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Newton's laws of motion give us a complete description of the behavior moving objects at low speeds. The laws are different at speeds reached by the particles at SLAC. Einstein's Special Theory of Relativity describes the motion of particles moving at close to the speed of light. In fact, it gives the correct laws of motion for any particle. This doesn't mean Newton was wrong, his equations are contained within the relativistic equations. Newton's "laws" provide a very good approximate form, valid when v is much less than c. For particles moving at slow speeds (very much less than the speed of light), the differences between Einstein's laws of motion and those derived by Newton are tiny. That's why relativity doesn't play a large role in everyday life. Einstein's theory supercedes Newton's, but Newton's theory provides a very good approximation for objects moving at everyday speeds. Einstein's theory is now very well established as the correct description of motion of relativistic objects, that is those traveling at a significant fraction of the speed of light. Because most of us have little experience with objects moving at speeds near the speed of light, Einstein's predictions may seem strange. However, many years of high energy physics experiments have thoroughly tested Einstein's theory and shown that it fits all results to date.
Theoretical Basis for Special Relativity. Einstein's theory of special relativity results from two statements -- the two basic postulates of special relativity: 1. The speed of light is the same for all observers, no matter what their relative speeds. 2. The laws of physics are the same in any inertial (that is, non-accelerated) frame of reference. This means that the laws of physics observed by a hypothetical observer traveling with a relativistic particle must be the same as those observed by an observer who is stationary in the laboratory. Given these two statements, Einstein showed how definitions of momentum and energy must be refined and how quantities such as length and time must change from one observer to another in order to get consistent results for physical quantities such as particle half-life. To decide whether his postulates are a correct theory of nature, physicists test whether the predictions of Einstein's theory match observations. Indeed many such tests have been made -- and the answers Einstein gave are right every time! The Speed of Light is the same for all observers. The first postulate -- the speed of light will be seen to be the same relative to any observer, independent of the motion of the observer -- is the crucial idea that led Einstein to formulate his theory. It means we can define a quantity c, the speed of light, which is a fundamental constant of nature. Note that this is quite different from the motion of ordinary, massive objects. If I am driving down the freeway at 50 miles per hour relative to the road, a car traveling in the same direction at 55 mph has a speed of only 5 mph relative to me, while a car coming in the opposite direction at 55 mph approaches me at a rate of 105 mph. Their speed relative to me depends on my motion as well as on theirs.
Physics is the same for all inertial observers. This second postulate is really a basic though unspoken assumption in all of science -- the idea that we can formulate rules of nature which do not depend on our particular observing situation. This does not mean that things behave in the same way on the earth and in space, e.g. an observer at the surface of the earth is affected by the earth's gravity, but it does mean that the effect of a force on an object is the same independent of what causes the force and also of where the object is or what its speed is. Einstein developed a theory of motion that could consistently contain both the same speed of light for any observer and the familiar addition of velocities described above for slow-moving objects. This is called the special theory of relativity, since it deals with the relative motions of objects. Note that Einstein's General Theory of Relativity is a separate theory about a very different topic -- the effects of gravity. Relativistic Definitions Physicists call particles with v/c comparable to 1 "relativistic" particles. Particles with v/c << 1 (very much less than one) are "non-relativistic." At SLAC, we are almost always dealing with relativistic particles. Below we catalogue some essential differences between the relativistic quantities the more familiar non-relativistic or low-speed approximate definitions and behaviors.
