DDonald Lynden-Bell CBE5th April 1935 – 6th February 2018Elected FRS 1978
Neil Wyn Evans
Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
30 March 2020
Abstract
Donald Lynden-Bell’s many contributions to astrophysics encompass generalrelativity, galactic dynamics, telescope design and observational astronomy. In the1960s, his papers on stellar dynamics led to fundamental insights into the equilibriaof elliptical galaxies, the growth of spiral patterns in disc galaxies and the stabilityof differentially rotating, self-gravitating flows. Donald introduced the ideas of‘violent relaxation’ and ‘the gravothermal catastrophe’ in pioneering work on thethermodynamics of galaxies and negative heat capacities. He shared the inauguralKavli Prize in Astrophysics in 2008 for his contributions to our understanding ofquasars. His prediction that dead quasars or supermassive black holes may reside inthe nuclei of nearby galaxies has been confirmed by multiple pieces of independentevidence. His work on accretion discs led to new insights into their workings, as wellas the realisation that the infrared excess in T Tauri stars was caused by protostellardiscs around these young stars. He introduced the influential idea of monolithiccollapse of a gas cloud as a formation mechanism for the Milky Way Galaxy. As thisgave way to modern ideas of merging and accretion as drivers of galaxy formation,Donald was the first to realise the importance of tidal streams as measures of thepast history and present day gravity field of the Galaxy. Though primarily a theorist,Donald participated in one of the first observational programs to measure the large-scale streaming of nearby galaxies. This led to the discovery of the ‘Great Attractor’.The depth and versatility of his contributions mark Donald out as one of the mostinfluential and pre-eminent astronomers of his day.
Donald Lynden-Bell was born on 5th April 1935 in Dover and christened in the Castle’schapel, one of the most ancient churches in England. His father, Lieutenant ColonelLachlan Arthur Lynden-Bell, fought on the Western Front and in the Middle East duringWorld War I and received a Military Cross. Donald’s very early childhood was spent inFort George, near Inverness, where his father was in command. It was a peripatetic life, as1 a r X i v : . [ phy s i c s . h i s t - ph ] J u l he family moved wherever his father was posted. Spells in Glasgow, Nairn and BroughtyFerry followed, before the outbreak of the Second World War. Despite being a professionalsoldier for 25 years, Donald’s father was now too poorly sighted for active service, andso he was despatched around the country to plan coastal defences. Donald spent theWar years at his grandparents’ house in Stonebarrow Hill, east of Charmouth in Dorset.The strongest influences on his early life came from his mother, Monica Rose, and hiselder sister, Jean. As a boy, he loved to collect fossils, shells, pressed flowers, moths andbutterflies, supplementing his earlier interests in mechanical toys. Donald found readingdifficult and did not share his mother’s love of literature and poetry. He finally managedto read at age nine, when his mother plotted a graph of number of words read each nightand would not let him stop until the graph was high enough. Donald remained a slow andreluctant reader throughout his life, but he always loved graphs!In 1944, Donald’s father joined the John Lewis Partnership and settled in ThamesDitton. After a couple of terms at Down House, a prep school in Esher, Donald was sentto a boarding school in Somerset to avoid the dangers of the doodlebugs. Once he hadovercome homesickness, Donald blossomed, enjoying early morning horse riding on theQuantocks. He became very interested in how things work, and took mechanical things topieces, building models with his Meccano. Donald also found out that he was very goodat mathematics, but his poor reading skills held him back in many other subjects.Nonetheless, he passed Common Entrance and went to Marlborough College in 1948.In those days, it was safe to bicycle on main roads, so it was possible to visit places up to10 miles away on a weekend afternoon. Donald got to know the flowers and butterfliesof the sweeping escarpment from Milk Hill down to Pewsey Vale, as well as the grandneolithic monuments of Avebury and Silbury Hill. Under the Labour Government, theSchool Certificate was replaced with O Levels, which at first could not be attempteduntil over the age of 16. As a result, Marlborough College decided that Donald’s yearshould be accelerated to take the last School Certificate at age 14. This was a greatblessing, as it allowed Donald to specialise in mathematics and science a year earlier,and give up subjects that he found uncongenial. At that time, Marlborough had somewonderful teachers in mathematics, including A.R.D. Ramsey (co-author of textbooks onmechanics) and E.G.H. Kempson (an Everest mountaineer who also taught climbing).Both had profound influence on Donald. Ramsey convinced his pupils that the gift ofmathematics was very special. His pupils should regard the great mathematicians of thepast as people to emulate and surpass. The best mathematicians were an elite, and itwas their moral duty to make the most of their gifts. Kempson taught Donald not justmathematics at a high level, but also rock climbing in Wales at the Idwal Slabs and theGlyderau. At the end of his time at Marlborough College, Kempson leant Donald a copyof James Hopwood Jeans’ book, Problems of Cosmogony and Stellar Dynamics . It wasa premonitory choice, as Donald’s thesis work was to make startling contributions to thissubject. A contemporary of Donald’s, Terry Wall FRS 1969, recalls that Donald wasalready devoted to astronomy and very active in the school’s Science Society.
Donald came up to Clare College, Cambridge in 1953. He convinced his Director ofStudies, R.J. Eden, that he should be allowed to skip Part I of the Mathematical Tripos,2ost of which he had already covered in school. Donald joined the MountaineeringClub (CUMC), the Astronomical Society, the Archimedeans and the Trinity MathematicalSociety, which at that time allowed Clare mathematicians to join. CUMC arrangedclimbing meets in Wales, the Lake District, Ben Nevis and the Alps. Donald felt awonderful sense of companionship with those on the same rope, as well as a love of thefreedom and serenity of the mountains.In those days, Heffers bookshop had a secondhand section on the top floor of theirshop in the old Petty Cury. There, Donald bought copies of the books of Arthur StanleyEddington FRS 1914, including
Space Time and Gravitation , The Mathematical Theoryof Relativity and
The Internal Constitution of the Stars . He was attracted to theoreticalastrophysics as mathematically challenging and physically fascinating. Donald’s Ph. D.thesis would later be dedicated to Eddington and his inspirational writing. Donald alsofound the quantum theory intriguing, but the interpretation of the mathematics deeplyunsatisfying. It gave up the idea of following what was happening in detail and substitutedmerely a calculation of the probabilities of the possible outcomes of an experiment. Tohim, it seemed the reverse of the disassembling of mechanical things to see how theyworked.Donald followed an eclectic choice of courses in his first year. Discovering thatHermann Bondi FRS 1959 was giving a Part III course on Relativity, he eagerly listenedin. Though a freshman, he found Bondi’s graduate-level lectures understandable, as theywere physically based and beautifully delivered. Donald also discovered that lectureson Observational Astronomy were held at the Observatories in the afternoon. So, heattended well-delivered and authoritative lectures by Donald Blackwell on the intricacies ofphenomenon observed on the solar surface. Perhaps because Donald was characteristicallyfollowing his own path, he only achieved an upper Second at the end of the first year.In his second year, Donald was supervised by two men who become very famous,namely Sir Michael Atiyah PRS 1990 and Abdus Salam (1979 Nobel prize winner inphysics). Salam held the strong belief that one could only really learn physics fromdirect contact with those carrying out experiments, and firmly advised Donald to leavethe Mathematical Tripos and switch to Physics Part II in the Natural Sciences Tripos forhis third year. Brian Pippard FRS 1956, who was Director of Studies in Physics at ClareCollege, tried to persuade Donald to take two years over Part II, but Donald insisted ondoing it in a year. He attended stimulating lectures by Otto Frisch FRS 1948 on nucleartheory, Martin Ryle FRS 1952 on Radio Astronomy and Nevill Mott FRS 1936 on quantummechanics. He learnt much from Joe Vinen FRS 1973, who had just performed his famousexperiments on the quantised vortices of liquid helium. At the end of the examinations,Donald went climbing on Ben Nevis, ascending the difficult route over the mighty cliffbelow the summit. There was little communication with the outside world, so when hethought the results would be out, he descended to Fort William and bought a newspaper.He discovered that he had an upper second again, and so was unsure whether he couldcontinue to research. Back in Cambridge, he was offered Ph. D. positions to work onexperimental studies of turbulence or on radioastronomy, but he eschewed both as tooexperimental for his tastes. He preferred the idea of taking Mathematics Part III with aview to doing a Ph. D. in Theoretical Physics.Although Donald’s Part III year was full of interesting courses, he did not have thesame freedom of spirit to take lectures on what he found most interesting. He had to learn3he main courses really well for the examinations – Fred Hoyle FRS 1957 on Cosmology,Leon Mestel FRS 1977 on Cosmical Electrodynamics, Paul Dirac FRS 1930 on QuantumField Theory and Nevill Mott on Atomic Collisions. This blunted his joy in finding thingsout. In the end, he obtained the desired distinction, but it was drudgery rather than love.He then found that most of the interesting potential supervisors in his preferred subject ofquantum theory were about to go on leave. So, Donald decided to turn his long-standinghobby of astronomy into his work, by becoming a graduate student of Mestel, investigatingthe role of magnetism in astronomyDuring his Part III year, Donald attended meetings of the Cambridge University NaturalSciences Club (CUNSC). This was limited to 13 people – the maximum that would fit intoan undergraduate room. Members read papers to one another about anything they foundinteresting, with the proviso that the lecture and discussion must be in terms everyonecould understand. Donald found it interesting to learn more of the life sciences, especiallybotany and evolutionary biology. Through the club, he met Ruth Truscott FRS 2006,who he found full of energy and fun. He was also greatly delighted to find someone who“enjoyed mathematics rather than feared it". Donald and Ruth were subsequently marriedfour years later on 1 July 1961 in Mosely Parish Church, and remained great supportersof CUNSC which played an important part in both their lives.
