Philosophiæ Naturalis Principia Mathematica
1687 work by Isaac Newton
' (English: "The Mathematical Principles of Natural Philosophy"), often referred to as simply the ' (), is a book by Isaac Newton that expounds Newton's laws of motion and his law of universal gravitation. The "Principia" is written in Latin and comprises three volumes, and was authorized, imprimatur, by Samuel Pepys, then-President of the Royal Society on 5 July 1686 and first published in 1687.
The is considered one of the most important works in the history of science. The French mathematical physicist Alexis Clairaut assessed it in 1747: "The famous book of "Mathematical Principles of Natural Philosophy" marked the epoch of a great revolution in physics. The method followed by its illustrious author Sir Newton ... spread the light of mathematics on a science which up to then had remained in the darkness of conjectures and hypotheses." The French scientist Joseph-Louis Lagrange described it as "the greatest production of the human mind". French polymath Pierre-Simon Laplace stated that "The "Principia" is pre-eminent above any other production of human genius". Newton's work has also been called "the greatest scientific work in history", and "the supreme expression in human thought of the mind's ability to hold the universe fixed as an object of contemplation".
A more recent assessment has been that while acceptance of Newton's laws was not immediate, by the end of the century after publication in 1687, "no one could deny that [out of the ] a science had emerged that, at least in certain respects, so far exceeded anything that had ever gone before that it stood alone as the ultimate exemplar of science generally".
The forms a mathematical foundation for the theory of classical mechanics. Among other achievements, it explains Johannes Kepler's laws of planetary motion, which Kepler had first obtained empirically. In formulating his physical laws, Newton developed and used mathematical methods now included in the field of calculus, expressing them in the form of geometric propositions about "vanishingly small" shapes. In a revised conclusion to the (<templatestyles src="Crossreference/styles.css" />), Newton emphasized the empirical nature of the work with the expression "Hypotheses non fingo" ("I frame/feign no hypotheses").
After annotating and correcting his personal copy of the first edition, Newton published two further editions, during 1713 with errors of the 1687 corrected, and an improved version of 1726.
Contents.
Expressed aim and topics covered.
The Preface of the work states:
<templatestyles src="Template:Blockquote/styles.css" />... Rational Mechanics will be the sciences of motion resulting from any forces whatsoever, and of the forces required to produce any motion, accurately proposed and demonstrated ... And therefore we offer this work as mathematical principles of his philosophy. For all the difficulty of philosophy seems to consist in this—from the phenomena of motions to investigate the forces of Nature, and then from these forces to demonstrate the other phenomena ...
Newton situates himself within the contemporary scientific movement which had "omit[ed] substantial forms and the occult qualities" and instead endeavoured to explain the world by empirical investigation and outlining of empirical regularities.
The "Principia" deals primarily with massive bodies in motion, initially under a variety of conditions and hypothetical laws of force in both non-resisting and resisting media, thus offering criteria to decide, by observations, which laws of force are operating in phenomena that may be observed. It attempts to cover hypothetical or possible motions both of celestial bodies and of terrestrial projectiles. It explores difficult problems of motions perturbed by multiple attractive forces. Its third and final book deals with the interpretation of observations about the movements of planets and their satellites.
The book:
The opening sections of the "Principia" contain, in revised and extended form, nearly all of the content of Newton's 1684 tract "De motu corporum in gyrum".
The "Principia" begin with "Definitions" and "Axioms or Laws of Motion", and continues in three books:
Book 1, "De motu corporum".
Book 1, subtitled "De motu corporum" ("On the motion of bodies") concerns motion in the absence of any resisting medium. It opens with a collection of mathematical lemmas on "the method of first and last ratios", a geometrical form of infinitesimal calculus.
The second section establishes relationships between centripetal forces and the law of areas now known as Kepler's second law (Propositions 1–3), and relates circular velocity and radius of path-curvature to radial force (Proposition 4), and relationships between centripetal forces varying as the inverse-square of the distance to the center and orbits of conic-section form (Propositions 5–10).
Propositions 11–31 establish properties of motion in paths of eccentric conic-section form including ellipses, and their relationship with inverse-square central forces directed to a focus and include Newton's theorem about ovals (lemma 28).
Propositions 43–45 are demonstration that in an eccentric orbit under centripetal force where the apse may move, a steady non-moving orientation of the line of apses is an indicator of an inverse-square law of force.
Book 1 contains some proofs with little connection to real-world dynamics. But there are also sections with far-reaching application to the solar system and universe:
Propositions 57–69 deal with the "motion of bodies drawn to one another by centripetal forces". This section is of primary interest for its application to the Solar System, and includes Proposition 66 along with its 22 corollaries: here Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions, a problem which later gained name and fame (among other reasons, for its great difficulty) as the three-body problem.
Propositions 70–84 deal with the attractive forces of spherical bodies. The section contains Newton's proof that a massive spherically symmetrical body attracts other bodies outside itself as if all its mass were concentrated at its centre. This fundamental result, called the Shell theorem, enables the inverse square law of gravitation to be applied to the real solar system to a very close degree of approximation.
Book 2, part 2 of "De motu corporum".
Part of the contents originally planned for the first book was divided out into a second book, which largely concerns motion through resisting mediums. Just as Newton examined consequences of different conceivable laws of attraction in Book 1, here he examines different conceivable laws of resistance; thus discusses resistance in direct proportion to velocity, and goes on to examine the implications of resistance in proportion to the square of velocity. Book 2 also discusses (in ) hydrostatics and the properties of compressible fluids; Newton also derives Boyle's law. The effects of air resistance on pendulums are studied in , along with Newton's account of experiments that he carried out, to try to find out some characteristics of air resistance in reality by observing the motions of pendulums under different conditions. Newton compares the resistance offered by a medium against motions of globes with different properties (material, weight, size). In Section 8, he derives rules to determine the speed of waves in fluids and relates them to the density and condensation (Proposition 48; this would become very important in acoustics). He assumes that these rules apply equally to light and sound and estimates that the speed of sound is around 1088 feet per second and can increase depending on the amount of water in air.
Less of Book 2 has stood the test of time than of Books 1 and 3, and it has been said that Book 2 was largely written to refute a theory of Descartes which had some wide acceptance before Newton's work (and for some time after). According to Descartes's theory of vortices, planetary motions were produced by the whirling of fluid vortices that filled interplanetary space and carried the planets along with them. Newton concluded Book 2 by commenting that the hypothesis of vortices was completely at odds with the astronomical phenomena, and served not so much to explain as to confuse them.
Book 3, "De mundi systemate".
Book 3, subtitled "De mundi systemate" ("On the system of the world"), is an exposition of many consequences of universal gravitation, especially its consequences for astronomy. It builds upon the propositions of the previous books and applies them with further specificity than in Book 1 to the motions observed in the Solar System. Here (introduced by Proposition 22, and continuing in Propositions 25–35) are developed several of the features and irregularities of the orbital motion of the Moon, especially the variation. Newton lists the astronomical observations on which he relies, and establishes in a stepwise manner that the inverse square law of mutual gravitation applies to Solar System bodies, starting with the satellites of Jupiter and going on by stages to show that the law is of universal application. He also gives starting at Lemma 4 and Proposition 40 the theory of the motions of comets, for which much data came from John Flamsteed and Edmond Halley, and accounts for the tides, attempting quantitative estimates of the contributions of the Sun and Moon to the tidal motions; and offers the first theory of the precession of the equinoxes. Book 3 also considers the harmonic oscillator in three dimensions, and motion in arbitrary force laws.
In Book 3 Newton also made clear his heliocentric view of the Solar System, modified in a somewhat modern way, since already in the mid-1680s he recognised the "deviation of the Sun" from the centre of gravity of the Solar System. For Newton, "the common centre of gravity of the Earth, the Sun and all the Planets is to be esteem'd the Centre of the World", and that this centre "either is at rest, or moves uniformly forward in a right line". Newton rejected the second alternative after adopting the position that "the centre of the system of the world is immoveable", which "is acknowledg'd by all, while some contend that the Earth, others, that the Sun is fix'd in that centre". Newton estimated the mass ratios Sun:Jupiter and Sun:Saturn, and pointed out that these put the centre of the Sun usually a little way off the common center of gravity, but only a little, the distance at most "would scarcely amount to one diameter of the Sun".
Commentary on the "Principia".
The sequence of definitions used in setting up dynamics in the "Principia" is recognisable in many textbooks today. Newton first set out the definition of mass
<templatestyles src="Template:Blockquote/styles.css" />The quantity of matter is that which arises conjointly from its density and magnitude. A body twice as dense in double the space is quadruple in quantity. This quantity I designate by the name of body or of mass.
This was then used to define the "quantity of motion" (today called momentum), and the principle of inertia in which mass replaces the previous Cartesian notion of "intrinsic force". This then set the stage for the introduction of forces through the change in momentum of a body. Curiously, for today's readers, the exposition looks dimensionally incorrect, since Newton does not introduce the dimension of time in rates of changes of quantities.
He defined space and time "not as they are well known to all". Instead, he defined "true" time and space as "absolute" and explained:
<templatestyles src="Template:Blockquote/styles.css" />Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to perceptible objects. And it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common. ... instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs; but in philosophical discussions, we ought to step back from our senses, and consider things themselves, distinct from what are only perceptible measures of them.
To some modern readers it can appear that some dynamical quantities recognised today were used in the "Principia" but not named. The mathematical aspects of the first two books were so clearly consistent that they were easily accepted; for example, Locke asked Huygens whether he could trust the mathematical proofs and was assured about their correctness.
However, the concept of an attractive force acting at a distance received a cooler response. In his notes, Newton wrote that the inverse square law arose naturally due to the structure of matter. However, he retracted this sentence in the published version, where he stated that the motion of planets is consistent with an inverse square law, but refused to speculate on the origin of the law. Huygens and Leibniz noted that the law was incompatible with the notion of the aether. From a Cartesian point of view, therefore, this was a faulty theory. Newton's defence has been adopted since by many famous physicists—he pointed out that the mathematical form of the theory had to be correct since it explained the data, and he refused to speculate further on the basic nature of gravity. The sheer number of phenomena that could be organised by the theory was so impressive that younger "philosophers" soon adopted the methods and language of the "Principia".
Rules of Reason.
Perhaps to reduce the risk of public misunderstanding, Newton included at the beginning of Book 3 (in the second (1713) and third (1726) editions) a section titled "Rules of Reasoning in Philosophy". In the four rules, as they came finally to stand in the 1726 edition, Newton effectively offers a methodology for handling unknown phenomena in nature and reaching towards explanations for them. The four Rules of the 1726 edition run as follows (omitting some explanatory comments that follow each):
This section of Rules for philosophy is followed by a listing of "Phenomena", in which are listed a number of mainly astronomical observations, that Newton used as the basis for inferences later on, as if adopting a consensus set of facts from the astronomers of his time.
Both the "Rules" and the "Phenomena" evolved from one edition of the "Principia" to the next. Rule 4 made its appearance in the third (1726) edition; Rules 1–3 were present as "Rules" in the second (1713) edition, and predecessors of them were also present in the first edition of 1687, but there they had a different heading: they were not given as "Rules", but rather in the first (1687) edition the predecessors of the three later "Rules", and of most of the later "Phenomena", were all lumped together under a single heading "Hypotheses" (in which the third item was the predecessor of a heavy revision that gave the later Rule 3).
From this textual evolution, it appears that Newton wanted by the later headings "Rules" and "Phenomena" to clarify for his readers his view of the roles to be played by these various statements.
In the third (1726) edition of the "Principia", Newton explains each rule in an alternative way and/or gives an example to back up what the rule is claiming. The first rule is explained as a philosophers' principle of economy. The second rule states that if one cause is assigned to a natural effect, then the same cause so far as possible must be assigned to natural effects of the same kind: for example, respiration in humans and in animals, fires in the home and in the Sun, or the reflection of light whether it occurs terrestrially or from the planets. An extensive explanation is given of the third rule, concerning the qualities of bodies, and Newton discusses here the generalisation of observational results, with a caution against making up fancies contrary to experiments, and use of the rules to illustrate the observation of gravity and space.
General Scholium.
The "General Scholium" is a concluding essay added to the second edition, 1713 (and amended in the third edition, 1726). It is not to be confused with the "General Scholium" at the end of Book 2, Section 6, which discusses his pendulum experiments and resistance due to air, water, and other fluids.
Here Newton used the expression hypotheses non fingo, "I formulate no hypotheses", in response to criticisms of the first edition of the "Principia". ("Fingo" is sometimes nowadays translated "feign" rather than the traditional "frame," although "feign" does not properly translate "fingo"). Newton's gravitational attraction, an invisible force able to act over vast distances, had led to criticism that he had introduced "occult agencies" into science. Newton firmly rejected such criticisms and wrote that it was enough that the phenomena implied gravitational attraction, as they did; but the phenomena did not so far indicate the cause of this gravity, and it was both unnecessary and improper to frame hypotheses of things not implied by the phenomena: such hypotheses "have no place in experimental philosophy", in contrast to the proper way in which "particular propositions are inferr'd from the phenomena and afterwards rendered general by induction".
Newton also underlined his criticism of the vortex theory of planetary motions, of Descartes, pointing to its incompatibility with the highly eccentric orbits of comets, which carry them "through all parts of the heavens indifferently".
Newton also gave theological argument. From the system of the world, he inferred the existence of a god, along lines similar to what is sometimes called the argument from intelligent or purposive design. It has been suggested that Newton gave "an oblique argument for a unitarian conception of God and an implicit attack on the doctrine of the Trinity". The General Scholium does not address or attempt to refute the church doctrine; it simply does not mention Jesus, the Holy Ghost, or the hypothesis of the Trinity.
Publishing the book.
Halley and Newton's initial stimulus.
In January 1684, Edmond Halley, Christopher Wren and Robert Hooke had a conversation in which Hooke claimed to not only have derived the inverse-square law but also all the laws of planetary motion. Wren was unconvinced, Hooke did not produce the claimed derivation although the others gave him time to do it, and Halley, who could derive the inverse-square law for the restricted circular case (by substituting Kepler's relation into Huygens' formula for the centrifugal force) but failed to derive the relation generally, resolved to ask Newton.
Halley's visits to Newton in 1684 thus resulted from Halley's debates about planetary motion with Wren and Hooke, and they seem to have provided Newton with the incentive and spur to develop and write what became "Philosophiae Naturalis Principia Mathematica". Halley was at that time a Fellow and Council member of the Royal Society in London (positions that in 1686 he resigned to become the Society's paid Clerk). Halley's visit to Newton in Cambridge in 1684 probably occurred in August. When Halley asked Newton's opinion on the problem of planetary motions discussed earlier that year between Halley, Hooke and Wren, Newton surprised Halley by saying that he had already made the derivations some time ago; but that he could not find the papers. (Matching accounts of this meeting come from Halley and Abraham De Moivre to whom Newton confided.) Halley then had to wait for Newton to "find" the results, and in November 1684 Newton sent Halley an amplified version of whatever previous work Newton had done on the subject. This took the form of a 9-page manuscript, "De motu corporum in gyrum" ("Of the motion of bodies in an orbit"): the title is shown on some surviving copies, although the (lost) original may have been without a title.