Gamma(Gamma Symbol) The measurable effects of relativity are based on gamma. Gamma depends only on the speed of a particle and is always larger than 1. By definition: Equation relating speed of light, speed of object and constant, gamma c is the speed of light v is the speed of the object in question For example, when an electron has traveled ten feet along the accelerator it has a speed of 0.99c, and the value of gamma at that speed is 7.09. When the electron reaches the end of the linac, its speed is 0.99999999995c where gamma equals 100,000. What do these gamma values tell us about the relativistic effects detected at SLAC? Notice that when the speed of the object is very much less than the speed of light (v << c), gamma is approximately equal to 1. This is a non-relativistic situation (Newtonian). Momentum For non-relativistic objects Newton defined momentum, given the symbol p, as the product of mass and velocity -- p = m v. When speed becomes relativistic, we have to modify this definition -- p = gamma (mv) Notice that this equation tells you that for any particle with a non-zero mass, the momentum gets larger and larger as the speed gets closer to the speed of light. Such a particle would have infinite momentum if it could reach the speed of light. Since it would take an infinite amount of force (or a finite force acting over an infinite amount of time) to accelerate a particle to infinite momentum, we are forced to conclude that a massive particle always travels at speeds less than the speed of light. Some text books will introduce the definition m0 for the mass of an object at rest, calling this the "rest mass" and define the quantity (M = gamma m0) as the mass of the moving object. This makes Newton's definition of momentum still true provided you choose the correct mass. In particle physics, when we talk about mass we always mean mass of an object at rest and we write it as m and keep the factor of gamma explicit in the equations. Energy Probably the most famous scientific equation of all time, first derived by Einstein is the relationship E = mc2. This tells us the energy corresponding to a mass m at rest. What this means is that when mass disappears, for example in a nuclear fission process, this amount of energy must appear in some other form. It also tells us the total energy of a particle of mass m sitting at rest. Einstein also showed that the correct relativistic expression for the energy of a particle of mass m with momentum p is E2 = m2c4 + p2c2. This is a key equation for any real particle, giving the relationship between its energy (E), momentum ( p), and its rest mass (m). If we substitute the equation for p into the equation for E above, with a little algebra, we get E = gamma mc2, so energy is gamma times rest energy. (Notice again that if we call the quantity M =gamma m the mass of the particle then E = Mc2 applies for any particle, but remember, particle physicists don't do that.) Let's do a calculation. The rest energy of an electron is 0.511 MeV. As we saw earlier, when an electron has gone about 10 feet along the SLAC linac, it has a speed of 0.99c and a gamma of 7.09. Therefore, using the equation E = gamma x the rest energy, we can see that the electron's energy after ten feet of travel is 7.09 x 0.511 MeV = 3.62 MeV. At the end of the linac, where gamma = 100,000, the energy of the electron is 100,000 x 0.511 MeV = 51.1 GeV. The energy E is the total energy of a freely moving particle. We can define it to be the rest energy plus kinetic energy (E = KE + mc2) which then defines a relativistic form for kinetic energy. Just as the equation for momentum has to be altered, so does the low-speed equation for kinetic energy (KE = (1/2)mv2). Let's make a guess based on what we saw for momentum and energy and say that relativistically KE = gamma(1/2)mv2. A good guess, perhaps, but it's wrong. Now here is an exercise for the interested reader. Calculate the quantity KE = E - mc2 for the case of v very much smaller than c, and show that it is the usual expression for kinetic energy (1/2 mv2) plus corrections that are proportional to (v/c)2 and higher powers of (v/c). The complicated result of this exercise points out why it is not useful to separate the energy of a relativistic particle into a sum of two terms, so when particle physicists say "the energy of a moving particle" they mean the total energy, not the kinetic energy. Another interesting fact about the expression that relates E and p above (E2 = m2c4 + p2c2), is that it is also true for the case where a particle has no mass (m=0). In this case, the particle always travels at a speed c, the speed of light. You can regard this equation as a definition of momentum for such a mass-less particle. Photons have kinetic energy and momentum, but no mass! In fact Einstein's relationship tells us more, it says Energy and mass are interchangeable. Or, better said, rest mass is just one form of energy. For a compound object, the mass of the composite is not just the sum of the masses of the constituents but the sum of their energies, including kinetic, potential, and mass energy. The equation E=mc2 shows how to convert between energy units and mass units. Even a small mass corresponds to a significant amount of energy. In the case of an atomic explosion, mass energy is released as kinetic energy of the resulting material, which has slightly less mass than the original material. In any particle decay process, some of the initial mass energy becomes kinetic energy of the products. Even in chemical processes there are tiny changes in mass which correspond to the energy released or absorbed in a process. When chemists talk about conservation