George Boole (boolean algebra)

From Wikipedia, the free encyclopedia

George Boole
Boole, c. 1860
Born	2 November 1815 Lincoln, Lincolnshire, England
Died	8 December 1864 (aged 49) Ballintemple, Cork, Ireland
Education	Bainbridge's Commercial Academy
Spouse(s)	Mary Everest Boole
Era	19th-century philosophy
Region	Western philosophy
School	Mathematical foundations of computing
Institutions	Lincoln Mechanics' Institute Queen's College, Cork
Main interests	Mathematics, Logic, Philosophy of mathematics
Notable ideas	Boolean algebra

George Boole (/buːl/; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland. He worked in the fields of differential equations and algebraic logic, and is best known as the author of The Laws of Thought (1854) which contains Boolean algebra. Boolean logic is credited with laying the foundations for the information age. Boole maintained that:

No general method for the solution of questions in the theory of probabilities can be established which does not explicitly recognize, not only the special numerical bases of the science, but also those universal laws of thought which are the basis of all reasoning, and which, whatever they may be as to their essence, are at least mathematical as to their form.

Early life

Boole's House and School at 3 Pottergate in Lincoln

 

Boole was born in Lincoln, Lincolnshire, England, the son of John Boole senior (1779–1848), a shoemaker and Mary Ann Joyce. He had a primary school education, and received lessons from his father, but due to a serious decline in business, he had little further formal and academic teaching. William Brooke, a bookseller in Lincoln, may have helped him with Latin, which he may also have learned at the school of Thomas Bainbridge. He was self-taught in modern languages. In fact, when a local newspaper printed his translation of a Latin poem, a scholar accused him of plagiarism under the pretense that he was not capable of such achievements. At age 16, Boole became the breadwinner for his parents and three younger siblings, taking up a junior teaching position in Doncaster at Heigham's School. He taught briefly in Liverpool.

Greyfriars, Lincoln, which housed the Mechanic's Institute

 

Boole participated in the Mechanics Institute, in the Greyfriars, Lincoln, which was founded in 1833. Edward Bromhead, who knew John Boole through the institution, helped George Boole with mathematics books and he was given the calculus text of Sylvestre François Lacroix by the Rev. George Stevens Dickson of St Swithin's, Lincoln. Without a teacher, it took him many years to master calculus.

At age 19, Boole successfully established his own school in Lincoln. He continued making his living by running schools until he was in his thirties. Four years later he took over Hall's Academy in Waddington, outside Lincoln, following the death of Robert Hall. In 1840 he moved back to Lincoln, where he ran a boarding school. Boole immediately became involved in the Lincoln Topographical Society, serving as a member of the committee, and presenting a paper entitled, On the origin, progress, and tendencies of Polytheism, especially amongst the ancient Egyptians and Persians, and in modern India. on 30 November 1841.

Boole became a prominent local figure, an admirer of John Kaye, the bishop. He took part in the local campaign for early closing. With Edmund Larken and others he set up a building society in 1847. He associated also with the Chartist Thomas Cooper, whose wife was a relation.

Plaque from the house in Lincoln

 

From 1838 onward Boole was making contacts with sympathetic British academic mathematicians and reading more widely. He studied algebra in the form of symbolic methods, as far as these were understood at the time, and began to publish research papers. After receiving positive feedback on his publications, he considered attending the University of Cambridge, but decided against attending when told he would have to start with the standard undergraduate courses and discontinue his own research.

Professor at Cork

The house at 5 Grenville Place in Cork, in which Boole lived between 1849 and 1855, and where he wrote The Laws of Thought (Picture taken during renovation.)

 

Boole's status as mathematician was recognized by his appointment in 1849 as the first professor of mathematics at Queen's College, Cork (now University College Cork (UCC)) in Ireland. He met his future wife, Mary Everest, there in 1850 while she was visiting her uncle John Ryall who was Professor of Greek. They married some years later in 1855. He maintained his ties with Lincoln, working there with E. R. Larken in a campaign to reduce prostitution.

Honors and awards

In 1844 Boole's paper On a General Method for Analysis won the first gold prize for mathematics awarded by the Royal Society. He was awarded the Keith Medal by the Royal Society of Edinburgh in 1855 and was elected a Fellow of the Royal Society (FRS) in 1857. He received honorary degrees of LL.D. from the University of Dublin and the University of Oxford.

Boole's gravestone in Blackrock, Cork, Ireland

 

Detail of stained glass window in Lincoln Cathedral dedicated to Boole

 

Plaque beneath Boole's window in Lincoln Cathedral

Works

Boole's first published paper was Researches in the theory of analytical transformations, with a special application to the reduction of the general equation of the second order, printed in the Cambridge Mathematical Journal in February 1840 (Volume 2, № 8, pp. 64–73), and it led to a friendship between Boole and Duncan Farquharson Gregory, the editor of the journal. His works are in about 50 articles and a few separate publications.

In 1841 Boole published an influential paper in early invariant theory. He received a medal from the Royal Society for his memoir of 1844, On A General Method of Analysis. It was a contribution to the theory of linear differential equations, moving from the case of constant coefficients on which he had already published, to variable coefficients. The innovation in operational methods is to admit that operations may not commute. In 1847 Boole published The Mathematical Analysis of Logic, the first of his works on symbolic logic.

Differential equations

Boole completed two systematic treatises on mathematical subjects during his lifetime. The Treatise on Differential Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, a sequel to the former work.

Analysis

In 1857, Boole published the treatise On the Comparison of Transcendents, with Certain Applications to the Theory of Definite Integrals, in which he studied the sum of residues of a rational function. Among other results, he proved what is now called Boole's identity:

${\displaystyle \mathrm {mes} \left\{x\in \mathbb {R} \,\mid \,\Re {\frac {1}{\pi }}\sum {\frac {a_{k}}{x-b_{k}}}\geq t\right\}={\frac {\sum a_{k}}{\pi t}}}$

for any real numbers a_k g.t. 0, b_k, and t g.t. 0. Generalizations of this identity play an important role in the theory of the Hilbert transform.

