Invitation to a Contemporary Physics (2004)
.pdf
7.7. Concluding Remarks: Harvesting Chaos |
265 |
There are still many open questions. For instance, we have discussed chaos for open dissipative systems only. There are closed dynamical systems that are conservative (non-dissipative) — the problem of chaos in a non-integrable Hamiltonian system. A celebrated example is that of a three-body problem in astronomy, the problem of three planets gravitationally attracting each other. This is a non-integrable system and chaos lurks there. This is, in fact, very relevant to the long-term stability of the solar system — on the time scale of hundreds of millions of years. Then there is the problem of two nonlinearly coupled oscillators for example. These can be treated quantum mechanically. Does chaos survive quantization? What is the signature of quantum chaos? There are other rather technical issues that remain still unresolved. But it can be fairly claimed that for most dissipative systems, the theoretical framework of chaos is complete. It is now time to harvest chaos.
Undoubtedly, the most outstanding application of our knowledge of chaos is to understanding fluid dynamical turbulence — the last unsolved problem of classical physics. For several types of closed flow, routes to turbulence are now understood. The problem of turbulence in open flows such as wakes and jets is, however, still unsettled. Do we have here, after all, the Landau scenario of confusion of incommensurate frequencies, appearing one at a time?
Next, comes the problem of short and intermediate range weather forecasting that started it all from the time of the early works of E. N. Lorenz. Chaos, of course, rules out long-range forecast. But can we extend our present 3–4 day forecasts to a 10–12 day forecast? One must estimate the time over which memory of the initial data is lost due to folding back of the phase-space trajectories. This ultimately limits the range of prediction.
Reconstruction of the strange attractor, if any, at the heart of an apparently random behavior, is a powerful diagnostic tool that is finding application in diverse fields of study including neurology, psychology, epidemiology, ecology, sociology and macro-economics.
There is yet another way in which we can make chaos really work for us. Thus, chaotic encryption can be used for secure communication — this is based on the idea that two similar chaotic oscillators can be synchronized through a fairly general kind of coupling. Also, if we want to explore a large portion of phase space e ciently, as for example, for mixing things in a chemical reactor, it may be best to use a chaotic trajectory. Nature seems to have made use of this chaotic access to, or exploration of, these possibilities in the vital matters of evolution, and in the workings of our immune system. Chaos in our neural network is also an important subject for study.
There are other highly speculative aspects of chaos. Chaos is physically intermediate between the ‘thermal bath’ and macroscopically ordered motion. What happens when a chaotic system is coupled to a ‘thermal bath’? Can we make ‘intelligent heat engines’? What is the thermodynamic cost of driving a system
266 |
Chaos: Chance Out of Necessity |
chaotic? Or an even more speculative question such as whether chaos is ultimately responsible for the well-known quantum uncertainties. We do not know.
It has been claimed that the scientific revolution, or rather the shift of paradigm caused by ‘chaos’ is comparable to that caused by quantum mechanics. To us this comparison does not seem very apt. In any case, judging by their respective impacts, the claim is overstated. But what can definitely be claimed is that chaos has definitely been a surprise of classical physics. It has the subtlety of number theory. You can certainly have fun with chaos if you have a personal computer, or even a hand-held calculator.
7.8 Summary
A dynamical system is said to be chaotic if its long time behavior cannot be computed and, therefore, cannot be predicted exactly even though the laws governing its motion are deterministic. Hence the name deterministic chaos. A common example is that of turbulence which develops in a fluid flowing through a pipe as the flow velocity exceeds a certain threshold value. The remarkable thing is that even a simple system, such as two coupled pendulums, having a small number of degrees of freedom can show chaos under appropriate conditions. Chaotic behavior results from the system’s sensitive dependence on the initial conditions. The latter amplifies uncontrollably even the smallest uncertainty, or error unavoidably present in the initial data as noise. This sensitivity derives essentially from the nonlinear feedback, and often dissipation inherent in the dynamical equations. The dynamical evolution may be described by continuous di erential equations or by discrete algebraic algorithms, and can be depicted graphically as the trajectory of a point in a phase space of appropriate dimensions. The representative point may converge to a stable fixed point, or a limit cycle, or a torus, and so on. These limiting sets of points in the phase space are called attractors and signify respectively a steady state, a singly periodic motion, or a doubly periodic motion, and so on. Chaos corresponds to a strange attractor, which is a region of phase space having a fractional dimension (a fractal) and its points are visited aperiodically.
