by Colin Pask
(Of course, the mathematics must be done for galaxies that are not spherical, like the common spiral galaxies, and readers may consult the chapter by Burbidge and Burbidge and the paper by Einasto for the technical details.)
Despite all the complications, the conclusion is clear: galaxies contain a great deal more mass that the visible components suggest, and they seem to have an extensive halo of dark matter around them. I define calculation 24, galaxy rotation and dark matter.
7.4.3 Today
Over fifty years have passed since the pioneering work of Oort, Zwicky, Rubin, and many others, and a vast amount of new work has been done on galaxy rotations and the need for dark matter to exist. But the conclusion remains the same. (For example, see the 2009 Nature review by Caldwell and Kamionkowski which is headed: “Observations continue to indicate that the Universe is dominated by invisible components—dark matter and dark energy.” A typical modern view is given in the Roos textbook, chapter 9, “Cosmic Structures and Dark Matter.”) The results of calculation 24, galaxy rotation and dark matter remain unchallenged and is a profound contribution to our knowledge of the universe.
Two questions may immediately come to mind. First, if dark matter (and energy) does exist, what is it? There is no definitive answer to that question in 2014. Second, is there a need to modify the underlying theory? The evidence for dark matter relies on the use of Newtonian mechanics, and there have been suggestions that modifications must be made. Yet, to date, there is no evidence that dark matter appears only as a failing of Newtonian mechanics.
After over forty years of personal effort, Vera Rubin could still write in her 2006 review:
What's spinning the stars and gas around so fast beyond the optical galaxy? What's keeping them from flying out into space? The current answer is, “Gravity, from matter that has no light.”14
At least dark matter (27 percent) and ordinary matter (5 percent) have normal gravitational properties. The other 68 percent of the universe in dark energy is even weirder (see section 7.2.4): it gives a “repulsive gravitational effect”!
7.5 ESCAPING GRAVITY AND MAKING BLACK HOLES
The force of gravity dominates our life on Earth and is often the factor deciding what is and what is not possible. For example, we must consider gravity when we ask whether humans can ever escape from the earth. In our lifetimes, we have seen truly amazing answers to that question.
Isaac Newton wrote a short book entitled The System of the World, which gave a simpler “popular” introduction to his Principia. In this book, he describes a thought experiment involving cannon balls shot off from a cannon on top of a mountain. As the muzzle speed is increased, the cannon balls travel further and further before they fall back to Earth. If the speed is great enough, a cannon ball “even might go quite round the whole Earth before it falls.” Thus he explains the possibility of the moon's orbit. However, Newton goes further: the muzzle speed is increased “lastly, so that it might never fall to Earth but go forward into the celestial spaces, and proceed in its spaces in infinitum.”
The minimum speed needed to escape from Earth and go into the “celestial spaces” is today called the escape velocity vesc. To find it, consider a body of mass m moving with speed v at distance r from the earth so that it has energy E:
In these equations, G is the usual gravitational constant, and here M is the mass of the earth. The condition for a velocity v to exist no matter how far away the body is from the earth (r becomes infinitely large, so the potential energy reduces to zero) is that E must be positive. The limiting case, when the speed is just enough for the body to keep going forever, will be when E = 0. Then we get the escape velocity at the surface of the earth, where r = Re, according to equation (7.5) as
At the surface of the earth, vesc is 11.2 km/sec. An object moving upward with this speed can escape from the gravitational pull of the earth.
7.5.1 Twentieth-Century Exploits
The ideas may have been clear a long time ago, but it was not until the twentieth century that mankind developed the technology to make space travel and exploration a reality. Rather than shoot up vehicles with speeds greater than the escape velocity, engineers built multistage rockets and launched satellites into orbit around the earth. The calculations showed how to optimize the rocket design and how to position or adjust the satellites’ orbits to give desired surveying and communication properties. Eventually humans were sent into orbit around the earth, still bound to the planet but now far above its surface. We may have become blasé about rockets and space vehicles, but the achievements have been monumental. Two examples illustrate these achievements and indicate the massive calculations involved.
On July 20, 1969, Americans Neil Armstrong and Buzz Aldrin stepped onto the moon, and man had escaped Earth. A now-famous diagram of the way this was achieved is shown in figure 7.6. There are some vital steps: a rocket must carry the astronauts into orbit around the earth; the space vehicle must then take the best path to the moon as it is influenced by gravitational forces of both the earth and the moon; the vehicle must remain in orbit around the moon and a landing craft must descend to the lunar surface; the process must then be done in reverse order and the astronauts returned safely to Earth. Notice that the journeys between Earth and the moon involve motion described by the three-body problem, one which has caused headaches ever since Newton first battled with it.
Figure 7.6. The main steps in the process of visiting the moon. From NASA, John C. Houbolt.
