular momentum does via rotational symmetry for the second law.

Formulary

The mathematical model of the kinematics of a planet subject to the laws allows a large range of further calculations.

First law

The orbit of every planet is an ellipse with the sun at one of the two foci.

Kepler's first law placing the Sun at the focus of an elliptical orbit

Heliocentric coordinate system (r, θ) for ellipse. Also shown are: semi-major axis a, semi-minor axis b and semi-latus rectum p; center of ellipse and its two foci marked by large dots. For θ = 0°, r = r_min and for θ = 180°, r = r_max.

Mathematically, an ellipse can be represented by the formula:

${\displaystyle r=\frac{p}{1+\epsilon \cos \theta },}$

where ${\displaystyle p}$ is the semi-latus rectum, ε is the eccentricity of the ellipse, r is the distance from the Sun to the planet, and θ is the angle to the planet's current position from its closest approach, as seen from the Sun. So (r, θ) are polar coordinates.

For an ellipse 0 < ε < 1 ; in the limiting case ε = 0, the orbit is a circle with the Sun at the centre (i.e. where there is zero eccentricity).

At θ = 0°, perihelion, the distance is minimum

${\displaystyle r_{min}=\frac{p}{1+\epsilon }}$

At θ = 90° and at θ = 270° the distance is equal to ${\displaystyle p}$.

At θ = 180°, aphelion, the distance is maximum (by definition, aphelion is – invariably – perihelion plus 180°)

${\displaystyle r_{max}=\frac{p}{1-\epsilon }}$

The semi-major axis a is the arithmetic mean between r_min and r_max:

${\displaystyle \begin{array}{rl}a& =\frac{r_{max}+r_{min}}{2}\\ a& =\frac{p}{1-{\epsilon }^2}\end{array}}$

The semi-minor axis b is the geometric mean between r_min and r_max:

${\displaystyle \begin{array}{rl}b& =\sqrt{r_{max}{r}_{min}}\\ b& =\frac{p}{\sqrt{1-{\epsilon }^2}}\end{array}}$

The semi-latus rectum p is the harmonic mean between r_min and r_max:

${\displaystyle \begin{array}{rl}p& ={\left(\frac{r_{max}^{-1}+r_{min}^{-1}}{2}\right)}^{-1}\\ pa& =r_{max}{r}_{min}=b^2\end{array}}$

The eccentricity ε is the coefficient of variation between r_min and r_max:

${\displaystyle \epsilon =\frac{r_{max}-r_{min}}{r_{max}+r_{min}}.}$

The area of the ellipse is

${\displaystyle A=\pi ab.}$

The special case of a circle is ε = 0, resulting in r = p = r_min = r_max = a = b and A = πr².

Second law

A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

The same (blue) area is swept out in a fixed time period. The green arrow is velocity. The purple arrow directed towards the Sun is the acceleration. The other two purple arrows are acceleration components parallel and perpendicular to the velocity.

The orbital radius and angular velocity of the planet in the elliptical orbit will vary. This is shown in the animation: the planet travels faster when closer to the Sun, then slower when farther from the Sun. Kepler's second law states that the blue sector has constant area. Kepler notably arrived at this law through assumptions that were either only approximately true or outright false. Nevertheless, the result of the Second Law is exactly true, as it is logically equivalent to the conservation of angular momentum, which is true for any body experiencing a radially symmetric force

In a small time ${\displaystyle dt}$ the planet sweeps out a small triangle having base line ${\displaystyle r}$ and height ${\displaystyle rd\theta }$ and area $dA=\frac{1}{2}\cdot r\cdot rd\theta$, so the constant areal velocity is

$${\displaystyle \frac{dA}{dt}=\frac{r^2}{2}\frac{d\theta }{dt}.}$$

The area enclosed by the elliptical orbit is ${\displaystyle \pi ab}$. So the period ${\displaystyle T}$ satisfies

${\displaystyle T\cdot \frac{r^2}{2}\frac{d\theta }{dt}=\pi ab}$

and the mean motion of the planet around the Sun

${\displaystyle n=\frac{2\pi }{T}}$

satisfies

${\displaystyle r^{2}d\theta =abndt.}$

And so,

$${\displaystyle \frac{dA}{dt}=\frac{abn}{2}=\frac{\pi ab}{T}.}$$

Orbits of planets with varying eccentricities.