While Donald learnt a great deal from Mestel, his thesis work did not go well. Hisproblem was to understand the effects of magnetic fields on star formation, but neitherDonald nor Mestel were familiar with the necessary computational techniques to solvethe awkward integro-differential equation that confronted them. Donald commented laterthat his “algorithm was numerically unstable, the problem was physically unstable andthe whole exercise not of prime importance for star formation”. After a frustrating year,Donald attended a summer course at the Royal Greenwich Observatory, then based atHerstmonceux Castle in Sussex. He struck up a friendship with the Astronomer Royal,Sir Richard Woolley FRS 1953, who set him some problems in stellar dynamics. Woolleywas a good scientist and a natural leader of men. He was very encouraging when Donaldmade some progress. When nearly a year later, the work with Mestel had completely runinto the sand, Donald already had an embryo paper based on Woolley’s problem. Thisencouraged Donald to shelve the work with Mestel completely, and concentrate on newproblems in stellar and galactic dynamics. This proved to be Donald’s salvation. Somenine months later, in an inspired burst of creativity, Donald had solved some important anduseful problems in stellar dynamics. He was able to present a research thesis for the JuniorResearch Fellowship competition at Clare College, which he won. Soon afterwards, hewas also awarded a Harkness Fellowship. It was a remarkable nine months turnaround.Donald’s 1960 thesis is entitled
Stellar and Galactic Dynamics . No first-rate math-ematician had really looked at this subject since the days of James Jeans FRS 1906 andEddington in the 1930s, so Donald was able to make substantial progress with analyticalmethods, always his forte. Jeans (anticipated by Poincaré) had already found that thedistribution function of a collisionless stellar system must be a function of the integrals ofmotion only (Binney and Tremaine, 2008). Donald developed a new method to identifypotentials admitting integrals of motion, and used it both to find new integrable systems4nd to categorise the ones already known (1). He laid emphasis on the importance of stel-lar systems with potentials separable in confocal ellipsoidal coordinates, first introducedinto astronomy in an informal way by Eddington. Donald called these systems Eddingtonpotentials, but they are now known in galactic dynamics texts as Stäckel potentials (Binneyand Tremaine, 2008) – "mistakenly . . . as he never derived them" (58). Donald discov-ered many of the properties of stellar systems with potentials separable in ellipsoidalcoordinates. Years later in the 1980s, these potentials were to attain a new vogue whenTim de Zeeuw, discovered the ‘perfect ellipsoid’ on a working visit to Donald in Cam-bridge (30). This is an exact model whose density law is similar to elliptical galaxies andwhose potential is separable in ellipsoidal coordinates (de Zeeuw, 1985). The work retainsrelevance to this day, as modern stellar dynamics lays heavy emphasis on action-anglecoordinates (Sanders and Binney, 2016). Fast numerical algorithms have been developedto approximate the potential by a nearby Stäckel potential and hence obtain actions andangles, and these form the basis of modern galactic dynamics software packages.Donald also attacked the self-consistent problem of stellar dynamics – that is, thefinding of the phase space distribution function of a stellar system (which can dependonly on the integrals of motion through the Jeans Theorem) consistent with the potentialand density. This necessitates the solution of a Volterra integral equation of the firstkind. If the stellar system is spherical, the distribution function can be extracted viaan Abel transform, as Eddington had discovered in 1916. Donald showed how to finddistribution functions for axisymmetric systems – a much harder problem – using Laplacetransforms (2). He provided a flattened Plummer model (suggested by Maarten Schmidt)for which the calculations are all analytic and so created the first axisymmetric, self-consistent galaxy model with a distribution function. These ideas were later developedand extended by others (Hunter and Qian, 1993; Evans, 1994). Although Woolley deservesthe credit for inspiring and encouraging Donald, it is clear that Donald’s mind had alreadybeen prepared for these speedy discoveries in galactic dynamics by his wide reading ofthe books of Jeans and Eddington. As his nominal supervisor Mestel remarked years laterto the author, “Donald did not really need supervising".
Donald took his Harkness Fellowship to CalTech. There he found congenial companywith two other English postdocs, Wallace Sargent FRS 1981 and Roger Griffin. Togetherwith Neville Woolf, the four organised a number of great camping and hiking trips in theAmerican Southwest. This included trips to Death Valley, Meteor Crater and RainbowBridge. A famous photograph (reproduced in Donald’s own biographical memoir of WalSargent) shows the four of them, joined by John Hazlehurst, having Christmas dinner in1960 at the bottom of the Meteor crater, Arizona, surrounded by a Union Jack. In April2011, the four men – now grey with age – reconvened on the 50th anniversary of theoriginal expedition to Rainbow Bridge. The reunion was the basis of the film
Star Men byAlison Rose, a graceful tribute to scientific passion, collegiality and ageing.In Caltech, Donald wrote one of his most famous papers (3), together with Olin Eggenand Allan Sandage and often referred to as ELS. They saw a strong correlation betweenthe ultraviolet excess and the orbital eccentricity for a sample of high velocity stars inthe solar neighbourhood. Stars with large ultraviolet excess have low metallicities. ELS5igure 1: Donald Lynden-Bell pictured in 2011 near Rainbow Bridge, Utah during thefilming of Star Men, directed by Alison Rose. This recreated an adventure Donald hadundertaken with Wal Sergent, Nic Woolf and Roger Griffin as postdocs in the 1960s.(Photo Credit: Inigo Films) 6ound that – locally – the most metal-poor stars have the largest orbital eccentricities, thehighest velocity dispersions and the lowest systemic rotation. In a provoking hypothesis,ELS suggested that the metal-deficient stars formed first in an approximately sphericalconfiguration. This original gas cloud then collapsed quickly, giving birth to subsequentgenerations of stars. The contraction of the Galaxy not only yielded a population ofyounger stars on nearly circular orbits, but also made the orbits of the halo stars highlyeccentric, thus neatly explaining the chemo-kinematical properties of their local stellarsample. This became the standard picture of galaxy formation in the 1960s and 1970s.However, two aspects of the ELS picture became troubling. First, the timescale forcollapse was estimated by ELS as the free-fall time or ≈ × years. This became difficultto reconcile with age spreads of Gigayears among the halo stars, suggesting a much moreprotracted formation. Second, the progressive chemical enrichment of the halo envisagedby ELS implied a strong correlation between metallicity and eccentricity for all halo stars,as well as a metallicity gradient with distance. Subsequent surveys, coupled with moreprecise determinations of stellar ages and elemental abundances, have revealed the chemo-dynamical history of the Galaxy in much greater detail than was available to ELS (Searleand Zinn, 1978; Beers and Sommer-Larsen, 1995). There is only a weak relationshipbetween metallicity and eccentricity for halo stars, and there is no detectable metallicitygradient. It is now believed that the correlation between ultraviolet excess and eccentricityseen by ELS in the local sample may have been caused by a selection bias affecting thedata (Norris et al., 1985). Notwithstanding this, the theoretical insights into the youngGalaxy propounded by ELS remain as shrewd today as they were over 50 years ago.The ELS picture of the formation of the Milky Way galaxy has been superseded byscenarios in which structure is accumulated hierarchically from smaller fragments. AlarToomre (1977) was already speculating “It seems almost inconceivable that there wasn’ta great deal of merging of sizeable bits and pieces (including quite a few lesser galaxies)early in the career of every major galaxy no matter what it now looks like. The processwould obviously have yielded halos from the stars already born, whereas any leftover gaswould have settled quickly into new discs embedded within such piles of stars". If so, thenthe properties of the halo stars depend on their progenitor satellites, and no tight correlationbetween dynamics and chemistry is expected. Recent discoveries with astrometric datafrom the Gaia satellite provide irrefutable corroboration with the identification of the earlymergers that built the Milky Way Galaxy (Belokurov et al., 2018; Myeong et al., 2019).Although no longer viable as a mechanism, ELS has proven to be very influential. It wasthe first paper to emphasise that halo stars take very long times to exchange energy andangular momentum with the rest of the Galaxy. Hence, they retain memory of early timesand so can be used to study the process of galaxy formation. It was also the first paper onthe formation of the Galaxy that provided testable and quantitative predictions, and so itplayed its pioneering role in opening up a new discipline to scientific scrutiny.As part of his contribution to ELS, Donald also developed a supremely elegant galaxymodel, as recounted in (38). Unbeknowst to him, it had been independently found a fewyears earlier by Michel Hénon (1959) and named ‘L’Amas Isochrone’. The isochrone isthe most general spherical potential for which the orbits, actions and angles of stars arecalculable analytically as closed form expressions (4). It is so named because the periodof any star depends only on its energy, and not on its angular momentum. The model isstill widely used as a simple, reasonably realistic representation of a galaxy and earns a7ightful place in the standard texts on galactic dynamics (Binney and Tremaine, 2008).