There are several routes to chaos. We have, for example, the period doubling route of Feigenbaum in which the dynamical system doubles its period at successive threshold values (bifurcation points) of the control parameter, until at a critical point, the period becomes infinite and the motion turns chaotic. These di erent routes define di erent universality classes of common behavior.
Chaos has become a paradigm for complex behavior. Many apparently random systems are actually chaotic. Powerful techniques have been developed to reconstruct the strange attractor from the monitoring of a single fluctuating variable. This helps us make limited predictions and diagnostics for diverse problems, such as the weather forecast and turbulence, epileptic seizures, populations of competing species, macroeconomic fluctuations, and the recurrence of epidemics.
7.9. Further Reading |
267 |
7.9 Further Reading
Books
•G. L. Baker and J. P. Gollub, Chaotic Dynamics: An Introduction
(Cambridge University Press, New York, 1990).
•P. Berg´e, Y. Pomeau and C. Vidal, Order Within Chaos (Wiley, New York, 1986).
•R. C. Hilborn, Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers (Oxford University Press, Oxford, 1994).
Semi-popular articles
•J. P. Crutchfield, J. D. Farmer, N. H. Packard and R. S. Shaw, Chaos, Scientific American 255 (December 1986), p. 38.
•Joseph Ford, How Random is a Coin Toss?, Physics Today 36 (April 1983), p. 40.
This page intentionally left blank
Bright Stars and Black Holes |
8 |
On a clear night, if you look up at the sky you will likely see a broad band of faint distant stars stretching from horizon to horizon which the ancient Greeks likened to a river of milk and accordingly called a galaxy. This band is an edge-on view of our galaxy, which we simply refer to as the Galaxy or Milky Way. If we could observe our galaxy far away from outside, it would appear as a gigantic thin disk with a central bulge. The disk is about 1021 m across, (0.5–1)×1019 m thick, packed mostly with stars traveling in nearly circular orbits about the galactic center with very small up-and-down motions. The central bulge itself contains stars moving with small circular speeds in random directions, which accounts for their roughly spherical distribution. Above and below the disk, a spherical swarm of globular clusters with large random velocities forms a halo concentrated toward the center of the Galaxy. If we could look at our galaxy face-on, as we can with external systems such as the galaxy NGC4622, we would see in the plane of the galactic disk a beautiful spiral pattern composed of several spiral arms outlined by very brilliant stars. These bright stars, strung along the arms like ‘beads on a string,’ contrast sharply with the very dark inside edges of the arms and the faint, more di use patches throughout the disk. These irregular formations suggest the presence of matter — gas and dust — in the space between the stars.
But it is the stars that play the most important role in a normal galaxy: they provide most of its mass and hence are responsible for the gravitational forces that bind the galaxies into stable associations. Not all stars are alike. They come in a wide variety of forms: massive and not so massive, large and small, bright and faint. Why is it so? This is a question central to all astronomy, which we will try to answer in this chapter. We will find that stars constantly change in composition and appearance: starting with di erent initial masses, they evolve at di erent rates from one form to another, and metamorphose into di erent final objects. Stars are born, live and die. Underpinning this study is the assumption that the laws of physics, discovered and verified here on earth, are universal: they can be applied to all the physical universe. That such an assumption is well founded has led to the creation of one of the most exact of the astrophysical sciences.