Low	High
Planet orbiting the Sun in a circular orbit (e=0.0)	Planet orbiting the Sun in an orbit with e=0.5
Planet orbiting the Sun in an orbit with e=0.2	Planet orbiting the Sun in an orbit with e=0.8
The red ray rotates at a constant angular velocity and with the same orbital time period as the planet, ${\displaystyle T=1}$. S: Sun at the primary focus, C: Centre of ellipse, S': The secondary focus. In each case, the area of all sectors depicted is identical.

Third law

The ratio of the square of an object's orbital period with the cube of the semi-major axis of its orbit is the same for all objects orbiting the same primary.

This captures the relationship between the distance of planets from the Sun, and their orbital periods.

Kepler enunciated in 1619 this third law in a laborious attempt to determine what he viewed as the "music of the spheres" according to precise laws, and express it in terms of musical notation. It was therefore known as the harmonic law. The original form of this law (referring to not the semi-major axis, but rather a "mean distance") holds true only for planets with small eccentricities near zero. 

Using Newton's law of gravitation (published 1687), this relation can be found in the case of a circular orbit by setting the centripetal force equal to the gravitational force:

${\displaystyle mr{\omega }^2=G\frac{mM}{r^2}}$

Then, expressing the angular velocity ω in terms of the orbital period ${\displaystyle T}$ and then rearranging, results in Kepler's Third Law:

${\displaystyle mr{\left(\frac{2\pi }{T}\right)}^2=G\frac{mM}{r^2}\to T^2=\left(\frac{4{\pi }^2}{GM}\right)r^3\to T^2\propto r^3}$

A more detailed derivation can be done with general elliptical orbits, instead of circles, as well as orbiting the center of mass, instead of just the large mass. This results in replacing a circular radius, ${\displaystyle r}$, with the semi-major axis, ${\displaystyle a}$, of the elliptical relative motion of one mass relative to the other, as well as replacing the large mass ${\displaystyle M}$ with ${\displaystyle M+m}$. However, with planet masses being so much smaller than the Sun, this correction is often ignored. The full corresponding formula is:

${\displaystyle \frac{a^3}{T^2}=\frac{G(M+m)}{4{\pi }^2}\approx \frac{GM}{4{\pi }^2}\approx 7.496\times {10}^{-6}\frac{{\text{AU}}^3}{{\text{days}}^2}\text{ is constant}}$

where ${\displaystyle M}$ is the mass of the Sun, ${\displaystyle m}$ is the mass of the planet, ${\displaystyle G}$ is the gravitational constant, ${\displaystyle T}$ is the orbital period and ${\displaystyle a}$ is the elliptical semi-major axis, and ${\displaystyle \text{AU}}$ is the astronomical unit, the average distance from earth to the sun.

Table

The following table shows the data used by Kepler to empirically derive his law:

Data used by Kepler (1618)

Planet	Mean distance to sun (AU)	Period (days)	$\frac{R^3}{T^2}$ (10-6 AU3/day2)
Mercury	0.389	87.77	7.64
Venus	0.724	224.70	7.52
Earth	1	365.25	7.50
Mars	1.524	686.95	7.50
Jupiter	5.20	4332.62	7.49
Saturn	9.510	10759.2	7.43

Upon finding this pattern Kepler wrote:

I first believed I was dreaming... But it is absolutely certain and exact that the ratio which exists between the period times of any two planets is precisely the ratio of the 3/2th power of the mean distance.