Symbolic logic

In 1847 Boole published the pamphlet Mathematical Analysis of Logic. He later regarded it as a flawed exposition of his logical system, and wanted An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities to be seen as the mature statement of his views. Contrary to widespread belief, Boole never intended to criticize or disagree with the main principles of Aristotle's logic. Rather he intended to systematize it, to provide it with a foundation, and to extend its range of applicability. Boole's initial involvement in logic was prompted by a current debate on quantification, between Sir William Hamilton who supported the theory of "quantification of the predicate", and Boole's supporter Augustus De Morgan who advanced a version of De Morgan duality, as it is now called. Boole's approach was ultimately much further reaching than either sides' in the controversy. It founded what was first known as the "algebra of logic" tradition.

Among his many innovations is his principle of wholistic reference, which was later, and probably independently, adopted by Gottlob Frege and by logicians who subscribe to standard first-order logic. A 2003 article provides a systematic comparison and critical evaluation of Aristotelian logic and Boolean logic; it also reveals the centrality of wholistic reference in Boole's philosophy of logic.

1854 definition of universe of discourse

In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined. The most unfettered discourse is that in which the words we use are understood in the widest possible application, and for them the limits of discourse are co-extensive with those of the universe itself. But more usually we confine ourselves to a less spacious field. Sometimes, in discoursing of men we imply (without expressing the limitation) that it is of men only under certain circumstances and conditions that we speak, as of civilized men, or of men in the vigor of life, or of men under some other condition or relation. Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. Furthermore, this universe of discourse is in the strictest sense the ultimate subject of the discourse.

Treatment of addition in logic

Boole conceived of "elective symbols" of his kind as an algebraic structure. But this general concept was not available to him: he did not have the segregation standard in abstract algebra of postulated (axiomatic) properties of operations, and deduced properties. His work was a beginning to the algebra of sets, again not a concept available to Boole as a familiar model. His pioneering efforts encountered specific difficulties, and the treatment of addition was an obvious difficulty in the early days.

Boole replaced the operation of multiplication by the word "and" and addition by the word "or". But in Boole's original system, + was a partial operation: in the language of set theory it would correspond only to disjoint union of subsets. Later authors changed the interpretation, commonly reading it as exclusive or, or in set theory terms symmetric difference; this step means that addition is always defined.

In fact there is the other possibility, that + should be read as disjunction. This other possibility extends from the disjoint union case, where exclusive or and non-exclusive or both give the same answer. Handling this ambiguity was an early problem of the theory, reflecting the modern use of both Boolean rings and Boolean algebras (which are simply different aspects of one type of structure). Boole and Jevons struggled over just this issue in 1863, in the form of the correct evaluation of x + x. Jevons argued for the result x, which is correct for + as disjunction. Boole kept the result as something undefined. He argued against the result 0, which is correct for exclusive or, because he saw the equation x + x = 0 as implying x = 0, a false analogy with ordinary algebra.

Probability theory

The second part of the Laws of Thought contained a corresponding attempt to discover a general method in probabilities. Here the goal was algorithmic: from the given probabilities of any system of events, to determine the consequent probability of any other event logically connected with those events.

Death

In late November 1864, Boole walked, in heavy rain, from his home at Lichfield Cottage in Ballintemple to the university, a distance of three miles, and lectured wearing his wet clothes. He soon became ill, developing pneumonia. As his wife believed that remedies should resemble their cause, she wrapped him in wet blankets – the wet having brought on his illness. Boole's condition worsened and on 8 December 1864, he died of fever-induced pleural effusion. 

He was buried in the Church of Ireland cemetery of St Michael's, Church Road, Blackrock (a suburb of Cork). There is a commemorative plaque inside the adjoining church.

Legacy

Bust of Boole at University College Cork

 

Boolean algebra is named after him, as is the crater Boole on the Moon. The keyword Bool represents a Boolean datatype in many programming languages, though Pascal and Java, among others, both use the full name Boolean. The library, underground lecture theater complex and the Boole Centre for Research in Informatics at University College Cork are named in his honor. A road called Boole Heights in Bracknell, Berkshire is named after him.

19th-century development

Boole's work was extended and refined by a number of writers, beginning with William Stanley Jevons. Augustus De Morgan had worked on the logic of relations, and Charles Sanders Peirce integrated his work with Boole's during the 1870s. Other significant figures were Platon Sergeevich Poretskii, and William Ernest Johnson. The conception of a Boolean algebra structure on equivalent statements of a propositional calculus is credited to Hugh MacColl (1877), in work surveyed 15 years later by Johnson. Surveys of these developments were published by Ernst Schröder, Louis Couturat, and Clarence Irving Lewis.

20th-century development

In modern notation, the free Boolean algebra on basic propositions p and q arranged in a Hasse diagram. The Boolean combinations make up 16 different propositions, and the lines show which are logically related.

 

In 1921 the economist John Maynard Keynes published a book on probability theory, A Treatise of Probability. Keynes believed that Boole had made a fundamental error in his definition of independence which vitiated much of his analysis. In his book The Last Challenge Problem, David Miller provides a general method in accord with Boole's system and attempts to solve the problems recognized earlier by Keynes and others. Theodore Hailperin showed much earlier that Boole had used the correct mathematical definition of independence in his worked out problems.

Boole's work and that of later logicians initially appeared to have no engineering uses. Claude Shannon attended a philosophy class at the University of Michigan which introduced him to Boole's studies. Shannon recognized that Boole's work could form the basis of mechanisms and processes in the real world and that it was therefore highly relevant. In 1937 Shannon went on to write a master's thesis, at the Massachusetts Institute of Technology, in which he showed how Boolean algebra could optimize the design of systems of electromechanical relays then used in telephone routing switches. He also proved that circuits with relays could solve Boolean algebra problems. Employing the properties of electrical switches to process logic is the basic concept that underlies all modern electronic digital computers. Victor Shestakov at Moscow State University (1907–1987) proposed a theory of electric switches based on Boolean logic even earlier than Claude Shannon in 1935 on the testimony of Soviet logicians and mathematicians Sofya Yanovskaya, Gaaze-Rapoport, Roland Dobrushin, Lupanov, Medvedev and Uspensky, though they presented their academic theses in the same year, 1938. But the first publication of Shestakov's result took place only in 1941 (in Russian). Hence, Boolean algebra became the foundation of practical digital circuit design; and Boole, via Shannon and Shestakov, provided the theoretical grounding for the Information Age.