Donald returned to Cambridge in 1962 as a University Assistant Lecturer at the Departmentof Applied Mathematics and Theoretical Physics (DAMTP), as well as Director of Studiesin Mathematics at Clare College. In his Ph. D. thesis, he had early ideas on regenerativetheories of spiral structure. He returned to this problem, with an engaging collaborator,Peter Goldreich, then a young NSF postdoctoral fellow in Cambridge. Together, theywrote two important papers (hereafter GLB) on the stability of gaseous disks, in theprocess discovering “swing amplification” (5,6). Since the Earl of Rosse’s discovery in1851 of spiral patterns in the nearby disc galaxy M51, the persistence of spiral structureagainst differential rotation was a major unsolved problem. By the 1960s, the riddle ofthe spirals was attracting the attention of an impressive range of applied mathematicians,astrophysicists and fluid dynamicists, including C.C. Lin, Frank Shu, Chris Hunter, AgrisKalnajs and Alar Toomre. A swashbuckling recent history gives a flavour of those piraticaltimes in spiral structure theory (Pasha, 2004).GLB’s work on gaseous discs, together with the almost contemporaneous study onstellar discs by Julian and Toomre (1966), were major pieces in solving the puzzle. GLBfound the unlocking key in a new shearing coordinate system, set up in a local patch ofa differentially rotating disc. The co-moving axes are oriented radially and tangentiallyalong the shearing flow, and the wave harmonics are sought in these new coordinatesas solutions of a neat second-order ordinary differential equation. A leading spiral waveunwinds into a trailing wave thanks to the differential rotation. At early times, the intercrestspacing is small and gas pressure ensures stability. As the waves are swept round, thepressure loses its potency, the wavelength rises and the shear takes over. This amplifies theunwrapping wave into a vigorously growing, trailing spiral. The conspiracy between self-gravity, shearing due to differential rotation and gas pressure (or epicyclic motion for stars)was later christened swing amplification, thanks to a hat-tipping article by Alar Toomre(1981), who first demonstrated the stellar dynamical analogues. In this picture, spiralityarises from regenerative “sheared gravitational instabilities". Spiral arms in galaxies arebeautiful transients that bloom and fade. Tidal forcing by companions (such as NGC 5194in the case of the Earl of Rosse’s M51) or gravitational wakes around massive objects(like giant molecular clouds) provide the seeds. In fact, GLB’s machinery was powerfulenough to handle most of the later work of the Lin-Shu school, where density-wave theorywas worked out using different methods (Goldreich and Tremaine, 1978).Donald resumed collaboration with Goldreich some years later on a very differenttopic – the then recently discovered bursts of decametric radiation from Jupiter, whichare correlated with the phase of Io, its closest moon (12). Jupiter’s magnetic field movesrapidly past Io as it orbits. Io is a good conductor, so a current is produced by its motionthrough the Jovian magnetic field. The plasma enclosed by the satellite’s flux tube movesas though it were rigidly attached to Io. The electromotive force developed across itsdiameter due to its motion is transmitted along the flux tube which passes through thesatellite and drives a current across each foot of the tube in the ionosphere. In Goldreich& Lynden-Bell’s model, it is the motion of the current up and down the flux tube thatgenerates the decametric bursts through beam instabilities. This model broke new ground8n showing how Io may play the role of a unipolar conductor and so control the bursts.The hot spot where Io’s flux tube meets Jupiter was finally found. It pointed 15 ◦ ahead ofthe orbital position of Io, just as they had predicted. As Donald recounted ruefully later, “it was not very long before I discovered that threehalf-jobs, lecturing, researching and teaching for one’s college, added up to one anda half rather than one" (56). To get more time for research, Donald decided to leaveCambridge and work under Woolley at the Royal Greenwich Observatory (RGO), then inHerstmonceux Castle in Sussex. No doubt Donald was also influenced by the inspirationalrole that Woolley played during his PhD years. His wife Ruth was appointed to a half-timelectureship in Chemistry at the new University of Sussex. They welcomed their daughterMarion to the family in 1965, and son Edward in 1968. Donald’s research blossomed atthe RGO in an intensely creative period under the stimulus of daily contact with observersin a working observatory.
Throughout the 1950s and 1960s, astronomers had been puzzled by the smooth light dis-tributions of elliptical galaxies, given that the timescale for two-body relaxation exceededtheir ages. This paradox is forcefully articulated in some of the textbooks of the time,such as
Dynamics of Stellar Systems (Ogorodnikov, 1965). Donald solved this problemby showing that collective relaxation effects can result in a very rapid or “violent relax-ation" (8). This proceeds through collisionless interactions, and thus the fine-grainedphase-space distribution of galaxies is conserved. However, if the relaxation is suffi-ciently violent (for example, if the energy of stars is substantially changed by a fluctuatingpotential), then initially adjoining units of phase-space become widely separated in thefinal state through phase mixing. Averaging the fine-grained phase-space density overany observable volume gives a coarse-grained distribution, which can be an equilibriumdistribution.Donald showed by an ingenious argument in statistical physics that the endpointof violent relaxation is independent of the details of the initial state. In doing this,he discovered a new type of statistical mechanics. Particles that are indistinguishablefollow Fermi-Dirac or Bose-Einstein statistics according to whether they obey an exclusionprinciple or not. Particles that are distinguishable without an exclusion principle obeyMaxwell-Boltzmann statistics. The remaining case of distinguishable particles obeyingan exclusion principle is now known as “Lynden-Bell statistics" in standard texts (TerHaar, 1977). It provides the equilibrium distribution function of a collisionless system inthe idealized limit when violent relaxation proceeds to completion.Nowadays, the Lynden-Bell distribution function is not believed to describe the equi-librium properties of elliptical galaxies. Since the 1990s, increasingly powerful numericalsimulations have mimicked the process of violent relaxation as part of the recipe of galaxyformation. Violent relaxation is more ineffective than envisaged in Donald’s statisticalargument, and some memory of environment and initial conditions is retained. Donaldrecognized this shortcoming in later work on H -functions (31) with Scott Tremaine FRS9igure 2: Donald Lynden-Bell pictured at Herstmonceux Castle in the mid 1960s (Courtesyof Ruth Lynden-Bell) 10994 and Hénon (THL). Here, the aim was to constrain the direction of evolution of acollisionless system by the principle of increase in entropy, a more modest goal than in(8). THL showed any equilibrium stellar system can be described as the stationary pointof some H -function – analogous to Boltzmann’s H -function in the kinetic theory of gases– provided its phase space distribution function is a decreasing function of energy alone.All such H -functions increase during violent relaxation, and the distribution function be-comes more and more mixed. THL used this to demonstrate a violently relaxed galaxyresembles observed elliptical galaxies only if the initial state is cold and clumpy. Donald received through the mail an astonishing 1962 paper by Vadim Antonov from thenLeningrad in the Soviet Union. It was finally translated and made widely available inEnglish over twenty years later (Antonov, 1985). Donald was unable to read Russian, buthe could follow the mathematics. Antonov took an assembly of self-gravitating particlesin a spherical container and maximized their entropy at fixed total energy. He found thatthe entropy was only a (local) maximum if the ratio of the density at the centre to thatat the container is less than 708. Above this value, no maximum exists. Donald carriedout his own investigations of the thermodynamics of a self-gravitating gas in a sphericalcontainer in collaboration with Roger Wood (11). At a density contrast of 389, Donaldfound that the heat