of mass, they mean that the sum of the masses of the atoms involved does not change. However, the masses of molecules are slightly smaller than the sum of the masses of the atoms they contain (which is why molecules do not just fall apart into atoms). If we look at the actual molecular masses, we find tiny mass changes do occur in any chemical reaction. At SLAC, and in any particle physics facility, we also see the reverse effect -- energy producing new matter. In the presence of charged particles a photon (which only has kinetic energy) can change into a massive particle and its matching massive antiparticle. The extra charged particle has to be there to absorb a little energy and more momentum, otherwise such a process could not conserve both energy and momentum. This process is one more confirmation of Einstein's special theory of relativity. It also is the process by which antimatter (for example the positrons accelerated at SLAC) is produced. Units of Mass, Energy, and Momentum Instead of using kilograms to measure mass, physicists use a unit of energy -- the electron volt. It is the energy gained by one electron when it moves through a potential difference of one volt. By definition, one electron volt (eV) is equivalent to 1.6 x 10-19 joules. Lets look at an example of how this energy unit works. The rest mass of an electron is 9.11 x 10-31 kg. Using E = mc2 and a calculator we get: E = 9.11 x 10-31 kg x (3 x 108 m/s)2 = 8.199 x 10-14 joules This gives us the energy equivalent of one electron. So, whether we say we have 9.11 x 10-31 kg or 8.199 x 10-14 joules, we really talking about the same thing -- an electron. Physicists go one stage further and convert the joules to electron volts. This gives the mass of an electron as 0.511 MeV (about half a million eV). So if you ask a high energy physicist what the mass of an electron is, you'll be told the answer in units of energy. You can blame Einstein for that! Eagle-eyed readers will notice that if you solve E=mc2 for m, you get m=E/c2, so the unit of energy should be eV/c2. What happened to the c2? It's very simple, particle physicists choose units of length so that the speed of light = 1! How can we do that? Quite easily, as long as everyone understands the system. All we have to do is use a conversion factor to get back the "real" (i.e. everyday) units, if we want them. Not only are mass and energy measured in eV, so is momentum. It makes life so much easier than dividing by c2 or c all the time. There is more information available on units in relativistic physics. Peculiar Relativistic Effects Length Contraction and Time Dilation One of the strangest parts of special relativity is the conclusion that two observers who are moving relative to one another, will get different measurements of the length of a particular object or the time that passes between two events. Consider two observers, each in a space-ship laboratory containing clocks and meter sticks. The space ships are moving relative to each other at a speed close to the speed of light. Using Einstein's theory: Each observer will see the meter stick of the other as shorter than their own, by the same factor gamma (gamma- defined above). This is called length contraction. Each observer will see the clocks in the other laboratory as ticking more slowly than the clocks in his/her own, by a factor gamma. This is called time dilation. In particle accelerators, particles are moving very close to the speed of light where the length and time effects are large. This has allowed us to clearly verify that length contraction and time dilation do occur. Time Dilation for Particles Particle processes have an intrinsic clock that determines the half-life of a decay process. However, the rate at which the clock ticks in a moving frame, as observed by a static observer, is slower than the rate of a static clock. Therefore, the half-life of a moving particles appears, to the static observer, to be increased by the factor gamma. For example, let's look at a particle sometimes created at SLAC known as a tau. In the frame of reference where the tau particle is at rest, its lifetime is known to be approximately 3.05 x 10-13 s. To calculate how far it travels before decaying, we could try to use the familiar equation distance equals speed times time. It travels so close to the speed of light that we can use c = 3x108 m/sec for the speed of the particle. (As we will see below, the speed of light in a vacuum is the highest speed attainable.) If you do the calculation you find the distance traveled should be 9.15 x 10-5 meters.
d = v t
d = (3 x 108 m/sec)( 3.05 x 10-13 s) = 9.15 x 10-5 m
Here comes the weird part - we measure the tau particle to travel further than this! Pause to think about that for a moment. This result is totally contradictory to everyday experience. If you are not puzzled by it, either you already know all about relativity or you have not been reading carefully. What is the resolution of this apparent paradox? The answer lies in time dilation. In our laboratory, the tau particle is moving. The decay time of the tau can be seen as a moving clock. According to relativity, moving clocks tick more slowly than static clocks. We use this fact to multiply the time of travel in the taus moving frame by gamma, this gives the time that we will measure. Then this time times c, the approximate speed of the tau, will give us the distance we expect a high energy tau to travel. I'm just saying that I've heard the phrase quantum gravity but I'm not sure what it means. How does it relate to other aspects of quantum physics, like unified field theory? What exactly is quantum gravity? Quantum gravity is an overall term