21st-century celebration

Boole's legacy surrounds us everywhere, in the computers, information storage and retrieval, electronic circuits and controls that support life, learning and communications in the 21st century. His pivotal advances in mathematics, logic and probability provided the essential groundwork for modern mathematics, microelectronic engineering and computer science." —University College Cork.

2015 saw the 200th anniversary of George Boole's birth. To mark the bicentenary year, University College Cork joined admirers of Boole around the world to celebrate his life and legacy. 

UCC's George Boole 200 project, featured events, student outreach activities and academic conferences on Boole's legacy in the digital age, including a new edition of Desmond MacHale's 1985 biography The Life and Work of George Boole: A Prelude to the Digital Age, 2014). 

The search engine Google marked the 200th anniversary of his birth on 2 November 2015 with an algebraic reimaging of its Google Doodle.

5 Grenville Place in 2017 following restoration by UCC

 

Litchfield Cottage in Ballintemple, Cork, where Boole lived for the last two years of his life, bears a memorial plaque. His former residence, in Grenville Place, is being restored through a collaboration between UCC and Cork City Council, as the George Boole House of Innovation, after the city council acquired the premises under the Derelict Sites Act.

Views

Boole's views were given in four published addresses: The Genius of Sir Isaac Newton; The Right Use of Leisure; The Claims of Science; and The Social Aspect of Intellectual Culture. The first of these was from 1835, when Charles Anderson-Pelham, 1st Earl of Yarborough gave a bust of Newton to the Mechanics' Institute in Lincoln. The second justified and celebrated in 1847 the outcome of the successful campaign for early closing in Lincoln, headed by Alexander Leslie-Melville, of Branston Hall. The Claims of Science was given in 1851 at Queen's College, Cork. The Social Aspect of Intellectual Culture was also given in Cork, in 1855 to the Cuvierian Society.

Though his biographer Des MacHale describes Boole as an "agnostic deist", Boole read a wide variety of Christian theology. Combining his interests in mathematics and theology, he compared the Christian trinity of Father, Son, and Holy Ghost with the three dimensions of space, and was attracted to the Hebrew conception of God as an absolute unity. Boole considered converting to Judaism but in the end was said to have chosen Unitarianism. Boole came to speak against a what he saw as "prideful" skepticism, and instead, favored the belief in a "Supreme Intelligent Cause." He also declared "I firmly believe, for the accomplishment of a purpose of the Divine Mind." In addition, he stated that he perceived "teeming evidences of surrounding design" and concluded that "the course of this world is not abandoned to chance and inexorable fate."

Two influences on Boole were later claimed by his wife, Mary Everest Boole: a universal mysticism tempered by Jewish thought, and Indian logic. Mary Boole stated that an adolescent mystical experience provided for his life's work:

My husband told me that when he was a lad of seventeen a thought struck him suddenly, which became the foundation of all his future discoveries. It was a flash of psychological insight into the conditions under which a mind most readily accumulates knowledge [...] For a few years he supposed himself to be convinced of the truth of "the Bible" as a whole, and even intended to take orders as a clergyman of the English Church. But by the help of a learned Jew in Lincoln he found out the true nature of the discovery which had dawned on him. This was that man's mind works by means of some mechanism which "functions normally towards Monism."

In Ch. 13 of Laws of Thought Boole used examples of propositions from Baruch Spinoza and Samuel Clarke. The work contains some remarks on the relationship of logic to religion, but they are slight and cryptic. Boole was apparently disconcerted at the book's reception just as a mathematical toolset:

George afterwards learned, to his great joy, that the same conception of the basis of Logic was held by Leibnitz, the contemporary of Newton. De Morgan, of course, understood the formula in its true sense; he was Boole's collaborator all along. Herbert Spencer, Jowett, and Robert Leslie Ellis understood, I feel sure; and a few others, but nearly all the logicians and mathematicians ignored [953] the statement that the book was meant to throw light on the nature of the human mind; and treated the formula entirely as a wonderful new method of reducing to logical order masses of evidence about external fact.

Mary Boole claimed that there was profound influence – via her uncle George Everest – of Indian thought in general and Indian logic, in particular, on George Boole, as well as on Augustus De Morgan and Charles Babbage:

Think what must have been the effect of the intense Hinduizing of three such men as Babbage, De Morgan, and George Boole on the mathematical atmosphere of 1830–65. What share had it in generating the Vector Analysis and the mathematics by which investigations in physical science are now conducted?

Family

In 1855 he married Mary Everest (niece of George Everest), who later wrote several educational works on her husband's principles. 

The Booles had five daughters:

• Mary Ellen (1856–1908) who married the mathematician and author Charles Howard Hinton and had four children: George (1882–1943), Eric (*1884), William (1886–1909) and Sebastian (1887–1923), inventor of the Jungle gym. After the sudden death of her husband, Mary Ellen committed suicide in Washington, D.C. in May 1908. Sebastian had three children:
  • Jean Hinton (married name Rosner) (1917–2002), a peace activist.
  • William H. Hinton (1919–2004) visited China in the 1930s and 40s and wrote an influential account of the Communist land reform.
  • Joan Hinton (1921–2010) worked for the Manhattan Project and lived in China from 1948 until her death on 8 June 2010; she was married to Sid Engst.
• Margaret (1858–1935), married Edward Ingram Taylor, an artist.
  • Their elder son Geoffrey Ingram Taylor became a mathematician and a Fellow of the Royal Society.
  • Their younger son Julian was a professor of surgery.
• Alicia (1860–1940), who made important contributions to four-dimensional geometry.
• Lucy Everest (1862–1904), who was the first female professor of chemistry in England.
• Ethel Lilian (1864–1960), who married the Polish scientist and revolutionary Wilfrid Michael Voynich and was the author of the novel The Gadfly.

