As explained in the preface, I have used high-school mathematics to present some of the material in this book. If you want to know what that means, if you want to learn whether you have the background necessary to do the mathematics, then scan through this introductory material. But remember, it is not necessary to follow all the derivations, particularly the ones in the boxes, if you just want to learn what the main ideas in modern gravity and astronomy are. So if you find your mathematics too old or rusty, then see how you get along without it.

High-school mathematics

The mathematics used is basic numeracy, algebra, and a tiny bit of trigonometry (which you can skip).

It is essential to understand scientific notation for numbers, that is how to write numbers in the form 3.2 × 106 and know what the factor 106 means. Scientists use this notation all the time, because otherwise they would be writing out long confusing strings of zeros. The number 3.2 × 106 means 3 200 000, obtained by moving the decimal point in 3.2 six places to the right. Similarly, the number 5.9 × 10−3 is 0.0059, obtained by moving the decimal point three places to the left.

The study of cosmology presents today's physicists with the biggest challenges to their understanding of gravity and of fundamental physics in general. Both on theoretical and on observational grounds, it seems that we will not be able to understand cosmology well until we understand physics better than we do today. But it also seems that cosmology could provide us with the keys to that deeper understanding of physics.

The biggest gap in physics is quantum gravity: we do not yet possess a consistent way of representing gravity as a quantum theory. There is no uncertainty principle in general relativity, no quantization of gravitational effects, no need to use probabilities in making predictions about the outcome of gravitational experiments. This seems inconsistent with the fact that all material systems that create gravity are quantum systems: if we can't say exactly where an electron is, how can we say exactly where its gravitational field is?

The cycle of birth, aging, death, and re-birth of stars dominates the activity of ordinary galaxies like our own Milky Way. The cycle generates the elements of which our own bodies are made, produces spectacular explosions called supernovae, and leaves behind “cinders”: remnants of stars that will usually no longer participate in the cycle. We call these white dwarfs, neutron stars, and black holes.

Governing this cycle is, as everywhere, gravity. An imbalance between gravity and heat in a transparent gas cloud leads to star formation. The long stable life of a star is a robust balance between nuclear energy generation and gravity. This balance is finally lost when the star runs out of nuclear fuel, leading to a quiet death as a white dwarf or to a violent death as a supernova.

We have seen how the Sun's gravity holds the planets in their orbits. The Sun's gravity also holds itself together. Like all stars, the Sun is a seething cauldron, its center a huge continuous hydrogen bomb trying to blow itself apart, restrained only by the immense force of its own gravity. In this chapter, we will see how the Sun has managed to maintain an impressively steady balance for billions of years. In the course of our study, we will learn about how light carries energy and we will build a computer model of the Sun.

Sunburn shows that light comes in packets, called photons

The Sun glows so brightly because it is hot. We can infer just how hot it is from its color. The color and temperature of the Sun are related to each other in just the same way as for hot objects on the Earth.

Black holes. No term evokes the mystery of modern gravity more than this one. The mystery of black holes is more than an invention of popularizers of astronomy and relativity. Black holes were certainly a mystery to Einstein and his contemporaries. Yet today black holes are everywhere: in X-ray binaries, in the centers of galaxies, and of course in books, like this one, on relativity and gravity!

Theorists attacked the problem of understanding black holes, not by using astronomical evidence, but by using lessons they had learned from quantum mechanics. Quantum thinking demanded that physicists ask only questions about things that could be measured, not about what is hidden from experiment. Thus, they can measure that light behaves sometimes as a particle (the photon) and sometimes as a wave, but they find it useless to ask what is a wave–particle.

In the last two chapters we have made a lot of progress in exploring the future and past of the Universe, basically just by using local Newtonian gravity. We argued that the dynamics of an expanding, homogeneous and isotropic cosmology can be calculated from Newtonian gravity, at least if the pressure in the Universe is negligible, because all we need to look at is the local Universe, the part nearest us. The assumption that the Universe is homogeneous guarantees that the rest of the Universe will behave the same as our local region.

1. ▷ The drawing under the text on this page illustrates how complicated three-dimensional solid objects could be. Why is the Universe apparently so simple?