One of the major debates for the mission designers concerned the actual process for landing on the moon. Should the vehicle that goes away from the earth land on the moon, or is some more elaborate scheme required? After considering various factors (such as calculating the fuel needed to take a rocket down to the lunar surface and—more importantly—escaping the moon's gravity again) the lunar-orbit rendezvous concept championed by engineer John C. Houbolt (1919–2014) was chosen. (See the book by Hansen for the story of the debate.) The space vehicle orbits the moon while a module peels off and descends to the lunar surface. On return, the module must rendezvous with the orbiting parent vehicle. Imagine the calculations that went into designing such a scheme! In fact, Armstrong and Aldrin landed some miles from the targeted landing site, but thankfully the fuel reserves were as planned.
It has now been discovered that spacecraft orbits around the moon cannot be calculated by assuming the usual gravitating sphere model. There are massive dense bodies—“mascons”—embedded in the moon's crust, and they cause significant perturbations to orbits, especially the low-lying ones. In fact, there appear to be just four “frozen orbits” at inclinations 27º, 50º, 76º, and 86º; other orbits will be perturbed and wander off course. (See the article by Konopliv.)
The second example of space exploration is the Voyager program. The space probes Voyager 1 and Voyager 2 were launched in 1977 and have become the first man-made objects to set off for interstellar space, thus escaping even the gravitational pull of the sun. The launch dates were chosen so that the probes could take a tour of the outer planets and provide the first close-up reports and extensive data on them and their moons. The orbits are shown in figure 7.7.
The calculations of the spacecraft paths have to take account of gravitational pulls by the planets to be visited in order to determine how visiting these planets can be done in an optimum manner. (For details, the textbook by Barger and Olsson is a good place to start, and Roy provides a comprehensive account of orbital dynamics.) One intriguing result from these calculations is that a spacecraft can gain a “gravity assist” if it flies by a planet in a suitable orbit so that it gains speed as it leaves the region of that planet (see Barger and Olsson, and the tutorial by Van Allen).
Figure 7.7. Paths in space for the Voyager 1 and Voyager 2 probes. From NASA.
We should not forget that the success of all missions flying beyond the earth and into space relies on our detailed understanding of the gravitational forces involved and our ability to use co
mputers to discover the best possible trajectories.
7.5.2 Most Intriguing of All: No Escape
The universe is full of mysterious objects—quasars, pulsars, red giants, white dwarfs, dark matter, and so on. But in a popular vote for the most intriguing, I think the black hole might win. In fact, the idea of a “dark star” is an old one going back to the Reverend John Michell (1724–1793). (Laplace also put forward a similar idea in 1796.) Michell is little known today, but he was a geologist of some note. It was he who designed the torsion balance method for measuring gravitational effects, although today it is associated with Henry Cavendish who first performed the experiments.
Equation (7.6) lets us ask about the size and mass of a body for which the escape velocity is the speed of light c. Substituting vesc = c, we find that
This tells us that if a body is massive enough—or small and dense enough—the escape velocity may be as large as the speed of light. Here, in John Michell's own words, are the consequences:
If the semi-diameter of a sphere of the same density with the sun were to exceed that of the sun by 500 to 1, a body falling from an infinite height towards it, would have acquired at its surface a greater velocity than that of light, and consequently, supposing light to be attracted by the same force in proportion to its vis inertiae, with other bodies, all light emitted from such a body would be made to return towards it, by its own proper gravity.15
Michell is telling us that for the right kind of massive body, no light will be able to entirely escape from it and so it will be dark; the escape velocity is greater than the speed of light.
The simple argument above is very clear, but it rests on Newton's idea that light consists of particles—which he called “corpuscles”—that behave like other bodies under the action of gravity. (More on this subject will come in chapter 9.) This theory is now not accepted, and we must turn to Einstein's general theory of relativity for a different approach. In Einstein's theory, gravity manifests itself by changing the structure of space-time around massive bodies. Then light follows the geodesics located in that structure. In turns out that for a spherically symmetric, static body, the differential element in space-time is given by the Schwarzchild metric:
The space coordinates are the spherical polars (r,θ,φ). It is not necessary to appreciate these equations in detail, but note that B(r) indicates how gravity (through the body mass M and the gravitational constant G) affects the intervals. In particular, things become very strange when B becomes zero, which happens at the Schwarzchild radius rs:
The conditions for and behaviors of black holes are obtained using arguments based on these metric properties (see Roos, chapter 3, for example). Notice that the critical Schwarzchild radius rs is just the same as R in equation (7.7) for Michell's dark star.
Karl Schwarzchild published his solution to Einstein's equations in 1916, and his calculation has had a major impact on cosmology. In 1939, Robert Oppenheimer and Harland Snyder discussed the behavior of collapsing neutron stars that become extremely dense. The importance and neglect of this work is summed up in Freeman Dyson's comment revealing a most unfortunate coincidence:
The neglect of Oppenheimer's greatest contribution to science was mostly due to an accident of history. His paper with Snyder, establishing in four pages the physical reality of black holes, was published in the Physical Review on 1 September, 1939, the same day Adolph Hitler sent his armies into Poland and began World War II.16
Ironically, it is for his work at Los Alamos, building the atomic bomb (more on this is in chapter 11), that Oppenheimer is known today. In 1958, David Finkelstein identified the Schwarzchild radius as defining a boundary which allowed things to pass only one way and thus the building up of black holes was explained. There is now much evidence for the existence of black holes, which gained their name from John Wheeler in 1967.