269
270 |
Bright Stars and Black Holes |
8.1 The Basic Properties of Stars
In the night sky, the ‘dog star’ (Sirius A) appears to be the brightest star, not because it radiates the most energy, but because it is among the closest to us. What we perceive in our observations of the sky is the apparent brightness, which depends on the star’s distance from us and does not express its true energy output. This true energy output, a property intrinsic to the star, is measured by the absolute brightness, or luminosity — the total amount of energy radiated each second in all forms at all wavelengths.1 As a star radiates in all directions, the fraction of its power reaching a detector with receiving surface area A a distance d away is A/4πd2. So the star’s luminosity L is given in terms of its apparent brightness L and distance d by L = 4πd2L. In SI metric units, L is given in watts, or W (1 W is 1 joule (J) of energy radiated each second), d in meters (m), and L in W/m2. A star’s apparent brightness can be measured by photometry.
Distances to nearby stars are measured by triangulation, whereas distances to nearby galaxies can be determined by the brightness and periodicity of their Cepheid variable stars.2 For these vast distances, astronomers use two basic units: the light-year, abbreviated as ly, which is the distance light travels in one year, i.e., 9.46×1015 m; and the parallax-second, or parsec, abbreviated to pc (1 pc = 3.26 ly). The nearest star beyond the sun, Proxima Centauri, is about 4.3 ly or 1.3 pc away, whereas the brightest star, Sirius A, is 8.8 ly or 2.7 pc away. The sun could be said to be eight light-minutes away. We see that once the distance d and apparent brightness L of a star are determined, we also have its intrinsic luminosity L.
Some stars, like Sirius A, are bluish; others, such as Antares in α Scorpii and Arcturus in α Bootes, tend to be reddish or orangish. Although sunlight appears to us as yellow, we have learned elsewhere (e.g., in Chapter 2) that it also contains, to a lesser extent, all other colors, or wavelengths, spread over a continuous spectrum. Let us now see how this applies to stars and what starlight can tell us.
In studying its general features, we may regard a star during most of its lifetime as a ball of gas in thermal equilibrium — a large collection of electrons and atoms in random motion, jostling each other, at a uniform temperature throughout. The random motions of the free particles in matter give rise to a form of energy we call heat, which is measured by the temperature of the object: the faster the thermal motions, the higher the temperature. As the electrons (and atoms and molecules) are constantly and randomly disturbed by thermal motions, getting pushed and pulled by other electrons and atoms, they emit a photon at each deflection with an energy precisely matching the kinetic energy lost by the interacting electron. Since the electrons move at random speeds and the radiative processes occur randomly, photons may be produced practically at any wavelengths. What we have then is a continuum spectrum of thermal radiation.
1It is more precisely called bolometric luminosity, to distinguish it from visible luminosity, which is the energy radiated each second in the visible part of the spectrum.
2More details on the definitions and measurements of cosmic distances can be found in Chapter 10.
8.1. The Basic Properties of Stars |
271 |
As a thermal object, our ball of gas obeys known statistical physical laws.3 In particular, the thermal radiation it emits exhibits an energy distribution over wavelengths with a profile uniquely defined by its temperature T (see Fig. C.3 of Appendix C). This distribution curve has a single maximum located at wavelength λmax such that λmaxT = 0.0029 m K, a relation known as Wien’s displacement law, which we have stated in SI metric units. Hotter sources radiate more blue light than cooler sources.
Another important consequence of the photon distribution is the Stefan– Boltzmann law, which says that the energy flux f (or the energy of radiation emitted at all wavelengths per unit area and per unit time) depends only on the temperature T of the radiating object and is given by f = σT 4, where σ is the Stefan–Boltzmann constant.4 As the temperature increases, the energy flux rises even more steeply. If we regard a star as a sphere of radius R, its luminosity, or radiant power, is given by the energy flux f generated over its whole surface area, 4πR2 — that is, L = 4πR2f, or L R2T 4. The hotter an object is, the brighter it shines.
In applying these laws to stars, we must remember that T is the star’s e ective surface temperature, which is the e ective average temperature of the various lightemitting layers of gas. So, once we’ve determined a star’s color, we use Wien’s law to obtain its surface temperature T . With T and L known, we can calculate the stellar radius R from the Stefan–Boltzmann law written in the form L R2T 4.