— translated from Harmonies of the World by Kepler (1619)

Log-log plot of period T vs semi-major axis a (average of aphelion and perihelion) of some Solar System orbits (crosses denoting Kepler's values) showing that a³/T² is constant (green line)

For comparison, here are modern estimates:

Modern data

Planet	Semi-major axis (AU)	Period (days)	$\frac{R^3}{T^2}$ (10-6 AU3/day2)
Mercury	0.38710	87.9693	7.496
Venus	0.72333	224.7008	7.496
Earth	1	365.2564	7.496
Mars	1.52366	686.9796	7.495
Jupiter	5.20336	4332.8201	7.504
Saturn	9.53707	10775.599	7.498
Uranus	19.1913	30687.153	7.506
Neptune	30.0690	60190.03	7.504

Planetary acceleration

Isaac Newton computed in his Philosophiæ Naturalis Principia Mathematica the acceleration of a planet moving according to Kepler's first and second laws.

1. The direction of the acceleration is towards the Sun.
2. The magnitude of the acceleration is inversely proportional to the square of the planet's distance from the Sun (the inverse square law).

This implies that the Sun may be the physical cause of the acceleration of planets. However, Newton states in his Principia that he considers forces from a mathematical point of view, not a physical, thereby taking an instrumentalist view. Moreover, he does not assign a cause to gravity.

Newton defined the force acting on a planet to be the product of its mass and the acceleration (see Newton's laws of motion). So:

1. Every planet is attracted towards the Sun.
2. The force acting on a planet is directly proportional to the mass of the planet and is inversely proportional to the square of its distance from the Sun.

The Sun plays an unsymmetrical part, which is unjustified. So he assumed, in Newton's law of universal gravitation:

1. All bodies in the Solar System attract one another.
2. The force between two bodies is in direct proportion to the product of their masses and in inverse proportion to the square of the distance between them.

As the planets have small masses compared to that of the Sun, the orbits conform approximately to Kepler's laws. Newton's model improves upon Kepler's model, and fits actual observations more accurately. (See two-body problem.)

Below comes the detailed calculation of the acceleration of a planet moving according to Kepler's first and second laws.

Acceleration vector

See also: Polar coordinate § Vector calculus, and Mechanics of planar particle motion

From the heliocentric point of view consider the vector to the planet ${\displaystyle \mathbf{r}=r\hat{\mathbf{r}}}$ where ${\displaystyle r}$ is the distance to the planet and ${\displaystyle \hat{\mathbf{r}}}$ is a unit vector pointing towards the planet.

$${\displaystyle \frac{d\hat{\mathbf{r}}}{dt}=\dot{\hat{\mathbf{r}}}=\dot{\theta }\hat{\mathit{\theta }},\frac{d\hat{\mathit{\theta }}}{dt}=\dot{\hat{\mathit{\theta }}}=-\dot{\theta }\hat{\mathbf{r}}}$$

where ${\displaystyle \hat{\mathit{\theta }}}$ is the unit vector whose direction is 90 degrees counterclockwise of ${\displaystyle \hat{\mathbf{r}}}$, and ${\displaystyle \theta }$ is the polar angle, and where a dot on top of the variable signifies differentiation with respect to time.

Differentiate the position vector twice to obtain the velocity vector and the acceleration vector:

$${\displaystyle \begin{array}{rl}\dot{\mathbf{r}}& =\dot{r}\hat{\mathbf{r}}+r\dot{\hat{\mathbf{r}}}=\dot{r}\hat{\mathbf{r}}+r\dot{\theta }\hat{\mathit{\theta }},\\ \ddot{\mathbf{r}}& =\left(\ddot{r}\hat{\mathbf{r}}+\dot{r}\dot{\hat{\mathbf{r}}}\right)+\left(\dot{r}\dot{\theta }\hat{\mathit{\theta }}+r\ddot{\theta }\hat{\mathit{\theta }}+r\dot{\theta }\dot{\hat{\mathit{\theta }}}\right)=\left(\ddot{r}-r{\dot{\theta }}^2\right)\hat{\mathbf{r}}+\left(r\ddot{\theta }+2\dot{r}\dot{\theta }\right)\hat{\mathit{\theta }}.\end{array}}$$