Andromeda Galaxy (updated)

From Wikipedia, the free encyclopedia

Andromeda Galaxy
The Andromeda Galaxy with satellite galaxies M32 (center left above the galactic nucleus) and M110 (center left below the galaxy)
Observation data (J2000 epoch)
Pronunciation	/ænˈdrɒmɪdə/
Constellation	Andromeda
Right ascension	00h 42m 44.3s
Declination	+41° 16′ 9″
Redshift	z = −0.001001 (minus sign indicates blueshift)
Helio radial velocity	−301 ± 1 km/s
Distance	2.54 ± 0.11 Mly (778 ± 33 kpc)
Apparent magnitude (V)	3.44
Absolute magnitude (V)	−21.5
Characteristics
Type	SA(s)b
Mass	~0.8-1.5×1012  M☉
Number of stars	~1 trillion (1012)
Size	~220 kly (diameter)
Apparent size (V)	3.167° × 1°
Other designations
M31, NGC 224, UGC 454, PGC 2557, 2C 56 (Core), CGCG 535-17, MCG +07-02-016, IRAS 00400+4059, 2MASX J00424433+4116074, GC 116, h 50, Bode 3, Flamsteed 58, Hevelius 32, Ha 3.3, IRC +40013

The Andromeda Galaxy, also known as Messier 31, M31, or NGC 224, is a spiral galaxy approximately 780 kiloparsecs (2.5 million light-years) from Earth, and the nearest major galaxy to the Milky Way. Its name stems from the area of the Earth's sky in which it appears, the constellation of Andromeda.

The 2006 observations by the Spitzer Space Telescope revealed that the Andromeda Galaxy contains approximately one trillion stars, more than twice the number of the Milky Way's estimated 200 to 400 billion stars. The Andromeda Galaxy's mass is estimated to be around 1.76 times that of the Milky Way Galaxy (~0.8-1.5×10¹² solar masses  vs the Milky Way's 8.5×10¹¹ solar masses), though a 2018 study found that the Andromeda Galaxy's mass is roughly the same as the Milky Way's. The Andromeda Galaxy, spanning approximately 220,000 light-years, is the largest galaxy in the Local Group, which is also home to the Triangulum Galaxy and other minor galaxies.

The Milky Way and Andromeda galaxies are expected to collide in ~4.5 billion years, merging to form a giant elliptical galaxy or a large disc galaxy. With an apparent magnitude of 3.4, the Andromeda Galaxy is among the brightest of the Messier objects making it visible to the naked eye from Earth on moonless nights, even when viewed from areas with moderate light pollution.

Observation history

Great Andromeda Nebula by Isaac Roberts, 1899.

 

Around the year 964, the Persian astronomer Abd al-Rahman al-Sufi described the Andromeda Galaxy, in his Book of Fixed Stars as a "nebulous smear".

Star charts of that period labeled it as the Little Cloud. In 1612, the German astronomer Simon Marius gave an early description of the Andromeda Galaxy based on telescopic observations. The German philosopher Immanuel Kant in 1755 in his work Universal Natural History and Theory of the Heavens conjectured that the blurry spot was an island universe. In 1764, Charles Messier cataloged Andromeda as object M31 and incorrectly credited Marius as the discoverer despite it being visible to the naked eye. In 1785, the astronomer William Herschel noted a faint reddish hue in the core region of Andromeda. He believed Andromeda to be the nearest of all the "great nebulae", and based on the color and magnitude of the nebula, he incorrectly guessed that it is no more than 2,000 times the distance of Sirius. In 1850, William Parsons, 3rd Earl of Rosse, saw and made the first drawing of Andromeda's spiral structure.

In 1864, William Huggins noted that the spectrum of Andromeda differs from a gaseous nebula. The spectra of Andromeda displays a continuum of frequencies, superimposed with dark absorption lines that help identify the chemical composition of an object. Andromeda's spectrum is very similar to the spectra of individual stars, and from this, it was deduced that Andromeda has a stellar nature. In 1885, a supernova (known as S Andromedae) was seen in Andromeda, the first and so far only one observed in that galaxy. At the time Andromeda was considered to be a nearby object, so the cause was thought to be a much less luminous and unrelated event called a nova, and was named accordingly; "Nova 1885".

In 1887, Isaac Roberts took the first photographs of Andromeda, which was still commonly thought to be a nebula within our galaxy. Roberts mistook Andromeda and similar spiral nebulae as solar systems being formed. In 1912, Vesto Slipher used spectroscopy to measure the radial velocity of Andromeda with respect to our Solar System—the largest velocity yet measured, at 300 kilometres per second (190 miles per second).

Island universe

Location of the Andromeda Galaxy (M31) in the Andromeda constellation.

 

In 1917, Heber Curtis observed a nova within Andromeda. Searching the photographic record, 11 more novae were discovered. Curtis noticed that these novae were, on average, 10 magnitudes fainter than those that occurred elsewhere in the sky. As a result, he was able to come up with a distance estimate of 500,000 light-years (3.2×10¹⁰ AU). He became a proponent of the so-called "island universes" hypothesis, which held that spiral nebulae were actually independent galaxies.

Andromeda Galaxy above the Very Large Telescope. The Triangulum Galaxy is visible on the top.

