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Astronomy: Edwin Hubble Essay, Research Paper





Vol. 83, No.6 December 1989 Whole No. 621

EDWIN HUBBLE 1889-1953

By Allan Sandage

The Observatories of the Carnegie Institution, Pasadena, California, U.S.A.

(Received September 22, 1989)

Hubble’s role. This year marks the centennial of the birth of Edwin Hubble. There can be no doubt that future historians, writing about the scientific advances of this age will describe the 20th century as epoch-changing in giving us the first correct view of how the universe is organized. The principal cosmological problem of discovering the large scale content of the universe was solved observationally between 1920 and 1936. Hubble was a major figure in this development. Knowledge that galaxies mark the space and provide the means to measure it was gained by the first convincing analysis of new data on the nature of the nebulae – knowledge that came directly from the sky rather than by dialectic discussion or revelation.

In Hubble’s time, the centre of observational work on the new astrophysics, and later on what we know as cosmology, was the Mount Wilson Observatory. The two largest telescopes in the world were there and could be regularly used on these problems. With his appointment to the Mount Wilson staff in 1919, Hubble had continuing access to both the 60-inch and the 100-inch Hooker reflectors.

He also had a most remarkable ability to cut to the core of unsolved problems concerning the nature of the nebulae. He would invariably proceed to the essence of a problem without stopping at the many lovely resting places that usually accompany the road to solutions, becoming the leading astronomer in the 1920s concerned with problems of the nebulae. In the 12 years from 1824 to 1936 he had set down the foundations upon which observational cosmology rests. From his central role in the solution of so grand a problem, Hubble has become a legend. But because part of his life has also become a myth, it is only from a study of his published papers that can we obtain a reasonable understanding of his enormous influence on the development of cosmology.

Hubble’s name is attached to many things of everyday astronomical life. There is Hubble’s zone of avoidance, the Hubble galaxy type, the Hubble sequence, the Hubble luminosity law for reflection nebulae, the Hubble luminosity profile for E galaxies, the Hubble constant, the Hubble time, the Hubble diagram, the Hubble redshift-distance relation, the Hubble radius for the universe, and now the Hubble Space Telescope. It seems appropriate in this centennial year to celebrate the memory of a scientist whom some have called the greatest astronomer (in changing paradigms) since the times of Galileo, Kepler and Newton. What did he do, and how did he do it?

Characteristics. It will be difficult for historians to write an accurate personal biography of Hubble using some of the extant archive sources. Known facts contradict part of the recollections set out in materials in the Hubble collection in the Huntington Library in San Marino, California, making it difficult to know which other parts are accurate. After his death, Mrs. Hubble, who survived him by 26 years, organized archive materials, also in the Huntington collection, around commentaries of his own, some of which glorify him in ways larger than life. In view of this, the authoritative historical essay by Osterbrock, Brashear, and Gwinn (1990) on Hubble’s education and career to about 1922 should be consulted. Their history is based on sources that are, as far as possible, independent of personal recollections, largely from letters and documents in the archives of the Yerkes and the Lick observatories.

But interesting as the personal aspects of the life of great scientists are in understanding how they arrive at solutions, the solutions themselves must be independent of the personality. Otherwise, the results have no objective reality. Yet the internal excitement in arriving at solutions is never this cold within the personality itself. Every scientist lives in a world of imagination. The grander the problem, the more wonderful must be the imagination. And Hubble lived with an ineffable problem – the discovery of the structure of the World on the largest scale. From this work, by him and by others of his generation, it is widely believed that some glimpse of a “creation event of the universe” became available to science by an objective method, not, as in other times, by metaphysics or speculation.

Hubble must have understood more clearly than anyone what he was dealing with and what he had accomplished. The realization that an understanding had finally been reached of the way the universe is organized, and above all, the discovery of its expansion, must have influenced in some way how Hubble lived with everyday events. His apparent aloofness, often mentioned by his critics, would, one suspects, not be so unusual in those who themselves could have been among the first to have had such a vision.

But we, in fact, know little of Hubble’s inner world. He was remarkably silent about the meaning of what he and Humason had found with their velocity-distance relation. Neither in his personal conversations nor in his writings did he discuss its implications for ideas about either the evolution of the universe from a primitive state or its meaning concerning a “creation event”. What little we can find of his thoughts about these matters – so common in the modern literature – must be inferred from his published papers.