capacity of the gas at constant volume becomes infinite and, for slightlygreater contrasts, it attains large negative values. Henceforth, the heat capacity increasesas the contrast increases, finally reaching zero at Antonov’s critical value (Binney andTremaine, 2008). Donald discovered that there is a range of density contrasts from 389 to708 at which maximum entropy systems exhibit negative heat capacity. Although it hadlong been known from the virial theorem that negative heat capacities occur for isolatedself-gravitating systems, the result was believed to have limited implications. Isolatedsystems cannot be in thermal equilibrium under gravity because particles can escapethrough evaporation or encounters. Antonov’s and Lynden-Bell & Wood’s gas spheres incontainers were the first true equilibrium systems displaying negative heat capacity.Donald quickly realised that this was the basis for an runaway instability, which hecoined “the gravothermal catastrophe". At high temperature, a negative heat capacitysystem in contact with a cold sink will give up heat until it is entirely exhausted, gettinghotter, rather than colder, as it does so. So, if a high temperature and a low temperaturegravitational system are in contact, energy is transferred out of the hot portion, whichcontracts, and into the cold portion, which expands. This naturally builds self-gravitatingsystems with tight cores and extended haloes or coronae. Donald originally envisagedapplications to the contracting phase of a red giant star. We now believe that the catastropheoccurs when the core of a globular cluster shrinks and heats up, causing it to transfer energyto stars in the cluster’s halo, and leading to core collapse. Donald, in later work withPeter Eggleton, followed up the implications of the gravothermal catastrophe for globularclusters (26). They showed not just the evolution of the core, but also the surroundings,are self-similar. The debris of material expelled from the core follows a power law in thedensity with exponent in the range -2 to -2.5.Negative specific heats fascinated Donald and he returned to the subject repeatedly.There is a standard thermodynamic proof that heat capacities are always positive, repeated11y Schrödinger (1968) in
Statistical Thermodynamics . So, Donald’s ideas were initiallyreceived with scepticism by theoretical physicists. It was later demonstrated by Thirring(1970) that the standard proof only holds for systems that can be in equilibrium in canonicalensembles. Microcanonical systems can have negative heat capacities. Together withRuth Lynden-Bell, Donald showed that phase transitions may be viewed as being dueto microscopic elements with negative heat capacity (21,47). He identified such unitsin the theories of ionization, chemical dissociation and the Van der Waals gas. Donaldtherefore believed that negative heat capacities are of general applicability in physics, andnot confined to self-gravitating bodies, like star clusters and black holes.
Donald started by studying cooperative phenomena in homogeneous stellar systems (9).Although not realistic, such models can still do useful work (as Chandrasekhar’s (1942)calculation of the dynamical friction formula in
Principles of Stellar Dynamics showed).Donald proved that two-stream instabilities do not exist in a homogeneous stellar systemconsisting of two Maxwellian streams. Using Nyquist diagrams, he found that instabilityoccurs only when at least one of the streams has a flat-topped velocity distribution. WithNigel Sanitt, Donald showed that a stellar system is stable whenever the correspondingbarotropic gaseous system is secularly stable (13). Their Schrödinger operator method(essentially an energy principle) showed that a large class of spherical stellar systemsare stable to all non-spherical modes of vibration, but stability to spherical modes wasfrustratingly harder to assess. Donald concluded, sadly but correctly, that “more generaland more powerful methods are needed in the field of galactic stability". In fact, progressin identifying the instabilities of spherical or elliptical galaxies has not come from energyprinciples, but had to wait until the advent of powerful numerical simulations in the 1990s.Donald also developed a secular stability criteria for stars and galaxies in work withJeremiah Ostriker, then an NSF Fellow in Cambridge. In their paper (henceforth LBO),they derived a variational principle to determine the secular stability of any differentially-rotating, self-gravitating fluid flow. With some modification (Bardeen et al., 1977), thishas proved useful in many areas of astronomy – for example, helioseismology, the stabilityof rotating stars and the effects of tides in extrasolar planets. LBO applied their result to thenormal modes of oscillation in galactic discs and formulated the “Anti-spiral Theorem".This states that, if in linear theory there exists a global spiral mode of trailing planformthat rotates without growing or decaying, then a similar mirror-image leading mode mustalso exist as well. The Anti-spiral Theorem was criticism of the then prevalent idea oflong-lived quasi-static spiral structure promulgated by C.C. Lin and co-workers. This wasthe very antithesis of GLB’s view of spirality as sheared gravitational instabilities. Donaldremarked later: “I always held the view that angular momentum transfer is the drivingforce behind spiral structure. In part the Anti-spiral Theorem was there because it seemedto point out that what C.C. Lin said was much less than the whole story."Donald returned to the fretfulness of spiral structure theory when Agris Kalnajs arrivedas a Research Fellow at the RGO. Together, Lynden-Bell & Kalnajs (LBK) laid one ofthe cornerstones of stellar dynamics, studying the role of resonances in the growth anddecay of spiral waves (16). LBK showed that there is absorption of angular momentumby stars that resonate with the wave at the outer Lindblad resonance. Emission of angular12omentum occurs at the inner Lindblad resonance. LBK derived a formulae for angularmomentum transport and showed that the role of spiral structure is to carry angularmomentum from the inner to the outer parts. LBK likened this advective transport toa fleet of trucks carrying coal. The trucks travel outward full of coal and return empty,so there is an outward flow of coal but no net flow of trucks. Similarly, stars can carryangular momentum outward, deposit it near the apocenter of their orbits, and return toacquire more angular momentum near pericenter, leading to an outward flow of angularmomentum with no outward flow of mass. LBK also noted that an orbiting star subject toweak tangential forces can exhibit unusual behaviour. Orbits close to corotation have verysmall mean motions in the corotating frame. Those on the inside drift slowly forwards,those on the outside drift slowly backwards. On feeling the forward tug of a spiral arm, aforward moving star gets onto an epicycle with a slightly greater angular momentum andits mean motion, far from speeding up, will slow down. Thus, uncooperative stars act “likedonkeys slowing down when pulled forwards and speeding up when held back." Donaldextended these ideas into a lucid treatment of resonant orbits (17), borrowing techniquesthat Max Born FRS 1939 originally developed in
The Mechanics of the Atom during thedays of the old quantum theory.LBK also suggested that galactic bars are standing waves which orient and trap themajor axes of orbits with two lobes so that they lie along the bar. (This is physicallyidentical to the radial orbit instability in elliptical galaxies). In this picture (25), bars canform only in regions in which the rotation curve rises slowly to a maximum. Once therotation curve turns over, the major axis of an elliptic orbit behaves like a ‘donkey’. Whensubject to a torque pulling forward, its precession slows down, so it anti-aligns with thebar’s potential well and weakens it. This elegant idea may hold true for some slow bars, butit is not the whole story, as numerical simulations and observational data both favour fastbar pattern speeds. As in the Jeans instability, gravity will have its way in organising orbitsonly provided the system is not too hot. So the mechanism of aligning and anti-aligningorbits has to be taken hand-in-hand with the population of orbits. Just as a hot gas canresist Jeans collapse, so a wide dispersion of orbital precessions can inhibit the formationof structures like bars through orbital cooperation.
