Great Calculations: A Surprising Look Behind 50 Scientific Inquiries

by Colin Pask

  The ancient Greeks placed the stars on a crystal sphere defining the outer limits of the universe. But already by the time of Titus Lucretius Carus (around 99 BCE to 55 BCE) there were strong arguments against such a picture. Lucretius wrote a wonderful poem summarizing all knowledge: De Rerum Natura, translated as The Poem on Nature or The Way Things Are. In the poem he writes:

  The universe itself is without limit of any kind,

  For if it had it would have to have an outside.

  Nothing can have an outside unless there is something beyond it

  So the point can be seen at which it ceases to be

  And beyond which the senses could follow it.

  There can be no such point for the whole of creation;

  If one thinks of the whole there can be nothing outside it,

  It can have no limit or measure, you could not conceive it.

  It does not matter what position you occupy,

  Space must stretch an infinite distance in every direction.1

  Here we have the great dilemma set out. Lucretius uses no mathematics, but he does give a lovely argument involving a spear thrower at the edge of the universe: Does the spear bounce back, or go on forever? Asking about the size of the universe, and about its age, origin, and composition, are among mankind's most profound questions.

  7.1 THE DARK SKY

  Go outside at night and you may see a sky covered in stars. You might say that the brilliant stars shine out of a great blackness. For some people, including Kepler and Halley, that turned into a puzzle: the question is not about the stars we see, but rather, why is the sky dark around them? The nature of this puzzle and the reactions it provoked are an important part of the history of cosmology. This is a good example of a problem arising from a simple calculation, and it is a problem that is solved by carrying out a series of other calculations. (To explore the full story in detail the books and papers of E. R. Harrison are recommended.)

  7.1.1 Posing the Problem

  Thirty years after the death of Copernicus, the astronomer Thomas Digges (1543–1595) published A Perfit Description of the Caelestiall Orbes in which he further reduced the special nature of mankind's status by placing the solar system amongst an infinite number of stars spreading out into an unbounded space. See figure 7.1. Digges was a modern day Lucretius.

  Figure 7.1. The solar system in an infinite universe as proposed by Thomas Digges in 1576. From Wikipedia, user Paddy.

  Digges obviously realized that there might be problems in assuming an endless, infinite number of stars, but he concluded that “the greatest part rest by reason of their wonderful distance invisible unto us.”2

  Kepler reacted to this idea of an infinite universe with terror and believed that it implied the whole “celestial vault would be as luminous as the Sun” and so “this world of ours does not belong to an undifferentiated swarm of countless others.”3 Thus the puzzle and a possible solution began to take shape: the universe cannot be infinite because that implies a bright night sky (Kepler), unless in some way the most distant stars are “invisible unto us” (Digges).

  Edmond Halley recognized the problem and gave it a more mathematical basis, which we can set out formally as follows. Suppose we assume the space around us is divided into spherical shells each filled uniformly with stars, n per unit volume, as in figure 7.2. Let a star contribute brightness B (in some measure) which will reduce inversely with the square of the distance from the observer according to the well-known laws of optics. For the shell of thickness dr at distance r the volume is πr2dr and so the brightness at the observer Bobs from a total sphere of radius R is given by summing or integrating over all the shells:

  Thus the r2 in the volume cancels with the 1/ r2 in the optical factor, and all shells contribute in the same way; the total brightness over all the spherical shells is proportional to the radius R of the whole universe. Now, if R is infinite, the brightness of the sky would become infinite, and, as Kepler feared, we would be burnt up in the hot radiation falling on the earth.

  Figure 7.2. Spherical shells of stars shining onto a central observer. A typical shell is at radial distance r, and the final shell has radius R that tends to infinity in an infinite universe. Figure created by Annabelle Boag.

  The German astronomer Heinrich Olbers (1758–1840) wrote later about this dark-sky puzzle, which has become known as Olbers's paradox—named rather unfairly to those who studied the puzzle before Olbers's time.

  It is interesting to note that one simple calculation and the simple observation that the night sky is mostly dark force us to confront some of the great questions about our universe.

  7.1.2 The Solution

  Halley followed Digges, suggesting that light from the most remote stars would not be detectable by us for some unspecified reason. In 1744, the Swiss astronomer Jean-Phillippe Loys de Chéseaux gave careful calculations about star numbers and effects but still concluded that the earth would receive light far brighter than sunlight. Chéseaux (and Olbers) suggested that the solution was absorption of light by interstellar matter. Herschel later pointed out that the interstellar material would become heated and hence radiate, so the overall problem remained.

  One of the first things to realize is that the radius R in equation (7.1) should really be the “look-out limit.” As we move out from the observer, the number of stars increases and gradually fills the sky so that eventually the light from stars further out is blocked by those nearer to the observer. (The analogy of looking through a forest is often used; after a certain distance all further tree trunks will be blocked from sight.) This limit can be calculated and is still very large—about 1023 or 1024 light years according to Edward Harrison.