The calculations dealing with escape from the effects of gravity—whether in Newton's framework and for space travel, or in Einstein's terms and for explaining strange objects in our universe—have been of enormous importance in mankind's travels and picture of the universe. It is essential to add calculation 25, escaping gravity to my list.
7.6 JUST TOO BIG
I expect some readers will be horrified by the work in cosmology not included in this chapter. I have tried to give a selection indicating some of the important advances and also those that are reasonably accepted and supported by observational evidence. Speculative theories involving string theory and multiverses are excluded. I did waiver about including something more on the big bang and the standard model; in particular, the calculations leading to the prediction of the cosmic microwave background radiation could have found a place. (Readers wanting a simple, general introduction might try Coles's book and Weinberg's The First Three Minutes.)
in which we see calculations about us and how our bodies work.
The natural, or biological, world tends to be complex, and it is not as easy to give the types of definitive laws and their precise mathematical expressions, which characterize the inanimate world of the physical sciences. Nevertheless, calculations have played an important part in the development of biology, and a few of my favorite examples are given in this chapter. They are chosen to illustrate advances made using calculations in six areas: processes in our bodies, the variability of humans across a population, the way in which populations grow, common patterns in animal physiology, advances in genetics, and modern medical diagnostic tools. It is only a small selection, but the results were truly revolutionary for medicine, genetics, and medical diagnostics.
(I remind the reader that the very first calculation—calculation 1, Malthus on population growth—discussed as a leading example in chapter 1, really belongs in this chapter, under the third of these areas.)
8.1 THE CIRCULATION OF THE BLOOD
From the Renaissance onward there was an emergence of learning and investigation in the arts and the sciences. Although the knowledge of the ancient world, preserved and extended by the Arabs, was still taught and revered, there was a gradual acceptance of the need to carefully scrutinize it and sometimes replace it entirely. Copernicus's move to a sun-centered solar system in 1543 is a classic example. The work of William Harvey discussed in this section provides another example, one that was to prove to be both radical and central to the development of biology and medicine.
William Harvey (1578–1657) was educated at universities in Cambridge and Padua. In 1602, he was made doctor of medicine at Cambridge, and he went on to become an eminent physician, anatomist, and surgeon. He was physician to King Charles I for fifteen years. Harvey embodied the new approaches to learning, as shown by his statement that in his lectures on anatomy he professed “to learn and teach anatomy, not from books, but from dissections, not from the positions of the philosophers but from the fabric of nature.”1 His special interest was blood, and by overthrowing ancient viewpoints, he created our modern approach to the subject. Those criticizing the old teachings often found themselves under attack, especially as the Church had incorporated much of that ancient thinking into its own teachings and dogma. It is remarkable then that Harvey's friend, the philosopher Thomas Hobbes, could write that Harvey was “the only one I know who has overcome public odium and established a new doctrine during his own lifetime.”2
The circulation of the blood through the body and its pumping by the heart are essential for life in animals to continue. It might seem strange then that the description of what was involved, given by Galen (about 129–200 CE), was accepted for over a thousand years, even though it is completely wrong. It becomes less strange when we appreciate two points. First, as mentioned earlier, the views of the ancients were revered, and few people were willing to challenge them. Second, it is extremely difficult to physically investigate blood flow and observe the beating heart—interfere with the system and death soon follows.
Blood in the body appears to come in two forms, one scarlet and one purple. We now know that they are the
oxygenated and deoxygenated forms of the same blood. Galen believed that there were indeed two types of blood. One was generated by the liver and then transported in the veins to various parts of the body where it was consumed. The arterial system originated in the lungs and carried a vivified blood and life-giving spirit to the rest of the body. These ideas were used to devise medical treatments, such as bloodletting, for dealing with different diseases and illnesses.
8.1.1 Harvey and Blood Circulation
To understand blood flow in the body is to understand the role played by the heart, lungs, liver, veins, and arteries. Harvey investigated all of these organs in a long series of experiments, many of them involving animals and gruesome procedures. The result was his book Exercitatio Anatomica de Motu Cordis et Sanguinis in Animalibus or An Anatomical Disquisition on the Motion of the Heart and Blood in Animals, published in 1628. (See the bibliography where the details of Andrew Gregory's valuable history Harvey's Heart are also given.) Harvey discovered the blood circulation system as we know it today. This was a revolutionary achievement, made all the more remarkable by the fact that Harvey was unaware of the capillaries (since they are so small), which connect veins and arteries to complete the circuit.
Galen's theory that blood is produced by the liver had to be totally discredited, and Harvey does this in chapter 9 of his book. Here, in Harvey's words, is the way the system works:
First, the blood is incessantly transmitted by the action of the heart from the vena cava to the arteries in such quantity that it cannot be supplied from the ingesta, and in such wise that the whole mass must very quickly pass through the organ; second, the blood under the influence of the arterial pulse enters and is impelled in a continuous, equable, and incessant stream through every part and member of the body, in much larger quantity than were sufficient for nutrition, or than the whole mass of fluids could supply; third; the veins in like manner return this blood incessantly to the heart from all parts and members of the body.3