 

In 1920, the Great Debate between Harlow Shapley and Curtis took place concerning the nature of the Milky Way, spiral nebulae, and the dimensions of the Universe. To support his claim of the Great Andromeda Nebula being, in fact, an external galaxy, Curtis also noted the appearance of dark lanes within Andromeda which resembled the dust clouds in our own galaxy, as well as historical observations of Andromeda Galaxy's significant Doppler shift. In 1922 Ernst Öpik presented a method to estimate the distance of Andromeda using the measured velocities of its stars. His result placed the Andromeda Nebula far outside our galaxy at a distance of about 450,000 parsecs (1,500,000 light-years). Edwin Hubble settled the debate in 1925 when he identified extragalactic Cepheid variable stars for the first time on astronomical photos of Andromeda. These were made using the 2.5-metre (8 ft 2 in) Hooker telescope, and they enabled the distance of Great Andromeda Nebula to be determined. His measurement demonstrated conclusively that this feature was not a cluster of stars and gas within our own galaxy, but an entirely separate galaxy located a significant distance from the Milky Way.

In 1943, Walter Baade was the first person to resolve stars in the central region of the Andromeda Galaxy. Baade identified two distinct populations of stars based on their metallicity, naming the young, high-velocity stars in the disk Type I and the older, red stars in the bulge Type II. This nomenclature was subsequently adopted for stars within the Milky Way, and elsewhere. (The existence of two distinct populations had been noted earlier by Jan Oort.) Baade also discovered that there were two types of Cepheid variables, which resulted in a doubling of the distance estimate to Andromeda, as well as the remainder of the Universe.

In 1950, radio emission from the Andromeda Galaxy was detected by Hanbury Brown and Cyril Hazard at Jodrell Bank Observatory. The first radio maps of the galaxy were made in the 1950s by John Baldwin and collaborators at the Cambridge Radio Astronomy Group. The core of the Andromeda Galaxy is called 2C 56 in the 2C radio astronomy catalog. In 2009, the first planet may have been discovered in the Andromeda Galaxy. This was detected using a technique called microlensing, which is caused by the deflection of light by a massive object.

General

The estimated distance of the Andromeda Galaxy from our own was doubled in 1953 when it was discovered that there is another, dimmer type of Cepheid. In the 1990s, measurements of both standard red giants as well as red clump stars from the Hipparcos satellite measurements were used to calibrate the Cepheid distances.

Formation and history

The Andromeda Galaxy as seen by NASA's Wide-field Infrared Survey Explorer.

 

The Andromeda Galaxy was formed roughly 10 billion years ago from the collision and subsequent merger of smaller protogalaxies.

This violent collision formed most of the galaxy's (metal-rich) galactic halo and extended disk. During this epoch, its rate of star formation would have been very high, to the point of becoming a luminous infrared galaxy for roughly 100 million years. Andromeda and the Triangulum Galaxy had a very close passage 2–4 billion years ago. This event produced high rates of star formation across the Andromeda Galaxy's disk—even some globular clusters—and disturbed M33's outer disk.

Over the past 2 billion years, star formation throughout Andromeda's disk is thought to have decreased to the point of near-inactivity. There have been interactions with satellite galaxies like M32, M110, or others that have already been absorbed by Andromeda Galaxy. These interactions have formed structures like Andromeda's Giant Stellar Stream. A galactic merger roughly 100 million years ago is believed to be responsible for a counter-rotating disk of gas found in the center of Andromeda as well as the presence there of a relatively young (100 million years old) stellar population.

Distance estimate

At least four distinct techniques have been used to estimate distances from Earth to the Andromeda Galaxy. In 2003, using the infrared surface brightness fluctuations (I-SBF) and adjusting for the new period-luminosity value and a metallicity correction of −0.2 mag dex⁻¹ in (O/H), an estimate of 2.57 ± 0.06 million light-years (1.625×10¹¹ ± 3.8×10⁹ astronomical units) was derived. A 2004 Cepheid variable method estimated the distance to be 2.51 ± 0.13 million light-years (770 ± 40 kpc). In 2005, an eclipsing binary star was discovered in the Andromeda Galaxy. The binary is two hot blue stars of types O and B. By studying the eclipses of the stars, astronomers were able to measure their sizes. Knowing the sizes and temperatures of the stars, they were able to measure their absolute magnitude. When the visual and absolute magnitudes are known, the distance to the star can be measured. The stars lie at a distance of 2.52×10⁶ ± 0.14×10⁶ ly (1.594×10¹¹ ± 8.9×10⁹ AU) and the whole Andromeda Galaxy at about 2.5×10⁶ ly (1.6×10¹¹ AU). This new value is in excellent agreement with the previous, independent Cepheid-based distance value. The TRGB method was also used in 2005 giving a distance of 2.56×10⁶ ± 0.08×10⁶ ly (1.619×10¹¹ ± 5.1×10⁹ AU). Averaged together, these distance estimates give a value of 2.54×10⁶ ± 0.11×10⁶ ly (1.606×10¹¹ ± 7.0×10⁹ AU). And, from this, the diameter of Andromeda at the widest point is estimated to be 220 ± 3 kly (67,450 ± 920 pc). Applying trigonometry (angular diameter), this is equivalent to an apparent 4.96° angle in the sky.

Mass and luminosity estimates

Mass

The Andromeda Galaxy pictured in ultraviolet light by GALEX.

 

Illustration showing both the size of each galaxy and the distance between the two galaxies, to scale.

 

Giant halo around Andromeda Galaxy.

 

Until 2018, mass estimates for the Andromeda Galaxy's halo (including dark matter) gave a value of approximately 1.5×10¹² M_☉ (or 1.5 trillion solar masses) compared to 8×10¹¹ M_☉ for the Milky Way. This contradicted earlier measurements, that seemed to indicate that Andromeda Galaxy and the Milky Way are almost equal in mass. In 2018, the equality of mass was re-established by radio results as approximately 8×10¹¹ M_☉ In 2006, Andromeda Galaxy's spheroid was determined to have a higher stellar density than that of the Milky Way and its galactic stellar disk was estimated at about twice the diameter of that of the Milky Way. The total stellar mass of Andromeda Galaxy is estimated to be between 8×10¹¹ M_☉ and 1.1×10¹¹ M_☉., (i.e., around twice as massive as that of the Milky Way) and 1.5×10¹¹ M_☉, with around 30% of that mass in the central bulge, 56% in the disk, and the remaining 14% in the halo. The radio results (similar mass to Milky Way galaxy) should be taken as likeliest as of 2018, although clearly this matter is still under active investigation by a number of research groups worldwide.

