It is often the case that we need to do sample comparison: we have someone else's data to compare with ours; or someone else's model to compare with our data; or even our data to compare with our model. We need to make the comparison and to decide something. We are doing hypothesis testing – are our data consistent with a model, with somebody else's data? In searching for correlations as we were in Chapter 4, we were hypothesis testing; in the model fitting of Chapter 6 we are involved in data modelling and parameter estimation.

Classical methods of hypothesis testing may be either parametric or non-parametric, distribution-free as it is sometimes called. Bayesian methods necessarily involve a known distribution. We have described the concepts of Bayesian versus frequentist and parametric versus non-parametric in the introductory Chapters 1 and 2. Table 5.1 summarizes these apparent dichotomies and indicates appropriate usage.

That non-parametric Bayesian tests do not exist appears self-evident, as the key Bayesian feature is the probability of a particular model in the face of the data. However, it is not quite this clear-cut, and there has been consideration of non-parametric methods in a Bayesian context (Gull & Fielden 1986). If we understand the data so that we can model its collection process, then the Bayesian route beckons (see Chapter 2 and its examples).

The distribution of objects on the celestial sphere, or on an imaged patch of this sphere, has ever been a major preoccupation of astronomers. Avoiding here the science of image processing, the province of thousands of books and papers, we consider some of the common statistical approaches used to quantify sky distributions in order to permit contact with theory. Before we turn to the adopted statistical weaponry of galaxy distribution, we discuss some general statistics applicable to the spherical surface.

Statistics on a spherical surface

Abstractly, the distribution of objects on the celestial sphere is simply the distribution of directions of a set of unit vectors. In this respect, other three-dimensional spaces may be of interest, like the Poincaré sphere with unit vectors indicating the state of polarization of radiation.

This is a thriving subfield of statistics and there is an excellent hand-book (Fisher, Lewis & Embleton 1987). Much of the motivation comes from geophysical topics (orientation of palaeomagnetism, for instance) but many other ‘spaces’ are of interest. The emphasis is on statistical modelling and a variety of distributions is available. The Fisher distribution, one of the most popular, plays a similar role in spherical statistics to that played by the Gaussian in ordinary statistics.

Peter Scheuer started this. In 1977 he walked into JVW's office in the Cavendish Lab and quietly asked for advice on what further material should be taught to the new intake of Radio Astronomy graduate students (that year including the hapless CRJ). JVW, wrestling with simple chi-square testing at the time, blurted out ‘They know nothing about practical statistics’. Peter left thoughtfully. A day later he returned. ‘Good news! The Management Board has decided that the students are going to have a course on practical statistics.’ Can I sit in, JVW asked innocently. ‘Better news! The Management Board has decided that you're going to teach it’.

So, for us, began the notion of practical statistics. A subject that began with gambling is not an arcane academic pursuit, but it is certainly subtle as well. It is fitting that Peter Scheuer was involved at the beginning of this (lengthy) project; his style of science exemplified both subtlety and pragmatism. We hope that we can convey something of both. If an echo of Peter's booming laugh is sometimes heard in these pages, it is because we both learned from him that a useful answer is often much easier – and certainly much more entertaining – than you at first think.

After the initial course, the material for this book grew out of various further courses, journal articles, and the abundant personal experience that results from understanding just a little of any field of knowledge that counts Gauss and Laplace amongst its originators.

There is a vast literature. Here we point to a few works which we have found useful, binning these into five types: popular, the basic text, the rigorous text, the data analysis manual, and the books of specialist interest to astronomers.

1. classic popular books have legendary titles: How to Lie with Statistics (Huff 1973), Facts from Figures (Moroney 1965), Statistics in Action (Sprent 1977) and Statistics without Tears (Rowntree 1981). They are all fun. A modern version with a twist in the title is Seeing through Statistics (Utts 1996), which entertains, serves as a statistics primer, and is almost a member of the next group.