So

$${\displaystyle \ddot{\mathbf{r}}=a_r\hat{\mathit{r}}+a_{\theta }\hat{\mathit{\theta }}}$$

where the radial acceleration is

$${\displaystyle a_r=\ddot{r}-r{\dot{\theta }}^2}$$

and the transversal acceleration is

$${\displaystyle a_{\theta }=r\ddot{\theta }+2\dot{r}\dot{\theta }.}$$

Inverse square law

Kepler's second law says that

$${\displaystyle r^2\dot{\theta }=nab}$$

is constant.

The transversal acceleration ${\displaystyle a_{\theta }}$ is zero:

$${\displaystyle \frac{d\left(r^2\dot{\theta }\right)}{dt}=r\left(2\dot{r}\dot{\theta }+r\ddot{\theta }\right)=r{a}_{\theta }=0.}$$

So the acceleration of a planet obeying Kepler's second law is directed towards the Sun.

The radial acceleration ${\displaystyle a_{\text{r}}}$ is

$${\displaystyle a_{\text{r}}=\ddot{r}-r{\dot{\theta }}^2=\ddot{r}-r{\left(\frac{nab}{r^2}\right)}^2=\ddot{r}-\frac{n^2{a}^2{b}^2}{r^3}.}$$

Kepler's first law states that the orbit is described by the equation:

$${\displaystyle \frac{p}{r}=1+\epsilon \cos (\theta ).}$$

Differentiating with respect to time

$${\displaystyle -\frac{p\dot{r}}{r^2}=-\epsilon \sin (\theta )\dot{\theta }}$$

or

$${\displaystyle p\dot{r}=nab\epsilon \sin (\theta ).}$$

Differentiating once more

$${\displaystyle p\ddot{r}=nab\epsilon \cos (\theta )\dot{\theta }=nab\epsilon \cos (\theta )\frac{nab}{r^2}=\frac{n^2{a}^2{b}^2}{r^2}\epsilon \cos (\theta ).}$$

The radial acceleration ${\displaystyle a_{\text{r}}}$ satisfies

$${\displaystyle p{a}_{\text{r}}=\frac{n^2{a}^2{b}^2}{r^2}\epsilon \cos (\theta )-p\frac{n^2{a}^2{b}^2}{r^3}=\frac{n^2{a}^2{b}^2}{r^2}\left(\epsilon \cos (\theta )-\frac{p}{r}\right).}$$

Substituting the equation of the ellipse gives

$${\displaystyle p{a}_{\text{r}}=\frac{n^2{a}^2{b}^2}{r^2}\left(\frac{p}{r}-1-\frac{p}{r}\right)=-\frac{n^2{a}^2}{r^2}{b}^2.}$$

The relation ${\displaystyle b^2=pa}$ gives the simple final result

$${\displaystyle a_{\text{r}}=-\frac{n^2{a}^3}{r^2}.}$$

This means that the acceleration vector ${\displaystyle \ddot{\mathbf{r}}}$ of any planet obeying Kepler's first and second law satisfies the inverse square law

$${\displaystyle \ddot{\mathbf{r}}=-\frac{\alpha }{r^2}\hat{\mathbf{r}}}$$

where

$${\displaystyle \alpha =n^2{a}^3}$$

is a constant, and ${\displaystyle \hat{\mathbf{r}}}$ is the unit vector pointing from the Sun towards the planet, and ${\displaystyle r}$ is the distance between the planet and the Sun.

Since mean motion ${\displaystyle n=\frac{2\pi }{T}}$ where ${\displaystyle T}$ is the period, according to Kepler's third law, ${\displaystyle \alpha }$ has the same value for all the planets. So the inverse square law for planetary accelerations applies throughout the entire Solar System.