Four central accomplishments. From 1922 to 1936 Hubble solved four of the central problems in cosmology, any one of which would have guaranteed him a position of the first rank in history.

(a). From 1922 to 1926 Hubble proposed a classification system for nebulae, both galactic (diffuse) and extragalactic. The galaxy classification system has become the Hubble morphological sequence of galaxy types.

(b). With his discovery of Cepheids in NGC 6822 in 1924, with parallel work in M33 and M31, Hubble settled decisively the question of the nature of the galaxies, whose correct solution, to be sure, had previously been given using what many believed to be inconclusive arguments, by Curtis, Lundmark, and ?pik.

(c). From 1926 to 1936 the distribution of galaxies, averaged over many solid angles, was determined to be homogeneous in distance. The test was made by showing that the coefficient of the log N(m) count distribution with magnitude has a value of about 0.6 at bright magnitudes. This proved that galaxies truly mark a space which is significant to the universe itself. Galaxy counts to the magnitude limit of the Mount Wilson 100-inch telescope were then used to attempt a measurement of the radius of curvature of space by finding deviations of the coefficient from the Euclidean value of faint magnitudes.

(d). The linear velocity-distance relation was set out in a discovery paper in 1929, followed by a series of papers with Humason between 1931 and 1936 that verified and extended the relation to large (i.e. 60,000 km s[-1] redshifts. This discovery lead to the notion of the expanding universe which is the centre-piece (being the necessary condition) for the cosmological models of the present day.

The published papers.

(a) The Hubble classification sequence.

No satisfactory classification system for galaxies existed before 1926, at which time two similar systems appeared in the literature, following discussions at the 1925 meeting of the International Astronomical Union in Cambridge, England. Before this time a purely descriptive system set up by Wolf (1908) had been used, but the classification showed no continuity between the Wolf types and was generally considered to be in need of revision. Hubble (1920) in his Ph.D. publication had remarked that the Wolf classification “while admittedly formal, offers an excellent scheme for temporary filing until a significant system shall be constructed” (emphasis added here), and later in the same publication “[the Wolf system] is wholly empirical and probably without physical significance, yet offers the best available system of filing away data and will later be on great service when a significant order is established” (again emphasis added here).

One cannot help but note that the construction of these two quotations shows that Hubble already possessed the confidence to accomplish what lay ahead when he would enter the field. This surety of language characterized much of his later writings – a surety which tended (and was intended) to conquer the field by prose as well as by the technical results. It was Hubble’s mastery of the language that gave some of his papers such dominance over prior work by others. Often the problem had in fact been solved, but without the same elegance of style, power of presentation, and excellence of summary possessed by Hubble when he was at his best. Clearly, the lesson for students is learn to write at the same time that you learn to do great science.

The barest outline of a new galaxy classification scheme had been given by Hubble, almost as an aside, in his fundamental paper on the nature of diffuse galactic nebulae (Hubble 1922a). The scheme was expanded four years later and illustrated by two photographic panels (Hubble 1926b). Clarifying comments (Hubble 1927) that answered criticisms by Reynolds (1927a, b), and a slightly expanded explanation set out in Chapter V of The Realm of the Nebulae (Hubble 1936d) was the extent to which Hubble amplified his 1926 discussion of the galaxy classification system. But the Hubble classification sequence has become so widely used that it is a curiosity to note that Hubble, in his reply to the comments by Reynolds, describes his principal announcement of the 1926 system merely as “a preface to some general statistical investigations.”

Hubble, nevertheless, took his “preface” seriously. He guarded its priority in a revealing footnote in part I of his 1926 paper. There he comments on a classification system proposed at about the same time by Lundmark (1926, 1927). Some of Hubble’s complaints, which he rarely made public, were unfounded, showing a sensitivity he generally kept hidden. Some of Hubble’s accusations are addressed in a partially justifiably acerbic reply by Lundmark (1927), also in a footnote, in Lundmark’s near great but largely neglected paper.

The quite meagre descriptions of his powerful classification scheme which Hubble gave in the literature were, however, sufficient to teach the system to the community for which it was intended, showing the system’s simplicity, yet its power. Finally, note that the famous tuning fork diagram which summarizes the system at a glance did not appear in Hubble’s 1926 technical paper in The Astrophysical Journal, but rather only in his popular book The Realm of the Nebulae.