But this line of reasoning has its limitations. Even if we calculate the dynamics of the Universe this way, we don't learn what the distant parts of the Universe will look like in our telescopes. The curvature of space, which is not part of a Newtonian discussion, will affect the paths of photons as they move through the Universe. Moreover, if we want to ask deeper questions about the Universe, such as those we pose in the next chapter, then we should know something more about its the larger-scale structure.

There would be no life as we know it on Earth without the atmosphere. Even life in the oceans would not exist: without the atmosphere's thermal “blanket”, the oceans would freeze. Yet in the beginning, the Earth probably had a very different atmosphere from its present one. The other planets, with their different masses and different distances from the Sun, all have vastly different atmospheres from the Earth's. In the retention of the atmosphere, and in the subsequent evolution of the atmosphere and of life itself, gravity has played a crucial role.

In this chapter, as we look at the role that gravity has played in this story, we shall encounter fundamental ideas about the nature of matter itself: how temperature and pressure can be explained by the random motions of atoms, why there is an absolute zero to the temperature, and even why atoms cannot quite settle down even at absolute zero.

Gravity is the engine that drives the Universe. But it does not work alone, of course. In fact, one of the most satisfying aspects of studying astronomy is that there is a role for essentially every branch of physics when one tries to explain the huge variety of phenomena that the Universe displays. One branch of physics, however, stands out from the rest because of its absolutely central place in helping us to learn about the Universe, and that is the study of the way hot bodies give off light.

Almost all of the information we have from astronomical bodies is carried to us by light, and almost all the light originates as radiation from some sort of hot region. The great breakthrough in physicists' understanding of such thermal radiation was made by the German physicist Max Planck (1858–1947) at the start of the twentieth century. (See Figure 10.2 on page 112.) The story of this breakthrough is the story of physicists' first steps toward quantum theory. It is also the story of the beginnings of a real understanding of the heavens.

Astrophysical spectral lines offer two important insights into the workings of our Universe. First, they are probes of the fundamental (QM) nature of matter because they originate from subatomic, atomic and molecular systems. Second, they provide, via the Doppler effect, critical dynamical information on astrophysical systems ranging in scale from planetary systems to superclusters of galaxies. Examples of major contemporary problems in astrophysics that can be addressed through spectral line studies and the associated quantum mechanics include.

Missing mass and the halos of galaxies The most common element in the Universe is hydrogen and much of it is in a cold state. Given the 10 eV gap between the ground state and the first excited state of the simple Bohr atom, we should have little direct knowledge of this gas, yet it is the best studied gaseous component of the Universe. The reason is the 21 cm line corresponding to the hyperfine splitting of the ground state. The extremely low transition probability of this transition and the consequently narrow width of this line have led to its widespread use in measuring galaxy dynamics and kinematics. Studies of galaxy rotation have shown evidence for missing matter and point to the possibility of dark-matter halos. The nature of the dark matter and the implication on the long-term fate of the Universe remain contentious issues in astrophysics. The nature of this line and its use in these studies is discussed.

In contrast to previous chapters, we now consider data transformation, how to transform data in order to produce better statistics, either to extract signal or to enhance signal.

There are many observations consisting of sequential data, such as intensity as a function of position as a radio telescope is scanned across the sky or as signal varies across a row on a CCD detector, single-slit spectra, time-measurements of intensity (or any other property). What sort of issues might concern us?

1. baseline detection and/or assessment, so that signal on this baseline can be analysed;

2. signal detection, identification for example of a spectral line or source in sequential data for which the noise may be comparable in magnitude to the signal;

3. filtering to improve signal-to-noise ratio;

4. quantifying the noise;

5. period-finding; searching the data for periodicities;

6. trend-finding; can we predict the future behaviour of subsequent data?

7. correlation of time series to find correlated signal between antenna pairs or to find spectral lines;

8. modelling; many astronomical systems give us our data convolved with some more-or-less known instrumental function, and we need to take this into account to get back to the true data.

The distinctive aspect of these types of analysis is that the feature of interest only emerges after a transformation.