The most famous piece of research that Donald did at Herstmonceux Castle is his startlingidea that quasars are powered by supermassive black holes and that most large galaxies,including our own, could host a dead quasar in their nucleus (14). Years later, this wasto win him the 2008 Kavli Prize in Astrophysics, jointly with Maarten Schmidt, “for theirseminal contributions to understanding the nature of quasars".Quasars were identified as ultraluminous active galactic nuclei following Cyril Haz-ard’s accurate radio position of the quasar 3C 273 in 1963, followed by Maarten Schmidt’sdetermination of its redshift of 0.16. The physical mechanism that powered the quasarsremained obscure. The nuclear region of 3C 273 is then less than 1 kpc in diameter, whilstits associated radio and optical jet is about 50 kpc away, implying a timescale greater than10 years. The optical luminosity of 3C 273 is enormous, at least 10 times brighter thanthe brightest giant elliptical galaxy. The idea that black holes may provide the energysource for quasars had been mooted in the literature – though the term “black hole" only13ecame commonplace after 1967. Hoyle and Fowler (1963) suggested that "only throughthe contraction of a mass of ≈ M (cid:12) to the relativistic limit can the energies of thestrongest sources be obtained". Soon after, Salpeter (1964) and Zel’dovich (1964) pro-posed that accretion onto a supermassive black hole may be the source of the luminosity.If material gradually spirals on to the innermost stable orbit of a nonrotating black hole at r = GM / c , the energy released per unit mass is ≈ . c , enough to provide the energyof a luminous quasar from a plausible mass.Donald was aware of the frenzy amongst observers about quasars from his time asHarkness Fellow at CalTech. A freak of numerology now spurred him on. “It was nowfour years since I had left Cambridge for the RGO and from our home in Barcombe I droveto Herstmonceux with Bernard Pagel FRS 1992. Part of our route lay along the A273road, and this daily reminded me that there was still no accepted theory of the quasar 3C273" (56). Donald knew that quasars have active periods and only rarely are they verybright. He made a rough estimate of how common dead quasars would be and concludedthat the nearest one to an average point in the Universe would be about 3 Mpc away. He thenargued that dead quasars in the form of “collapsed bodies" (or supermassive black holes)should be ubiquitous in galactic nuclei, given the lifetime energy output of quasars andtheir abundance at early times. He suggested that dead quasars may be detectable throughmeasurement of the mass-to-light ratios of nearby galactic nuclei. Donald explored thethermal radiation and particle emission produced by a disk of gas orbiting the hole, withenergy dissipation related to magnetic and turbulent processes. He found that “withdifferent values of the [black hole mass and accretion rate] these disks are capable ofproviding an explanation for a large fraction of the incredible phenomena of high energyastrophysics, including galactic nuclei, Seyfert galaxies, quasars and cosmic rays." Thisvery influential paper made plausible to most astronomers that the black hole model –which had hitherto received scant attention – was the correct explanation of quasars andactive galactic nuclei. Together with Martin Rees, he explored the idea that the nearestsupermassive black hole to us may even be at the centre of the Milky Way Galaxy (15).Donald’s predictions were confirmed within his lifetime. High resolution spectroscopy,especially with the Hubble Space Telescope , of galactic nuclei produced an abundanceof evidence for compact dark mass concentrations of up to 10 M (cid:12) pc − (Kormendy andRichstone, 1995). Even stronger evidence exists for NGC 4258, for which the rotationalkinematics of water masers show the existence of a central supermassive black hole withmass of 3 . × M (cid:12) (Miyoshi et al., 1995). The black hole at the centre of the Milky Waywas confirmed by spectroscopic and proper motion studies of stars, exploiting adaptiveoptics (Schödel, 2002; Ghez, 2008). The compact radio source Sagittarius A (cid:63) is the MilkyWay’s supermassive black hole. It has a mass ≈ × M (cid:12) and a Schwarzschild radius of0 .
08 astronomical units, as judged from the accelerations of stars orbiting around it. Finalconfirmation of Donald’s prediction was provided by the
Event Horizon Telescope , whichsynthesized radio images of M87 to give a dramatic picture of the shadow of its centralsupermassive black hole – perhaps the single most famous image of 2019, but one thatDonald did not live to see. 14
Return to Cambridge (1972)
Sir Richard Woolley retired as Director of the RGO in 1971 and this precipitated Donaldto look for a professorship elsewhere. Donald was particularly interested in developinghis ideas in mathematical relativity, and so wanted stimulus and encouragement fromother theoreticians, whereas the RGO was almost wholly observational. As chance wouldhave it, two astronomical professorships were vacant in Cambridge, the Jacksonian Chairin Natural Philosophy at the Cavendish Laboratory and the Chair of Astrophysics at theObservatories, formerly occupied by the observational astronomer Roderick Redman.Donald consulted Ryle on which position to apply for. Ryle recommended Donald applyfor the Chair of Astrophysics.What then happened is entangled in the confusing story of the enmity between Ryle andHoyle, then the Plumian Professor of Astronomy and Natural Philosophy at the Instituteof Theoretical Astronomy, Cambridge. Donald’s recollection of events is given in (56),whereas Hoyle’s lively memories are recounted in his rollicking autobiography,
Homeis Where the Wind Blows (Hoyle, 1994). Although Donald behaved with good gracethroughout, events sadly so conspired as to cause Hoyle to resign his Professorship andleave Cambridge forever.Unbeknownst to Donald, Hoyle had hoped to encourage a leading observational as-tronomer to occupy the Chair of Astrophysics. His preferred candidates were Sargent orGeoffrey Burbidge FRS 1968. The other electors dissented, and after much wrangling,Donald was elected against Hoyle’s wishes.This all occurred whilst astronomy in Cambridge was undergoing reorganization.Hoyle had founded the Institute of Theoretical Astronomy (IoTA) in 1968, thanks to grantsfrom Lords Wolfson and Nuffield. These provided funds for a building, a powerful com-puter, and 12 posts for five years. Hoyle attracted many young and gifted astronomersto IoTA, but the long-term funding was always precarious. Anticipating the expiry ofthe starting grants, the University produced a report on the future of Cambridge astron-omy, which advocated that IoTA be merged with the old Observatories. This was verymuch Hoyle’s vision, as he hoped Cambridge could complement its strong theoreticalresearch with observational programs exploiting new large-scale facilities, like the Anglo-Australian Telescope. This merger created a new University department, the present-dayInstitute of Astronomy (IoA).Hoyle assumed that he would be the Director of the newly created IoA, to which Donaldwas happy to assent having no great desire to take on administrative duties (56). However,in a situation of muddled complexity, Hoyle came to believe that Donald was being chosenas the Director behind his back by the University. Hoyle perceived this as the culminationof many slights, following years of prolonged feuding with some senior members of theBritish astronomy community (Hoyle, 1994). He resigned from the Plumian chair inSeptember 1971 and left Cambridge academic life forever for the remote Cockley Moorin the fells above Ullswater.So, when Donald arrived in Cambridge in 1972, he faced a huge task as directorof a department which had been created under disputatious conditions and which had12 staff members whose jobs would expire at the end of the year! Donald’s pragmaticreaction was to strike a deal with George Batchelor FRS 1957, fluid dynamicist andHead of the Department of Applied Mathematics and Theoretical Physics (DAMTP).15athematical relativity and theoretical cosmology were henceforth transferred to DAMTP,who provided permanent lectureships. This was the nucleus of DAMTP’s general relativityand gravitation group, which was to became world-famous under Stephen Hawking’sintellectual leadership in the 1980s and 1990s. Two further lectureships, one shared withDAMTP, were created and offered to Douglas Gough FRS 1997 and Peter Eggleton. Thenew department was beginning to stabilize, when, in a bold coup, Donald enticed MartinRees FRS 1979 to return from the University of Sussex as a youthful Plumian Professor.Donald and Martin were to run the IoA amicably over the next twenty years, alternatingthe Directorship for five year periods from 1972 to the mid 1990s. The number of staff,fellows and students increased from 70 in 1972 to 138 at the end of Donald’s reign asDirector in 1994. Donald and Martin together oversaw the growth of the IoA into one ofthe premier research institutes in astronomy in the world.Despite a difficult induction as Director, Donald was generously forgiving. His opinionof Hoyle’s many scientific achievements remained extremely high (48). Donald alwaysregarded Hoyle as possessing a very remarkable combination of scientific imagination,mathematical ability and physical insight (as well as Yorkshire bluntness). He stronglyregretted the omission of Hoyle from the award of the Nobel Prize in Physics in 1983, whichwent to Hoyle’s long-time collaborator William Fowler for studies of the formation of thechemical elements in the universe. The author recalls Donald remarking forcefully, “Thework would never have happened without Fred".