In addition to stars, Andromeda Galaxy's interstellar medium contains at least around 7.2×10⁹ M_☉ in the form of neutral hydrogen, at least 3.4×10⁸ M_☉ as molecular hydrogen (within its innermost 10 kiloparsecs), and 5.4×10⁷ M_☉ of dust.

Andromeda Galaxy is surrounded by a massive halo of hot gas that is estimated to contain half the mass of the stars in the galaxy. The nearly invisible halo stretches about a million light-years from its host galaxy, halfway to our Milky Way galaxy. Simulations of galaxies indicate the halo formed at the same time as the Andromeda Galaxy. The halo is enriched in elements heavier than hydrogen and helium, formed from supernovae and its properties are those expected for a galaxy that lies in the "green valley" of the Galaxy color–magnitude diagram. Supernovae erupt in Andromeda Galaxy's star-filled disk and eject these heavier elements into space. Over Andromeda Galaxy's lifetime, nearly half of the heavy elements made by its stars have been ejected far beyond the galaxy's 200,000-light-year-diameter stellar disk.

Luminosity

Compared to the Milky Way, the Andromeda Galaxy appears to have predominantly older stars with ages >7×10⁹ years. The estimated luminosity of Andromeda Galaxy, ~2.6×10¹⁰ L_☉, is about 25% higher than that of our own galaxy. However, the galaxy has a high inclination as seen from Earth and its interstellar dust absorbs an unknown amount of light, so it is difficult to estimate its actual brightness and other authors have given other values for the luminosity of the Andromeda Galaxy (some authors even propose it is the second-brightest galaxy within a radius of 10 mega-parsecs of the Milky Way, after the Sombrero Galaxy, with an absolute magnitude of around -22.21 or close). 

An estimation done with the help of Spitzer Space Telescope published in 2010 suggests an absolute magnitude (in the blue) of −20.89 (that with a color index of +0.63 translates to an absolute visual magnitude of −21.52, compared to −20.9 for the Milky Way), and a total luminosity in that wavelength of 3.64×10¹⁰ L_☉.

The rate of star formation in the Milky Way is much higher, with Andromeda Galaxy producing only about one solar mass per year compared to 3–5 solar masses for the Milky Way. The rate of novae in the Milky Way is also double that of Andromeda Galaxy. This suggests that the latter once experienced a great star formation phase, but is now in a relative state of quiescence, whereas the Milky Way is experiencing more active star formation. Should this continue, the luminosity of the Milky Way may eventually overtake that of Andromeda Galaxy.

According to recent studies, the Andromeda Galaxy lies in what in the galaxy color–magnitude diagram is known as the "green valley", a region populated by galaxies like the Milky Way in transition from the "blue cloud" (galaxies actively forming new stars) to the "red sequence" (galaxies that lack star formation). Star formation activity in green valley galaxies is slowing as they run out of star-forming gas in the interstellar medium. In simulated galaxies with similar properties to Andromeda Galaxy, star formation is expected to extinguish within about five billion years from the now, even accounting for the expected, short-term increase in the rate of star formation due to the collision between Andromeda Galaxy and the Milky Way.

Structure

The Andromeda Galaxy seen in infrared by the Spitzer Space Telescope, one of NASA's four Great Space Observatories.

 

Image of the Andromeda Galaxy taken by Spitzer in infrared, 24 micrometers (Credit:NASA/JPL–Caltech/Karl D. Gordon, University of Arizona).

A Galaxy Evolution Explorer image of the Andromeda Galaxy. The bands of blue-white making up the galaxy's striking rings are neighborhoods that harbor hot, young, massive stars. Dark blue-grey lanes of cooler dust show up starkly against these bright rings, tracing the regions where star formation is currently taking place in dense cloudy cocoons. When observed in visible light, Andromeda Galaxy's rings look more like spiral arms. The ultraviolet view shows that these arms more closely resemble the ring-like structure previously observed in infrared wavelengths with NASA's Spitzer Space Telescope. Astronomers using the latter interpreted these rings as evidence that the galaxy was involved in a direct collision with its neighbor, M32, more than 200 million years ago.

 

Based on its appearance in visible light, the Andromeda Galaxy is classified as an SA(s)b galaxy in the de Vaucouleurs–Sandage extended classification system of spiral galaxies. However, data from the 2MASS survey showed that Andromeda is actually a barred spiral galaxy, like the Milky Way, with Andromeda's bar oriented along its long axis.

In 2005, astronomers used the Keck telescopes to show that the tenuous sprinkle of stars extending outward from the galaxy is actually part of the main disk itself. This means that the spiral disk of stars in the Andromeda Galaxy is three times larger in diameter than previously estimated. This constitutes evidence that there is a vast, extended stellar disk that makes the galaxy more than 220,000 light-years (67,000 parsecs) in diameter. Previously, estimates of the Andromeda Galaxy's size ranged from 70,000 to 120,000 light-years (21,000 to 37,000 pc) across.

The galaxy is inclined an estimated 77° relative to Earth (where an angle of 90° would be viewed directly from the side). Analysis of the cross-sectional shape of the galaxy appears to demonstrate a pronounced, S-shaped warp, rather than just a flat disk. A possible cause of such a warp could be gravitational interaction with the satellite galaxies near the Andromeda Galaxy. The Galaxy M33 could be responsible for some warp in Andromeda's arms, though more precise distances and radial velocities are required.