for theories that attempt to unify gravity with the other fundamental forces of physics (which are already unified together). It generally posits a theoretical entity, a graviton, which is a virtual particle that mediates the gravitational force. This is what distinguishes quantum gravity from certain other unified field theories ... although, in fairness, some theories that are typically classified as quantum gravity don't necessary require a graviton. What's a graviton? The standard model of quantum mechanics (developed between 1970 & 1973) postulates that the other three fundamental forces of physics are mediated by virtual bosons. Photons mediate the electromagnetic force, W & Z bosons mediate the weak nuclear force, and gluons (such as quarks) mediate the strong nuclear force. The graviton, therefore, would mediate the gravitational force. If found, the graviton is expected to be massless (because it acts instantaneously at long distances) and have spin 2 (because gravity is a second-rank tensor field). Is quantum gravity proven? The major problem in experimentally testing any theory of quantum gravity is that the energy levels required to observe the conjectures are unattainable in current laboratory experiments. Even theoretically, quantum gravity runs into serious problems. Gravitation is currently explained through the theory of general relativity, which makes very different assumptions about the universe at the macroscopic scale than those made by quantum mechanics at the microscopic scale. Attempts to combine them generally run into the "renormalization problem," in which the sum of all of the forces do not cancel out and result in an infinite value. In quantum electrodynamics, this happened occasionally, but one could renormalize the mathematics to remove these issues. Such renormalization does not work in a quantum interpretation of gravity. The assumptions of quantum gravity are generally that such a theory will prove to be both simple and elegant, so many physicists attempt to work backward, predicting a theory that they feel might account for the symmetries observed in current physics and then seeing if those theories work. Some unified field theories that are classified as quantum gravity theories include String theory, Superstring theory, M-theory, Supergravity, Loop quantum gravity, Twistor theory, Noncommutative geometry, Euclidean quantum gravity, Wheeler-deWitt equation. Of course, it's fully possible that if quantum gravity does exist, it will be neither simple nor elegant, in which case these attempts are being approached with faulty assumptions and, likely, would be inaccurate. Only time, and experimentation, will tell for sure. It is also possible, as some of the above theories predict, that an understanding of quantum gravity will not merely consolidate the theories, but will rather introduce a fundamentally new understanding of space and time. What is gamma in this case? It depends on the tau's energy. A typical SLAC tau particle has a gamma = 20. Therefore, we detect the tau to decay in an average distance of 20 x (9.15 x 10-5 m) = 1.8 x 10-3 m or approximately 1.8 millimeters. This is 20 times further than we expect it to go if we use classical rather than relativistic physics. (Of course, we actually observe a spread of decay times according to the exponential decay law and a corresponding spread of distances. In fact, we use the measured distribution of distances to find the tau half-life.) Observations particles with a variety of velocities have shown that time dilation is a real effect. In fact the only reason cosmic ray muons ever reach the surface of the earth before decaying is the time dilation effect. Length Contraction Instead of analyzing the motion of the tau from our frame of reference, we could ask what the tau would see in its reference frame. Its half-life in its reference frame is 3.05 x 10-13 s. This does not change. The tau goes nowhere in this frame.How far would an observer, sitting in the tau rest frame, see an observer in our laboratory frame move while the tau lives? We just calculated that the tau would travel 1.8 mm in our frame of reference. Surely we would expect the observer in the tau frame to see us move the same distance relative to the tau particle. Not so says the tau-frame observer -- you only moved 1.8 mm/gamma = 0.09 mm relative to me. This is length contraction. How long did the tau particle live according to the observer in the tau frame? We can rearrange d = v x t to read t = d/v. Here we use the same speed, Because the speed of the observer in the lab relative to the tau is just equal to (but in the opposite direction) of the speed of the tau relative to the observer in the lab, so we can use the same speed. So time = 0.09 x 10-3 m/(3 x 10 cool m/sec = 3.0 x 10-13 sec. This is the half-life of the tau as seen in its rest frame, just as it should be! Mathematics can be used to make predictions about physical systems. But economists and social scientists can likewise use mathematics to make predictions about economic and social systems. The contrast lies only in the details of the mathematical descriptions, not in the language of mathematics itself. For example, the study of linear motion involves equations which are, by definition, linear. The change in position, for example, is given simply by multiplying velocity by time in the case of no acceleration. Where constant, non-zero acceleration exists, the velocity changes linearly with time and it is relatively simply to incorporate that effect into determining the change in displacement with time. There are other observations of nature which appear just as simple, but are not linear. For