Newton's tract "De motu corporum in gyrum", which he sent to Halley in late 1684, derived what is now known as the three laws of Kepler, assuming an inverse square law of force, and generalised the result to conic sections. It also extended the methodology by adding the solution of a problem on the motion of a body through a resisting medium. The contents of "De motu" so excited Halley by their mathematical and physical originality and far-reaching implications for astronomical theory, that he immediately went to visit Newton again, in November 1684, to ask Newton to let the Royal Society have more of such work. The results of their meetings clearly helped to stimulate Newton with the enthusiasm needed to take his investigations of mathematical problems much further in this area of physical science, and he did so in a period of highly concentrated work that lasted at least until mid-1686.
Newton's single-minded attention to his work generally, and to his project during this time, is shown by later reminiscences from his secretary and copyist of the period, Humphrey Newton. His account tells of Isaac Newton's absorption in his studies, how he sometimes forgot his food, or his sleep, or the state of his clothes, and how when he took a walk in his garden he would sometimes rush back to his room with some new thought, not even waiting to sit before beginning to write it down. Other evidence also shows Newton's absorption in the "Principia": Newton for years kept up a regular programme of chemical or alchemical experiments, and he normally kept dated notes of them, but for a period from May 1684 to April 1686, Newton's chemical notebooks have no entries at all. So, it seems that Newton abandoned pursuits to which he was formally dedicated and did very little else for well over a year and a half, but concentrated on developing and writing what became his great work.
The first of the three constituent books was sent to Halley for the printer in spring 1686, and the other two books somewhat later. The complete work, published by Halley at his own financial risk, appeared in July 1687. Newton had also communicated "De motu" to Flamsteed, and during the period of composition, he exchanged a few letters with Flamsteed about observational data on the planets, eventually acknowledging Flamsteed's contributions in the published version of the "Principia" of 1687.
Preliminary version.
The process of writing that first edition of the "Principia" went through several stages and drafts: some parts of the preliminary materials still survive, while others are lost except for fragments and cross-references in other documents.
Surviving materials show that Newton (up to some time in 1685) conceived his book as a two-volume work. The first volume was to be titled "De motu corporum, Liber primus", with contents that later appeared in extended form as Book 1 of the "Principia".
A fair-copy draft of Newton's planned second volume "De motu corporum, Liber Secundus" survives, its completion dated to about the summer of 1685. It covers the application of the results of "Liber primus" to the Earth, the Moon, the tides, the Solar System, and the universe; in this respect, it has much the same purpose as the final Book 3 of the "Principia", but it is written much less formally and is more easily read.
It is not known just why Newton changed his mind so radically about the final form of what had been a readable narrative in "De motu corporum, Liber Secundus" of 1685, but he largely started afresh in a new, tighter, and less accessible mathematical style, eventually to produce Book 3 of the "Principia" as we know it. Newton frankly admitted that this change of style was deliberate when he wrote that he had (first) composed this book "in a popular method, that it might be read by many", but to "prevent the disputes" by readers who could not "lay aside the[ir] prejudices", he had "reduced" it "into the form of propositions (in the mathematical way) which should be read by those only, who had first made themselves masters of the principles established in the preceding books". The final Book 3 also contained in addition some further important quantitative results arrived at by Newton in the meantime, especially about the theory of the motions of comets, and some of the perturbations of the motions of the Moon.
The result was numbered Book 3 of the "Principia" rather than Book 2 because in the meantime, drafts of "Liber primus" had expanded and Newton had divided it into two books. The new and final Book 2 was concerned largely with the motions of bodies through resisting mediums.
But the "Liber Secundus" of 1685 can still be read today. Even after it was superseded by Book 3 of the "Principia", it survived complete, in more than one manuscript. After Newton's death in 1727, the relatively accessible character of its writing encouraged the publication of an English translation in 1728 (by persons still unknown, not authorised by Newton's heirs). It appeared under the English title "A Treatise of the System of the World". This had some amendments relative to Newton's manuscript of 1685, mostly to remove cross-references that used obsolete numbering to cite the propositions of an early draft of Book 1 of the "Principia". Newton's heirs shortly afterwards published the Latin version in their possession, also in 1728, under the (new) title "De Mundi Systemate", amended to update cross-references, citations and diagrams to those of the later editions of the "Principia", making it look superficially as if it had been written by Newton after the "Principia", rather than before. The "System of the World" was sufficiently popular to stimulate two revisions (with similar changes as in the Latin printing), a second edition (1731), and a "corrected" reprint of the second edition (1740).
Halley's role as publisher.
The text of the first of the three books of the "Principia" was presented to the Royal Society at the close of April 1686. Hooke made some priority claims (but failed to substantiate them), causing some delay. When Hooke's claim was made known to Newton, who hated disputes, Newton threatened to withdraw and suppress Book 3 altogether, but Halley, showing considerable diplomatic skills, tactfully persuaded Newton to withdraw his threat and let it go forward to publication. Samuel Pepys, as president, gave his imprimatur on 30 June 1686, licensing the book for publication. The Society had just spent its book budget on "De Historia piscium", and the cost of publication was borne by Edmund Halley (who was also then acting as publisher of the "Philosophical Transactions of the Royal Society"): the book appeared in summer 1687. After Halley had personally financed the publication of "Principia", he was informed that the society could no longer afford to provide him the promised annual salary of £50. Instead, Halley was paid with leftover copies of "De Historia piscium".
Historical context.
Beginnings of the Scientific Revolution.
Nicolaus Copernicus had moved the Earth away from the center of the universe with the heliocentric theory for which he presented evidence in his book "De revolutionibus orbium coelestium" ("On the revolutions of the heavenly spheres") published in 1543. Johannes Kepler wrote the book "Astronomia nova" ("A new astronomy") in 1609, setting out the evidence that planets move in elliptical orbits with the Sun at one focus, and that planets do not move with constant speed along this orbit. Rather, their speed varies so that the line joining the centres of the sun and a planet sweeps out equal areas in equal times. To these two laws he added a third a decade later, in his 1619 book "Harmonices Mundi" ("Harmonies of the world"). This law sets out a proportionality between the third power of the characteristic distance of a planet from the Sun and the square of the length of its year.
The foundation of modern dynamics was set out in Galileo's book "Dialogo sopra i due massimi sistemi del mondo" ("Dialogue on the two main world systems") where the notion of inertia was implicit and used. In addition, Galileo's experiments with inclined planes had yielded precise mathematical relations between elapsed time and acceleration, velocity or distance for uniform and uniformly accelerated motion of bodies.
Descartes' book of 1644 "Principia philosophiae" ("Principles of philosophy") stated that bodies can act on each other only through contact: a principle that induced people, among them himself, to hypothesize a universal medium as the carrier of interactions such as light and gravity—the aether. Newton was criticized for apparently introducing forces that acted at distance without any medium. Not until the development of particle theory was Descartes' notion vindicated when it was possible to describe all interactions, like the strong, weak, and electromagnetic fundamental interactions, using mediating gauge bosons and gravity through hypothesized gravitons.
Newton's role.
Newton had studied these books, or, in some cases, secondary sources based on them, and taken notes entitled "Quaestiones quaedam philosophicae" ("Questions about philosophy") during his days as an undergraduate. During this period (1664–1666) he created the basis of calculus and performed the first experiments in the optics of colour. At this time, his proof that white light was a combination of primary colours (found via prismatics) replaced the prevailing theory of colours and received an overwhelmingly favourable response and occasioned bitter disputes with Robert Hooke and others, which forced him to sharpen his ideas to the point where he already composed sections of his later book "Opticks" by the 1670s in response. Work on calculus is shown in various papers and letters, including two to Leibniz. He became a fellow of the Royal Society and the second Lucasian Professor of Mathematics (succeeding Isaac Barrow) at Trinity College, Cambridge.
Newton's early work on motion.
In the 1660s Newton studied the motion of colliding bodies and deduced that the centre of mass of two colliding bodies remains in uniform motion. Surviving manuscripts of the 1660s also show Newton's interest in planetary motion and that by 1669 he had shown, for a circular case of planetary motion, that the force he called "endeavour to recede" (now called centrifugal force) had an inverse-square relation with distance from the center. After his 1679–1680 correspondence with Hooke, described below, Newton adopted the language of inward or centripetal force. According to Newton scholar J. Bruce Brackenridge, although much has been made of the change in language and difference of point of view, as between centrifugal or centripetal forces, the actual computations and proofs remained the same either way. They also involved the combination of tangential and radial displacements, which Newton was making in the 1660s. The difference between the centrifugal and centripetal points of view, though a significant change of perspective, did not change the analysis. Newton also clearly expressed the concept of linear inertia in the 1660s: for this Newton was indebted to Descartes' work published 1644.
Controversy with Hooke.
Hooke published his ideas about gravitation in the 1660s and again in 1674. He argued for an attracting principle of gravitation in "Micrographia" of 1665, in a 1666 Royal Society lecture "On gravity", and again in 1674, when he published his ideas about the "System of the World" in somewhat developed form, as an addition to "An Attempt to Prove the Motion of the Earth from Observations". Hooke clearly postulated mutual attractions between the Sun and planets, in a way that increased with nearness to the attracting body, along with a principle of linear inertia. Hooke's statements up to 1674 made no mention, however, that an inverse square law applies or might apply to these attractions. Hooke's gravitation was also not yet universal, though it approached universality more closely than previous hypotheses. Hooke also did not provide accompanying evidence or mathematical demonstration. On these two aspects, Hooke stated in 1674: "Now what these several degrees [of gravitational attraction] are I have not yet experimentally verified" (indicating that he did not yet know what law the gravitation might follow); and as to his whole proposal: "This I only hint at present", "having my self many other things in hand which I would first compleat, and therefore cannot so well attend it" (i.e., "prosecuting this Inquiry").
In November 1679, Hooke began an exchange of letters with Newton, of which the full text is now published. Hooke told Newton that Hooke had been appointed to manage the Royal Society's correspondence, and wished to hear from members about their researches, or their views about the researches of others; and as if to whet Newton's interest, he asked what Newton thought about various matters, giving a whole list, mentioning "compounding the celestial motions of the planets of a direct motion by the tangent and an attractive motion towards the central body", and "my hypothesis of the lawes or causes of springinesse", and then a new hypothesis from Paris about planetary motions (which Hooke described at length), and then efforts to carry out or improve national surveys, the difference of latitude between London and Cambridge, and other items. Newton's reply offered "a fansy of my own" about a terrestrial experiment (not a proposal about celestial motions) which might detect the Earth's motion, by the use of a body first suspended in air and then dropped to let it fall. The main point was to indicate how Newton thought the falling body could experimentally reveal the Earth's motion by its direction of deviation from the vertical, but he went on hypothetically to consider how its motion could continue if the solid Earth had not been in the way (on a spiral path to the centre). Hooke disagreed with Newton's idea of how the body would continue to move. A short further correspondence developed, and towards the end of it Hooke, writing on 6 January 1680 to Newton, communicated his "supposition ... that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall, and Consequently that the Velocity will be in a subduplicate proportion to the Attraction and Consequently as Kepler Supposes Reciprocall to the Distance." (Hooke's inference about the velocity was actually incorrect.)
In 1686, when the first book of Newton's "Principia" was presented to the Royal Society, Hooke claimed that Newton had obtained from him the "notion" of "the rule of the decrease of Gravity, being reciprocally as the squares of the distances from the Center". At the same time (according to Edmond Halley's contemporary report) Hooke agreed that "the Demonstration of the Curves generated therby" was wholly Newton's.
A recent assessment about the early history of the inverse square law is that "by the late 1660s", the assumption of an "inverse proportion between gravity and the square of distance was rather common and had been advanced by a number of different people for different reasons". Newton himself had shown in the 1660s that for planetary motion under a circular assumption, force in the radial direction had an inverse-square relation with distance from the center. Newton, faced in May 1686 with Hooke's claim on the inverse square law, denied that Hooke was to be credited as author of the idea, giving reasons including the citation of prior work by others before Hooke. Newton also firmly claimed that even if it had happened that he had first heard of the inverse square proportion from Hooke, which it had not, he would still have some rights to it in view of his mathematical developments and demonstrations, which enabled observations to be relied on as evidence of its accuracy, while Hooke, without mathematical demonstrations and evidence in favour of the supposition, could only guess (according to Newton) that it was approximately valid "at great distances from the center".
The background described above shows there was basis for Newton to deny deriving the inverse square law from Hooke. On the other hand, Newton did accept and acknowledge, in all editions of the "Principia", that Hooke (but not exclusively Hooke) had separately appreciated the inverse square law in the Solar System. Newton acknowledged Wren, Hooke and Halley in this connection in the Scholium to Proposition 4 in Book 1. Newton also acknowledged to Halley that his correspondence with Hooke in 1679–80 had reawakened his dormant interest in astronomical matters, but that did not mean, according to Newton, that Hooke had told Newton anything new or original: "yet am I not beholden to him for any light into that business but only for the diversion he gave me from my other studies to think on these things & for his dogmaticalness in writing as if he had found the motion in the Ellipsis, which inclined me to try it ...".) Newton's reawakening interest in astronomy received further stimulus by the appearance of a comet in the winter of 1680/1681, on which he corresponded with John Flamsteed.
In 1759, decades after the deaths of both Newton and Hooke, Alexis Clairaut, mathematical astronomer eminent in his own right in the field of gravitational studies, made his assessment after reviewing what Hooke had published on gravitation. "One must not think that this idea ... of Hooke diminishes Newton's glory", Clairaut wrote; "The example of Hooke" serves "to show what a distance there is between a truth that is glimpsed and a truth that is demonstrated".
Location of early edition copies.
It has been estimated that as many as 750 copies of the first edition were printed by the Royal Society, and "it is quite remarkable that so many copies of this small first edition are still in existence ... but it may be because the original Latin text was more revered than read". A survey published in 1953 located 189 surviving copies with nearly 200 further copies located by the most recent survey published in 2020, suggesting that the initial print run was larger than previously thought. However, more recent book historical and bibliographical research has examined those prior claims, and concludes that Macomber's earlier estimation of 500 copies is likely correct.
In 2016, a first edition sold for $3.7 million.
The second edition (1713) were printed in 750 copies, and the third edition (1726) were printed in 1,250 copies.
A facsimile edition (based on the 3rd edition of 1726 but with variant readings from earlier editions and important annotations) was published in 1972 by Alexandre Koyré and I. Bernard Cohen.
Later editions.
Second edition, 1713.