  There are two other numbers that must be noted. First, the speed of light is finite (300,000 km/sec), so we must consider light that has had time to reach us. Second, stars have a finite lifetime, say 1010 years. Harrison claims that Lord Kelvin was the first person to bring all of these sorts of facts together in calculations reported in a paper published in 1901. Harrison goes on to describe ideas set out by Edgar Allan Poe that also make sense of the dark-sky mystery. The final viewpoint on the matter, according to Harrison, is summarized as follows:

  Darkness of the night sky is due not to the absorption of starlight, not to hierarchical clustering of stars, not to the finiteness of the universe, not to expansion of the universe, and not to many other proposed causes. The explanation is quite simple and can be stated in various equivalent ways. Because of the finite luminous age of stars and the finite speed of light, the number of visible stars is too few to cover the entire sky; most stars needed to cover the sky are so far away that their light has not reached us; the light travel-time from the most distant stars is greater than their luminous lifetime;…Why is the sky dark at night? Because starlight is too feeble to fill the dark universe.4

  The dark-sky problem is intriguing, and despite Harrison's views (which are often taken as definitive), it still stirs up controversy. The dark-sky problem played an important role in making cosmologists consider the universe and its properties and is a worthy addition to my list is calculation 21, why the night sky is dark.

  7.2 WHAT SORT OF UNIVERSE

  When we look up into the sky, we only see a miniscule part of the universe, and we are drawn to certain particular visible structures such as the planets and the Milky Way. To make progress on a larger scale, we need to find a guiding principle. The one generally assumed is the cosmological principle: the universe is homogeneous and isotropic—it is the same everywhere and in any direction we care to look. Obviously at first sight the universe does not appear to be the same in every direction, but that only emphasizes the enormous scale on which we must operate. We know there is a mix of stars, galaxies, and all sorts of strange objects, and we assume that mix continues equally everywhere in the universe when we extend our considerations to the largest scales. Observational evidence and astronomical surveys support the cosmological principle.

 
Using the cosmological principle allows us to use observations to infer the most general properties of the universe. How that is done using the combination of observation and theory—the use of calculations to make sense of data—is the subject of this section.

  This section provides a nice example of three important uses of calculations:

  To help deal with experimental data and their presentation

  To deduce new results from a given theory

  To calculate values for physical parameters which may be used to describe the universe and promote further experimental work

  7.2.1 Calculating with Observational Data: Hubble's Discovery

  Studies by early workers such as V. M. Slipher and C. Wirtz indicated that spiral nebulae were moving away from us at great speeds, and those speeds seemed to be correlated with the brightness of the spiral nebulae. The speeds are calculated from the amount of redshift in the observed spectral lines. Concepts and observations were developed that allowed properties such as brightness and its variations to be used to calculate estimates of the distances to those nebulae. (For a simple introduction to these matters see the books by Coles and Weinberg.) The stage was set for one of the most famous papers in cosmology: A Relation between the Distance and Radial Velocity among Extra-Galactic Nebulae, published by Edwin Hubble in 1929. Hubble presented his data in a diagram as in figure 7.3.

  Figure 7.3. Hubble's 1929 plot of velocity versus distance for extragalactic nebulae. From Edwin P. Hubble, “A Relationship between Distance and Radial Velocity among Extra-Galactic Nebulae,” Proceedings of the National Academy of Science 15, no. 3 (March 15, 1929).

  Hubble's results say that the observed nebulae are receding from us with velocities that increase with the distance. This result has now been generalized and interpreted using the cosmological principle to give a result central to all cosmology:

  the universe is expanding; if observed from any point, elements in the universe are receding from that point with speeds which are increasing with the separation distance.

  But Hubble went further than that with his statement: “the results establish a roughly linear relation between velocities and distances among nebulae.”5 This may well be the most significant result in cosmology, and it is now written as

  Here V is the speed of recession of the object being observed at distance r. Naturally, such a basic finding about the absolute nature of the whole universe has led to an enormous effort to verify and extend Hubble's original results. (For a detailed, comprehensive study see the 2010 review by Freedman and Madore). A more recent diagram of the data is shown in figure 7.4.

  This is a good example of how a calculation (here a data fit and suggesting a simple linear relationship) identifies an important parameter. Hubble's law as stated in equation (7.2) identifies H0 as a (if not the) key parameter in cosmology. I return to this point in section 7.2.3.

  (A historical aside: there is some debate about who deserves credit for the work discussed in this section; as is often the case, several people, probably independently, came to conclusions about the expanding universe similar to those reached by Hubble. See the letter by Way and Nussbaumer for a summary and references, and the later response by Livio.)

  Figure 7.4. More recent data supporting Hubble's law. The small box at the origin indicates the extent of Hubble's original data. Reprinted with permission, © Cambridge University Press, from E. R. Harrison, Cosmology: The Science of the Universe, 2nd ed. (Cambridge: Cambridge University Press, 2000).

