How Did Life Arise on Earth?

How Did Life Arise on Earth?

7:38:00 PM   Amazing, Education, History, Nature, Science,

Was it a magnificent stroke of luck, or was it inevitable? 

Earth is estimated to be about 4.5 billion years old, and for much of that history it has been home to life in one weird form or another.

Indeed, some scientists think life appeared the moment our planet's environment was stable enough to support it.

The earliest evidence for life on Earth comes from fossilized mats of cyanobacteria called stromatolites in Greenland that are about 3.7 billion years old. Ancient as their origins are, these bacteria (which are still around today) are already biologically complex—they have cell walls protecting their protein-producing DNA, so scientists think life must have begun much earlier. In fact, there are hints of life in even more primeval rocks: 4.1-billion-year-old zircons from Western Australia contain high amounts of a form of carbon typically used in biological processes. [7 Theories on the Origin of Life]

But despite knowing approximately when life first appeared on Earth, scientists are still far from answering how it appeared.

"Many theories of the origin of life have been proposed, but since it's hard to prove or disprove them, no fully accepted theory exists," said Diana Northup, a cave biologist at the University of New Mexico.

The answer to this question would not only fill one of the largest gaps in scientists' understanding of nature, but also would have important implications for the likelihood of finding life elsewhere in the universe.

Lots of ideas

Today, there are several competing theories for how life arose on Earth. Some question whether life began on Earth at all, asserting instead that it came from a distant world or the heart of a fallen comet or asteroid. Some even say life might have arisen here more than once.

"There may have been several origins," said David Deamer, a biochemist at the University of California, Santa Cruz. "We usually make 'origins' plural just to indicate that we don't necessarily claim there was just a single origin, but just an origin that didn't happen to get blasted by giant [asteroid] impacts."

Most scientists agree that life went through a period when RNA was the head-honcho molecule, guiding life through its nascent stages. According to this "RNA World" hypothesis, RNA was the crux molecule for primitive life and only took a backseat when DNA and proteins—which perform their jobs much more efficiently than RNA—developed.

"A lot of the most clever and most talented people in my field have accepted that the RNA World was not just possible, but probable," Deamer said.

RNA is very similar to DNA, and today carries out numerous important functions in each of our cells, including acting as a transitional-molecule between DNA and protein synthesis, and functioning as an on-and-off switch for some genes. [Extreme Life on Earth: 8 Bizarre Creatures]

But the RNA World hypothesis doesn't explain how RNA itself first arose. Like DNA, RNA is a complex molecule made of repeating units of thousands of smaller molecules called nucleotides that link together in very specific, patterned ways. While there are scientists who think RNA could have arisen spontaneously on early Earth, others say the odds of such a thing happening are astronomical.

"The appearance of such a molecule, given the way chemistry functions, is incredibly improbable. It would be a once-in-a-universe long shot," said Robert Shapiro, a chemist at New York University. "To adopt this [view], you have to believe we were incredibly lucky."

The anthropic principle

But "astronomical" is a relative term. In his book, The God Delusion, biologist Richard Dawkins entertains another possibility, inspired by work in astronomy and physics.

Suppose, Dawkins says, the universe contains a billion billion planets (a conservative estimate, he says), then the chances that life will arise on one of them is not really so remarkable.

Furthermore, if, as some physicists say, our universe is just one of many, and each universe contained a billion billion planets, then it's nearly a certainty that life will arise on at least one of them.

As Dawkins writes, "There may be universes whose skies have no stars: but they also have no inhabitants to notice the lack."

Shapiro doesn't think it's necessary to invoke multiple universes or life-laden comets crashing into ancient Earth. Instead, he thinks life started with molecules that were smaller and less complex than RNA, which performed simple chemical reactions that eventually led to a self-sustaining system involving the formation of more complex molecules.

"If you fall back to a simpler theory, the odds aren't astronomical anymore," Shapiro told Live Science.

Trying to recreate an event that happened billions of years ago is a daunting task, but many scientists believe that, like the emergence of life itself, it is still possible.

"The solution of a mystery of this magnitude is totally unpredictable," said Freeman Dyson, a professor emeritus of physics at Princeton University in New Jersey. "It might happen next week or it might take a thousand years."