‘Detection’ is one of the commonest words in the practising astronomers' vocabulary. It is the preliminary to much else that happens in astronomy, whether it means locating a spectral line, a faint star or a gamma-ray burst. Indeed of its wide range of meanings, here we take the location, and confident measurement, of some sort of feature in a fixed region of an image or spectrum.

When a detection is obvious to even the most sceptical referee, statistical questions usually do not arise in the first instance. The parameters that result from such a detection have a signal-to-noise ratio so high that the detection finds its way into the literature as fact. However, elusive objects or features at the limit of detectability tend to become the focus of interest in any branch of astronomy. Then, the notion of detection (and non-detection) requires careful examination and definition.

Non-detections are especially important because they define how representative any catalogue of objects may be. This set of non-detections can represent vital information in deducing the properties of a population of objects; if something is never detected, that too is a fact, and can be exploited statistically. Every observation potentially contains information.

Science is about decision. Building instruments, collecting data, reducing data, compiling catalogues, classifying, doing theory – all of these are tools, techniques or aspects which are necessary. But we are not doing science unless we are deciding something; only decision counts. Is this hypothesis or theory correct? If not, why not? Are these data self-consistent or consistent with other data? Adequate to answer the question posed? What further experiments do they suggest?

We decide by comparing. We compare by describing properties of an object or sample, because lists of numbers or images do not present us with immediate results enabling us to decide anything. Is the faint smudge on an image a star or a galaxy? We characterize its shape, crudely perhaps, by a property, say the full-width half-maximum, the FWHM, which we compare with the FWHM of the point-spread function. We have represented a dataset, the image of the object, by a statistic, and in so doing we reach a decision.

Statistics are there for decision and because we know a background against which to take a decision. To this end, every measurement we make, and every parameter or value we derive, requires an error estimate, a measure of range (expressed in terms of probability) that encompasses our belief of the true value of the parameter. We are taught this by our masters in the course of interminable undergrad lab experiments.

Whether He does or not, the concepts of probability are important in astronomy for two reasons.

1. Astronomical measurements are subject to random measurement error, perhaps more so than most physical sciences because of our inability to rerun experiments and our perpetual wish to observe at the extreme limit of instrumental capability. We have to express these errors as precisely and usefully as we can. Thus when we say ‘an interval of 10−6 units, centred on the measured mass of the Moon, has a 95 per cent chance of containing the true value’, it is a much more quantitative statement than ‘the mass of the Moon is 1±10−6 units’. The second statement really only means anything because of some unspoken assumption about the distribution of errors. Knowing the error distribution allows us to assign a probability, or measure of confidence, to the answer.

2. The inability to do experiments on our subject matter leads us to draw conclusions by contrasting properties of controlled samples. These samples are often small and subject to uncertainty in the same way that a Gallup poll is subject to ‘sampling error’. In astronomy we draw conclusions such as ‘the distributions of luminosity in X-ray-selected Type I and Type II objects differ at the 95 per cent level of significance’. Very often the strength of this conclusion is dominated by the number of objects in the sample and is virtually unaffected by observational error.

In embarking on statistics we are entering a vast area, enormously developed for the Gaussian distribution in particular. This is classical territory; historically, statistics were developed because the approach now called Bayesian had fallen out of favour. Hence direct probabilistic inferences were superseded by the indirect and conceptually different route, going through statistics and intimately linked to hypothesis testing. The use of statistics is not particularly easy. The alternatives to Bayesian methods are subtle and not very obvious; they are also associated with some fairly formidable mathematical machinery. We will avoid this, presenting only results and showing the use of statistics, while trying to make clear the conceptual foundations.

Statistics

Statistics are designed to summarize, reduce or describe data. The formal definition of a statistic is that it is some function of the data alone. For a set of data X1, X2, …, some examples of statistics might be the average, the maximum value or the average of the cosines. Statistics are therefore combinations of finite amounts of data. In the following discussion, and indeed throughout, we try to distinguish particular fixed values of the data, and functions of the data alone, by upper case (except for Greek letters). Possible values, being variables, we will denote in the usual algebraic spirit by lower case.

The summarizing aspect of statistics is exemplified by those describing (1) location and (2) spread or scatter.