Two later editions were published by Newton: Newton had been urged to make a new edition of the "Principia" since the early 1690s, partly because copies of the first edition had already become very rare and expensive within a few years after 1687. Newton referred to his plans for a second edition in correspondence with Flamsteed in November 1694. Newton also maintained annotated copies of the first edition specially bound up with interleaves on which he could note his revisions; two of these copies still survive, but he had not completed the revisions by 1708. Newton had almost severed connections with one would-be editor, Nicolas Fatio de Duillier, and another, David Gregory seems not to have met with his approval and was also terminally ill, dying in 1708. Nevertheless, reasons were accumulating not to put off the new edition any longer. Richard Bentley, master of Trinity College, persuaded Newton to allow him to undertake a second edition, and in June 1708 Bentley wrote to Newton with a specimen print of the first sheet, at the same time expressing the (unfulfilled) hope that Newton had made progress towards finishing the revisions. It seems that Bentley then realised that the editorship was technically too difficult for him, and with Newton's consent he appointed Roger Cotes, Plumian professor of astronomy at Trinity, to undertake the editorship for him as a kind of deputy (but Bentley still made the publishing arrangements and had the financial responsibility and profit). The correspondence of 1709–1713 shows Cotes reporting to two masters, Bentley and Newton, and managing (and often correcting) a large and important set of revisions to which Newton sometimes could not give his full attention. Under the weight of Cotes' efforts, but impeded by priority disputes between Newton and Leibniz, and by troubles at the Mint, Cotes was able to announce publication to Newton on 30 June 1713. Bentley sent Newton only six presentation copies; Cotes was unpaid; Newton omitted any acknowledgement to Cotes.
Among those who gave Newton corrections for the Second Edition were: Firmin Abauzit, Roger Cotes and David Gregory. However, Newton omitted acknowledgements to some because of the priority disputes. John Flamsteed, the Astronomer Royal, suffered this especially.
The Second Edition was the basis of the first edition to be printed abroad, which appeared in Amsterdam in 1714.
Third edition, 1726.
After his serious illness in 1722 and after the appearance of a reprint of the second edition in Amsterdam in 1723, the 80-year-old Newton began to revise once again the Principia in the fall of 1723. The third edition was published 25 March 1726, under the stewardship of "Henry Pemberton, M.D., a man of the greatest skill in these matters..."; Pemberton later said that this recognition was worth more to him than the two hundred guinea award from Newton.
In 1739–1742, two French priests, Pères Thomas LeSeur and François Jacquier (of the Minim order, but sometimes erroneously identified as Jesuits), produced with the assistance of J.-L. Calandrini an extensively annotated version of the "Principia" in the 3rd edition of 1726. Sometimes this is referred to as the "Jesuit edition": it was much used, and reprinted more than once in Scotland during the 19th century.
Émilie du Châtelet also made a translation of Newton's "Principia" into French. Unlike LeSeur and Jacquier's edition, hers was a complete translation of Newton's three books and their prefaces. She also included a Commentary section where she fused the three books into a much clearer and easier to understand summary. She included an analytical section where she applied the new mathematics of calculus to Newton's most controversial theories. Previously, geometry was the standard mathematics used to analyse theories. Du Châtelet's translation is the only complete one to have been done in French and hers remains the standard French translation to this day.
Translations.
Four full English translations of Newton's "Principia" have appeared, all based on Newton's 3rd edition of 1726. The first, from 1729, by Andrew Motte, was described by Newton scholar I. Bernard Cohen (in 1968) as "still of enormous value in conveying to us the sense of Newton's words in their own time, and it is generally faithful to the original: clear, and well written". The 1729 version was the basis for several republications, often incorporating revisions, among them a widely used modernised English version of 1934, which appeared under the editorial name of Florian Cajori (though completed and published only some years after his death). Cohen pointed out ways in which the 18th-century terminology and punctuation of the 1729 translation might be confusing to modern readers, but he also made severe criticisms of the 1934 modernised English version, and showed that the revisions had been made without regard to the original, also demonstrating gross errors "that provided the final impetus to our decision to produce a wholly new translation".
The second full English translation, into modern English, is the work that resulted from this decision by collaborating translators I. Bernard Cohen, Anne Whitman, and Julia Budenz; it was published in 1999 with a guide by way of introduction.
The third such translation is due to Ian Bruce, and appears, with many other translations of mathematical works of the 17th and 18th centuries, on his website.
The fourth complete English translation is due to Charles Leedham-Green, professor emeritus of mathematics at Queen Mary University of London, and was published in 2021 by Cambridge University Press. Prof. Leedham-Green was motivated to produce that translation, on which he worked for twenty years, in part because of his dissatisfaction with the work of Cohen, Whitman, and Budenz, whose translation of the "Principia" he found unnecessarily obscure. Leedham-Green's aim was to convey Newton's own reasoning and arguments in a way intelligible to a modern mathematical scientist. His translation is heavily annotated and his explanatory notes make use of the modern secondary literature on some of the more difficult technical aspects of Newton's work.
Dana Densmore and William H. Donahue have published a translation of the work's central argument, published in 1996, along with expansion of included proofs and ample commentary. The book was developed as a textbook for classes at St. John's College and the aim of this translation is to be faithful to the Latin text.
Varia.
In 1977, the spacecraft Voyager 1 and 2 left earth for the interstellar space carrying a picture of a page from Newton's "Principia Mathematica", as part of the Golden Record, a collection of messages from humanity to extraterrestrials.
In 2014, British astronaut Tim Peake named his upcoming mission to the International Space Station "Principia" after the book, in "honour of Britain's greatest scientist". Tim Peake's "Principia" launched on 15 December 2015 aboard Soyuz TMA-19M.
References.
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Further reading.
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External links.
Latin versions.
First edition (1687)
Second edition (1713)
Third edition (1726)
Later Latin editions

Mercury (planet)
First planet from the Sun
Mercury is the first planet from the Sun and the smallest in the Solar System. In English, it is named after the ancient Roman god (Mercury), god of commerce and communication, and the messenger of the gods. Mercury is classified as a terrestrial planet, with roughly the same surface gravity as Mars. The surface of Mercury is heavily cratered, as a result of countless impact events that have accumulated over billions of years. Its largest crater, Caloris Planitia, has a diameter of , which is about one-third the diameter of the planet (). Similarly to the Earth's Moon, Mercury's surface displays an expansive rupes system generated from thrust faults and bright ray systems formed by impact event remnants.
Mercury's sidereal year (88.0 Earth days) and sidereal day (58.65 Earth days) are in a 3:2 ratio. This relationship is called spin–orbit resonance, and "sidereal" here means "relative to the stars". Consequently, one solar day (sunrise to sunrise) on Mercury lasts for around 176 Earth days: twice the planet's sidereal year. This means that one side of Mercury will remain in sunlight for one Mercurian year of 88 Earth days; while during the next orbit, that side will be in darkness all the time until the next sunrise after another 88 Earth days.
Combined with its high orbital eccentricity, the planet's surface has widely varying sunlight intensity and temperature, with the equatorial regions ranging from at night to during sunlight. Due to the very small axial tilt, the planet's poles are permanently shadowed. This strongly suggests that water ice could be present in the craters. Above the planet's surface is an extremely tenuous exosphere and a faint magnetic field that is strong enough to deflect solar winds. Mercury has no natural satellites.
As of the early 2020s, many broad details of Mercury's geological history are still under investigation or pending data from space probes. Like other planets in the Solar System, Mercury was formed approximately 4.5 billion years ago. Its mantle is highly homogeneous, which suggests that Mercury had a magma ocean early in its history, like the Moon. According to current models, Mercury may have a solid silicate crust and mantle overlying a solid outer core, a deeper liquid core layer, and a solid inner core. There are many competing hypotheses about Mercury's origins and development, some of which incorporate collision with planetesimals and rock vaporization.
Nomenclature.
Historically, humans knew Mercury by different names depending on whether it was an evening star or a morning star. By about 350 BC, the ancient Greeks had realized the two stars were one. They knew the planet as , meaning "twinkling", and , for its fleeting motion, a name that is retained in modern [[Greek language|Greek]] ( ). The Romans named the planet after the swift-footed Roman messenger god, [[Mercury (mythology)|Mercury]] (Latin ), whom they equated with the Greek Hermes, because it moves across the sky faster than any other planet, though some associated the planet with [[Apollo]] instead, as detailed by [[Pliny the Elder]]. The [[astronomical symbol]] for Mercury is a stylized version of Hermes' [[caduceus]]; a [[Christian cross]] was added in the 16th century:[[File:Mercury symbol (fixed width).svg|16px|☿]].
Physical characteristics.
[[File:Terrestrial planet size comparisons-2.jpg|thumb|upright=1.4|Mercury to scale among the [[Inner Solar System]] [[planetary-mass object]]s beside the Sun, arranged by the order of their orbits outward from the Sun (from left: Mercury, [[Venus]],
[[Earth]], the [[Moon]], [[Mars]] and [[Ceres (dwarf planet)|Ceres]])]]
Mercury is one of four [[terrestrial planet]]s in the [[Solar System]], which means it is a rocky body like Earth. It is the smallest planet in the Solar System, with an [[equator]]ial [[radius]] of . Mercury is also [[list of Solar System objects by radius|smaller]]—albeit more massive—than the largest [[natural satellite]]s in the Solar System, [[Ganymede (moon)|Ganymede]] and [[Titan (moon)|Titan]]. Mercury consists of approximately 70% metallic and 30% [[silicate]] material.
Internal structure.
[[File:Mercury with magnetic field.svg|left|thumb|upright=1.6|Mercury's internal structure and magnetic field]]
Mercury appears to have a solid silicate [[Crust (geology)|crust]] and mantle overlying a solid, metallic outer core layer, a deeper liquid core layer, and a solid inner core. The composition of the iron-rich core remains uncertain, but it likely contains nickel, silicon and perhaps sulfur and carbon, plus trace amounts of other elements. The planet's density is the second highest in the Solar System at 5.427 g/cm3, only slightly less than Earth's density of 5.515 g/cm3. If the effect of [[gravitational compression]] were to be factored out from both planets, the materials of which Mercury is made would be denser than those of Earth, with an uncompressed density of 5.3 g/cm3 versus Earth's 4.4 g/cm3. Mercury's density can be used to infer details of its inner structure. Although Earth's high density results appreciably from gravitational compression, particularly at the [[planetary core|core]], Mercury is much smaller and its inner regions are not as compressed. Therefore, for it to have such a high density, its core must be large and rich in iron.
The radius of Mercury's core is estimated to be , based on interior models constrained to be consistent with a [[moment of inertia factor]] of . Hence, Mercury's core occupies about 57% of its volume; for Earth this proportion is 17%. Research published in 2007 suggests that Mercury has a molten core. The mantle-crust layer is in total thick. Projections differ as to the size of the crust specifically; data from the "Mariner 10" and "MESSENGER" probes suggests a thickness of , whereas an [[Airy isostacy]] model suggests a thickness of . One distinctive feature of Mercury's surface is the presence of numerous narrow ridges, extending up to several hundred kilometers in length. It is thought that these were formed as Mercury's core and mantle cooled and contracted at a time when the crust had already solidified.
Mercury's core has a higher iron content than that of any other planet in the Solar System, and several theories have been proposed to explain this. The most widely accepted theory is that Mercury originally had a metal–silicate ratio similar to common [[chondrite]] meteorites, thought to be typical of the Solar System's rocky matter, and a mass approximately 2.25 times its current mass. Early in the Solar System's history, Mercury may have been struck by a [[planetesimal]] of approximately <templatestyles src="Fraction/styles.css" />1⁄6 Mercury's mass and several thousand kilometers across. The impact would have stripped away much of the original crust and mantle, leaving the core behind as a relatively major component. A similar process, known as the [[giant impact hypothesis]], has been proposed to explain the formation of Earth's Moon.
Alternatively, Mercury may have formed from the [[solar nebula]] before the Sun's energy output had stabilized. It would initially have had twice its present mass, but as the [[protostar|protosun]] contracted, temperatures near Mercury could have been between 2,500 and 3,500 K and possibly even as high as 10,000 K. Much of Mercury's surface rock could have been vaporized at such temperatures, forming an atmosphere of "rock vapor" that could have been carried away by the [[solar wind]]. A third hypothesis proposes that the solar nebula caused [[drag (physics)|drag]] on the particles from which Mercury was [[Accretion (astrophysics)|accreting]], which meant that lighter particles were lost from the accreting material and not gathered by Mercury.
Each hypothesis predicts a different surface composition, and two space missions have been tasked with making observations of this composition. The first "[[MESSENGER]]", which ended in 2015, found higher-than-expected potassium and sulfur levels on the surface, suggesting that the giant impact hypothesis and vaporization of the crust and mantle did not occur because said potassium and sulfur would have been driven off by the extreme heat of these events. "[[BepiColombo]]", which will arrive at Mercury in 2025, will make observations to test these hypotheses. The findings so far would seem to favor the third hypothesis; however, further analysis of the data is needed.
Surface geology.
Mercury's surface is similar in appearance to that of the Moon, showing extensive [[Lunar mare|mare]]-like plains and heavy cratering, indicating that it has been geologically inactive for billions of years. It is more [[heterogeneous]] than the surface of [[Mars]] or the Moon, both of which contain significant stretches of similar geology, such as [[Lunar mare|maria]] and plateaus. [[Albedo]] features are areas of markedly different reflectivity, which include impact craters, the resulting ejecta, and [[ray system]]s. Larger albedo features correspond to higher reflectivity plains. Mercury has "[[wrinkle-ridge]]s" (dorsa), Moon-like [[highland]]s, mountains (montes), plains (planitiae), escarpments (rupes), and valleys ([[Vallis (planetary geology)|valles]]).
[[File:Unmasking the Secrets of Mercury.jpg|thumb|left|[[MESSENGER#Scientific instruments|MASCS]] spectrum scan of Mercury's surface by "MESSENGER"]]
The planet's mantle is chemically heterogeneous, suggesting the planet went through a [[magma ocean]] phase early in its history. Crystallization of minerals and convective overturn resulted in a layered, chemically heterogeneous crust with large-scale variations in chemical composition observed on the surface. The crust is low in iron but high in sulfur, resulting from the stronger early [[Chemical reduction|chemically reducing]] conditions than is found on other terrestrial planets. The surface is dominated by iron-poor [[pyroxene]] and [[olivine]], as represented by [[enstatite]] and [[forsterite]], respectively, along with sodium-rich [[plagioclase]] and minerals of mixed magnesium, calcium, and iron-sulfide. The less reflective regions of the crust are high in carbon, most likely in the form of graphite.
Names for features on Mercury come from a variety of sources and are set according to the [[IAU]] [[planetary nomenclature]] system. Names coming from people are limited to the deceased. Craters are named for artists, musicians, painters, and authors who have made outstanding or fundamental contributions to their field. Ridges, or dorsa, are named for scientists who have contributed to the study of Mercury. Depressions or [[fossa (geology)|fossae]] are named for works of architecture. Montes are named for the word "hot" in a variety of languages. [[Plain]]s or planitiae are named for [[Mercury (god)|Mercury]] in various languages. [[Escarpment]]s or [[rupēs]] are named for ships of scientific expeditions. Valleys or valles are named for abandoned cities, towns, or settlements of antiquity.
Impact basins and craters.
[[File:PIA19421-Mercury-Craters-MunchSanderPoe-20150416.jpg|thumb|left|Enhanced-color image of craters [[Munch (crater)|Munch]] (left), [[Sander (crater)|Sander]] (center), and [[Poe (crater)|Poe]] (right) amid volcanic plains (orange) near [[Caloris Basin]]]]
Mercury was heavily bombarded by comets and [[asteroid]]s during and shortly following its formation 4.6 billion years ago, as well as during a possibly separate subsequent episode called the [[Late Heavy Bombardment]] that ended 3.8 billion years ago. Mercury received impacts over its entire surface during this period of intense crater formation, facilitated by the lack of any [[atmosphere]] to slow impactors down. During this time Mercury was [[volcano|volcanically]] active; basins were filled by [[magma]], producing smooth plains similar to the maria found on the Moon. One of the most unusual craters is [[Apollodorus (crater)|Apollodorus]], or "the Spider", which hosts a series of radiating troughs extending outwards from its impact site.