The inverse square law is a differential equation. The solutions to this differential equation include the Keplerian motions, as shown, but they also include motions where the orbit is a hyperbola or parabola or a straight line. (See Kepler orbit.)

Newton's law of gravitation

By Newton's second law, the gravitational force that acts on the planet is:

$${\displaystyle \mathbf{F}=m_{\text{planet}}\ddot{\mathbf{r}}=-m_{\text{planet}}\alpha r^{-2}\hat{\mathbf{r}}}$$

where ${\displaystyle m_{\text{planet}}}$ is the mass of the planet and ${\displaystyle \alpha }$ has the same value for all planets in the Solar System. According to Newton's third law, the Sun is attracted to the planet by a force of the same magnitude. Since the force is proportional to the mass of the planet, under the symmetric consideration, it should also be proportional to the mass of the Sun, ${\displaystyle m_{\text{Sun}}}$. So

$${\displaystyle \alpha =G{m}_{\text{Sun}}}$$

where ${\displaystyle G}$ is the gravitational constant.

The acceleration of Solar System body number i is, according to Newton's laws:

$${\displaystyle {\ddot{\mathbf{r}}}_i=G\sum _{j\ne i}{m}_j{r}_{ij}^{-2}{\hat{\mathbf{r}}}_{ij}}$$

where ${\displaystyle m_j}$ is the mass of body j, ${\displaystyle r_{ij}}$ is the distance between body i and body j, ${\displaystyle {\hat{\mathbf{r}}}_{ij}}$ is the unit vector from body i towards body j, and the vector summation is over all bodies in the Solar System, besides i itself.

In the special case where there are only two bodies in the Solar System, Earth and Sun, the acceleration becomes

$${\displaystyle {\ddot{\mathbf{r}}}_{\text{Earth}}=G{m}_{\text{Sun}}{r}_{\text{Earth},\text{Sun}}^{-2}{\hat{\mathbf{r}}}_{\text{Earth},\text{Sun}}}$$

which is the acceleration of the Kepler motion. So this Earth moves around the Sun according to Kepler's laws.

If the two bodies in the Solar System are Moon and Earth the acceleration of the Moon becomes

$${\displaystyle {\ddot{\mathbf{r}}}_{\text{Moon}}=G{m}_{\text{Earth}}{r}_{\text{Moon},\text{Earth}}^{-2}{\hat{\mathbf{r}}}_{\text{Moon},\text{Earth}}}$$

So in this approximation, the Moon moves around the Earth according to Kepler's laws.

In the three-body case the accelerations are

$${\displaystyle \begin{array}{rl}{\ddot{\mathbf{r}}}_{\text{Sun}}& =G{m}_{\text{Earth}}{r}_{\text{Sun},\text{Earth}}^{-2}{\hat{\mathbf{r}}}_{\text{Sun},\text{Earth}}+G{m}_{\text{Moon}}{r}_{\text{Sun},\text{Moon}}^{-2}{\hat{\mathbf{r}}}_{\text{Sun},\text{Moon}}\\ {\ddot{\mathbf{r}}}_{\text{Earth}}& =G{m}_{\text{Sun}}{r}_{\text{Earth},\text{Sun}}^{-2}{\hat{\mathbf{r}}}_{\text{Earth},\text{Sun}}+G{m}_{\text{Moon}}{r}_{\text{Earth},\text{Moon}}^{-2}{\hat{\mathbf{r}}}_{\text{Earth},\text{Moon}}\\ {\ddot{\mathbf{r}}}_{\text{Moon}}& =G{m}_{\text{Sun}}{r}_{\text{Moon},\text{Sun}}^{-2}{\hat{\mathbf{r}}}_{\text{Moon},\text{Sun}}+G{m}_{\text{Earth}}{r}_{\text{Moon},\text{Earth}}^{-2}{\hat{\mathbf{r}}}_{\text{Moon},\text{Earth}}\end{array}}$$

These accelerations are not those of Kepler orbits, and the three-body problem is complicated. But Keplerian approximation is the basis for perturbation calculations. (See Lunar theory.)