(b). The convincing proof that galaxies are island universes.

Here the story is too well known, needing no detail in its retelling. Hubble’s

discovery of Cepheids in NGC 6822, M33, and M31 was “simply” the final, albeit conclusive, demonstration that the arguments by Lundmark, Curtis, and ?pik that galaxies are external to the Milky Way were correct. But the situation was not, of course, so simple, given van Maanen’s measurements of proper motions.

In any investigation, the greatness of any synthesis of data lies in knowing what clues to ignore. All who have ever tried know there is always a plenum of false clues in any work. Hubble’s ability to know which clues to trust, which to discard, and then which to use to tie up the facts to make a case, was superior. By ignoring van Maanen’s result, Hubble’s demonstration of the presence of Cepheids with their period-luminosity relation was complete and final.

Only three papers were written by Hubble on the problem, yet even with the first on NGC 6822 (Hubble 1925) he had brought the debate to a close. The papers on M33 (Hubble 1926a) and M31 (Hubble 1929a) showed the generality of the result, but no one seriously criticised the 1925 initial result.

(c). Distribution of Galaxies in Space.

With Hubble’s final proof that galaxies are beyond the Milky Way, the major problem then became whether they are fair markers of the universe, or if they are merely part of a hierarchical structure in a next rung up in the organization of matter. The solution rested on the way galaxies are distributed in distance. If they increase in numbers in proportion to the surveyed volume (with no indication of an edge, as with stars in our galaxy), they would, then, clearly be the basic unit of the distribution.

The obvious test could be made using galaxy counts to various magnitude limits. The purpose of the counts would be to find the rate of increase in galaxy numbers with increasing volume. From his work on the galaxy luminosity function (a recurring calibration throughout many of the papers, cf. Hubble 1926b, 1934b, 1936a, 1936b, 1936d) Hubble knew that galaxies have a spread in absolute magnitude. Nevertheless, as long as the luminosity function does not diverge at the faint end, the counts to different limits of magnitude will exhibit a distribution that varies as

log N(m) ~ 0.6m,

regardless of the form of the luminosity function, provided that the objects are distributed homogeneously in distance.

With this knowledge, the early aim of Hubble’s work on counts was to determine the numerical value of the coefficient of the magnitude term. In the first discussion in his remarkable 1926 paper, Hubble shows that the data then known were consistent with the required value of 0.6, indicating homogeneity (his equation 10 from Table XVII of Hubble 1926b). He used data from the standard sources of counts then available, including the classical work on galaxy distribution by Seares (1925) – a generally neglected major discussion of what is now known as Hubble’s zone of avoidance.

But it was clear that the data could be fundamentally improved and carried to fainter magnitudes by using the enormous power of the 100-inch reflector. Building on the experience of his Ph.D. work, Hubble began a massive observing programme to do just that. The results began to appear in a series of papers that was to culminate in 1936 in the attempt to measure the curvature of space.

In his first paper, Hubble (1931) gives no hint of the direction which the problem would take toward the curvature determination when he teamed with R.C. Tolman in 1935. The 1931 announcement was simply an abstract of preliminary results from his new survey of galaxy counts made with the Mount Wilson telescopes.

The detailed paper on the distribution appeared three years later (Hubble 1934a). As in the Cepheid work ten years earlier, this paper was so thoroughly convincing that it brought the problem of the mean galaxy distribution, which by then was more than 100 years old, to a close. The paper has become a classic. Its power lies in the large amount of new data presented, and in Hubble’s straightforward, seemingly simple analysis of them – a trait characteristic of much of Hubble’s work. After presenting the data and the technical methods of reducing the material to “uniform plate conditions”, Hubble treats (1) the distribution in galactic latitude outlining the “zone of avoidance”, recovering Seares’ (1925) prior result, (2) the extinction in the poles (the famous cosecant distribution of the counts which has so confused modern discussions; see Noonan 1971 for a critique), (3) the tendency to cluster, based on the nature of the count residuals, field-to-field, the residuals being Gaussian in log N(m) rather than in N(m) itself, (4) the space density of galaxies, (5) the mean mass of galaxies, and (6) the mean density of matter in space of the order of 10[-30] g cm[-3]. Curiously, no mention of space curvature was made in this paper nor in the account of his Halley lecture (Hubble 1934b), although it was to be the major theme from then on.