Nowadays, there is huge interest in accretion disks, driven by their ubiquity in astrophysicsfrom supermassive black holes in active galactic nuclei to star and planet formation. Thishas ensured that Donald’s next paper (18) is now his most highly cited of all.Given a source of dissipation, an accretion disc causes angular momentum to betransported outward, leading to an expansion of the outer parts, accompanied by theaccumulation of more and more mass towards the centre. This observation is at least asold as 1927, appearing in Sir Harold Jeffreys’ discussion of the solar nebula in the firstedition of his textbook
The Earth . In his
Nature paper on quasars, Donald assumed thatthe main mechanism was magnetic viscosity, or the transfer of angular momentum bymagnetic stresses. But, Donald had forgotten his own earlier work! In his 1960 Ph. D.thesis, he had already derived – but not published – a set of solutions for thin accretion discswith kinematic viscosity. A graduate student at the IoA, Jim Pringle then re-discoveredthem in his 1974 Ph. D. thesis! And, although they did not know it then, both had beenpartly anticipated by Rainer Lüst (1952), a student of Carl von Weizsäcker (see Pringle(1981) for the historical details).Together, Donald and Pringle (hereafter LBP) joined forces to write an important paperon accretion discs (18). The equation for the evolution of the surface density of an discadmits exact analytic solutions when the kinematic viscosity depends only on a power-lawof the radius, and not on time or surface density. The solutions fall into two classes, one16igure 3: Donald Lynden-Bell during the 1970s while Professor of Astrophysics at Cam-bridge University (Courtesy of Ruth Lynden-Bell).17or which the torque at the centre is zero and all the matters flows to the centre (accretiondiscs), the other for which the torque at the centre is finite, and the matter is expelled toinfinity (decretion discs). LBP also provided the Green function – that is, the evolutionof a viscous ring of material – from which these more general solutions of the linearparabolic equation are constructed. Turbulent flows and the resulting eddies certainlyoffer a transport mechanism for angular momentum and mass, which is independent ofthe existence of magnetic fields. LBP provided a generic model for the evolution of thinaccretion discs, though detailed specification of the angular momentum transport and anymass loss processes at work is needed for predictive power in any particular application.The main application in LBP is to what were then called “the nebular variables",nowadays T Tauri or pre-main sequence stars. At the time, it was known that T Tauristars have both an ultraviolet excess and an infrared excess. LBP’s explanation is thatthe excesses are caused by accretion discs around these young stars. The flux from thedisc has a flatter spectrum than Rayleigh-Jeans and so dominates the spectrum at longerwavelengths and provides the infrared excess. The blue end is dominated by emissionfrom the boundary layer where the disc and star meet and high temperatures prevail. Withsome modifications, this has remained the standard picture till today. The infrared excessis one of the primary methods used to identify young stellar objects and proto-planetarydiscs in astronomical surveys.
In the 1980s, Donald was one of seven astronomers – the Seven Samurai – who conductedan ambitious program to measure the distances and recession velocities for 400 ellipticalgalaxies. The aim was to understand the large scale peculiar motions of nearby galaxies(including our own) with respect to the Hubble flow. The other Samurai were ProfessorsSandy Faber, Roger Davies, David Burstein, Alan Dressler, Roberto Terlevich and GaryWegner. The Samurai devised a new distance indicator for elliptical galaxies (32), theso-called D n − σ relation where σ is the central velocity dispersion and D n is the diameterof the central region within which the average surface brightness exceeds 20.75 magarcsec − . This distance estimator was a substantial advance on the earlier Faber-Jacksonrelation, but nowadays it has itself been superseded by the Fundamental Plane (Binneyand Merrifield, 1998). Unexpectedly, the Samurai discovered a coherent, large amplitudeflow in the direction of Hydra-Centaurus, which they named the “Great Attractor” (33).It is situated at a distance of roughly 80 Mpc away from the Milky Way in the directionof the constellation Norma. The Great Attractor is pulling in millions of galaxies ina region of the Universe that includes the Milky Way, the Local Group and the largerVirgo Supercluster, and the still larger Hydra-Centaurus Supercluster, at velocities of upto a thousand kilometers per second (32,33). Based on the kinematics, the unseen massinhabiting the voids between the clusters of galaxies was estimated to be ≈
10 timesmore than the visible matter. The total mass of the Great Attractor was reckoned as ≈ × M (cid:12) .At the time, this was one of the most ambitious galaxy redshift surves ever attempted.It led to the discovery of bulk motions comparable in magnitude to the motion of theLocal Group through the cosmic microwave background (CMB). Together with OferLahav, Donald showed that the optical dipole lies within 7 ◦ of the direction of the Local18roup’s motion through the CMB. The directions of the optical, infrared and CMBdipoles are all consistent with each other (34). Donald, Lahav and others used this forcosmological parameter estimation (37), though this work predates the discovery of thenon-zero cosmological constant Λ. Calculations of the mass of the Great Attractor havesubsequently been revised downward by about an order of magnitude following infraredand X-ray studies (Kocevski et al., 2007). Galaxies located on the other side of the GreatAttractor are no longer thought to be pulled in its direction, as further more massiveagglomerations of galaxies lie behind it. These comprise the Shapley Supercluster, aboutfour times more distant than the Great Attractor (Kocevski and Ebeling, 2006).Galaxy surveys proved to be a productive research area over the next decades, but thefield took a different turn. With the development of plate scanning machines, computersreplaced humans in identifying galaxies, as in the APM Galaxy Survey. Using this asan input source catalogue, the Two-Degree Field Galaxy Survey obtained redshifts for ≈
220 000 galaxies. These were used to measure the two-dimensional two-point correlationfunction, from which the projected and redshift space correlation functions are derived.Donald continued to be interested in the area, especially in techniques for recovering thecosmological density, velocity and potential fields from all-sky redshift catalogues. Hedeveloped a method based on expansion of the fields in orthogonal basis functions (44).Peculiar velocities introduce a coupling of the radial harmonics describing the densityfield in redshift space, but leave the angular modes unaffected. In linear theory, the radialcoupling is described by an analytic distortion matrix which can be inverted to give the realspace values. Statistical or ‘shot’ noise is mitigated by regularizing the matrix inversionwith a Wiener filter.The notion that there may be galaxies hidden behind the Zone of Avoidance, wheredust and gas in the Milky Way obscure about a quarter of the Sky, was also fruitful. Itinspired Donald to participate in HI surveys of this region. This led to the discovery ofa large barred spiral galaxy, Dwingeloo 1, about 3 Mpc away (41). It is a member of agroup containing IC 342 and the Maffei galaxies, just beyond the Local Group in whichthe Milky Way resides.
By the late 1970s, it was clear that the monolithic collapse model for the formation ofthe Galaxy in ELS (3) was incorrect. The idea that mergers were major players in theongoing drama of the formation of the Milky Way was directly confirmed when firstthe Magellanic stream (Mathewson et al., 1974), and then the Sagittarius stream (Ibataet al., 1994), were discovered. It is hard to improve on Donald’s own description of thecuriosity that propelled him forward: “Some bad weather during an otherwise successfulphotometric observing run at SAAO, Sutherland, gave me time to ponder the Magellanicstream and to wonder whether, like the Magellanic Clouds, other near neighbours ofour Galaxy were associated with streams of neutral hydrogen. So, taking C. W. Allen’s
Astrophysical Quantities , I plotted the small satellites of the Milky Way on to R. D. Davies’recent map of high velocity hydrogen. Taking our satellites in order of distance I foundthat the Large and Small Magellanic Clouds, Draco, Ursa Minor and probably Sculptor alllay in or close to the directions of high velocity hydrogen streams" (19). The association ofDraco and Ursa Minor with the Magellanic Stream was strengthened by the discovery that19heir elongations lie along the line of the Magellanic Stream (28), also noted by Hunterand Tremaine (1977). Donald believed the fact that that Sculptor, although lying close tothe Magellanic Stream, nevertheless points at Fornax, and that this line passes on throughthe distant satellites Leo I and II, suggests that there may be another tidal stream, distinctfrom the Magellanic Stream (28,29). In this picture, the dwarf spheroidal satellites of theMilky Way lie on two streams and are elongated along them (29).Donald quickly realised the importance of tidal streams for measurements of theenclosed mass. He embarked on a suite of computer simulations of the Magellanic systemwith Doug Lin, then a hungry research fellow. The details of the simulations have notsurvived the test of time, as Donald assumed that the Magellanic Stream is produced bytidal interaction of the neutral hydrogen envelope of the Large and Small Magellanic Cloudwith the Galaxy. Nowadays, with the benefit of accurate proper motions, we know thatinteractions between the Large and Small Magellanic Clouds created the Stream. Thisthough does not lessen the significance of the first mass measurement with tidal streams.The conclusions (27) that the Galaxy must have a massive dark halo out to at least 70 kpcand that the circular velocity of the Galaxy is 244 ±
20 kms − are completely correct.The discovery of the Sagittarius stream in 1994 fired up Donald’s interest anew.Together with Ruth Lynden-Bell, he argued that objects belonging to the same tidal streamlie on the same great circle in the Galactocentric sky (42). This is exactly true if the streamfollows an orbit in a spherical potential. The Galactocentric direction to any object thendefines the pole of a great circle. So, the great circle joining any two objects has as itspole the intersection of the two great circles whose poles are at the objects. Thus, if thereare many objects belonging to a stream, there will be a multiple intersection point of allthe great circles whose poles point at the objects. Donald searched for such intersectionsand was able to recover the earlier streams associated with the Magellanic Clouds andwith Fornax, and to identify new ones associated with Sagittarius and its former globularclusters. The Sloan Digital Sky Survey provided multiband photometry on Galactic starscovering a quarter of the sky around the Galactic North Pole. This dataset proved to be atreasure trove in the hunt for streams. When Vasily Belokurov and Wyn Evans discovereda delicate track of stars crossing “the Field of Streams", they christened it “the OrphanStream" because of its lack of obvious progenitor. Donald took an immediate interest inthis parentless stream (54). He soon pointed out an astonishing fact. The track of theOrphan Stream on the sky exactly matches a distended high velocity stream of neutralhydrogen, Complex A. Subsequently, Donald showed that a solution exists in which thetwo streams share the same orbit, though they lie on different wraps (55).Donald was the first to see the great future of tidal streams as Galactic potentiometers.He assumed streams are orbits, enabling him to use his favourite workhorse of analyticmethods. This has been superseded by numerical codes that generate streams usingLagrange point stripping (Gibbons et al., 2014), revealing that streams are not exact orbits.In fact, deviations from orbits provide some of the most interesting results – such asthe recent measurement of the mass of the Large Magellanic Cloud through its subtleperturbations on the Orphan Stream that twist the latter’s orbital plane (Erkal et al., 2019).Donald’s work on the planes of dwarf satellites around the Milky Way has also evolvedinto a field of heated controversy. We now know that the distribution of satellites aroundthe Milky Way is highly anisotropic and that the Magellanic Clouds are surrounded bytheir own retinue of dwarf galaxies (Koposov et al., 2015). Whether this anisotropy reveals20igure 4: Donald Lynden-Bell at the conference on ‘Gravitational Dynamics’ in honourof his sixtieth birthday held in Cambridge in 1995. Shown left to right are Alice Duncan(Donald’s secretary), Prof Leon Mestel (Donald’s PhD supervisor), Donald himself, ProfRuth Lynden-Bell and Prof Joseph Katz (Donald’s long term collaborator on relativity).(Courtesy of Lafayette Photography.)fundamental flaws with our ideas of galaxy formation or whether such anisotropies followfrom group infall of satellites along preferential directions in the cosmic web remains acontested problem today.