In the above discussion, we considered radiation by free electrons.5 The nature of radiation changes completely when the radiating electrons are confined in atoms, because then they (and hence the atoms themselves) may exist only in a limited number of discrete states defined by the rules of quantum mechanics. The energies of these states, plotted in increasing order like rungs in a ladder, give us a graphical representation of the atomic energy spectrum, as unique to an atomic species as the fingerprint to an individual. When an atom interacts with an electromagnetic field, it may pass from one allowed state to another by absorbing or emitting a photon, provided the photon energy exactly matches the energy di erence of the two atomic states. As the atomic energy spectrum is discrete, only discrete values are allowed for the photon energy in emission or absorption. If a few of these values are somehow known, then we can identify the atomic element involved.
When a beam of light passes through a cloud of atoms, many photons of certain colors may be absorbed by the atoms and removed from the beam. The absence of these (‘line’) photons in the outgoing beam reveals itself in absorption lines, which appear as dark vertical lines on a photographic spectrum or valleys on a graph (Fig. 8.1). Similarly, when excited atoms de-excite by radiative transitions, photons of specific colors are emitted and added to the outgoing light, giving rise to emission lines, which appear as bright vertical lines or sharp peaks.
3See Appendix C for more details.
4In SI units, σ = 5.67 × 10−8 W/(m2K4).
5Of course, they are not really free since interaction fields exist all around. What we mean is that they are not bound in atoms.
272 |
Bright Stars and Black Holes |
Wavelength |
Wavelength |
Intensity |
Intensity |
(a) (b)
Figure 8.1: (a) Emission lines and (b) absorption lines on a background radiation continuum.
For stars, absorption spectra come about through the following processes. As thermal photons of all wavelengths rise up from the stellar depths to the surface, those with the ‘right’ energies (for which atomic transitions are allowed) will be absorbed by atoms in the cooler upper layers. The excited atoms then de-excite either by non-radiative collisions or by radiative transitions. In the first case, the surplus energy is converted into kinetic energy for the particles involved in the collisions, and no radiation is produced. In the second case, when radiation accompanies the de-excitation, the photon emitted is equally likely to travel in any direction. Because the hot stellar surface is surrounded by a dark absorbing space, there are more photons originally traveling toward us being scattered out of our line of sight than into it, and so there is a net loss of line photons traveling in our direction. In either case, photons are removed from the underlying radiation field at the allowed transition energies, leading to the appearance of absorption lines in the stellar spectrum.
The pattern of the spectral lines tells us what elements are present in the star, and their relative strengths (i.e., widths and degrees of darkness) tell us what fraction of atoms of an element is in each excited state. This is a crucial result. Why? Because with it we gain direct information about the composition of the star and, from the composition, determine the temperature, density and pressure conditions in the far-away gas.
In a classification system invented by astronomers Edward Pickering and Annie Cannon, stars are divided into spectral classes. These may run in any order, but the sequence finally adopted by astronomers6 is O, B, A, F, G, K and M, with the O stars showing ionized helium lines in their spectra, the B stars showing neutral helium lines, etc. But since the observed spectral properties of a star reflect its e ective temperature, spectral classes form a temperature sequence. The adopted
6Just remember — Oh, Be A Fine Girl, Kiss Me!
8.2. The Hertzsprung–Russell Diagram |
273 |
sequence corresponds to an order of decreasing temperatures. The O stars are so hot, at 40 000 K, that even the most tightly bound atoms, such as helium, become ionized, losing some of their electrons. In class G stars, at around 5 500 K, lines of ionized metals appear along with lines of neutral hydrogen. By M class, the temperature has dropped so low, at 3 000 K, that atoms can stick together to form molecules.
Finally, what about the star’s mass? Let us first consider a double-star system, consisting of two visible stars in orbit around each other. Classical mechanics gives the period of revolution (or the time it takes to complete an orbit) in terms of the distance between the two stars and the sum of their masses. So by measuring the period of revolution and the distance between the two stars, we can calculate the sum of their masses. Again, from classical mechanics, we know that each star orbits around an imaginary point called the center of mass. By measuring the distance of each star from the center of mass, we can determine the ratio of the masses. Once the sum and the ratio of the masses are known, we can get each individual mass. For example, the bright star Sirius A and the faint star Sirius B form a binary system, close enough to us to permit a detailed analysis of its motion. With the data available, the preceding steps show that the sum of the masses7 is 3 M , and A is twice as massive as B. It follows immediately that Sirius A has a mass of 2 M , and Sirius B, 1 M .