Hubble’s interest in what Gauss and Karl Schwarzschild called experimental geometry can be traced to his collaboration with Tolman that must have begun in 1934. Their joint paper (Hubble and Tolman 1935), sets out how galaxy counts, conceptually, could be used to find the curvature of space by direct measurement. The principle is to determine if the volume encompassed within various “distances”, appropriately defined, increases at the rate of r[3], or more rapidly or more slowly than this Euclidean value.

The observational problem is complicated by the delicate corrections required to the data for the effects of redshifts, etc. But the grandeur of the conception and the carrying out of the programme still provokes the modern reader, despite the fact that the attempt failed because of large errors in the magnitude scales and what we now know to be the overwhelming effects of galaxy evolution in the look-back times.

The technical aspects of the methods need not be discussed here (cf. Sandage 1988 for that) nor the criticisms of them. More useful is a chronicle of Hubble’s progress in the curvature programme following his initial collaboration with Tolman. The problem still remains as a principal goal of observational cosmology. But because of the effects of galaxy evolution, galaxy counts are no longer considered to be the main source of data with which to solve it. Rather, we now attempt, in one way or another, to measure the deceleration of the expansion from which spatial density can be derived and hence the curvature from Einstein’s relativity equations.

Hubble’s (1936c) major paper discussing his attempt contains two fainter points on the N(m) count curve determined at Mount Wilson plus the important N(m) additional data point from Mayall’s (1934) Ph.D. survey. A principal part of the analysis centres on the effects of red shifts on the observed N(m) distribution and the corrections required due to the redshift effect. These corrections have been mentioned in the Halley lecture (Hubble 1934b), but were not there discussed relative to the space curvature measurement. Part of the 1936 paper was concerned with the problem for the first time.

Hubble concluded that his observed log N(m) distribution showed a large departure from Euclidean geometry, provided that the effect of redshifts on the apparent magnitudes was calculated as if the redshifts were due to a real expansion. A different correction is required if no motion exists, the redshifts then being due to an unknown cause. Hubble believed that his count data gave a more reasonable result concerning spatial curvature if the redshift correction was made assuming no recession. To the very end of his writings he maintained this position, favouring (or at the very least keeping open) the model where no true expansion exists, and therefore that the redshift “represents a hitherto unrecognized principle of nature”. This viewpoint is emphasized (a) in The Realm of the Nebulae, (b) in his reply (Hubble 1937a) to the criticisms of the 1936 papers by Eddington and by McVittie, and (c) in his 1937 Rhodes Lectures published as The Observational Approach to Cosmology (Hubble 1937b). It also persists in his last published scientific paper which is an account of his Darwin Lecture (Hubble 1953).

From the beginning of his career Hubble was intrigued with the distribution of nebulae. His work on the problem began with his Ph.D. research (Hubble 1920), elementary as this now appears. He returned to the subject time and again with ever-increasing sophistication until the end of his career. He had even begun a major count programme anew in 1949 using the just-completed Palomar 48-inch Schmidt telescope (unpublished investigation) in an effort to use the modern magnitude scales then being set up photoelectrically. The aim was to investigate again from scratch the space curvature measurement.

In his paper of 1926 Hubble had used his estimate of the average space density of galaxies to calculate the space curvature of the static Einstein universe. This fact is of quite ironic interest because it was Hubble himself, more than anyone, who three years later set out the observational foundation for the non-static solutions to the Einstein field equations of gravity. His use, then, of a static model to calculate the space curvature shows that as late as 1926 he had believed in non-expanding models, despite the large velocities that had been observed by Slipher and the attempts by many astronomers to understand these velocities using particular cosmological models. Recall that the Friedmann non-static solutions had been discovered in 1922, evidently unknown at Mount wilson at the time. The next major observational development was the discovery of the redshift-distance relation in 1929.

d). The redshift-distance relations.

As is well known the Einstein field equations of gravity admit only three stationary solutions (Tolman 1929 and 1934 sections 133-145). By stationary is meant that the manifold is not expanding. The mathematical expression of this condition is that the coefficients of each of the spatial coordinates in the equation of the metric is not a function of time.