[[Craters on Mercury]] range in diameter from small bowl-shaped cavities to [[multi-ringed impact basin]]s hundreds of kilometers across. They appear in all states of degradation, from relatively fresh rayed craters to highly degraded crater remnants. Mercurian craters differ subtly from lunar craters in that the area blanketed by their ejecta is much smaller, a consequence of Mercury's stronger surface gravity. According to [[International Astronomical Union]] rules, each new crater must be named after an artist who was famous for more than fifty years, and dead for more than three years, before the date the crater is named.
The largest known crater is [[Caloris Planitia]], or Caloris Basin, with a diameter of . The impact that created the Caloris Basin was so powerful that it caused [[lava]] eruptions and left a concentric mountainous ring ~ tall surrounding the [[impact crater]]. The floor of the Caloris Basin is filled by a geologically distinct flat plain, broken up by ridges and fractures in a roughly polygonal pattern. It is not clear whether they were volcanic lava flows induced by the impact or a large sheet of impact melt.
At the [[antipodes|antipode]] of the Caloris Basin is a large region of unusual, hilly terrain known as the "Weird Terrain". One hypothesis for its origin is that shock waves generated during the Caloris impact traveled around Mercury, converging at the basin's antipode (180 degrees away). The resulting high stresses fractured the surface. Alternatively, it has been suggested that this terrain formed as a result of the convergence of ejecta at this basin's antipode.
[[File:EW1027346412Gnomap.png|thumb|Tolstoj basin is along the bottom of this image of Mercury's limb]]
Overall, 46 impact basins have been identified. A notable basin is the -wide, multi-ring [[Tolstoj Basin]] that has an ejecta blanket extending up to from its rim and a floor that has been filled by smooth plains materials. [[Beethoven Basin]] has a similar-sized ejecta blanket and a -diameter rim. Like the Moon, the surface of Mercury has likely incurred the effects of [[space weathering]] processes, including solar wind and [[micrometeorite]] impacts.
Plains.
There are two geologically distinct plains regions on Mercury. Gently rolling, hilly [[Inter-crater plains on Mercury|plains in the regions between craters]] are Mercury's oldest visible surfaces, predating the heavily cratered terrain. These inter-crater plains appear to have obliterated many earlier craters, and show a general paucity of smaller craters below about in diameter.
Smooth plains are widespread flat areas that fill depressions of various sizes and bear a strong resemblance to lunar maria. Unlike lunar maria, the smooth plains of Mercury have the same albedo as the older inter-crater plains. Despite a lack of unequivocally volcanic characteristics, the localization and rounded, lobate shape of these plains strongly support volcanic origins. All the smooth plains of Mercury formed significantly later than the Caloris basin, as evidenced by appreciably smaller crater densities than on the Caloris ejecta blanket.
Compressional features.
An unusual feature of Mercury's surface is the numerous compression folds, or [[rupes]], that crisscross the plains. These exist on the Moon, but are much more prominent on Mercury. As Mercury's interior cooled, it contracted and its surface began to deform, creating [[wrinkle ridge]]s and [[lobate scarp]]s associated with [[thrust fault]]s. The scarps can reach lengths of and heights of . These compressional features can be seen on top of other features, such as craters and smooth plains, indicating they are more recent. Mapping of the features has suggested a total shrinkage of Mercury's radius in the range of ~. Most activity along the major thrust systems probably ended about 3.6–3.7 billion years ago. Small-scale thrust fault scarps have been found, tens of meters in height and with lengths in the range of a few kilometers, that appear to be less than 50 million years old, indicating that compression of the interior and consequent surface geological activity continue to the present.
Volcanism.
[[File:Picasso crater.png|thumb|[[Picasso (crater)|Picasso crater]]—the large arc-shaped pit located on the eastern side of its floor is postulated to have formed when subsurface magma subsided or drained, causing the surface to collapse into the resulting void.]]
There is evidence for [[pyroclastic flow]]s on Mercury from low-profile [[shield volcano]]es. Fifty-one pyroclastic deposits have been identified, where 90% of them are found within impact craters. A study of the degradation state of the impact craters that host pyroclastic deposits suggests that pyroclastic activity occurred on Mercury over a prolonged interval.
A "rimless depression" inside the southwest rim of the Caloris Basin consists of at least nine overlapping volcanic vents, each individually up to in diameter. It is thus a "[[compound volcano]]". The vent floors are at least below their brinks and they bear a closer resemblance to volcanic craters sculpted by explosive eruptions or modified by collapse into void spaces created by magma withdrawal back down into a conduit. Scientists could not quantify the age of the volcanic complex system but reported that it could be on the order of a billion years.
Surface conditions and exosphere.
[[File:North pole of Mercury -- NASA.jpg|thumb|Composite of the north pole of Mercury, where NASA confirmed the discovery of a large volume of water ice, in permanently dark craters that are found there.]]
The surface temperature of Mercury ranges from . It never rises above 180 K at the poles, due to the absence of an atmosphere and a steep temperature gradient between the equator and the poles. At [[perihelion]], the equatorial [[subsolar point]] is located at latitude 0°W or 180°W, and it climbs to a temperature of about . During [[aphelion]], this occurs at 90° or 270°W and reaches only . On the dark side of the planet, temperatures average . The intensity of [[sunlight]] on Mercury's surface ranges between 4.59 and 10.61 times the [[solar constant]] (1,370 W·m−2).
Although daylight temperatures at the surface of Mercury are generally extremely high, observations strongly suggest that ice (frozen water) exists on Mercury. The floors of deep craters at the poles are never exposed to direct sunlight, and temperatures there remain below 102 K, far lower than the global average. This creates a [[Cold trap (astronomy)|cold trap]] where ice can accumulate. Water ice strongly reflects [[radar]], and observations by the 70-meter [[Goldstone Solar System Radar]] and the [[Very Large Array|VLA]] in the early 1990s revealed that there are patches of high radar [[Reflection (physics)|reflection]] near the poles. Although ice was not the only possible cause of these reflective regions, astronomers thought it to be the most likely explanation. The presence of [[ice|water ice]] was confirmed using "MESSENGER" images of craters at the north pole.
The icy crater regions are estimated to contain about 1014–1015 kg of ice, and may be covered by a layer of [[regolith]] that inhibits [[Sublimation (phase transition)|sublimation]]. By comparison, the [[Antarctica|Antarctic]] ice sheet on Earth has a mass of about 4×1018 kg, and Mars's south polar cap contains about 1016 kg of water. The origin of the ice on Mercury is not yet known, but the two most likely sources are from [[outgassing]] of water from the planet's interior and deposition by impacts of comets.
Mercury is too small and hot for its [[gravity]] to retain any significant [[atmosphere]] over long periods of time; it does have a tenuous surface-bounded [[exosphere]] at a surface pressure of less than approximately 0.5 nPa (0.005 picobars). It includes [[hydrogen]], [[helium]], [[oxygen]], [[sodium]], [[calcium]], [[potassium]], [[magnesium]], [[silicon]], and [[hydroxide]], among others. This exosphere is not stable—atoms are continuously lost and replenished from a variety of sources. [[Hydrogen atom]]s and [[helium atom]]s probably come from the solar wind, [[diffusion|diffusing]] into Mercury's [[magnetosphere]] before later escaping back into space. The [[radioactive decay]] of elements within Mercury's crust is another source of helium, as well as sodium and potassium. Water vapor is present, released by a combination of processes such as comets striking its surface, [[sputtering]] creating water out of hydrogen from the solar wind and oxygen from rock, and sublimation from reservoirs of water ice in the permanently shadowed polar craters. The detection of high amounts of water-related ions like O+, OH−, and [[hydronium|H3O+]] was a surprise. Because of the quantities of these ions that were detected in Mercury's space environment, scientists surmise that these molecules were blasted from the surface or exosphere by the solar wind.
Sodium, potassium, and calcium were discovered in the atmosphere during the 1980s–1990s, and are thought to result primarily from the vaporization of surface rock struck by micrometeorite impacts including presently from [[Comet Encke]]. In 2008, magnesium was discovered by "MESSENGER". Studies indicate that, at times, sodium emissions are localized at points that correspond to the planet's magnetic poles. This would indicate an interaction between the magnetosphere and the planet's surface.
According to NASA, Mercury is not a suitable planet for Earth-like life. It has a [[exosphere|surface boundary exosphere]] instead of a layered atmosphere, extreme temperatures, and high solar radiation. It is unlikely that any living beings can withstand those conditions. Some parts of the subsurface of Mercury may have been [[Planetary habitability|habitable]], and perhaps [[life form]]s, albeit likely primitive [[microorganism]]s, may have existed on the planet.
Magnetic field and magnetosphere.
[[File:Mercury Magnetic Field NASA.jpg|thumb|Graph showing relative strength of Mercury's magnetic field]]
Despite its small size and slow 59-day-long rotation, Mercury has a significant, and apparently global, [[magnetic field]]. According to measurements taken by "Mariner 10", it is about 1.1% the strength of [[Earth's magnetic field|Earth's]]. The magnetic-field strength at Mercury's equator is about 300 [[Tesla (unit)|nT]]. Like that of Earth, Mercury's magnetic field is [[dipolar]] and nearly aligned with the planet's spin axis (10° dipolar tilt, compared to 11° for Earth). Measurements from both the "Mariner 10" and "MESSENGER" space probes have indicated that the strength and shape of the magnetic field are stable.
It is likely that this magnetic field is generated by a [[Dynamo theory|dynamo]] effect, in a manner similar to the magnetic field of Earth. This dynamo effect would result from the circulation of the planet's iron-rich liquid core. Particularly strong [[tidal heating]] effects caused by the planet's high orbital eccentricity would serve to keep part of the core in the liquid state necessary for this dynamo effect.
Mercury's magnetic field is strong enough to deflect the solar wind around the planet, creating a magnetosphere. The planet's magnetosphere, though small enough to fit within Earth, is strong enough to trap solar wind [[plasma (physics)|plasma]]. This contributes to the space weathering of the planet's surface. Observations taken by the "Mariner 10" spacecraft detected this low energy plasma in the magnetosphere of the planet's nightside. Bursts of energetic particles in the planet's magnetotail indicate a dynamic quality to the planet's magnetosphere.
During its second flyby of the planet on October 6, 2008, "MESSENGER" discovered that Mercury's magnetic field can be extremely "leaky". The spacecraft encountered magnetic "tornadoes"—twisted bundles of magnetic fields connecting the planetary magnetic field to interplanetary space—that were up to 800 km wide or a third of the radius of the planet. These twisted magnetic flux tubes, technically known as [[flux transfer event]]s, form open windows in the planet's magnetic shield through which the solar wind may enter and directly impact Mercury's surface via [[magnetic reconnection]]. This also occurs in Earth's magnetic field. The "MESSENGER" observations showed the reconnection rate was ten times higher at Mercury, but its proximity to the Sun only accounts for about a third of the reconnection rate observed by "MESSENGER".
Orbit, rotation, and longitude.
Mercury has the most [[Orbital eccentricity|eccentric]] orbit of all the planets in the Solar System; its eccentricity is 0.21 with its distance from the Sun ranging from . It takes 87.969 Earth days to complete an orbit. The diagram illustrates the effects of the eccentricity, showing Mercury's orbit overlaid with a circular orbit having the same [[semi-major axis]]. Mercury's higher velocity when it is near perihelion is clear from the greater distance it covers in each 5-day interval. In the diagram, the varying distance of Mercury to the Sun is represented by the size of the planet, which is inversely proportional to Mercury's distance from the Sun.
This varying distance to the Sun leads to Mercury's surface being flexed by [[tidal bulge]]s raised by the [[Sun]] that are about 17 times stronger than the Moon's on Earth. Combined with a 3:2 [[#Spin-orbit resonance|spin–orbit resonance]] of the planet's rotation around its axis, it also results in complex variations of the surface temperature. The resonance makes a single [[solar day]] (the length between two [[meridian (astronomy)|meridian]] transits of the Sun) on Mercury last exactly two Mercury years, or about 176 Earth days.
Mercury's orbit is inclined by 7 degrees to the plane of Earth's orbit (the [[ecliptic]]), the largest of all eight known solar planets. As a result, [[transits of Mercury]] across the face of the Sun can only occur when the planet is crossing the plane of the ecliptic at the time it lies between Earth and the Sun, which is in May or November. This occurs about every seven years on average.
Mercury's [[axial tilt]] is almost zero, with the best measured value as low as 0.027 degrees. This is significantly smaller than that of [[Jupiter]], which has the second smallest axial tilt of all planets at 3.1 degrees. This means that to an observer at Mercury's poles, the center of the Sun never rises more than 2.1 [[arcminutes]] above the horizon. By comparison, the [[angular size]] of the Sun as seen from Mercury ranges from <templatestyles src="Fraction/styles.css" />1+1⁄4 to 2 degrees across.
At certain points on Mercury's surface, an observer would be able to see the Sun peek up a little more than two-thirds of the way over the horizon, then reverse and set before rising again, all within the same [[Extraterrestrial skies#Mercury|Mercurian day]]. This is because approximately four Earth days before perihelion, Mercury's angular [[orbital speed|orbital velocity]] equals its angular [[rotational velocity]] so that the Sun's [[improper motion|apparent motion]] ceases; closer to perihelion, Mercury's angular orbital velocity then exceeds the angular rotational velocity. Thus, to a hypothetical observer on Mercury, the Sun appears to move in a [[apparent retrograde motion|retrograde]] direction. Four Earth days after perihelion, the Sun's normal apparent motion resumes. A similar effect would have occurred if Mercury had been in synchronous rotation: the alternating gain and loss of rotation over a revolution would have caused a libration of 23.65° in longitude.
For the same reason, there are two points on Mercury's equator, 180 degrees apart in [[longitude]], at either of which, around perihelion in alternate Mercurian years (once a Mercurian day), the Sun passes overhead, then reverses its apparent motion and passes overhead again, then reverses a second time and passes overhead a third time, taking a total of about 16 Earth-days for this entire process. In the other alternate Mercurian years, the same thing happens at the other of these two points. The amplitude of the retrograde motion is small, so the overall effect is that, for two or three weeks, the Sun is almost stationary overhead, and is at its most brilliant because Mercury is at perihelion, its closest to the Sun. This prolonged exposure to the Sun at its brightest makes these two points the hottest places on Mercury. Maximum temperature occurs when the Sun is at an angle of about 25 degrees past noon due to [[Diurnal temperature variation#Temperature lag|diurnal temperature lag]], at 0.4 Mercury days and 0.8 Mercury years past sunrise. Conversely, there are two other points on the equator, 90 degrees of longitude apart from the first ones, where the Sun passes overhead only when the planet is at aphelion in alternate years, when the apparent motion of the Sun in Mercury's sky is relatively rapid. These points, which are the ones on the equator where the apparent retrograde motion of the Sun happens when it is crossing the horizon as described in the preceding paragraph, receive much less solar heat than the first ones described above.