2. come in types (a) and (b), both of which cover similar material for the first two-thirds of each book. They start with descriptive or summarizing statistics (mean, standard deviation), the distributions of these statistics, then moving to the concept of probability and hence statistical inference and hypothesis testing, including correlation of two variables. They then diverge, choosing from a menu including analysis of variance (ANOVA), regression analysis, non-parametric statistics, etc. Modern versions come in bright colours and flavours, perhaps to help presentation to undergraduates of a subject with which excitement is not always associated. The value of many such books is exceptional because of the sales they generate. They are complete with tables, ready summaries of tests and formulae inside covers or in coloured insets, and frequently arrive with CDs and floppy disks including test datasets. […]

When we make a set of measurements, it is instinct to try to correlate the observations with other results. One or more motives may be involved in this instinct: for instance we might wish (1) to check that other observers' measurements are reasonable, (2) to check that our measurements are reasonable, (3) to test a hypothesis, perhaps one for which the observations were explicitly made, or (4) in the absence of any hypothesis, any knowledge, or anything better to do with the data, to find if they are correlated with other results in the hope of discovering some new and universal truth.

4.1 The fishing trip

Take the last point first. Suppose that we have plotted something against something, on a fishing expedition of this type. There are grave dangers on this expedition, and we must ask ourselves the following questions.

The contributions of the Hubble Space Telescope to our understanding of starburst galaxies are reviewed. Over the past decade, HST's imagers and spectrographs have returned highquality data from the far-ultraviolet to the near-infrared at unprecedented spatial resolution. A representative set of HST key observations is used to address several relevant issues: Where are starbursts found? What is their stellar content? How do they evolve with time? How do the stars and the interstellar medium interact? The review concludes with a list of science highlights and a forecast for the second decade.

Overview

Almost exactly 10 years ago ST ScI hosted its annual symposium entitled Massive Stars in Starbursts (Leitherer et al. 1991). Those were the weeks immediately prior to HST's launch, and the conference organizers felt it appropriate to have a meeting on the subject of starbursts because HST had the potential for significant contributions. Starbursts are compact (10°—103 pc), young (∼ 106—108 yr) sites of star formation, often with high dust obscuration. These properties make starbursts ideal targets for HST, given its superior spatial resolution, ultraviolet (UV) sensitivity, and (later-on) infrared (IR) capabilities.

As we all know, the high hopes were not immediately fulfilled, and it was not until after the First Servicing Mission that HST lived up to the expectations.

One of the important topics of current astrophysical research is the role that supermassive black holes play in shaping the morphology of their host galaxies. There is increasing evidence for the presence of massive black holes at the centers of all galaxies and many efforts are directed at understanding the processes that lead to their formation, the duty cycle for the active phase and the question of the fueling mechanism. Related issues are the epoch of formation of the supermassive black holes, their time evolution and growth and the role they play in the early ionization of the Universe. Considerable observational and theoretical work has been carried out in this field over the last few years and I will review some of the recent key areas of progress.

Introduction

It is now widely accepted that quasars (QSOs) and Active Galactic Nuclei (AGN) are powered by accretion onto massive black holes. This has led to extensive theoretical and observational studies to elucidate the properties of the black holes, the characteristics of the accretion mechanisms and the mechanisms responsible for the production and transportation of the energy from the central regions to the extended radio lobes.

However, over the last few years there has been an increasing realization that Massive Dark Objects (MDOs) may actually reside at the centers of all galaxies (Ho 1998, Magorrian et al. 1998, Richstone et al. 1998, Gebhardt et al. 2000a, Gebhardt et al. 2000b, Merrit & Ferrarese 2001, van der Marel 1999).

Supernova 1987A has been a prime target for the Hubble Space Telescope since its launch, and it will remain so throughout the lifetime of HST. Here I review the observations of SN1987A, paying particular attention to the rapidly developing impact of the blast wave with the circumstellar matter as observed by HST and the Chandra Observatory.

Introduction

If there was ever a match made in heaven, it is the combination of SN1987A and the Hubble Space Telescope. Although the HST was not available to witness the first three years after outburst, it has been the primary instrument to observe SN1987A since then.

SN1987A in the Large Magellanic Cloud is the brightest supernova to be observed since SN1604 (Kepler), the first to be observed in every band of the electromagnetic spectrum, and the first to be detected through its initial burst of neutrinos. Although the bolometric luminosity of SN1987A today is ≈ 10-6 of its value at maximum light (Lmax ≈ 2.5 × 108 L⊙), it will remain bright enough to be observed for many decades in the radio, infrared, optical, UV, and X-ray bands.

SN1987A is classified as a Type II supernova (SNeII) by virtue of the strong hydrogen lines in its spectrum. It was atypical of SNeII in that its light curve did not reach maximum until three months after outburst and its maximum luminosity was about 1/10 the mean maximum luminosity of SNeII.

The Space Telescope Science Institute Symposium on “A Decade of HST Science” took place during 11–14 April 2000.