The two stationary solutions of historical importance are those of Einstein (1917) and of de Sitter (1916a, b; 1917), neither of which later proved to describe the true situation. Einstein’s did not because it contained matter but no redshift (it was truly static both in space and time). De Sitter’s did not because it had no matter, but curiously did have spectrum shifts (both red and blue) of test particles placed in the space which it described. This was due to a scandalous space-dependent factor in the metric coefficient of the time dimension , despite the static nature of the space coordinates.

The “de Sitter spectral shift effect” had been looked for by many astronomers (see Hubble’s history in his chapter V of The Realm of the Nebulae) without convincing success. Robertson (1928) had predicted a linear relation and believed he had found a suggestive effect that could be interpreted in this way. He had correlated Slipher’s redshifts with the distances he had estimated using apparent magnitudes. Robertson gave no details. His result was set out in a single paragraph in a highly theoretical paper, but he was clearly aware of the possibility of the Kr term in the velocity field and that the universe might not, after all, be static.

As with Hubble’s Cepheid paper 5 years before, and his space distribution paper to come 5 years in the future, Hubble’s (1929b) discovery paper of the expansion was written so convincingly that it was believed almost immediately. Despite its astonishing content and its few data points, Hubble must have been quite certain of the result. In the paper immediately preceding Hubble’s, Humason (1929) reported the very large (for the time) redshift for NGC7619 of 3779 km s[-1], far larger than any redshift known before. From this result Hubble must have been certain that a significant phenomenon was at hand.

All effort was then made at Mount Wilson to confirm and to extend the astounding possibility that the universe expands. By 1930 Humason (1931) had obtained redshifts of galaxies in clusters whose “velocities” were as high as 20,000 km s[-1]. In perhaps the most important paper on the series, Hubble and Humason (1931) showed beyond doubt (a) the existence of the effect, (b) that it was linear with distance, and (c) that the brightest members of clusters are predominantly E galaxies (a major discovery related to galaxy and cluster formation).

The work was extended to field galaxies soon thereafter (Hubble and Humason 1934), showing the generality of the phenomenon. By 1936 the work had been completed as far as it was to be done with the Mount wilson reflector, reaching redshifts of 40,000 km s[-1] for the Ursa Major No. 2 cluster (Humason 1936, Hubble 1936). Humason began the work again in 1949, using the Palomar 200-inch reflector, reaching 60,000 km s[-1] (Humason, Mayall and Sandage 1956) for the Hydra cluster, but was stopped from going further by the techniques of the time in the presence of the night sky air glow. Hubble (1953) symmarized the work finally in his Darwin Lecture.

(e). Other programmers.

The previous subsections have outlined the four major subjects in which Hubble’s results were dominant in the 1930s. But he produced other works of influence as well, the results of which are also part of modern astronomical culture.

(1). He solved the problem of the source of radiation and the nature of the spectra of diffuse nebulae, recognizing the difference between emission and reflection nebulae (Hubble 1922a, b), and proving that the source of radiation of reflection nebulae is an associated star. An elegant appreciation of the work is given by Greenstein (1951).

(2). The surface brightness profiles of E galaxies were measured accurately for the first time (Hubble 1930), providing the basic model from which later modifications and extensions of the profile laws would be derived by others.

(3). He began the detailed study of the stellar content of the nearby galaxies. Besides the identification and measurements of Cepheids and very bright irregular variables in members of the Local Group, he made the unprecedented identification of globular clusters in M31 (Hubble 1932), starting an activity that occupies many present-day astronomers.

(4). He discussed the sense of rotation of the spiral arms in individual galaxies. The most important papers, in which the solution of the problem was set out by identifying the near sides of galaxy images by the dust lane asymmetries, are Hubble (1935, 1943) and a paper with Mayall (Hubble and Mayall 1941).

(5). In a most important paper, Baade and Hubble (1939) found the nature of the Sculptor and Fornax dwarf E galaxies that had been announced by Shapley in 1938. Their discovery of RR Lyrae stars provided Baade with the crucial clue to his eventual population concept (cf. Sandage 1986 for a review).

An Assessment. The principal surprises in rereading Hubble’s papers from the vantage point of 1989, after the discoveries of the Gamow, Alpher, Herman 3 K radiation, the development of radio astronomy, the discovery of how to age-date the stars, and the invention of the new cosmology of grand unification, are (1) the nature of Hubble’s methods, and (2) those central items that he hardly discussed but which seem so much a part of the cosmology that he pioneered.