The study of binary systems shows that ordinary stars with nearly identical spectra usually have nearly identical masses (and vice versa). This fact allows the masses of many stars, not necessarily in binary systems, to be estimated from their spectral properties.
8.1.1Summary
A star has characteristic luminosity, temperature, mass and spectral lines. Once its luminosity and temperature are known, its size can be inferred from the Stefan– Boltzmann law, and its dominant color determined by Wien’s displacement law. There are no simple relations for the mass, which must be determined by observations or estimated from the star’s spectroscopic properties, which also tell us about the internal composition of the star.
8.2 The Hertzsprung–Russell Diagram
Observed stars come in a bewildering variety: masses range from 0.1 M to 60 M or more; luminosities from one millionth to a million times the sun’s luminosity
7It is sometimes convenient to express astronomical quantities in solar units. The solar mass is 1 M = 1.99 × 1030 kg; the solar radius, 1 R = 6.96 × 108 m; and the bolometric luminosity of
26 |
astronomical unit (AU) is the mean radius of the earth’s |
|||
the sun, 1 L = 3.84 × 10 W. The |
||||
|
11 |
m. |
||
orbit about the sun. 1 AU = 1.50 × 10 |
|
|||
274 |
Bright Stars and Black Holes |
|
|
|
Spectral class |
|
|
|
||
|
0 |
B |
|
A |
F |
G |
K |
M |
|
106 |
100R |
|
|
|
|
|
|
|
40 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
104 |
|
|
|
|
|
|
|
|
20 |
10R |
|
|
|
|
|
|
|
102 |
10 |
|
|
|
G |
|
|
) |
|
4 |
|
|
|
|
||
|
1R |
3 |
|
|
|
|
||
(L/L |
|
|
|
|
|
|||
|
|
2 |
|
|
|
|||
|
|
|
|
|
|
|
||
|
|
|
|
|
1 |
|
|
|
Luminosity |
101 |
|
|
|
|
|
|
|
|
|
|
|
Sun |
0.5 |
|||
|
0.1R |
|
|
|
||||
|
|
|
|
|
|
|||
10–2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0.01R |
|
WD |
|
|
|
0.1 |
|
|
10–4 |
|
|
|
|
|
|
|
|
10–6 |
14 |
|
10 |
7 |
5 |
|
3.5 |
|
20 |
|
|
|||||
|
|
Effective temperature (× 1000 K) |
|
|
||||
Figure 8.2: Hertzsprung–Russell diagram. The dashed line represents the main sequence; along this line, numbers give stellar masses in solar units. G indicates the general region of red giants, and WD that of white dwarfs.
(L ); surface temperatures from one third to ten times the solar temperature; and sizes from less than one hundredth to more than 100 times the solar size. And there are blue stars and red stars. But when we represent every star by a point on a twodimensional logarithmic plot, luminosity versus surface temperature, we find that the stars do not fall haphazardly on the diagram, but occupy quite specific regions, which implies definite relationships between the various forms of stars. This type of plot is called the Hertzsprung–Russell (H–R) diagram (Fig. 8.2) in honor of its creators.8
If such a two-dimensional plot is found to be useful, it is because we need only two parameters to di erentiate various types of stars: of the basic stellar properties, only two are independent. Let us try to understand this point.
First, the color and temperature of a star are tied together by Wien’s law. And so in an H–R plot with the temperature scale increasing from right to left, the right part of the diagram contains redder, cooler stars, and the left part, bluer, hotter stars.
Next, any point on an H–R diagram corresponds to a specific luminosity, temperature and radius, according to L R2T 4; and if the scales of the coor-
8Danish astronomer Ejnar Hertzsprung and American astronomer Henry Norris Russell.