example, we can ask how a population of organisms changes with time. The organisms could be yeast cells in a petri dish, chickens in a barnyard, or people on a planet. In the most simplistic view of these organisms, we find that their population increases with time. The reason is that organisms reproduce. Since only organisms of the same type or species can reproduce, the most naive way of quantitatively describing the reproduction is to say that the increase in the future depends on the number of organisms around today. For example, say that we are studying yeast cells in a dish. If we make the following assumptions, The petri dish contains only yeast cells, All yeast cells are the same, Each cell lives forever, The birth rate or probability that any cell will produce an offspring stays constant, Cells do not interfere with one another, There is an infinite supply of vital elements (e.g. food, space, etc.). Then the change in the population (call it P) should be where P depends on the time at which you count the population and b is the constant birth rate for each individual cell. Of course, this equation doesn't seem very realistic since yeast cells don't live forever. We might be able to supply them with an always adequate food supply and put them into bigger and bigger dishes as they expand, but eventually some have to die. How do we model death quantitatively? Death decreases the population and obviously we can't have more cells dying than there are cells, so the amount of decrease must also depend on the current population where d is the probability of death or death rate for each individual in the population (again assuming that all individual cells are the same). This equation also doesn't reflect reality since it doesn't include reproduction. A combination of the two, however, should be more realistic, so now we have something that seems more reasonable. The change in population depends on the population you start with multiplied by the rate of birth, b, minus the death rate, d. If the birth rate exceeds the death rate, then the population increases with time. If the death rate exceeds the birth rate, the population decreases with time. Now we can turn to Maple to solve the equation and look at the results. Before we do that, note that to be useful we will have to compare our solution to some real data. We don't happen to have any handy, but, the interesting history of the equation derived above is that the scientists who first applied this equation to biology thought that it should apply to human populations as readily as it does to yeast cells! Does that seem plausible? Let's try it out! We begin by considering our own United States. We have to choose some convenient time in which to start our observation. The first people to apply it did so around the turn of the 19th century. The first U.S. Census was 1790, so let's start from there. At that time the population in the U.S. was 3.929 million. That's our starting population; we will call it , so in units of millions of people. To start from 1790 instead of zero, let's define our time as t - 1790 instead of just t. That will make things easier to handle for calculation. Now we can turn to Maple. Note that in the following, the response from Maple is centered and in italic type. Also note that not all of Maple's responses are shown. (You can download the Maple file directly to your computer (select Maple as the application to handle this file) by clicking here.) #First start off by describing the change in population with time by #writing in Maple the equation we just derived, and asking Maple to
#solve it. We call the solution psol just to give it a name. The
#condition in 1790 is a constraint to our solution, so we let Maple
#know about it when we ask for a solution.
dsolve({diff(P(t),t)=a*P(t), P(0)=P0}, P(t)); #The solution gives us the population for any time t we care to use. If
#we want t to be a year like, e.g. 1860, then our solution is
psol := exp(a*(t - 1790))*P0; #with P0 being equal to 3.929 in units of millions of people. Now we are
#almost ready to compare to real data. The real data we'll use are the
#census figures from 1790 to 1970. Here they are:
uspop := [1790, 3.939, 1800, 5.308, 1810, 7.24, 1820, 9.638, 1830, 12.866, 1840, 17.069, 1850, 23.192, 1860, 31.443, 1870, 38.558, 1880, 50.156,
1890, 62.948, 1900, 75.995, 1910, 91.972, 1920, 105.711, 1930, 122.775,
1940, 131.669, 1950, 150.697, 1960, 179.323, 1970, 203.185];
#The final thing needed for a comparison is to determine the value of the
#constant a (remember that this is the birth rate - death rate). To get it,
#we have to look at the data for a year later than 1790. Since we are
#assuming that the birth and death rates do not change, we can pick any
#year we want. Let's choose 1830. In that year, the population was
#12.866 million people. That's the value we want our Maple equation to
#give. Let's let Maple tell us what value of a satisfies.
solve(exp(a*(1830 - 1790))*3.939 = 12.866, {a});
#The solution is a = 0.029655. Let's use that value and plot the solution
#given to us by Maple on top of the points from the census data in
#succeeding decades. A few data points is enough to tell whether all
#of this makes any sense at all. Let's look from 1790 to 1870.
plot({psol, uspop}, t=1790..1870, P=0..80, style=POINT); Doesn't look bad at all does it! Somehow, this naive little model with simple equations does a good job of predicting the U.S. population over a period of 80 years! Do we expect this success to continue for later years? Give your vote yes or no! If you vote no, what do you expect to happen? If you vote yes, what do you expect to see as the number of years increases?