Joe Vinen, his tutor at Clare College, encouraged Donald’s “first groping steps" to anunderstanding of inertia in an undergraduate essay (8). Donald liked the formulation ofMach’s principle in one of his favourite books,
Cosmology by Bondi (1952), who wrote“Local inertial frames are determined through the distributions of energy and momentumin the Universe by some weighted averages of the apparent motions". This led him tostudy the associated effect of “dragging of inertial frames", or “gravomagnetism". Themain consequence of the gravomagnetic field (or velocity-dependent acceleration) is thata moving body near a massive rotating object experiences an acceleration not predictedby Newtonian gravity. There are further subtle predictions, such as induced rotation of afalling object and precession of a spinning object. At the RGO, Donald already introduceda reformulation of the Einstein field equations as integral equations involving retardedGreen’s functions in Friedmann-Robertson-Walker (FRW) universes (8). This approachstalled, because it did not attribute any inertial influence to gravitational waves.21nce in Cambridge, general relativity became one of his favourite research pastimes.He struck up a productive and long-standing collaboration Joseph Katz and Jiri Bičák.They showed that, within the context of linearized perturbation theory of FRW universes,Mach’s principle follows from the constraint equations of general relativity, provided thatthe universe is closed (43). They studied frame dragging effects in a number of situations,such as inside a collapsing and slowly rotating spherical matter shell, or inside rotatingcylinders and investigated how rotations beyond the cosmological horizon affect the localinertial frame. They introduced “Machian gauges" to study general linear perturbationsof FRW universes (53). These admit much less freedom than the synchronous gaugescommonly used in cosmology, but they allow local inertial frames to be determinedinstantaneously via the perturbed Einstein field equations from the distributions of energyand momentum in the universe. This is directly inspired by Donald’s understanding ofMach’s principle. The study of Machian effects inspired the formulation of conservationlaws with respect to curved backgrounds and the introduction of the Katz-Bičák-Lynden-Bell (or KBL) superpotentials (46). These are unambiguous in spacetimes with or withouta cosmological constant, and have found numerous applications – for example, in thestudies of the back reactions in slow-roll inflation or in finding energy-mass and other totalcharges for black holes in asymptotically non-flat backgrounds. The construction of theKBL superpotentials is Donald’s most cited paper in general relativity.Donald realised that an infinite disc with a flat rotation curve can additionally onlyinvolve the constants of general relativity – Newton’s gravitational constant and the speedof light. No length scale can be made out of these constants, so the geometry is self-similar.Together with Serge Pineault, he constructed elegant solutions to the field equations for self-similar counter-rotating discs, as well as numerically computed rotating solutions (22,23).At the end of 1980s, Donald studied spherical self-similar solutions for cold collapse withJosé Lemos (35,36). The main achievement was the detailed illustration of self-similarNewtonian solutions for non-crossing, collapsing spherical shells of matter. In the firstpaper, each shell was assumed to fall from rest at infinity. Next, self-similar solutionsfor collapsing and expanding non-crossing shells were considered. They analyzed thegeneral relativistic analogues of the Newtonian solutions, which are the Lemaître-Tolman-Bondi metrics for dust. Among known finite-mass non-spherical solutions of Einstein’sequations, few have physical sources. From 1992, Donald worked on the idea of thinrelativistic discs as possible sources of static spacetimes. Together with Katz and Bičák,he showed that most vacuum Weyl solutions can arise as the metrics of counter-rotatingrelativistic discs (39). Inspired by new developments in Newtonian potential theory (Evansand de Zeeuw, 1992), he also constructed an infinite number of new static, relativisticdisc solutions, including disc sources for the Kerr metric (40). Many of Donald’s exactsolutions to the Einstein field equations have found their way into the standard referencebook (Stephani et al., 2003).Although Donald’s work in relativity was not as influential as his work in astrophysics,it gave him enormous pleasure. He did not favour the geometric approaches to generalrelativity common in many modern textbooks, but much preferred the physical treatmentin the celebrated book by Landau & Lifshitz on
The Classical Theory of Fields , repeatedlyextolling it as “the best book on relativity ever written".22igure 5: Donald’s telescope. Its mirror was constructed by Donald as an undergraduate.(Courtesy of Amanda Smith.) 23
Belfast, Cambridge (1996-2018)
In 1995, Ruth Lynden-Bell moved to Queen’s University Belfast as co-founder of the inter-disciplinary Atomistic Simulation Group and Professor of Condensed Matter Simulation.Between 1996 and 2003, Donald held a Senior Research Fellowship as Visiting Professorat Queen’s University, Belfast. He normally went to Armagh Observatory once a week,while spending the Summers in Cambridge to work with Katz and Bičák. Donald andRuth returned to Cambridge when she retired from Queen’s University in 2003. Donaldremained very active in research, especially in relativity, the collimation of jets, stellardynamics and – a new research theme – optics.
Donald kept a 6-inch reflector in his office. He had made the mirror himself during hisundergraduate days: "Grinding, polishing, and Foucault-testing my 6 1/2 inch telescopemirror while an undergraduate taught me that persistence pays, but my lack of patiencemakes me poorly suited to practical work" (56). In 1983, Donald was instrumental inurging Roderick Willstrop to explore the possibilities of optical systems that could giveas wide a field as the Schmidt Telescope, but with a much shorter tube. FollowingE. J. Kibblewhite’s suggestion to study 3-mirror Paul-Baker systems, Willstrop then foundthat a large field could be obtained by perforating the paraboloidal primary mirror, bringingthe convex spherical secondary mirror nearer to the primary, and putting the tertiaryconcave spherical mirror behind the primary (Willstrop, 1984). When by experimenting,Willstrop had reached a field of view of 4 ◦ , Donald was still not entirely satisfied andurged him to try to reach 5 ◦ . Variants of such three mirror designs are the basis for manymodern survey telescopes, including the Vera Rubin Telescope .In 2002, Donald began a series of papers on exact optics with assistance from Will-strop. Donald knew of Karl Schwarzschild’s papers of 1905, which give a completetheory of imagery in the field for any reflecting telescope with a single axis. Histori-cally, aspheric surfaces were very expensive, so Schwarzschild concentrated attention onspherical surfaces or those for which the profiles were conic sections. Without makingany such assumptions, Donald found all 2-mirror systems free of spherical aberration andcoma (49,50), including previously known telescope designs, such as Ritchey-Chretien orCouder. The difference between Exact Optics and all earlier papers on optical design isthat Donald asked: What shapes must two mirrors have to be free of spherical aberrationand coma? All previous opticians chose plausible mirror shapes and asked how goodwill the images be and what (small) changes in the shapes of the mirrors will improve theimages? Donald’s papers are mathematical, but they show an abiding interest in practicalconstruction of instruments that stretches back to Donald’s Meccano childhood toys.