  7.2.2 Calculating from Einstein's General Relativity Equation

  In Isaac Newton's theory of dynamics, bodies interact via forces, and their movements may be measured relative to a given coordinate system or reference frame. In particular, the force controlling the motion of large bodies in space is gravity acting over large distances according to equation (6.1). Newton was concerned about the stability of the universe, and in his Principia, he resolved the stability difficulty with a view that

  lest the system of fixed stars, by their gravity, fall on each other mutually, he [God] hath placed those systems at immense distance from one another.6

  Interestingly, the idea of the collapse of a finite universe was considered by Lucretius in his De Rerum Natura.

  It was not until 1916 that suitable ways to tackle these problems were set out by Albert Einstein in his general theory of relativity. The Newtonian concepts were replaced by a strange theory in which bodies change the properties of the space through which they move thus affecting how they move. In John Archibald Wheeler's famous words “space-time tells matter how to move; matter tells space-time how to curve.”7 Einstein's equations can be used to explore the properties of the universe on the very largest scale in which the cosmological principle holds. Instead of considering the discrete entities, like stars, which we observe, the universe is characterized by the properties of density ρ and pressure p corresponding to a continuum. Of course the observed entities lead to the density when suitably averaged. If these assumptions are built into the theory, the result is the Robertson-Walker metric expressing the properties of space-time. This metric contains a scale factor R(t), which describes how the spacing varies in any local coordinate system in use at time t varies. Bodies may be referred to—such as a Cartesian grid of coordinates, as usual—but over time the size of that grid changes. (This is described in every book on cosmology; I recommend the treatment in the books by Lambourne—see the very clear chapter 8—and Roos.)

  Using the continuum density and pressure model in Einstein's equations leads to the Friedman equations for the scale factor R(t):

  In these equations, G is the usual Newtonian gravitational constant, and k is a factor related to the spatial geometry involved. (At present it seems likely that k may be zero.) Readers not familiar with the mathematics involved should simply note that the first equation tells us the scale factor in terms of the density ρ (if k is indeed zero) and that the scale is increasing. The second equation tells us about the rate of change of the scaling factor: Is it a steady change (increase or decrease) or is the rate of change itself varying?

  The calculation in this case has taken us from Einstein's equations with certain input assumptions to new equations for the scale of the universe.

  7.2.3 Results

  We can now link the two calculations (one on Hubble-type experimental data, and the other on Einstein's equations under certain assumptions) because the Hubble factor H(t) and the scale factor R(t) are linked by

  Using the above calculations and available data, we can come to the remarkable conclusion that

  the Universe has a nearly flat spatial geometry with k = 0 and a total density that is close to 1 × 10–26kg m–3. Such a universe originated with a big bang…and has an expansion age of about 13.7 billion years.8

  Such a staggering result is why I add calculation 22, state of the universe to my list.

  7.2.4 A Weird Twist to the Story

  Recall that Lucretius and Newton were both concerned by the way gravitational effects would cause matter to move closer together and produce a collapsing universe. The second Friedman equation (7.3) tells us that the second derivative of the scale factor is negative, implying that the rate of expansion of the universe should be slowing down, in keeping with the contracting effect of gravitational forces. However, observations (for which the astronomers Saul Perlmutter, Adam Reiss, and Brian Schmidt were awarded a Nobel Prize in 2011) indicate that the rate of expansion is actually increasing.

  The suggested solution to this dilemma is to postulate the existence of something called dark energy, which contributes a density ρde and has the weird fluid property that its pressure is negative so that the right-hand side of equation (7.3) becomes positive. Dark energy is somehow driving an acceleration in the rate at which the universe is expanding. It acts something like a repulsive gravitational force. Current data suggest that at this time, dark energy contributes well over 68 percent to the density of matter in the universe. This is a wei
rd and disturbing result. The “dark” refers to the fact that this dark energy has not been seen (detected) in other ways, and here is an incredible mystery for future cosmologists to come to grips with.

  I must point out that this is connected with one of those wonderful stories in science: Einstein's worries over his cosmological constant Λ. In his early work Einstein introduced a constant Λ into his theory to satisfy a requirement that the universe could be static, as was believed to be the case at the time. He did not like the introduction of what seemed to be an arbitrary constant, and at various times Λ has disappeared. It now seems to have resurfaced in the guise of dark energy. (Interested readers might consult “Lambda: The Constant that Refuses to Die,” an entertaining review by John Earman.)

  7.3 CHEMICAL ELEMENTS, THE UNIVERSE, AND US

  I included this section for four reasons: it relates to why we can actually exist, it shows us some very clever and innovative thinking, it involves a remarkable prediction made by a remarkable astrophysicist, and it introduces the fascinating subject of a link between humanity and the nature of the universe.

  In the previous section, we saw how the nature of the universe has been revealed using a mixture of theory and observation. It is now widely accepted that the universe began with the big bang. In the earliest times of the universe there was a hot, dense mixture of photons, neutrons, protons, and electrons. The theory explains how the universe cooled and expanded, and how gravitational forces eventually pulled matter into clumps that became stars and galaxies. But there is no mention of the essential chemical elements that go to form us and our world. Where did they come from? The answer is one of the most extraordinary stories in all of science. It is also a story that inspires philosophical thoughts about the universe and our place in it.