To examine what happens, let's go back to Maple and plot more points.
plot({psol, uspop}, t=1790..1970, style=POINT); We see that at about 1870 things go very badly for our simple calculation. The simple formula predicts an exponentially increasing population while the real data shows a slow rise. It is obvious that no population of living creatures on earth continues to increase exponentially, however. We can consider what happens if such a thing were to take place with, say, cockroaches . The exponential curve quickly determines that cockroaches would overwhelm all the planet. There are other factors such as immigration which affect the population. In addition, the birth and death rates are not constant with time. Wars, disease, cultural habits (e.g. smoking, pollution of the environment, etc.) affect the death rate while improved medicine, food, working conditions, etc. increase the birth rate. Even for our simple yeast cells, we would have to say that there is some limit to the size of the petri dish we could use to hold them. Eventually, crowding and competition for resources would limit the exponential growth. In every environment, there is a maximum number of any given species that can be supported. This one fact must clearly be added into our equation to make it more accurate. With the inclusion of only this one additional constraint, let's answer the following questions. How would we account for limits to the size of the population in predicting changes to the population? In mathematical terms would we expect the change in population as the size approaches the maximum to be positive, negative, or zero? Sketch the shape of the curve that allows for limiting the number of individuals in the population. To get the correct behavior for a maximum population, we have to get very familiar with the properties of curves such as their slope or tangent at each point. For the exponentially rising curve, the slope is always positive; the curve is always increasing. Furthermore, the slope increases with time so that the rate at which the curve goes up increases. To get a limit at a finite number for the population we have to have our population curve approach a horizontal line on the graph. For that to happen, the slope of the curve must decrease. In mathematical terms, we say that the first derivative or change in population with respect to time must become negative after some time. The simplest way to make that happen is to write down the following equation:where a and c are positive constants. This equation is called the logistic equation or Verhulst-Pearl equation. It was first developed by Pierre-Francois Verhault and later used by Raymond Pearl and Lowell Reed to model the population of the United States.
The logistic equation describes a situation in which the change is positive for a while if a > cP but eventually the population gets large enough so that cP > a and the change becomes negative. This is the simplest way to flip the sign of the slope.The justification of this type of equation for our case of a limiting environment is relatively straightforward. If the constant c were zero, then we get our original equation in which the rate of growth is aP. This corresponds to an environment with unlimited space and resources. To correct for a non-infinite environment, we note that the maximum number of organisms in the environment is M. At any given time, the number of organisms that can still be born is M - P(t). The fraction of maximum attainable population that is still possible is therefore, The rate of growth allowed should be the unlimited rate times this fraction of growth allowable or In this way, the growth is the maximum value for small P (i.e. the
fraction is 1) and goes to zero as the fraction approaches 0. The
equation describing the finite environment model would then be
If we identify c = a/M, then we see that this is our logistic equation.
Now let's see how it fares against the census data. Let's return to Maple. #We consider here the solution to the Verhulst-Pearl or logistic equation
#
# First redefine the constants used in the equation as variable
a := 'a';
P0 := 'P0';
dsolve({diff(P(t),t)=P*(a - c*P), P(0) = P0}, P(t));
#There are two unknowns in this equation, the values of a and of c. If
#we wish to match our logistic expression to data, we have to find their
#values. Since there are two unknowns, we will need at least one more
#data point to find their values. Let's choose the data point at
#1800. Since the population is known for 1800 (we can call it P1), let's
#solve the logistic expression for the unknown constant c
solve(a*P0/(c*P0 + exp(-a*t)*(a - c*P0)) = P1, {c}}
#Now our constraint on the value of c is that it must be positive.
#We can plug in our expression for time as t - 1790, and evaluate which
#values of a make it possible for c to be greater than zero.
cconst := (a*P0 - P1*exp(-a*10)*a)/(-P1*P0 + P1*exp(-a*10)*P0);
P0 := 3.929;
P1 := 5.308;
plot(cconst, a=.025..0.035);
#
#The plot shows that we need a about 0.030, which is close to our old value
#for a with the exponential equation. In this case, we find that
evalf(subs(a=.030, cconst));
#We can actually fit several points of the data to find the optimal values
#of a = .031 and c = .00016. Now plot the result.
a := 0.031;
c := .00016;
P0 := 3.929;
pmax := a*P0/(c*P0 + exp(-a*(t-1790))*(a - c*P0));
plot({pmax, uspop}, t=1790..2000, style=POINT);
The result is pretty spectacular. We can fit the data over a much larger span of time. Of course, there is still a large discursion of the census data toward large values compared to theory at about 1940. Pearl and Reed did the theoretical curve shown in the mid-1920's, found a good fit, and predicted that the maximum size of the U.S. population would be less than about 200 million.