Position as a function of time

Kepler used his two first laws to compute the position of a planet as a function of time. His method involves the solution of a transcendental equation called Kepler's equation.

The procedure for calculating the heliocentric polar coordinates (r,θ) of a planet as a function of the time t since perihelion, is the following five steps:

1. Compute the mean motion n = (2π rad)/P, where P is the period.
2. Compute the mean anomaly M = nt, where t is the time since perihelion.
3. Compute the eccentric anomaly E by solving Kepler's equation:
   $${\displaystyle M=E-\epsilon \sin E,}$$

   
   where ${\displaystyle \epsilon }$ is the eccentricity.
4. Compute the true anomaly θ by solving the equation:
   $${\displaystyle (1-\epsilon ){\tan }^2\frac{\theta }{2}=(1+\epsilon ){\tan }^2\frac{E}{2}}$$

   
5. Compute the heliocentric distance r:
   $${\displaystyle r=a(1-\epsilon \cos E),}$$

   
   where ${\displaystyle a}$ is the semimajor axis.

The position polar coordinates (r,θ) can now be written as a Cartesian vector ${\displaystyle \mathbf{p}=r⟨\cos \theta ,\sin \theta ⟩}$ and the Cartesian velocity vector can then be calculated as ${\displaystyle \mathbf{v}=\frac{\sqrt{\mu a}}{r}⟨-\sin E,\sqrt{1-{\epsilon }^2}\cos E⟩}$, where ${\displaystyle \mu }$ is the standard gravitational parameter.

The important special case of circular orbit, ε = 0, gives θ = E = M. Because the uniform circular motion was considered to be normal, a deviation from this motion was considered an anomaly.

The proof of this procedure is shown below.

Mean anomaly, M

Main article: Mean anomaly

Geometric construction for Kepler's calculation of θ. The Sun (located at the focus) is labeled S and the planet P. The auxiliary circle is an aid to calculation. Line xd is perpendicular to the base and through the planet P. The shaded sectors are arranged to have equal areas by positioning of point y.

The Keplerian problem assumes an elliptical orbit and the four points:

• s the Sun (at one focus of ellipse);
• z the perihelion
• c the center of the ellipse
• p the planet

and

• ${\displaystyle a=|cz|,}$ distance between center and perihelion, the semimajor axis,
• ${\displaystyle \epsilon =\frac{|cs|}{a},}$ the eccentricity,
• ${\displaystyle b=a\sqrt{1-{\epsilon }^2},}$ the semiminor axis,
• ${\displaystyle r=|sp|,}$ the distance between Sun and planet.
• ${\displaystyle \theta =\mathrm{\angle }zsp,}$ the direction to the planet as seen from the Sun, the true anomaly.

The problem is to compute the polar coordinates (r,θ) of the planet from the time since perihelion, t.

It is solved in steps. Kepler considered the circle with the major axis as a diameter, and

• ${\displaystyle x,}$ the projection of the planet to the auxiliary circle
• ${\displaystyle y,}$ the point on the circle such that the sector areas |zcy| and |zsx| are equal,
• ${\displaystyle M=\mathrm{\angle }zcy,}$ the mean anomaly.

The sector areas are related by ${\displaystyle |zsp|=\frac{b}{a}\cdot |zsx|.}$

The circular sector area ${\displaystyle |zcy|=\frac{a^{2}M}{2}.}$

The area swept since perihelion,

$${\displaystyle |zsp|=\frac{b}{a}\cdot |zsx|=\frac{b}{a}\cdot |zcy|=\frac{b}{a}\cdot \frac{a^{2}M}{2}=\frac{abM}{2},}$$

is by Kepler's second law proportional to time since perihelion. So the mean anomaly, M, is proportional to time since perihelion, t.

$${\displaystyle M=nt,}$$

where n is the mean motion.