Mercury attains an inferior conjunction (nearest approach to Earth) every 116 Earth days on average, but this interval can range from 105 days to 129 days due to the planet's eccentric orbit. Mercury can come as near as to Earth, and that is slowly declining: The next approach to within is in 2679, and to within in 4487, but it will not be closer to Earth than until 28,622. Its period of retrograde motion as seen from Earth can vary from 8 to 15 days on either side of an inferior conjunction. This large range arises from the planet's high orbital eccentricity. Essentially, because Mercury is closest to the Sun, when taking an average over time, Mercury is most often the closest planet to the Earth, and—in that measure—it is the closest planet to each of the other planets in the Solar System.
Longitude convention.
The longitude convention for Mercury puts the zero of longitude at one of the two hottest points on the surface, as described above. However, when this area was first visited, by "Mariner 10", this zero meridian was in darkness, so it was impossible to select a feature on the surface to define the exact position of the meridian. Therefore, a small crater further west was chosen, called [[Hun Kal]], which provides the exact reference point for measuring longitude. The center of Hun Kal defines the 20° west meridian. A 1970 International Astronomical Union resolution suggests that longitudes be measured positively in the westerly direction on Mercury. The two hottest places on the equator are therefore at longitudes 0° W and 180° W, and the coolest points on the equator are at longitudes 90° W and 270° W. However, the "MESSENGER" project uses an east-positive convention.
Spin-orbit resonance.
[[File:Mercury's orbital resonance.svg|thumb|After one orbit, Mercury has rotated 1.5 times, so after two complete orbits the same hemisphere is again illuminated.]]
For many years it was thought that Mercury was synchronously [[tidally locked]] with the Sun, [[rotating]] once for each orbit and always keeping the same face directed towards the Sun, in the same way that the same side of the Moon always faces Earth. Radar observations in 1965 proved that the planet has a 3:2 spin-orbit resonance, rotating three times for every two revolutions around the Sun. The eccentricity of Mercury's orbit makes this resonance stable—at perihelion, when the solar tide is strongest, the Sun is nearly stationary in Mercury's sky.
The 3:2 resonant tidal locking is stabilized by the variance of the tidal force along Mercury's eccentric orbit, acting on a permanent dipole component of Mercury's mass distribution. In a circular orbit there is no such variance, so the only resonance stabilized in such an orbit is at 1:1 (e.g., Earth–Moon), when the tidal force, stretching a body along the "center-body" line, exerts a torque that aligns the body's axis of least inertia (the "longest" axis, and the axis of the aforementioned dipole) to always point at the center. However, with noticeable eccentricity, like that of Mercury's orbit, the tidal force has a maximum at perihelion and therefore stabilizes resonances, like 3:2, ensuring that the planet points its axis of least inertia roughly at the Sun when passing through perihelion.
The original reason astronomers thought it was synchronously locked was that, whenever Mercury was best placed for observation, it was always nearly at the same point in its 3:2 resonance, hence showing the same face. This is because, coincidentally, Mercury's rotation period is almost exactly half of its synodic period with respect to Earth. Due to Mercury's 3:2 spin-orbit resonance, a solar day lasts about 176 Earth days. A [[sidereal day]] (the period of rotation) lasts about 58.7 Earth days.
Simulations indicate that the orbital eccentricity of Mercury varies [[chaos theory|chaotically]] from nearly zero (circular) to more than 0.45 over millions of years due to [[Perturbation (astronomy)|perturbations]] from the other planets. This was thought to explain Mercury's 3:2 spin-orbit resonance (rather than the more usual 1:1), because this state is more likely to arise during a period of high eccentricity. However, accurate modeling based on a realistic model of tidal response has demonstrated that Mercury was captured into the 3:2 spin-orbit state at a very early stage of its history, within 20 (more likely, 10) million years after its formation.
Numerical simulations show that a future [[Secular resonance|secular]] [[Orbital resonance|orbital resonant]] interaction with the perihelion of Jupiter may cause the eccentricity of Mercury's orbit to increase to the point where there is a 1% chance that the orbit will be destabilized in the next five billion years. If this happens, Mercury may fall into the Sun, collide with Venus, be ejected from the Solar System, or even disrupt the rest of the inner Solar System.
Advance of perihelion.
[[File:Drehung der Apsidenlinie light.svg|right|thumb|[[Apsidal precession]] of Mercury's orbit]]
In 1859, the French mathematician and astronomer [[Urbain Le Verrier]] reported that the slow [[precession]] of Mercury's orbit around the Sun could not be completely explained by [[Newtonian mechanics]] and perturbations by the known planets. He suggested, among possible explanations, that another planet (or perhaps instead a series of smaller "corpuscules") might exist in an orbit even closer to the Sun than that of Mercury, to account for this perturbation. Other explanations considered included a slight oblateness of the Sun. The success of the search for [[Neptune]] based on its perturbations of the orbit of [[Uranus]] led astronomers to place faith in this possible explanation, and the hypothetical planet was named [[Vulcan (hypothetical planet)|Vulcan]], but no such planet was ever found.
The observed [[perihelion precession]] of Mercury is 5,600 [[arcseconds]] (1.5556°) per century relative to Earth, or per century relative to the inertial [[International Celestial Reference Frame|ICRF]]. Newtonian mechanics, taking into account all the effects from the other planets and including 0.0254 arcseconds per century due to the oblateness of the Sun, predicts a precession of 5,557 arcseconds (1.5436°) per century relative to Earth, or per century relative to ICRF. In the early 20th century, [[Albert Einstein]]'s [[general theory of relativity]] provided the explanation for the observed precession, by formalizing gravitation as being mediated by the curvature of spacetime. The effect is small: just per century (or 0.43 arcsecond per year, or 0.1035 arcsecond per orbital period) for Mercury; it therefore requires a little over 12.5 million orbits, or 3 million years, for a full excess turn. Similar, but much smaller, effects exist for other Solar System bodies: 8.6247 arcseconds per century for Venus, 3.8387 for Earth, 1.351 for Mars, and 10.05 for [[1566 Icarus]].
Observation.
[[File:Mercury.jpg|thumb|Image mosaic by "Mariner 10", 1974]]
Mercury's [[apparent magnitude]] is calculated to vary between −2.48 (brighter than [[Sirius]]) around [[superior conjunction]] and +7.25 (below the limit of naked-eye visibility) around [[inferior conjunction]]. The mean apparent magnitude is 0.23 while the standard deviation of 1.78 is the largest of any planet. The mean apparent magnitude at superior conjunction is −1.89 while that at inferior conjunction is +5.93. Observation of Mercury is complicated by its proximity to the Sun, as it is lost in the Sun's glare for much of the time. Mercury can be observed for only a brief period during either morning or evening twilight.
Ground-based telescope observations of Mercury reveal only an illuminated partial disk with limited detail. The [[Hubble Space Telescope]] cannot observe Mercury at all, due to safety procedures that prevent its pointing too close to the Sun. Because the shift of 0.15 revolutions of Earth in a Mercurian year makes up a seven-Mercurian-year cycle (0.15 × 7 ≈ 1.0), in the seventh Mercurian year, Mercury follows almost exactly (earlier by 7 days) the sequence of phenomena it showed seven Mercurian years before.
Like the Moon and Venus, Mercury exhibits [[Lunar phase|phases]] as seen from Earth. It is "new" at [[inferior conjunction]] and "full" at superior conjunction. The planet is rendered invisible from Earth on both of these occasions because of its being obscured by the Sun, except at its new phase during a transit. Mercury is technically brightest as seen from Earth when it is at a full phase. Although Mercury is farthest from Earth when it is full, the greater illuminated area that is visible and the [[opposition effect|opposition brightness surge]] more than compensates for the distance. The opposite is true for Venus, which appears brightest when it is a [[crescent]], because it is much closer to Earth than when [[gibbous]].
[[File:PIA19247-Mercury-NPolarRegion-Messenger20150316.jpg|thumb|left|False-color map showing the maximum temperatures of the north polar region]]
[[File:Mercury Venus Moon over San Jose 08 Jan 2024.png|thumb|right|Mercury (lower left) as seen from [[San Jose, California]] with Venus and the Moon.]]
Mercury is best observed at the first and last quarter, although they are phases of lesser brightness. The first and last quarter phases occur at greatest [[elongation (astronomy)|elongation]] east and west of the Sun, respectively. At both of these times, Mercury's separation from the Sun ranges anywhere from 17.9° at perihelion to 27.8° at aphelion. At greatest "western" elongation, Mercury rises at its earliest before sunrise, and at greatest "eastern" elongation, it sets at its latest after sunset.
[[File:PIA19422-Mercury-CarnegieRupes-MDIS-MLA-20150416.jpg|thumb|right|False-color image of [[Carnegie Rupes]], a tectonic landform—high terrain (red); low (blue).]]
Mercury is more often and easily visible from the [[Southern Hemisphere]] than from the [[Northern Hemisphere|Northern]]. This is because Mercury's maximum western elongation occurs only during early autumn in the Southern Hemisphere, whereas its greatest eastern elongation happens only during late winter in the Southern Hemisphere. In both of these cases, the angle at which the planet's orbit intersects the horizon is maximized, allowing it to rise several hours before sunrise in the former instance and not set until several hours after sundown in the latter from southern mid-latitudes, such as Argentina and South Africa.
An alternate method for viewing Mercury involves observing the planet with a [[telescope]] during daylight hours when conditions are clear, ideally when it is at its greatest elongation. This allows the planet to be found easily, even when using telescopes with apertures. However, great care must be taken to obstruct the Sun from sight because of the extreme risk for eye damage. This method bypasses the limitation of twilight observing when the ecliptic is located at a low elevation (e.g. on autumn evenings). The planet is higher in the sky and less atmospheric effects affect the view of the planet. Mercury can be viewed as close as 4° to the Sun near superior conjunction when it is almost at its brightest.
Mercury can, like several other planets and the brightest stars, be seen during a total [[solar eclipse]].
Observation history.
Ancient astronomers.
[[File:Mercury-bonatti.png|thumb|Mercury, from "Liber astronomiae", 1550]]
The earliest known recorded observations of Mercury are from the [[MUL.APIN]] tablets. These observations were most likely made by an [[Assyria]]n astronomer around the 14th century BC. The [[cuneiform]] name used to designate Mercury on the MUL.APIN tablets is transcribed as UDU.IDIM.GU\U4.UD ("the jumping planet"). [[Babylonian astronomy|Babylonian records]] of Mercury date back to the 1st millennium BC. The [[Babylonia]]ns called the planet [[Nabu]] after the messenger to the gods in [[Babylonian mythology|their mythology]].
The [[Greeks in Egypt|Greco]]-[[Ancient Egypt|Egyptian]] astronomer [[Ptolemy]] wrote about the possibility of planetary transits across the face of the Sun in his work "Planetary Hypotheses". He suggested that no transits had been observed either because planets such as Mercury were too small to see, or because transits were too infrequent.
[[File:Shatir500.jpg|thumb|left|[[Ibn al-Shatir]]'s model for the appearances of Mercury, showing the multiplication of [[epicycles]] using the [[Tusi couple]], thus eliminating the Ptolemaic eccentrics and [[equant]].]]
In [[ancient China]], Mercury was known as "the Hour Star" ("Chen-xing" ). It was associated with the direction north and the phase of water in the [[Five Phases]] system of metaphysics. Modern [[Chinese culture|Chinese]], [[Korean culture|Korean]], [[Japanese culture|Japanese]] and [[Vietnamese culture|Vietnamese]] cultures refer to the planet literally as the "water star" (), based on the [[Five elements (Chinese philosophy)|Five elements]]. [[Hindu mythology]] used the name [[Budha]] for Mercury, and this god was thought to preside over Wednesday. The god [[Odin]] (or Woden) of [[Germanic paganism]] was associated with the planet Mercury and Wednesday. The [[Maya civilization|Maya]] may have represented Mercury as an owl (or possibly four owls; two for the morning aspect and two for the evening) that served as a messenger to the [[underworld]]. Mercury was sometimes known as [[Stilbon (mythology)|Stilbon]] ([[Greek language|Greek]]: Στίλβων) meaning 'the shining, glittering'.
In [[medieval Islamic astronomy]], the [[Al-Andalus|Andalusian]] astronomer [[Abū Ishāq Ibrāhīm al-Zarqālī]] in the 11th century described the [[deferent]] of Mercury's [[geocentric orbit]] as being oval, like an egg or a [[Pine nut|pignon]], although this insight did not influence his astronomical theory or his astronomical calculations. In the 12th century, [[Ibn Bajjah]] observed "two planets as black spots on the face of the Sun", which was later suggested as the transit of Mercury and/or Venus by the [[Maragheh observatory|Maragha]] astronomer [[Qotb al-Din Shirazi]] in the 13th century. Most such medieval reports of transits were later taken as observations of [[sunspot]]s.
In India, the [[Kerala school of astronomy and mathematics|Kerala school]] astronomer [[Nilakantha Somayaji]] in the 15th century developed a partially heliocentric planetary model in which Mercury orbits the Sun, which in turn orbits Earth, similar to the [[Tychonic system]] later proposed by [[Tycho Brahe]] in the late 16th century.
Ground-based telescopic research.
The first telescopic observations of Mercury were made by [[Thomas Harriot]] and [[Galileo]] from 1610. In 1612, [[Simon Marius]] observed the brightness of Mercury varied with the planet's orbital position and concluded it had phases "in the same way as Venus and the Moon". In 1631, [[Pierre Gassendi]] made the first telescopic observations of the transit of a planet across the Sun when he saw a transit of Mercury predicted by [[Johannes Kepler]]. In 1639, [[Giovanni Zupi]] used a telescope to discover that the planet had orbital phases similar to Venus and the Moon. The observation demonstrated conclusively that Mercury orbited the Sun.
A rare event in astronomy is the passage of one planet in front of another ([[occultation]]), as seen from Earth. Mercury and Venus occult each other every few centuries, and the event of May 28, 1737, is the only one historically observed, having been seen by [[John Bevis]] at the [[Royal Greenwich Observatory]]. The next occultation of Mercury by Venus will be on December 3, 2133.
The difficulties inherent in observing Mercury meant that it was far less studied than the other planets. In 1800, [[Johann Schröter]] made observations of surface features, claiming to have observed mountains. [[Friedrich Bessel]] used Schröter's drawings to erroneously estimate the rotation period as 24 hours and an axial tilt of 70°. In the 1880s, [[Giovanni Schiaparelli]] mapped the planet more accurately, and suggested that Mercury's rotational period was 88 days, the same as its orbital period due to tidal locking. This phenomenon is known as [[synchronous rotation]]. The effort to map the surface of Mercury was continued by [[Eugenios Antoniadi]], who published a book in 1934 that included both maps and his own observations. Many of the planet's surface features, particularly the [[List of albedo features on Mercury|albedo features]], take their names from Antoniadi's map.