Spectroscopic studies have provided detailed measurements of the rotational velocity of the Andromeda Galaxy as a function of radial distance from the core. The rotational velocity has a maximum value of 225 kilometers per second (140 mi/s) at 1,300 light-years (82,000,000 astronomical units) from the core, and it has its minimum possibly as low as 50 kilometers per second (31 mi/s) at 7,000 light-years (440,000,000 AU) from the core. Further out, rotational velocity rises out to a radius of 33,000 light-years (2.1×10⁹ AU), where it reaches a peak of 250 kilometers per second (160 mi/s). The velocities slowly decline beyond that distance, dropping to around 200 kilometers per second (120 mi/s) at 80,000 light-years (5.1×10⁹ AU). These velocity measurements imply a concentrated mass of about 6×10⁹ M_☉ in the nucleus. The total mass of the galaxy increases linearly out to 45,000 light-years (2.8×10⁹ AU), then more slowly beyond that radius.

The spiral arms of the Andromeda Galaxy are outlined by a series of HII regions, first studied in great detail by Walter Baade and described by him as resembling "beads on a string". His studies show two spiral arms that appear to be tightly wound, although they are more widely spaced than in our galaxy. His descriptions of the spiral structure, as each arm crosses the major axis of the Andromeda Galaxy, are as follows:

Baade's spiral arms of M31

Arms (N=cross M31's major axis at north, S=cross M31's major axis at south)	Distance from center (arcminutes) (N*/S*)	Distance from center (kpc) (N*/S*)	Notes
N1/S1	3.4/1.7	0.7/0.4	Dust arms with no OB associations of HII regions.
N2/S2	8.0/10.0	1.7/2.1	Dust arms with some OB associations.
N3/S3	25/30	5.3/6.3	As per N2/S2, but with some HII regions too.
N4/S4	50/47	11/9.9	Large numbers of OB associations, HII regions, and little dust.
N5/S5	70/66	15/14	As per N4/S4 but much fainter.
N6/S6	91/95	19/20	Loose OB associations. No dust visible.
N7/S7	110/116	23/24	As per N6/S6 but fainter and inconspicuous.

Since the Andromeda Galaxy is seen close to edge-on, it is difficult to study its spiral structure. Rectified images of the galaxy seem to show a fairly normal spiral galaxy, exhibiting two continuous trailing arms that are separated from each other by a minimum of about 13,000 light-years (820,000,000 astronomical units) and that can be followed outward from a distance of roughly 1,600 light-years (100,000,000 AU) from the core. Alternative spiral structures have been proposed such as a single spiral arm or a flocculent pattern of long, filamentary, and thick spiral arms.

The most likely cause of the distortions of the spiral pattern is thought to be interaction with galaxy satellites M32 and M110. This can be seen by the displacement of the neutral hydrogen clouds from the stars.

In 1998, images from the European Space Agency's Infrared Space Observatory demonstrated that the overall form of the Andromeda Galaxy may be transitioning into a ring galaxy. The gas and dust within the galaxy is generally formed into several overlapping rings, with a particularly prominent ring formed at a radius of 32,000 light-years (2.0×10⁹ AU) (10 kiloparsecs) from the core, nicknamed by some astronomers the ring of fire. This ring is hidden from visible light images of the galaxy because it is composed primarily of cold dust, and most of the star formation that is taking place in the Andromeda Galaxy is concentrated there.

Later studies with the help of the Spitzer Space Telescope showed how Andromeda Galaxy's spiral structure in the infrared appears to be composed of two spiral arms that emerge from a central bar and continue beyond the large ring mentioned above. Those arms, however, are not continuous and have a segmented structure.

Close examination of the inner region of the Andromeda Galaxy with the same telescope also showed a smaller dust ring that is believed to have been caused by the interaction with M32 more than 200  million years ago. Simulations show that the smaller galaxy passed through the disk of the Andromeda Galaxy along the latter's polar axis. This collision stripped more than half the mass from the smaller M32 and created the ring structures in Andromeda. It is the co-existence of the long-known large ring-like feature in the gas of Messier 31, together with this newly discovered inner ring-like structure, offset from the barycenter, that suggested a nearly head-on collision with the satellite M32, a milder version of the Cartwheel encounter.

Studies of the extended halo of the Andromeda Galaxy show that it is roughly comparable to that of the Milky Way, with stars in the halo being generally "metal-poor", and increasingly so with greater distance. This evidence indicates that the two galaxies have followed similar evolutionary paths. They are likely to have accreted and assimilated about 100–200 low-mass galaxies during the past 12 billion years. The stars in the extended halos of the Andromeda Galaxy and the Milky Way may extend nearly one third the distance separating the two galaxies.

Nucleus

Hubble image of the Andromeda Galaxy core showing possible double structure. NASA/ESA photo.

 

M31 is known to harbor a dense and compact star cluster at its very center. In a large telescope it creates a visual impression of a star embedded in the more diffuse surrounding bulge. In 1991, the Hubble Space Telescope was used to image Andromeda Galaxy's inner nucleus. The nucleus consists of two concentrations separated by 1.5 parsecs (4.9 ly). The brighter concentration, designated as P1, is offset from the center of the galaxy. The dimmer concentration, P2, falls at the true center of the galaxy and contains a black hole measured at 3–5 × 10⁷ M_☉ in 1993, and at 1.1–2.3 × 10⁸ M_☉ in 2005. The velocity dispersion of material around it is measured to be ≈ 160 km/s.

Chandra X-ray telescope image of the center of Andromeda Galaxy. A number of X-ray sources, likely X-ray binary stars, within the galaxy's central region appear as yellowish dots. The blue source at the center is at the position of the supermassive black hole.

 

It has been proposed that the observed double nucleus could be explained if P1 is the projection of a disk of stars in an eccentric orbit around the central black hole. The eccentricity is such that stars linger at the orbital apocenter, creating a concentration of stars. P2 also contains a compact disk of hot, spectral-class A stars. The A stars are not evident in redder filters, but in blue and ultraviolet light they dominate the nucleus, causing P2 to appear more prominent than P1.

While at the initial time of its discovery it was hypothesized that the brighter portion of the double nucleus is the remnant of a small galaxy "cannibalized" by Andromeda Galaxy, this is no longer considered a viable explanation, largely because such a nucleus would have an exceedingly short lifetime due to tidal disruption by the central black hole. While this could be partially resolved if P1 had its own black hole to stabilize it, the distribution of stars in P1 does not suggest that there is a black hole at its center.