Eccentric anomaly, E

When the mean anomaly M is computed, the goal is to compute the true anomaly θ. The function θ = f(M) is, however, not elementary. Kepler's solution is to use

$${\displaystyle E=\mathrm{\angle }zcx,}$$

x as seen from the centre, the eccentric anomaly as an intermediate variable, and first compute E as a function of M by solving Kepler's equation below, and then compute the true anomaly θ from the eccentric anomaly E. Here are the details.

$${\displaystyle \begin{array}{rl}|zcy|& =|zsx|=|zcx|-|scx|\\ with|scx|& =\frac{|cs|.|dx|}{2}\\ \frac{a^{2}M}{2}& =\frac{a^{2}E}{2}-\frac{a\epsilon \cdot a\sin E}{2}\end{array}}$$

Division by a²/2 gives Kepler's equation

$${\displaystyle M=E-\epsilon \sin E.}$$

This equation gives M as a function of E. Determining E for a given M is the inverse problem. Iterative numerical algorithms are commonly used.

Having computed the eccentric anomaly E, the next step is to calculate the true anomaly θ.

But note: Cartesian position coordinates with reference to the center of ellipse are (a cos E, b sin E)

With reference to the Sun (with coordinates (c,0) = (ae,0) ), r = (a cos E – ae, b sin E)

True anomaly would be arctan(r_y/rx), magnitude of r would be √r · r.

True anomaly, θ

Note from the figure that

$${\displaystyle |cd|=|cs|+|sd|}$$

so that

$${\displaystyle a\cos E=a\epsilon +r\cos \theta .}$$

Dividing by ${\displaystyle a}$ and inserting from Kepler's first law

$${\displaystyle \frac{r}{a}=\frac{1-{\epsilon }^2}{1+\epsilon \cos \theta }}$$

to get

$${\displaystyle \cos E=\epsilon +\frac{1-{\epsilon }^2}{1+\epsilon \cos \theta }\cos \theta =\frac{\epsilon (1+\epsilon \cos \theta )+\left(1-{\epsilon }^2\right)\cos \theta }{1+\epsilon \cos \theta }=\frac{\epsilon +\cos \theta }{1+\epsilon \cos \theta }.}$$

The result is a usable relationship between the eccentric anomaly E and the true anomaly θ.

A computationally more convenient form follows by substituting into the trigonometric identity:

$${\displaystyle {\tan }^2\frac{x}{2}=\frac{1-\cos x}{1+\cos x}.}$$

Get

$${\displaystyle \begin{array}{rl}{\tan }^2\frac{E}{2}& =\frac{1-\cos E}{1+\cos E}=\frac{1-\frac{\epsilon +\cos \theta }{1+\epsilon \cos \theta }}{1+\frac{\epsilon +\cos \theta }{1+\epsilon \cos \theta }}\\ & =\frac{(1+\epsilon \cos \theta )-(\epsilon +\cos \theta )}{(1+\epsilon \cos \theta )+(\epsilon +\cos \theta )}=\frac{1-\epsilon }{1+\epsilon }\cdot \frac{1-\cos \theta }{1+\cos \theta }=\frac{1-\epsilon }{1+\epsilon }{\tan }^2\frac{\theta }{2}.\end{array}}$$

Multiplying by 1 + ε gives the result

$${\displaystyle (1-\epsilon ){\tan }^2\frac{\theta }{2}=(1+\epsilon ){\tan }^2\frac{E}{2}}$$

This is the third step in the connection between time and position in the orbit.

Distance, r

The fourth step is to compute the heliocentric distance r from the true anomaly θ by Kepler's first law:

$${\displaystyle r(1+\epsilon \cos \theta )=a\left(1-{\epsilon }^2\right)}$$

Using the relation above between θ and E the final equation for the distance r is:

$${\displaystyle r=a(1-\epsilon \cos E).}$$