In June 1962, Soviet scientists at the [[Institute of Radio-engineering and Electronics]] of the [[USSR Academy of Sciences]], led by [[Vladimir Kotelnikov]], became the first to bounce a radar signal off Mercury and receive it, starting radar observations of the planet. Three years later, radar observations by Americans [[Gordon H. Pettengill]] and Rolf B. Dyce, using the [[Arecibo radio telescope]] in [[Puerto Rico]], showed conclusively that the planet's rotational period was about 59 days. The theory that Mercury's rotation was synchronous had become widely held, and it was a surprise to astronomers when these radio observations were announced. If Mercury were tidally locked, its dark face would be extremely cold, but measurements of radio emission revealed that it was much hotter than expected. Astronomers were reluctant to drop the synchronous rotation theory and proposed alternative mechanisms such as powerful heat-distributing winds to explain the observations.
In 1965, Italian astronomer [[Giuseppe Colombo]] noted that the rotation value was about two-thirds of Mercury's orbital period, and proposed that the planet's orbital and rotational periods were locked into a 3:2 rather than a 1:1 resonance. Data from "Mariner 10" subsequently confirmed this view. This means that Schiaparelli's and Antoniadi's maps were not "wrong". Instead, the astronomers saw the same features during every "second" orbit and recorded them, but disregarded those seen in the meantime, when Mercury's other face was toward the Sun, because the orbital geometry meant that these observations were made under poor viewing conditions.
Ground-based optical observations did not shed much further light on Mercury, but radio astronomers using interferometry at microwave wavelengths, a technique that enables removal of the solar radiation, were able to discern physical and chemical characteristics of the subsurface layers to a depth of several meters. Not until the first space probe flew past Mercury did many of its most fundamental morphological properties become known. Moreover, technological advances have led to improved ground-based observations. In 2000, high-resolution [[lucky imaging]] observations were conducted by the [[Mount Wilson Observatory]] Hale telescope. They provided the first views that resolved surface features on the parts of Mercury that were not imaged in the "Mariner 10" mission. Most of the planet has been mapped by the Arecibo radar telescope, with resolution, including polar deposits in shadowed craters of what may be water ice.
Research with space probes.
[[File:MESSENGER Assembly.jpg|thumb|left|"MESSENGER" being prepared for launch]]
[[File:PIA18389-MarsCuriosityRover-MercuryTransitsSun-20140603.gif|thumb|Mercury transiting the [[Sun]] as viewed by the Mars rover [[Curiosity (rover)|"Curiosity"]] (June 3, 2014).]]
Reaching Mercury from Earth poses significant technical challenges, because it orbits so much closer to the Sun than Earth. A Mercury-bound spacecraft launched from Earth must travel over into the Sun's gravitational [[potential well]]. Mercury has an [[orbital speed]] of , whereas Earth's orbital speed is . Therefore, the spacecraft must make a larger change in [[velocity]] ([[delta-v]]) to get to Mercury and then enter orbit, as compared to the delta-v required for, say, [[Exploration of Mars|Mars planetary missions]].
The [[potential energy]] liberated by moving down the Sun's potential well becomes [[kinetic energy]], requiring a delta-v change to do anything other than pass by Mercury. Some portion of this [[delta-v budget]] can be provided from a [[gravity assist]] during one or more fly-bys of Venus. To land safely or enter a stable orbit the spacecraft would rely entirely on rocket motors. [[Aerobraking]] is ruled out because Mercury has a negligible atmosphere. A trip to Mercury requires more rocket fuel than that required to [[escape velocity|escape]] the Solar System completely. As a result, only three space probes have visited it so far. A proposed alternative approach would use a [[solar sail]] to attain a Mercury-synchronous orbit around the Sun.
"Mariner 10".
[[File:Mariner 10.jpg|thumb|"Mariner 10", the first probe to visit Mercury]]
The first spacecraft to visit Mercury was NASA's "Mariner 10" (1974–1975). The spacecraft used the gravity of Venus to adjust its orbital velocity so that it could approach Mercury, making it both the first spacecraft to use this gravitational "slingshot" effect and the first NASA mission to visit multiple planets. "Mariner 10" provided the first close-up images of Mercury's surface, which immediately showed its heavily cratered nature, and revealed many other types of geological features, such as the giant scarps that were later ascribed to the effect of the planet shrinking slightly as its iron core cools. Unfortunately, the same face of the planet was lit at each of "Mariner 10"'s close approaches. This made close observation of both sides of the planet impossible, and resulted in the mapping of less than 45% of the planet's surface.
The spacecraft made three close approaches to Mercury, the closest of which took it to within of the surface. At the first close approach, instruments detected a magnetic field, to the great surprise of planetary geologists—Mercury's rotation was expected to be much too slow to generate a significant [[dynamo]] effect. The second close approach was primarily used for imaging, but at the third approach, extensive magnetic data were obtained. The data revealed that the planet's magnetic field is much like Earth's, which deflects the solar wind around the planet. For many years after the "Mariner 10" encounters, the origin of Mercury's magnetic field remained the subject of several competing theories.
On March 24, 1975, just eight days after its final close approach, "Mariner 10" ran out of fuel. Because its orbit could no longer be accurately controlled, mission controllers instructed the probe to shut down. "Mariner 10" is thought to be still orbiting the Sun, passing close to Mercury every few months.
"MESSENGER".
[[File:Details of MESSENGER's Impact Location.jpg|thumb|Estimated details of the impact of "MESSENGER" on April 30, 2015]]
A second NASA mission to Mercury, named "MESSENGER" (MErcury Surface, Space ENvironment, GEochemistry, and Ranging), was launched on August 3, 2004. It made a fly-by of Earth in August 2005, and of Venus in October 2006 and June 2007 to place it onto the correct trajectory to reach an orbit around Mercury. A first fly-by of Mercury occurred on January 14, 2008, a second on October 6, 2008, and a third on September 29, 2009. Most of the hemisphere not imaged by "Mariner 10" was mapped during these fly-bys. The probe successfully entered an elliptical orbit around the planet on March 18, 2011. The first orbital image of Mercury was obtained on March 29, 2011. The probe finished a one-year mapping mission, and then entered a one-year extended mission into 2013. In addition to continued observations and mapping of Mercury, "MESSENGER" observed the 2012 [[solar maximum]].
[[File:MercuryTopo.png|thumb|right|Topography of Mercury based on MDIS (Mercury Dual Imaging System) data]] The mission was designed to clear up six key issues: Mercury's high density, its geological history, the nature of its magnetic field, the structure of its core, whether it has ice at its poles, and where its tenuous atmosphere comes from. To this end, the probe carried imaging devices that gathered much-higher-resolution images of much more of Mercury than "Mariner 10", assorted [[spectrometer]]s to determine the abundances of elements in the crust, and [[magnetometer]]s and devices to measure velocities of charged particles. Measurements of changes in the probe's orbital velocity were expected to be used to infer details of the planet's interior structure. "MESSENGER"'s final maneuver was on April 24, 2015, and it crashed into Mercury's surface on April 30, 2015. The spacecraft's impact with Mercury occurred at 3:26:01 p.m. [[Eastern Time Zone|EDT]] on April 30, 2015, leaving a crater estimated to be in diameter.
"BepiColombo".
The [[European Space Agency]] and the [[Japanese Space Agency]] developed and launched a joint mission called "BepiColombo", which will orbit Mercury with two probes: one to map the planet and the other to study its magnetosphere. Launched on October 20, 2018, "BepiColombo" is expected to reach Mercury in 2025. It will release a magnetometer probe into an elliptical orbit, then chemical rockets will fire to deposit the mapper probe into a circular orbit. Both probes will operate for one terrestrial year. The mapper probe carries an array of spectrometers similar to those on "MESSENGER", and will study the planet at many different wavelengths including [[infrared]], [[ultraviolet]], [[X-ray]] and [[gamma ray]]. "BepiColombo" conducted the first of its six planned Mercury flybys on October 1, 2021, and the sixth was completed on January 9, 2025. The spacecraft will enter the planet's orbit in 2026.
"Perseverance" rover.
On March 5, 2024, NASA released images of transits of the moon [[Deimos (moon)|Deimos]], the moon [[Phobos (moon)|Phobos]] and the planet Mercury as viewed by the "Perseverance" rover on the planet Mars.
See also.
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Notes.
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References.
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External links.
[[Category:Mercury (planet)| ]]
[[Category:Planets of the Solar System]]
[[Category:Terrestrial planets]]
[[Category:Astronomical objects known since antiquity]]
[[Category:Solar System]]

Force
Influence that can change motion of an object
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A force is an influence that can cause an object to change its velocity unless counterbalanced by other forces. The concept of force makes the everyday notion of pushing or pulling mathematically precise. Because the magnitude and direction of a force are both important, force is a vector quantity. The SI unit of force is the newton (N), and force is often represented by the symbol F.
Force plays an important role in classical mechanics. The concept of force is central to all three of Newton's laws of motion. Types of forces often encountered in classical mechanics include elastic, frictional, contact or "normal" forces, and gravitational. The rotational version of force is torque, which produces changes in the rotational speed of an object. In an extended body, each part applies forces on the adjacent parts; the distribution of such forces through the body is the internal mechanical stress. In the case of multiple forces, if the net force on an extended body is zero the body is in equilibrium.
In modern physics, which includes relativity and quantum mechanics, the laws governing motion are revised to rely on fundamental interactions as the ultimate origin of force. However, the understanding of force provided by classical mechanics is useful for practical purposes.
Development of the concept.
Philosophers in antiquity used the concept of force in the study of stationary and moving objects and simple machines, but thinkers such as Aristotle and Archimedes retained fundamental errors in understanding force. In part, this was due to an incomplete understanding of the sometimes non-obvious force of friction and a consequently inadequate view of the nature of natural motion. A fundamental error was the belief that a force is required to maintain motion, even at a constant velocity. Most of the previous misunderstandings about motion and force were eventually corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Newton formulated laws of motion that were not improved for over two hundred years.
By the early 20th century, Einstein developed a theory of relativity that correctly predicted the action of forces on objects with increasing momenta near the speed of light and also provided insight into the forces produced by gravitation and inertia. With modern insights into quantum mechanics and technology that can accelerate particles close to the speed of light, particle physics has devised a Standard Model to describe forces between particles smaller than atoms. The Standard Model predicts that exchanged particles called gauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known: in order of decreasing strength, they are: strong, electromagnetic, weak, and gravitational. High-energy particle physics observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction.
Pre-Newtonian concepts.
Since antiquity the concept of force has been recognized as integral to the functioning of each of the simple machines. The mechanical advantage given by a simple machine allowed for less force to be used in exchange for that force acting over a greater distance for the same amount of work. Analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was especially famous for formulating a treatment of buoyant forces inherent in fluids.
Aristotle provided a philosophical discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotle's view, the terrestrial sphere contained four elements that come to rest at different "natural places" therein. Aristotle believed that motionless objects on Earth, those composed mostly of the elements earth and water, were in their natural place when on the ground, and that they stay that way if left alone. He distinguished between the innate tendency of objects to find their "natural place" (e.g., for heavy bodies to fall), which led to "natural motion", and unnatural or forced motion, which required continued application of a force. This theory, based on the everyday experience of how objects move, such as the constant application of a force needed to keep a cart moving, had conceptual trouble accounting for the behavior of projectiles, such as the flight of arrows. An archer causes the arrow to move at the start of the flight, and it then sails through the air even though no discernible efficient cause acts upon it. Aristotle was aware of this problem and proposed that the air displaced through the projectile's path carries the projectile to its target. This explanation requires a continuous medium such as air to sustain the motion.
Though Aristotelian physics was criticized as early as the 6th century, its shortcomings would not be corrected until the 17th century work of Galileo Galilei, who was influenced by the late medieval idea that objects in forced motion carried an innate force of impetus. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the Aristotelian theory of motion. He showed that the bodies were accelerated by gravity to an extent that was independent of their mass and argued that objects retain their velocity unless acted on by a force, for example friction. Galileo's idea that force is needed to change motion rather than to sustain it, further improved upon by Isaac Beeckman, René Descartes, and Pierre Gassendi, became a key principle of Newtonian physics.
In the early 17th century, before Newton's "Principia", the term "force" () was applied to many physical and non-physical phenomena, e.g., for an acceleration of a point. The product of a point mass and the square of its velocity was named (live force) by Leibniz. The modern concept of force corresponds to Newton's (accelerating force).
Newtonian mechanics.
Sir Isaac Newton described the motion of all objects using the concepts of inertia and force. In 1687, Newton published his magnum opus, "Philosophiæ Naturalis Principia Mathematica". In this work Newton set out three laws of motion that have dominated the way forces are described in physics to this day. The precise ways in which Newton's laws are expressed have evolved in step with new mathematical approaches.
First law.
Newton's first law of motion states that the natural behavior of an object at rest is to continue being at rest, and the natural behavior of an object moving at constant speed in a straight line is to continue moving at that constant speed along that straight line. The latter follows from the former because of the principle that the laws of physics are the same for all inertial observers, i.e., all observers who do not feel themselves to be in motion. An observer moving in tandem with an object will see it as being at rest. So, its natural behavior will be to remain at rest with respect to that observer, which means that an observer who sees it moving at constant speed in a straight line will see it continuing to do so.
Second law.
According to the first law, motion at constant speed in a straight line does not need a cause. It is "change" in motion that requires a cause, and Newton's second law gives the quantitative relationship between force and change of motion.
Newton's second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. If the mass of the object is constant, this law implies that the acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the mass of the object.
A modern statement of Newton's second law is a vector equation:
formula_1
where formula_2 is the momentum of the system, and formula_3 is the net (vector sum) force.399 If a body is in equilibrium, there is zero "net" force by definition (balanced forces may be present nevertheless). In contrast, the second law states that if there is an "unbalanced" force acting on an object it will result in the object's momentum changing over time.
In common engineering applications the mass in a system remains constant allowing as simple algebraic form for the second law. By the definition of momentum,
formula_4
where "m" is the mass and formula_5 is the velocity. If Newton's second law is applied to a system of constant mass, "m" may be moved outside the derivative operator. The equation then becomes
formula_6
By substituting the definition of acceleration, the algebraic version of Newton's second law is derived:
formula_7
Third law.
Whenever one body exerts a force on another, the latter simultaneously exerts an equal and opposite force on the first. In vector form, if formula_8 is the force of body 1 on body 2 and formula_9 that of body 2 on body 1, then
formula_10
This law is sometimes referred to as the "action-reaction law", with formula_11 called the "action" and formula_12 the "reaction".
Newton's Third Law is a result of applying symmetry to situations where forces can be attributed to the presence of different objects. The third law means that all forces are "interactions" between different bodies. and thus that there is no such thing as a unidirectional force or a force that acts on only one body.
In a system composed of object 1 and object 2, the net force on the system due to their mutual interactions is zero:
formula_13
More generally, in a closed system of particles, all internal forces are balanced. The particles may accelerate with respect to each other but the center of mass of the system will not accelerate. If an external force acts on the system, it will make the center of mass accelerate in proportion to the magnitude of the external force divided by the mass of the system.