There is no doubt that the Hubble Space Telescope (HST) in its first decade of operation has had a profound impact on astronomical research. But HST did much more than that. It literally brought a glimpse of the wonders of the universe into millions of homes worldwide, thereby inspiring an unprecedented public curiosity and interest in science.

HST has seen farther and sharper than any optical/UV/IR telescope before it. Unlike astronomical experiments that were dedicated to a single, very specific goal, HST's achievements are generally not of the type of singular discoveries. More often, HST has taken what were existing hints and suspicions from ground-based observations and has turned them into certainty.

In other cases, the level of detail that HST has provided forced theorists to re-think previous broad-brush models, and to construct new ones that would be consistent with the superior emerging data. In a few instances, the availability of HST's razor-sharp vision at critical events provided unique insights into individual phenomena.

These proceedings represent a part of the invited talks that were presented at the symposium, in order of presentation. We thank the contributing authors for preparing their papers.

It is rare in astronomy to have a purely physics-based technique for studying the distant universe. Rooted in General Relativity, the image distortion and time delay of light from distant objects caused by foreground gravitational lenses offers such a window on the universe. Using only combinations of measured redshifts, angles, and arrival times of source intensity fluctuations, lensing observations can probe the mass distribution of the lens, the rate of expansion of the universe (the Hubble constant), the acceleration of expansion (dark energy), and the total amount of matter in the universe. The HST has made and will continue to make unique contributions to this new window on the universe.

Introduction

The universe is not as it seems: distant galaxies and quasars are in the wrong places. Their apparent positions on the sky have moved relative to where they would normally appear, and the culprit is mass-energy. Specifically, a massive object (a star, a galaxy, a cluster of galaxies) will warp space-time around it, causing light rays to bend as they pass by. If a mass concentration lies between us and a distant source, that source will appear in an altered location. The effect is called gravitational lensing, and it also systematically distorts the images of resolved sources like galaxies.

The high spectral resolution and high signal to noise capabilities of the Goddard High Resolution Spectrograph (GHRS) have permitted very accurate measurements of the gas phase abundances and physical conditions in interstellar clouds found in the Galactic disk and low halo and of the matter in several Galactic high velocity clouds. The interstellar gas phase abundances provide important clues about the composition of dust grain mantles and cores, and about the origins of intermediate and high velocity gas in the Galactic disk and halo. The processes that circulate gas from the disk into the low halo do not destroy dust grain cores. The gas in Complex C in the direction of Mrk 290 has a metallicity of 0.089 ± 0.024 solar, which implies the accretion of low metallicity gas by the Milky Way at a rate per unit area sufficient to solve the long standing Galactic G-dwarf problem. GHRS studies of interstellar Si IV, C IV, and N V absorption toward stars and AGNs have yielded measures of the 3 to 5 kpc extension of hot gas into the halo of the Milky Way. The GHRS results coupled with new measurements from the Far-Ultraviolet Spectroscopic Explorer (FUSE) satellite of O VI absorption by hot halo gas permit a study of the physical conditions in the hot Galactic Corona originally envisioned by Lyman Spitzer in his classic 1956 paper “On a Possible Interstellar Galactic Corona.”

HST observed Mars during all 5 oppositions between 1990 and 1999, providing unique new observations of the planet's atmosphere and surface during seasons which are typically poorly-observed telescopically and in wavelength regions or at spatial scales that are not at all observed by spacecraft. HST observations also filled a crucial gap in synoptic observations of Mars prior to 1998, during a time when no spacecraft were observing the planet. HST data have provided important new insights and understanding of the Martian atmosphere, surface, and satellites, and they continue to fulfill important spacecraft mission support functions, including atmospheric aerosol characterization, dust storm monitoring, and instrument cross-calibration.

Introduction

Mars has been the subject of intense telescopic observations for centuries (see, for example, reviews by Martin et al. 1992 and Sheehan 1988). Interest in the red planet stems partly from its prominent appearance in the night sky as a bright extended object roughly every 26 months, and also from historic telescopic observations and more recent spacecraft encounters that have revealed many similarities between Mars and the Earth in terms of surface and atmospheric characteristics and climatic histories. While cold and arid today and probably inhospitable to most forms of life, evidence exists indicating that Mars once may have had a much more clement climate, during a postulated “warm and wet” epoch early in solar system history (e.g. Pollack et al. 1987; Carr, 1998).