Fittingly, some of Donald’s last papers return to subjects associated with his Ph. D. thesis– both the stellar dynamics that was in it, and the magnetohydrodynamics that was meantto be in it! The proper motions of Milky Way stars supplied by the multi-epoch
SloanDigital Sky Survey data or by the
Gaia satellite meant that the six components of the stellar24elocity dispersion tensor in the Galaxy could finally be constructed directly from the data.Empirically, the velocity dispersion tensor is close to exact alignment in spherical polarcoordinates (57). Inspired by this, Donald and co-workers proved the theorem that if theeven part of the phase space distribution function is invariant under time reversal, thenthe velocity dispersion tensor must everywhere exactly aligned in ellipsoidal coordinates(which include spherical polars as a special case) and the potential must be of exactlyseparable or Stäckel form.Donald had always been puzzled as to how swirling discs of conducting fluid aroundyoung stars, quasars and micro-quasars are able to generate highly collimated jets overdistances hundreds or thousands of times the size of the disc. At first, he suspected theorigin may be vortices above and below the central object that cause the beaming (24).Later, he favoured magnetic collimation, and built a series of simplified models to gaininsight. In these, magnetic forces dominate over any gas pressure in the jet. Outside themagnetic cavity of the jet, there is pressure from an ambient coronal medium. The centralbody and accretion disc are both massive and conducting, so they drag the magnetic fieldwith their fluid motions. The build-up of magnetic energy due to differential motionsdrives the jet. Donald used the virial theorem to show that, as the upward magnetic fluxof an accretion disc is twisted relative to the downward flux, the height of the magneticfield configuration grows (45,52). It assumes a vertical cylindrical geometry in whicheach additional twist of the field produces an equal increment in the height of the cylinder.So, after many twists, the cylinder becomes very tall and thin. Donald thought that thiscollimates the narrow jets seen in quasars, radio galaxies, and young stars (Herbig-Haroobjects). Donald built sequences of force-free models to illustrate this, arguing that this ismore insightful than magnetohydrodynamical simulations.
10 Influence and Legacy
Donald passed away on 6th February 2018 at his home in Cambridge, after an earlierstroke from which he never recovered.Donald loved mathematics and he used his skills in diverse areas of astrophysicsand relativity. But, he always insisted that research must be, first and foremost, greatfun. Donald worked on whatever took his fancy, from data analysis to exact optics,from thermodynamics to general relativity, from stellar dynamics to large-scale structure.He loved to work with graduate students, densely covering the blackboard with chalkedequations; he loved to ask questions in seminars with his booming voice, often from anunusual perspective; he loved to talk astronomy over breakfast ... coffee, lunch or dinner... explaining his new ideas to sometimes bewildered listeners; he loved to lecture inhis buoyant and chaotic style, especially about his own research. He bubbled over withindefatigable enthusiasm for any topic in astronomy or related fields. He had somethingof the boisterousness, energy and exuberance of A.A. Milne’s Tigger.Donald was at the pinnacle of his creative powers in the 1960s and early 1970s.Amongst the welter of fine papers, three stand out as enduring achievements of the veryhighest calibre – the discovery of swing amplification in gas discs with Goldreich; the
Nature paper that won him the Kavli Prize for the suggestion that dead quasars residein the centres of galaxies; and the paper with Pringle that elucidated the workings ofaccretion discs. Further, the accumulated papers on stellar dynamics from his 1960 Ph. D25hesis to his 1973 Saas-Fee Lectures form a hugely impressive corpus of work. This wasthe first sustained and vigorous attack on the problems of the structure and dynamics ofgalaxies by a powerful thinker since the days of Eddington and Jeans – almost exclusivelyusing analytic methods. Galactic dynamics became more computational from the 1980sonwards with numerical simulations playing an increasingly important role. This wasnot quite to Donald’s taste, and so his imagination took him on to other astronomicalproblems – relativity, Mach’s principle, exact optics – where he could use pen, paper andmathematical prowess.Donald’s achievements were recognized by a succession of medals and awards fromboth national and international learned societies, but one that gave him great pleasure washis election to the Presidency of the Royal Astronomical Society (RAS). He regarded theRAS as playing a crucial role in UK astronomy and it was a very important part of hislife. Donald was a tremendous mentor and colleague. He supervised over fifty students,including Simon White FRS 1997 (“Simon just came and told me interesting things"), WynEvans, Ofer Lahav, Michael Penston, Jim Collett, Mike Hudson, Christophe Pichon andSomak Raychaudhury. Though not formally supervisor, Donald also played an importantrole in the Ph. D. work of Ken Freeman FRS 1998, Jim Pringle and Tim de Zeeuw.Donald did not enjoy administration, but he was a very conscientious individual. Hetook over the Directorship of the Institute of Astronomy at a perilous moment, yet thanksto his stewardship – and that of Martin Rees – the Institute has thrived over the followingdecades. Elsewhere, there are personal reminiscences of Donald from his friends andcolleagues (Rees, 2018; Evans, 2018; Lahav, 2018). Here, it suffices to say that very fewpeople will carve so distinctive and memorable a trail in astronomy – and in life – the waythat Donald Lynden-Bell did.
Acknowledgments
The author is very grateful to Ruth Lynden-Bell for advice, help and comments, aswell as access to Donald’s papers and private writings. She also made available herpersonal collection of photographs of Donald. The author is indebted to many people foruseful criticism of the text, including Jiri Bičák, Jim Collett, Douglas Gough, ElizabethGriffin, Mark Hurn, Ofer Lahav, Malcolm Longair, Jim Pringle, Martin Rees, Alison Rose,Amanda Smith, Scott Tremaine, Alar Toomre, Terry Wall and Roderick Willstrop.
AWARDS AND RECOGNITION
Balzan Prize (1983)RAS Eddington Medal (1984)AAS Brouwer Award (1990)Membership of National Academy of Sciences (1990)Oort Visiting Professorship, Leiden (1992)RAS Gold Medal (1993)Bruce Medal of Astronomical Society of the Pacific (1998)26enry Norris Russell Lectureship, AAS (2000)Commander of the British Empire (2000)John J Carty Award for the Advancement of Science (2000)Blauuw Visiting Professorship, Groningen (2007)Kavli Prize for Astrophysics (2008) 27 rief Author Profile
Wyn Evans is Professor of Astrophysics at the University of Cambridge. He was agraduate student of Donald Lynden-Bell from 1985-1988. He subsequently was a Juniorand Senior Research Fellow of King’s College, Cambridge and a Lindemann Fellow atthe Massachusetts Institute of Technology. He was Reader in Theoretical Physics, OxfordUniversity, before returning to Cambridge in 2003. His research interests include galacticstructure and dynamics, the dark matter problem, astroparticle physics, Survey science,solar system dynamics and gravitational lensing.28 eferences (1) 1962 Stellar dynamics. Potentials with isolating integrals,
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The Observatory , MonthlyNotices of the Royal Astronomical Society
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Relativity Theory and Astro-physics. Vol.2: Galactic Structure , Lectures in Applied Mathematics, Vol. 9.Edited by Jürgen Ehlers. Providence, Rhode Island: American MathematicalSociety.(10) 1967 (With J. P. Ostriker) On the stability of differentially rotating bodies.
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Monthly Notices f the Royal Astronomical Society MonthlyNotices of the Royal Astronomical Society , 1-30 (doi: 10.1093/mnras/157.1.1)(17) 1973 Topics in the Dynamics of Stellar Systems In
Saas-Fee Advanced Course:Dynamical Structure and Evolution of Stellar Systems , eds. L. Martinet, M.Mayor, Geneva Observatory.(18) 1974 (With J. E. Pringle) The evolution of viscous discs and the origin of thenebular variables.
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Physics Scripta Monthly Notices of the RoyalAstronomical Society
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Internal Kinematics and Dynamicsof Galaxies , IAU Symposium 100, 89-91. Edited by E. Athanassoula, Reidel,Dordrecht. 3030) 1985 (With P. T. de Zeeuw) Best approximate quadratic integrals in stellar dynam-ics.
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Spectroscopy and Photometry ofElliptical Galaxies. I. New Distance Estimator
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The Observatory , Monthly Notices of theRoyal Astronomical Society
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L64-L66. (doi: 10.1111/j.1745-3933.2007.00321.x)(56) 2010 Searching for Insight.
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