(1). Hubble’s methods were largely inductive – nearly pure Baconian. His usual procedure was to assemble massive data sets from which he generalized to reach conclusions of wide scope that had continuing applications in further advances. Occasionally he did employ analytical methods such as in his analysis of the source of the light from diffuse nebulae (Hubble 1922b), his analysis of the flattening distribution of E galaxies (Hubble 1926b), his use of the Emden gravitationally bound gas sphere in understanding the luminosity of E galaxies (Hubble 1930), and his analysis of the galaxy counts for the space curvature (Hubble 1936c using the formalism of Tolman). But the method used in his most important papers – those papers that convincingly changed a field – was that of nearly pure Baconian induction. His success was remarkable, and his proportionate influence nearly unparalleled in modern astronomy.

(2). The most curious impression we are left with is his lack of comment on the significance of the redshift phenomenon, which is surely one of the most important discoveries in science. In none of his writings did Hubble comment on the central importance that the form of the redshift-distance law is linear. This single feature is most crucial for the standard model. Heckmann (1942) was perhaps the first to emphasize the singular significance of the linear form.

A linear velocity field has two fundamental properties; (a) each observer sees the identical form and expansion rate from any vantage point, and (b) it is the only velocity field that permits all points in the manifold to be “together” at some time in the past. Discovery of the linear form is usually taken to be as important as the discovery of the expansion itself if the phenomenon has any relevance to “the creation of the universe”. But hardly any hint of this appears in Hubble’s writings, despite his discovery of the linearity. There is also a lack of discussion of how the expansion relates to “beginnings” – a topic emphasized so strongly in modern cosmological writings. We simply do not know if Hubble was impressed with his discovery in these ways.

The second puzzling omission is a lack of emphasis on the meaning of the numerical value of the expansion rate (i.e. the Hubble constant). In an expanding model with a singular point, the inverse Hubble rate is related to the age of the model, the exact function depending on the deceleration. Credence that we are dealing in the redshift phenomenon with an aspect of “a creation event” requires that the “Hubble time”, obtained from the inverse “Hubble constant”, be the same as the “age of the universe” determined in other ways. We are so used to talking in this way that it is surprising to see none of this in Hubble’s writings.

Of course, it is true that the three types of cosmological time scales were not well known in Hubble’s time. These are (a) the Hubble time, (b) the age of our Galaxy via its oldest stars, and (c) the age of the oldest chemical elements. The method to age-date the stars had to await the understanding of the Sch?nberg-Chandrasekhar (1942) limit as a departure of evolving stars off the main sequence of the Hertzsprung-Russell diagram, an understanding that came only by the developments in stellar evolution in the early 1950s. The age of the chemical elements, although known in principle in about 1910 by Rutherford based on the first understandings of radioactivity, was not worked out in detail until, also, the decade of the 1950s. What we take for granted in the current work that is organized to test the agreement of the three time scales to within say 20 per cent was not possible in Hubble’s time. Nevertheless, it remains a curiosity that Hubble did not strongly emphasize the problem publicly, or, if at all, even privately to himself.

There was, of course, the embarrassment that the inverse of the Hubble expansion rate (i.e. the Hubble time) was only two billion years on Hubble’s 1930 to 1953 distance scale whereas the Earth was believed to be a bit older than three billion years even in 1936. It was left to the inventors of the steady state cosmology to emphasize this discrepancy of time scales, pointing out that any of the Friedmann models (sans cosmological constant) that were used to espouse a “beginning” could not be true.

The influence of Hubble was so great that errors in his 1930/1936 distance scale were considered to be out of the question in his time. The discrepancies began to appear only when the 200-inch Palomar reflector was put into operation in late 1949 by the heroic and largely unheralded two-year effort by I. S. Bowen. Baade began obtaining data which showed that Hubble’s scale must be modified. We now know that the scale must be stretched by a factor of a least 5, more likely by a factor a bit larger than 10. But it must be fairly pointed out that some astronomers, not believing that the problem of the distance scale has been solved by the results of the 200-inch programme from 1950 to 1980, have suggested that the value of the Hubble constant can be determined to the satisfaction of the sceptics only by the future use of the Hubble Space Telescope. For this one suspects that Hubble might have been pleased

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