Discrete sources

The Andromeda Galaxy in high-energy X-ray and ultraviolet light (released 5 January 2016).

 

Artist's concept of the Andromeda Galaxy core showing a view across a disk of young, blue stars encircling a supermassive black hole. NASA/ESA photo.

 

Apparently, by late 1968, no X-rays had been detected from the Andromeda Galaxy. A balloon flight on October 20, 1970, set an upper limit for detectable hard X-rays from the Andromeda Galaxy. The Swift BAT all-sky survey successfully detected hard X-rays coming from a region centered 6 arc-seconds away from the galaxy center. The emission above 25 keV was later found to be originating from a single source named 3XMM J004232.1+411314, and identified as a binary system where a compact object (a neutron star or a black hole) accretes matter from a star.

Multiple X-ray sources have since been detected in the Andromeda Galaxy, using observations from the European Space Agency's (ESA) XMM-Newton orbiting observatory. Robin Barnard et al. hypothesized that these are candidate black holes or neutron stars, which are heating the incoming gas to millions of kelvins and emitting X-rays. Neutron stars and black holes can be distinguished mainly by measuring their masses. An observation campaign of NuSTAR space mission identified 40 objects of this kind in the galaxy.

There are approximately 460 globular clusters associated with the Andromeda Galaxy. The most massive of these clusters, identified as Mayall II, nicknamed Globular One, has a greater luminosity than any other known globular cluster in the Local Group of galaxies. It contains several million stars, and is about twice as luminous as Omega Centauri, the brightest known globular cluster in the Milky Way. Globular One (or G1) has several stellar populations and a structure too massive for an ordinary globular. As a result, some consider G1 to be the remnant core of a dwarf galaxy that was consumed by Andromeda in the distant past. The globular with the greatest apparent brightness is G76 which is located in the south-west arm's eastern half. Another massive globular cluster, named 037-B327 and discovered in 2006 as is heavily reddened by the Andromeda Galaxy's interstellar dust, was thought to be more massive than G1 and the largest cluster of the Local Group; however, other studies have shown it is actually similar in properties to G1.

Star clusters in the Andromeda Galaxy.

 

Unlike the globular clusters of the Milky Way, which show a relatively low age dispersion, Andromeda Galaxy's globular clusters have a much larger range of ages: from systems as old as the galaxy itself to much younger systems, with ages between a few hundred million years to five billion years.

In 2005, astronomers discovered a completely new type of star cluster in the Andromeda Galaxy. The new-found clusters contain hundreds of thousands of stars, a similar number of stars that can be found in globular clusters. What distinguishes them from the globular clusters is that they are much larger—several hundred light-years across—and hundreds of times less dense. The distances between the stars are, therefore, much greater within the newly discovered extended clusters.

In 2012, a microquasar, a radio burst emanating from a smaller black hole, was detected in the Andromeda Galaxy. The progenitor black hole is located near the galactic center and has about 10 ${\displaystyle {\begin{smallmatrix}M_{\odot }\end{smallmatrix}}}$. Discovered through a data collected by the European Space Agency's XMM-Newton probe, and subsequently observed by NASA's Swift Gamma-Ray Burst Mission and Chandra X-Ray Observatory, the Very Large Array, and the Very Long Baseline Array, the microquasar was the first observed within the Andromeda Galaxy and the first outside of the Milky Way Galaxy.

Satellites

Messier 32 is to the left of the center, Messier 110 is to the bottom-right of the center.

 

Like the Milky Way, the Andromeda Galaxy has satellite galaxies, consisting of 14 known dwarf galaxies. The best known and most readily observed satellite galaxies are M32 and M110. Based on current evidence, it appears that M32 underwent a close encounter with the Andromeda Galaxy in the past. M32 may once have been a larger galaxy that had its stellar disk removed by M31, and underwent a sharp increase of star formation in the core region, which lasted until the relatively recent past.

M110 also appears to be interacting with the Andromeda Galaxy, and astronomers have found in the halo of the latter a stream of metal-rich stars that appear to have been stripped from these satellite galaxies. M110 does contain a dusty lane, which may indicate recent or ongoing star formation. M32 have young population as well.

In 2006, it was discovered that nine of the satellite galaxies lie in a plane that intersects the core of the Andromeda Galaxy; they are not randomly arranged as would be expected from independent interactions. This may indicate a common tidal origin for the satellites.

Collision with the Milky Way

The Andromeda Galaxy is approaching the Milky Way at about 110 kilometers per second (68 miles per second). It has been measured approaching relative to our Sun at around 300 km/s (190 mi/s) as the Sun orbits around the center of our galaxy at a speed of approximately 225 km/s (140 mi/s). This makes the Andromeda Galaxy one of about 100 observable blueshifted galaxies. Andromeda Galaxy's tangential or sideways velocity with respect to the Milky Way is relatively much smaller than the approaching velocity and therefore it is expected to collide directly with the Milky Way in about 4 billion years. A likely outcome of the collision is that the galaxies will merge to form a giant elliptical galaxy or perhaps even a large disc galaxy. Such events are frequent among the galaxies in galaxy groups. The fate of the Earth and the Solar System in the event of a collision is currently unknown. Before the galaxies merge, there is a small chance that the Solar System could be ejected from the Milky Way or join the Andromeda Galaxy.

Amateur observing

The Andromeda Galaxy is bright enough to be seen with the naked eye, even with some light pollution. Andromeda is best seen during autumn nights in the Northern Hemisphere, when from mid-latitudes the galaxy reaches zenith (its highest point at midnight) so can be seen almost all night. From the Southern Hemisphere, it is most visible at the same months, that is in spring, and away from our equator does not reach a high altitude over the northern horizon, making it difficult to observe. Binoculars can reveal some larger structures and its two brightest satellite galaxies, M32 and M110. An amateur telescope can reveal Andromeda's disk, some of its brightest globular clusters, dark dust lanes and the large star cloud NGC 206.