The Galactic Center has been the subject of a variety of HST observing programs, mainly since the installation of NICMOS. The observational strengths of NICMOS lie with its sensitivity and very stable point spread function which enables a variety of studies including sensitive searches for variable sources and accurate colors across the 1 to 2.5 µm region. The emission line filters in NICMOS enable studies of the interstellar medium and a search for [SiVI] emission as a ‘smoking gun’ for gas clouds near a black hole powered accretion disk.

Introduction

The center of the Milky Way is of course the closest galaxy nucleus and is a natural area to choose to study in detail. The discovery of a peculiar radio source, SgrA*, and the subsequent demonstration that it is a black hole has only heightened interest in the center. Figure 1 shows a contour plot at 1.04 µm compared to a NICMOS image at 1.45 µm which clearly shows why the Galactic Center requires use of infrared instrument like NICMOS with Av ∼ 30 while AK ∼ 3.3.

The Galactic Center has been studied with HST from the first observing cycle using WFPC proposed in an era where the nature of many of the stars was not understood, and the existence of a cluster in very close proximity to the black hole, SgrA*, was unknown.

This review aims to give an overview of the contribution of the Hubble Space Telescope to our understanding of the detailed properties of Local Group dwarf galaxies and their older stellar populations. The exquisite stable high spatial resolution combined with photometric accuracy of images from the Hubble Space Telescope have allowed us to probe further back into the history of star formation of a large variety of different galaxy types with widely differing star formation properties. It has allowed us to extend our studies out to the edges of the Local Group and beyond with greater accuracy than ever before. We have learned several important things about dwarf galaxy evolution from these studies. Firstly we have found that no two galaxies have identical star formation histories; some galaxies may superficially look the same today, but they have invariably followed different paths to this point. Now that we have managed to probe deep into the star formation history of dwarf irregular galaxies in the Local Group it is obvious that there are a number of similarities with the global properties of dwarf elliptical/spheroidal type galaxies, which were previously thought to be quite distinct. The elliptical/spheroidals tend to have one or more discrete episodes of star formation through-out their history and dwarf irregulars are characterized by quasi-continuous star-formation.

The angular resolution of HST has provided stunning images of star forming regions, circumstellar disks, protostellar jets, and outflows from young stars. HST has resolved the cooling layers behind shocks, and enabled the determination of outflow proper motions on time scales less than the post-shock coolingtime. Observations of the best studied region of star formation, the Orion Nebula, has produced many surprises. HST's superior resolution led to the identification of many new outflow systems based on their proper motions, the discovery of dozens of microjets from young stars, and the detection of wide-angle wind-wind collision fronts. HST has also produced spectacular images of circumstellar disks which have led to a rethinking of some aspects of planet formation. It now appears that most stars in the sky are born in environments similar to the Orion Nebula where within a few hundred thousand years after their formation, proto-planetary disks are subjected to the intense radiation fields of nearby massive stars. As a result, Orion's disks are rapidly evaporating. But at the same time their dust grains appear to be growing. Multi-wavelength images indicate that most of the solid mass in these disks may already be in large grains, possibly larger than a millimeter in size. The formation frequency of planets and the architectures of planetary systems will be determined by the competition between grain growth and photo-evaporation.

Globular clusters represent only a small fraction of the total mass in their parent galaxy, but provide a vast array of tests for stellar physics, dynamics, and galaxy formation. This review discusses the prominent accomplishments of HST-based programs:

1. The definition of precise fiducial sequences in the HR diagram, extending down to the hydrogen-burning limit,

2. Discovery of the upper white dwarf cooling sequence in several clusters,

3. Discovery of a highly consistent IMF on the lower main sequence,

4. Definitive age measurements for the oldest clusters in the outermost halo of the Milky Way, the Magellanic Clouds, and the dwarf elliptical satellites of the Milky Way,

5. Elucidation of the innermost structure of M15 and other core-collapsed clusters,

6. Discovery of surprisingly large “anomalous” populations of stars within dense cluster cores: extended blue horizontal-branch stars, blue stragglers, and others,

7. The first reliable color-magnitude studies for globular clusters in M31, M33, and other outlying Local Group members,

8. Discovery of massive young clusters in starburst galaxies with ages as small as 1 Myr,

9. Measurement of metallicity distribution functions among globular cluster systems in many giant E galaxies—bimodality is common, but details differ strongly, and

10. Deep imaging of cluster luminosity distributions in gE galaxies in Virgo, Fornax, and other Abell clusters as distant as Coma.