Combining Newton's Second and Third Laws, it is possible to show that the linear momentum of a system is conserved in any closed system. In a system of two particles, if formula_14 is the momentum of object 1 and formula_15 the momentum of object 2, then
formula_16
Using similar arguments, this can be generalized to a system with an arbitrary number of particles. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define a system such that net momentum is never lost nor gained.ch.12
Defining "force".
Some textbooks use Newton's second law as a "definition" of force. However, for the equation formula_17 for a constant mass formula_18 to then have any predictive content, it must be combined with further information. Moreover, inferring that a force is present because a body is accelerating is only valid in an inertial frame of reference. The question of which aspects of Newton's laws to take as definitions and which to regard as holding physical content has been answered in various ways, which ultimately do not affect how the theory is used in practice. Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of force include Ernst Mach and Walter Noll.
Combining forces.
Forces act in a particular direction and have sizes dependent upon how strong the push or pull is. Because of these characteristics, forces are classified as "vector quantities". This means that forces follow a different set of mathematical rules than physical quantities that do not have direction (denoted scalar quantities). For example, when determining what happens when two forces act on the same object, it is necessary to know both the magnitude and the direction of both forces to calculate the result. If both of these pieces of information are not known for each force, the situation is ambiguous.
Historically, forces were first quantitatively investigated in conditions of static equilibrium where several forces canceled each other out. Such experiments demonstrate the crucial properties that forces are additive vector quantities: they have magnitude and direction. When two forces act on a point particle, the resulting force, the "resultant" (also called the "net force"), can be determined by following the parallelogram rule of vector addition: the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector that is equal in magnitude and direction to the transversal of the parallelogram. The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action.ch.12
Free-body diagrams can be used as a convenient way to keep track of forces acting on a system. Ideally, these diagrams are drawn with the angles and relative magnitudes of the force vectors preserved so that graphical vector addition can be done to determine the net force.
As well as being added, forces can also be resolved into independent components at right angles to each other. A horizontal force pointing northeast can therefore be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Resolving force vectors into components of a set of basis vectors is often a more mathematically clean way to describe forces than using magnitudes and directions. This is because, for orthogonal components, the components of the vector sum are uniquely determined by the scalar addition of the components of the individual vectors. Orthogonal components are independent of each other because forces acting at ninety degrees to each other have no effect on the magnitude or direction of the other. Choosing a set of orthogonal basis vectors is often done by considering what set of basis vectors will make the mathematics most convenient. Choosing a basis vector that is in the same direction as one of the forces is desirable, since that force would then have only one non-zero component. Orthogonal force vectors can be three-dimensional with the third component being at right angles to the other two.ch.12
Equilibrium.
When all the forces that act upon an object are balanced, then the object is said to be in a state of equilibrium. Hence, equilibrium occurs when the resultant force acting on a point particle is zero (that is, the vector sum of all forces is zero). When dealing with an extended body, it is also necessary that the net torque be zero. A body is in "static equilibrium" with respect to a frame of reference if it at rest and not accelerating, whereas a body in "dynamic equilibrium" is moving at a constant speed in a straight line, i.e., moving but not accelerating. What one observer sees as static equilibrium, another can see as dynamic equilibrium and vice versa.
Static.
Static equilibrium was understood well before the invention of classical mechanics. Objects that are not accelerating have zero net force acting on them.
The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, an object on a level surface is pulled (attracted) downward toward the center of the Earth by the force of gravity. At the same time, a force is applied by the surface that resists the downward force with equal upward force (called a normal force). The situation produces zero net force and hence no acceleration.
Pushing against an object that rests on a frictional surface can result in a situation where the object does not move because the applied force is opposed by static friction, generated between the object and the table surface. For a situation with no movement, the static friction force "exactly" balances the applied force resulting in no acceleration. The static friction increases or decreases in response to the applied force up to an upper limit determined by the characteristics of the contact between the surface and the object.
A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as weighing scales and spring balances. For example, an object suspended on a vertical spring scale experiences the force of gravity acting on the object balanced by a force applied by the "spring reaction force", which equals the object's weight. Using such tools, some quantitative force laws were discovered: that the force of gravity is proportional to volume for objects of constant density (widely exploited for millennia to define standard weights); Archimedes' principle for buoyancy; Archimedes' analysis of the lever; Boyle's law for gas pressure; and Hooke's law for springs. These were all formulated and experimentally verified before Isaac Newton expounded his three laws of motion.ch.12
Dynamic.
Dynamic equilibrium was first described by Galileo who noticed that certain assumptions of Aristotelian physics were contradicted by observations and logic. Galileo realized that simple velocity addition demands that the concept of an "absolute rest frame" did not exist. Galileo concluded that motion in a constant velocity was completely equivalent to rest. This was contrary to Aristotle's notion of a "natural state" of rest that objects with mass naturally approached. Simple experiments showed that Galileo's understanding of the equivalence of constant velocity and rest were correct. For example, if a mariner dropped a cannonball from the crow's nest of a ship moving at a constant velocity, Aristotelian physics would have the cannonball fall straight down while the ship moved beneath it. Thus, in an Aristotelian universe, the falling cannonball would land behind the foot of the mast of a moving ship. When this experiment is actually conducted, the cannonball always falls at the foot of the mast, as if the cannonball knows to travel with the ship despite being separated from it. Since there is no forward horizontal force being applied on the cannonball as it falls, the only conclusion left is that the cannonball continues to move with the same velocity as the boat as it falls. Thus, no force is required to keep the cannonball moving at the constant forward velocity.
Moreover, any object traveling at a constant velocity must be subject to zero net force (resultant force). This is the definition of dynamic equilibrium: when all the forces on an object balance but it still moves at a constant velocity. A simple case of dynamic equilibrium occurs in constant velocity motion across a surface with kinetic friction. In such a situation, a force is applied in the direction of motion while the kinetic friction force exactly opposes the applied force. This results in zero net force, but since the object started with a non-zero velocity, it continues to move with a non-zero velocity. Aristotle misinterpreted this motion as being caused by the applied force. When kinetic friction is taken into consideration it is clear that there is no net force causing constant velocity motion.ch.12
Examples of forces in classical mechanics.
Some forces are consequences of the fundamental ones. In such situations, idealized models can be used to gain physical insight. For example, each solid object is considered a rigid body.
Gravitational force or Gravity.
What we now call gravity was not identified as a universal force until the work of Isaac Newton. Before Newton, the tendency for objects to fall towards the Earth was not understood to be related to the motions of celestial objects. Galileo was instrumental in describing the characteristics of falling objects by determining that the acceleration of every object in free-fall was constant and independent of the mass of the object. Today, this acceleration due to gravity towards the surface of the Earth is usually designated as formula_19 and has a magnitude of about 9.81 meters per second squared (this measurement is taken from sea level and may vary depending on location), and points toward the center of the Earth. This observation means that the force of gravity on an object at the Earth's surface is directly proportional to the object's mass. Thus an object that has a mass of formula_18 will experience a force:
formula_21
For an object in free-fall, this force is unopposed and the net force on the object is its weight. For objects not in free-fall, the force of gravity is opposed by the reaction forces applied by their supports. For example, a person standing on the ground experiences zero net force, since a normal force (a reaction force) is exerted by the ground upward on the person that counterbalances his weight that is directed downward.ch.12
Newton's contribution to gravitational theory was to unify the motions of heavenly bodies, which Aristotle had assumed were in a natural state of constant motion, with falling motion observed on the Earth. He proposed a law of gravity that could account for the celestial motions that had been described earlier using Kepler's laws of planetary motion.
Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that the acceleration of the Moon around the Earth could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an inverse square law. Further, Newton realized that the acceleration of a body due to gravity is proportional to the mass of the other attracting body. Combining these ideas gives a formula that relates the mass (formula_22) and the radius (formula_23) of the Earth to the gravitational acceleration:
formula_24
where formula_25 is the distance between the two objects' centers of mass and formula_26 is the unit vector pointed in the direction away from the center of the first object toward the center of the second object.
This formula was powerful enough to stand as the basis for all subsequent descriptions of motion within the Solar System until the 20th century. During that time, sophisticated methods of perturbation analysis were invented to calculate the deviations of orbits due to the influence of multiple bodies on a planet, moon, comet, or asteroid. The formalism was exact enough to allow mathematicians to predict the existence of the planet Neptune before it was observed.
Electromagnetic.
The electrostatic force was first described in 1784 by Coulomb as a force that existed intrinsically between two charges. The properties of the electrostatic force were that it varied as an inverse square law directed in the radial direction, was both attractive and repulsive (there was intrinsic polarity), was independent of the mass of the charged objects, and followed the superposition principle. Coulomb's law unifies all these observations into one succinct statement.
Subsequent mathematicians and physicists found the construct of the "electric field" to be useful for determining the electrostatic force on an electric charge at any point in space. The electric field was based on using a hypothetical "test charge" anywhere in space and then using Coulomb's Law to determine the electrostatic force. Thus the electric field anywhere in space is defined as
formula_27
where formula_28 is the magnitude of the hypothetical test charge. Similarly, the idea of the "magnetic field" was introduced to express how magnets can influence one another at a distance. The Lorentz force law gives the force upon a body with charge formula_28 due to electric and magnetic fields:
formula_30
where formula_3 is the electromagnetic force, formula_32 is the electric field at the body's location, formula_33 is the magnetic field, and formula_5 is the velocity of the particle. The magnetic contribution to the Lorentz force is the cross product of the velocity vector with the magnetic field.
The origin of electric and magnetic fields would not be fully explained until 1864 when James Clerk Maxwell unified a number of earlier theories into a set of 20 scalar equations, which were later reformulated into 4 vector equations by Oliver Heaviside and Josiah Willard Gibbs. These "Maxwell's equations" fully described the sources of the fields as being stationary and moving charges, and the interactions of the fields themselves. This led Maxwell to discover that electric and magnetic fields could be "self-generating" through a wave that traveled at a speed that he calculated to be the speed of light. This insight united the nascent fields of electromagnetic theory with optics and led directly to a complete description of the electromagnetic spectrum.
Normal.
When objects are in contact, the force directly between them is called the normal force, the component of the total force in the system exerted normal to the interface between the objects. The normal force is closely related to Newton's third law. The normal force, for example, is responsible for the structural integrity of tables and floors as well as being the force that responds whenever an external force pushes on a solid object. An example of the normal force in action is the impact force on an object crashing into an immobile surface.ch.12
Friction.
Friction is a force that opposes relative motion of two bodies. At the macroscopic scale, the frictional force is directly related to the normal force at the point of contact. There are two broad classifications of frictional forces: static friction and kinetic friction.
The static friction force (formula_35) will exactly oppose forces applied to an object parallel to a surface up to the limit specified by the coefficient of static friction (formula_36) multiplied by the normal force (formula_37). In other words, the magnitude of the static friction force satisfies the inequality:
formula_38
The kinetic friction force (formula_39) is typically independent of both the forces applied and the movement of the object. Thus, the magnitude of the force equals:
formula_40
where formula_41 is the coefficient of kinetic friction. The coefficient of kinetic friction is normally less than the coefficient of static friction.
Tension.
Tension forces can be modeled using ideal strings that are massless, frictionless, unbreakable, and do not stretch. They can be combined with ideal pulleys, which allow ideal strings to switch physical direction. Ideal strings transmit tension forces instantaneously in action–reaction pairs so that if two objects are connected by an ideal string, any force directed along the string by the first object is accompanied by a force directed along the string in the opposite direction by the second object. By connecting the same string multiple times to the same object through the use of a configuration that uses movable pulleys, the tension force on a load can be multiplied. For every string that acts on a load, another factor of the tension force in the string acts on the load. Such machines allow a mechanical advantage for a corresponding increase in the length of displaced string needed to move the load. These tandem effects result ultimately in the conservation of mechanical energy since the work done on the load is the same no matter how complicated the machine.ch.12
Spring.
A simple elastic force acts to return a spring to its natural length. An ideal spring is taken to be massless, frictionless, unbreakable, and infinitely stretchable. Such springs exert forces that push when contracted, or pull when extended, in proportion to the displacement of the spring from its equilibrium position. This linear relationship was described by Robert Hooke in 1676, for whom Hooke's law is named. If formula_42 is the displacement, the force exerted by an ideal spring equals:
formula_43
where formula_44 is the spring constant (or force constant), which is particular to the spring. The minus sign accounts for the tendency of the force to act in opposition to the applied load.ch.12
Centripetal.
For an object in uniform circular motion, the net force acting on the object equals:
formula_45
where formula_18 is the mass of the object, formula_47 is the velocity of the object and formula_25 is the distance to the center of the circular path and formula_49 is the unit vector pointing in the radial direction outwards from the center. This means that the net force felt by the object is always directed toward the center of the curving path. Such forces act perpendicular to the velocity vector associated with the motion of an object, and therefore do not change the speed of the object (magnitude of the velocity), but only the direction of the velocity vector. More generally, the net force that accelerates an object can be resolved into a component that is perpendicular to the path, and one that is tangential to the path. This yields both the tangential force, which accelerates the object by either slowing it down or speeding it up, and the radial (centripetal) force, which changes its direction.ch.12
Continuum mechanics.
Newton's laws and Newtonian mechanics in general were first developed to describe how forces affect idealized point particles rather than three-dimensional objects. In real life, matter has extended structure and forces that act on one part of an object might affect other parts of an object. For situations where lattice holding together the atoms in an object is able to flow, contract, expand, or otherwise change shape, the theories of continuum mechanics describe the way forces affect the material. For example, in extended fluids, differences in pressure result in forces being directed along the pressure gradients as follows:
formula_50
where formula_51 is the volume of the object in the fluid and formula_52 is the scalar function that describes the pressure at all locations in space. Pressure gradients and differentials result in the buoyant force for fluids suspended in gravitational fields, winds in atmospheric science, and the lift associated with aerodynamics and flight.ch.12
A specific instance of such a force that is associated with dynamic pressure is fluid resistance: a body force that resists the motion of an object through a fluid due to viscosity. For so-called "Stokes' drag" the force is approximately proportional to the velocity, but opposite in direction:
formula_53
where:
More formally, forces in continuum mechanics are fully described by a stress tensor with terms that are roughly defined as
formula_56
where formula_57 is the relevant cross-sectional area for the volume for which the stress tensor is being calculated. This formalism includes pressure terms associated with forces that act normal to the cross-sectional area (the matrix diagonals of the tensor) as well as shear terms associated with forces that act parallel to the cross-sectional area (the off-diagonal elements). The stress tensor accounts for forces that cause all strains (deformations) including also tensile stresses and compressions.
Fictitious.
There are forces that are frame dependent, meaning that they appear due to the adoption of non-Newtonian (that is, non-inertial) reference frames. Such forces include the centrifugal force and the Coriolis force. These forces are considered fictitious because they do not exist in frames of reference that are not accelerating.ch.12 Because these forces are not genuine they are also referred to as "pseudo forces".
In general relativity, gravity becomes a fictitious force that arises in situations where spacetime deviates from a flat geometry.
Concepts derived from force.
Rotation and torque.
Forces that cause extended objects to rotate are associated with torques. Mathematically, the torque of a force formula_3 is defined relative to an arbitrary reference point as the cross product:
formula_59
where formula_60 is the position vector of the force application point relative to the reference point.
Torque is the rotation equivalent of force in the same way that angle is the rotational equivalent for position, angular velocity for velocity, and angular momentum for momentum. As a consequence of Newton's first law of motion, there exists rotational inertia that ensures that all bodies maintain their angular momentum unless acted upon by an unbalanced torque. Likewise, Newton's second law of motion can be used to derive an analogous equation for the instantaneous angular acceleration of the rigid body:
formula_61
where
This provides a definition for the moment of inertia, which is the rotational equivalent for mass. In more advanced treatments of mechanics, where the rotation over a time interval is described, the moment of inertia must be substituted by the tensor that, when properly analyzed, fully determines the characteristics of rotations including precession and nutation.
Equivalently, the differential form of Newton's Second Law provides an alternative definition of torque:
formula_64
where formula_65 is the angular momentum of the particle.
Newton's Third Law of Motion requires that all objects exerting torques themselves experience equal and opposite torques, and therefore also directly implies the conservation of angular momentum for closed systems that experience rotations and revolutions through the action of internal torques.
Yank.
The yank is defined as the rate of change of force
formula_66
The term is used in biomechanical analysis, athletic assessment and robotic control. The second ("tug"), third ("snatch"), fourth ("shake"), and higher derivatives are rarely used.
Kinematic integrals.
Forces can be used to define a number of physical concepts by integrating with respect to kinematic variables. For example, integrating with respect to time gives the definition of impulse:
formula_67
which by Newton's Second Law must be equivalent to the change in momentum (yielding the Impulse momentum theorem).
Similarly, integrating with respect to position gives a definition for the work done by a force:
formula_68
which is equivalent to changes in kinetic energy (yielding the work energy theorem).
Power "P" is the rate of change d"W"/d"t" of the work "W", as the trajectory is extended by a position change formula_69 in a time interval d"t":
formula_70
so
formula_71
with formula_72 the velocity.
Potential energy.
Instead of a force, often the mathematically related concept of a potential energy field is used. For instance, the gravitational force acting upon an object can be seen as the action of the gravitational field that is present at the object's location. Restating mathematically the definition of energy (via the definition of work), a potential scalar field formula_73 is defined as that field whose gradient is equal and opposite to the force produced at every point:
formula_74
Forces can be classified as conservative or nonconservative. Conservative forces are equivalent to the gradient of a potential while nonconservative forces are not.ch.12
Conservation.
A conservative force that acts on a closed system has an associated mechanical work that allows energy to convert only between kinetic or potential forms. This means that for a closed system, the net mechanical energy is conserved whenever a conservative force acts on the system. The force, therefore, is related directly to the difference in potential energy between two different locations in space, and can be considered to be an artifact of the potential field in the same way that the direction and amount of a flow of water can be considered to be an artifact of the contour map of the elevation of an area.ch.12
Conservative forces include gravity, the electromagnetic force, and the spring force. Each of these forces has models that are dependent on a position often given as a radial vector formula_60 emanating from spherically symmetric potentials. Examples of this follow:
For gravity:
formula_76
where formula_77 is the gravitational constant, and formula_78 is the mass of object "n".
For electrostatic forces:
formula_79
where formula_80 is electric permittivity of free space, and formula_81 is the electric charge of object "n".
For spring forces:
formula_82
where formula_44 is the spring constant.ch.12
For certain physical scenarios, it is impossible to model forces as being due to a simple gradient of potentials. This is often due a macroscopic statistical average of microstates. For example, static friction is caused by the gradients of numerous electrostatic potentials between the atoms, but manifests as a force model that is independent of any macroscale position vector. Nonconservative forces other than friction include other contact forces, tension, compression, and drag. For any sufficiently detailed description, all these forces are the results of conservative ones since each of these macroscopic forces are the net results of the gradients of microscopic potentials.ch.12
The connection between macroscopic nonconservative forces and microscopic conservative forces is described by detailed treatment with statistical mechanics. In macroscopic closed systems, nonconservative forces act to change the internal energies of the system, and are often associated with the transfer of heat. According to the Second law of thermodynamics, nonconservative forces necessarily result in energy transformations within closed systems from ordered to more random conditions as entropy increases.ch.12
Units.
The SI unit of force is the newton (symbol N), which is the force required to accelerate a one kilogram mass at a rate of one meter per second squared, or kg·m·s−2.The corresponding CGS unit is the dyne, the force required to accelerate a one gram mass by one centimeter per second squared, or g·cm·s−2. A newton is thus equal to 100,000 dynes.
The gravitational foot-pound-second English unit of force is the pound-force (lbf), defined as the force exerted by gravity on a pound-mass in the standard gravitational field of 9.80665 m·s−2. The pound-force provides an alternative unit of mass: one slug is the mass that will accelerate by one foot per second squared when acted on by one pound-force. An alternative unit of force in a different foot–pound–second system, the absolute fps system, is the poundal, defined as the force required to accelerate a one-pound mass at a rate of one foot per second squared.
The pound-force has a metric counterpart, less commonly used than the newton: the kilogram-force (kgf) (sometimes kilopond), is the force exerted by standard gravity on one kilogram of mass. The kilogram-force leads to an alternate, but rarely used unit of mass: the metric slug (sometimes mug or hyl) is that mass that accelerates at 1 m·s−2 when subjected to a force of 1 kgf. The kilogram-force is not a part of the modern SI system, and is generally deprecated, sometimes used for expressing aircraft weight, jet thrust, bicycle spoke tension, torque wrench settings and engine output torque.
 See also "Ton-force".
Revisions of the force concept.
At the beginning of the 20th century, new physical ideas emerged to explain experimental results in astronomical and submicroscopic realms. As discussed below, relativity alters the definition of momentum and quantum mechanics reuses the concept of "force" in microscopic contexts where Newton's laws do not apply directly.
Special theory of relativity.
In the special theory of relativity, mass and energy are equivalent (as can be seen by calculating the work required to accelerate an object). When an object's velocity increases, so does its energy and hence its mass equivalent (inertia). It thus requires more force to accelerate it the same amount than it did at a lower velocity. Newton's Second Law,
formula_84
remains valid because it is a mathematical definition. But for momentum to be conserved at relativistic relative velocity, formula_47, momentum must be redefined as:
formula_86
where formula_87 is the rest mass and formula_88 the speed of light.
The expression relating force and acceleration for a particle with constant non-zero rest mass formula_18 moving in the formula_90 direction at velocity formula_47 is:
formula_92
where
formula_93
is called the Lorentz factor. The Lorentz factor increases steeply as the relative velocity approaches the speed of light. Consequently, the greater and greater force must be applied to produce the same acceleration at extreme velocity. The relative velocity cannot reach formula_88.§15–8
If formula_47 is very small compared to formula_88, then formula_97 is very close to 1 and
formula_98
is a close approximation. Even for use in relativity, one can restore the form of
formula_99
through the use of four-vectors. This relation is correct in relativity when formula_100 is the four-force, formula_18 is the invariant mass, and formula_102 is the four-acceleration.
The "general" theory of relativity incorporates a more radical departure from the Newtonian way of thinking about force, specifically gravitational force. This reimagining of the nature of gravity is described more fully below.
Quantum mechanics.
Quantum mechanics is a theory of physics originally developed in order to understand microscopic phenomena: behavior at the scale of molecules, atoms or subatomic particles. Generally and loosely speaking, the smaller a system is, the more an adequate mathematical model will require understanding quantum effects. The conceptual underpinning of quantum physics is different from that of classical physics. Instead of thinking about quantities like position, momentum, and energy as properties that an object "has", one considers what result might "appear" when a measurement of a chosen type is performed. Quantum mechanics allows the physicist to calculate the probability that a chosen measurement will elicit a particular result. The expectation value for a measurement is the average of the possible results it might yield, weighted by their probabilities of occurrence.
In quantum mechanics, interactions are typically described in terms of energy rather than force. The Ehrenfest theorem provides a connection between quantum expectation values and the classical concept of force, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, the Born rule is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over time; that is, depending on the time at which (for example) a position measurement is performed, the probabilities for its different possible outcomes will vary. The Ehrenfest theorem says, roughly speaking, that the equations describing how these expectation values change over time have a form reminiscent of Newton's second law, with a force defined as the negative derivative of the potential energy. However, the more pronounced quantum effects are in a given situation, the more difficult it is to derive meaningful conclusions from this resemblance.
Quantum mechanics also introduces two new constraints that interact with forces at the submicroscopic scale and which are especially important for atoms. Despite the strong attraction of the nucleus, the uncertainty principle limits the minimum extent of an electron probability distribution and the Pauli exclusion principle prevents electrons from sharing the same probability distribution. This gives rise to an emergent pressure known as degeneracy pressure. The dynamic equilibrium between the degeneracy pressure and the attractive electromagnetic force give atoms, molecules, liquids, and solids stability.
Quantum field theory.
In modern particle physics, forces and the acceleration of particles are explained as a mathematical by-product of exchange of momentum-carrying gauge bosons. With the development of quantum field theory and general relativity, it was realized that force is a redundant concept arising from conservation of momentum (4-momentum in relativity and momentum of virtual particles in quantum electrodynamics). The conservation of momentum can be directly derived from the homogeneity or symmetry of space and so is usually considered more fundamental than the concept of a force. Thus the currently known fundamental forces are considered more accurately to be "fundamental interactions".
While sophisticated mathematical descriptions are needed to predict, in full detail, the result of such interactions, there is a conceptually simple way to describe them through the use of Feynman diagrams. In a Feynman diagram, each matter particle is represented as a straight line (see world line) traveling through time, which normally increases up or to the right in the diagram. Matter and anti-matter particles are identical except for their direction of propagation through the Feynman diagram. World lines of particles intersect at interaction vertices, and the Feynman diagram represents any force arising from an interaction as occurring at the vertex with an associated instantaneous change in the direction of the particle world lines. Gauge bosons are emitted away from the vertex as wavy lines and, in the case of virtual particle exchange, are absorbed at an adjacent vertex. The utility of Feynman diagrams is that other types of physical phenomena that are part of the general picture of fundamental interactions but are conceptually separate from forces can also be described using the same rules. For example, a Feynman diagram can describe in succinct detail how a neutron decays into an electron, proton, and antineutrino, an interaction mediated by the same gauge boson that is responsible for the weak nuclear force.
Fundamental interactions.
All of the known forces of the universe are classified into four fundamental interactions. The strong and the weak forces act only at very short distances, and are responsible for the interactions between subatomic particles, including nucleons and compound nuclei. The electromagnetic force acts between electric charges, and the gravitational force acts between masses. All other forces in nature derive from these four fundamental interactions operating within quantum mechanics, including the constraints introduced by the Schrödinger equation and the Pauli exclusion principle. For example, friction is a manifestation of the electromagnetic force acting between atoms of two surfaces. The forces in springs, modeled by Hooke's law, are also the result of electromagnetic forces. Centrifugal forces are acceleration forces that arise simply from the acceleration of rotating frames of reference.
The fundamental theories for forces developed from the unification of different ideas. For example, Newton's universal theory of gravitation showed that the force responsible for objects falling near the surface of the Earth is also the force responsible for the falling of celestial bodies about the Earth (the Moon) and around the Sun (the planets). Michael Faraday and James Clerk Maxwell demonstrated that electric and magnetic forces were unified through a theory of electromagnetism. In the 20th century, the development of quantum mechanics led to a modern understanding that the first three fundamental forces (all except gravity) are manifestations of matter (fermions) interacting by exchanging virtual particles called gauge bosons. This Standard Model of particle physics assumes a similarity between the forces and led scientists to predict the unification of the weak and electromagnetic forces in electroweak theory, which was subsequently confirmed by observation.
Gravitational.
Newton's law of gravitation is an example of "action at a distance": one body, like the Sun, exerts an influence upon any other body, like the Earth, no matter how far apart they are. Moreover, this action at a distance is "instantaneous." According to Newton's theory, the one body shifting position changes the gravitational pulls felt by all other bodies, all at the same instant of time. Albert Einstein recognized that this was inconsistent with special relativity and its prediction that influences cannot travel faster than the speed of light. So, he sought a new theory of gravitation that would be relativistically consistent. Mercury's orbit did not match that predicted by Newton's law of gravitation. Some astrophysicists predicted the existence of an undiscovered planet (Vulcan) that could explain the discrepancies. When Einstein formulated his theory of general relativity (GR) he focused on Mercury's problematic orbit and found that his theory added a correction, which could account for the discrepancy. This was the first time that Newton's theory of gravity had been shown to be inexact.
Since then, general relativity has been acknowledged as the theory that best explains gravity. In GR, gravitation is not viewed as a force, but rather, objects moving freely in gravitational fields travel under their own inertia in straight lines through curved spacetime – defined as the shortest spacetime path between two spacetime events. From the perspective of the object, all motion occurs as if there were no gravitation whatsoever. It is only when observing the motion in a global sense that the curvature of spacetime can be observed and the force is inferred from the object's curved path. Thus, the straight line path in spacetime is seen as a curved line in space, and it is called the "ballistic trajectory" of the object. For example, a basketball thrown from the ground moves in a parabola, as it is in a uniform gravitational field. Its spacetime trajectory is almost a straight line, slightly curved (with the radius of curvature of the order of few light-years). The time derivative of the changing momentum of the object is what we label as "gravitational force".
Electromagnetic.
Maxwell's equations and the set of techniques built around them adequately describe a wide range of physics involving force in electricity and magnetism. This classical theory already includes relativity effects. Understanding quantized electromagnetic interactions between elementary particles requires quantum electrodynamics (or QED). In QED, photons are fundamental exchange particles, describing all interactions relating to electromagnetism including the electromagnetic force.
Strong nuclear.
There are two "nuclear forces", which today are usually described as interactions that take place in quantum theories of particle physics. The strong nuclear force is the force responsible for the structural integrity of atomic nuclei, and gains its name from its ability to overpower the electromagnetic repulsion between protons.
The strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD). The strong force is the fundamental force mediated by gluons, acting upon quarks, antiquarks, and the gluons themselves. The strong force only acts "directly" upon elementary particles. A residual is observed between hadrons (notably, the nucleons in atomic nuclei), known as the nuclear force. Here the strong force acts indirectly, transmitted as gluons that form part of the virtual pi and rho mesons, the classical transmitters of the nuclear force. The failure of many searches for free quarks has shown that the elementary particles affected are not directly observable. This phenomenon is called color confinement.232
Weak nuclear.
Unique among the fundamental interactions, the weak nuclear force creates no bound states. The weak force is due to the exchange of the heavy W and Z bosons. Since the weak force is mediated by two types of bosons, it can be divided into two types of interaction or "vertices" — charged current, involving the electrically charged W+ and W− bosons, and neutral current, involving electrically neutral Z0 bosons. The most familiar effect of weak interaction is beta decay (of neutrons in atomic nuclei) and the associated radioactivity. This is a type of charged-current interaction. The word "weak" derives from the fact that the field strength is some 1013 times less than that of the strong force. Still, it is stronger than gravity over short distances. A consistent electroweak theory has also been developed, which shows that electromagnetic forces and the weak force are indistinguishable at a temperatures in excess of approximately . Such temperatures occurred in the plasma collisions in the early moments of the Big Bang.
References.
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