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Does the Soul Exist?

Does the Soul Exist?

7:23:00 PM   Amazing, Education, Nature,

Many people today believe they possess a soul. While conceptions of the soul differ, many would describe it as an "invisible force that appears to animate us."

It's often believed the soul can survive death and is intimately associated with a person’s memories, passions and values. Some argue the soul has no mass, takes no space and is localised nowhere.

But as a neuroscientist and psychologist, I have no use for the soul. On the contrary, all functions attributable to this kind of soul can be explained by the workings of the brain.

Psychology is the study of behaviour. To carry out their work of modifying behaviour, such as in treating addiction, phobia, anxiety and depression, psychologists do not need to assume people have souls. For the psychologists, it is not so much that souls do not exist, it is that there is no need for them.

It is said psychology lost its soul in the 1930s. By this time, the discipline fully became a science, relying on experimentation and control rather than introspection.

What is the soul?

It is not only religious thinkers who have proposed that we possess a soul. Some of the most notable proponents have been philosophers, such as Plato (424-348 BCE) and René Descartes in the 17th century.

Plato believed we do not learn new things but recall things we knew before birth. For this to be so, he concluded, we must have a soul.

Centuries later, Descartes wrote his thesis Passions of the Soul, where he argued there was a distinction between the mind, which he described as a "thinking substance," and the body, "the extended substance." He wrote:

… because we have no conception of the body as thinking in any way, we have reason to believe that every kind of thought which exists in us belongs to the soul.

One of the many arguments Descartes advanced for the existence of the soul was that the brain, which is a part of the body, is mortal and divisible – meaning it has different parts – and the soul is eternal and indivisible – meaning it is an inseparable whole. Therefore, he concluded they must be different things.

But advances in neuroscience have shown these arguments to be false.

Stripping humans of the soul

In the 1960s, Nobel laureate Roger Sperry showed that the mind and our consciousness are divisible, therefore disproving that aspect of Descartes' theory.

Sperry studied patients whose corpus callosum, the superhighway connecting the right and left hemispheres, had been severed by surgery aiming to control the spread of epileptic seizures. The surgery blocked or reduced the transfer of perceptual, sensory, motor and cognitive information between the two hemispheres.

Sperry showed each hemisphere could be trained to perform a task, but this experience was not available to the untrained hemisphere. That is, each hemisphere could process information outside the awareness of the other. In essence, this meant the operation produced a double consciousness.

Thus, Descartes cannot be correct in his assertion the brain is divisible but the soul, which can be read as the mind or consciousness, is not. In his effort to prove the existence of the soul in humans, Descartes actually provided an argument against it.

Rather than investigating rats with souls, psychologists stripped humans of theirs. In 1949, psychologist D.O. Hebb claimed the mind is the integration of the activity of the brain.

Many neurophilosophers have come to the same conclusion as the psychologists, with Patricia Churchland more recently claiming there is no ghost in the machine.

The brain does it all

If the soul is where emotion and motivation reside, where mental activity occurs, sensations are perceived, memories are stored, reasoning takes place and decisions are taken, then there is no need to hypothesise its existence. There is an organ that already performs these functions: the brain.

This idea goes back to the ancient physician Hippocrates (460-377 BCE) who said:

Men ought to know that from nothing else but the brain come joys, delights, laughter and sports, and sorrows, griefs, despondency and lamentations. And by this … we acquire wisdom and knowledge, and see and hear, and know what are foul and what are fair, what are bad and what are good, what are sweet and what are unsavoury…

The brain is the organ with a map of our body, the outside world and our experience. Damage to the brain, as in accidents, dementias or congenital malformations, produces a commensurate damage to personality.

Consider one of the functions supposedly – if we listen to Plato – carried out by the soul: memory. A major knock on the head can make you lose your memories of the past several years. If the soul is an immaterial substance separate from our physical being, it should not be injured by the knock. If memory were stored in the soul, it should not have been lost.

The neuronal activity in the brain is responsible for the cognitive and emotional dysfunctions in people with autism; it would be cruel and unethical to blame their hypothetical souls.

Manipulation of the brain is sufficient to alter emotion and mood. The soul is totally superfluous to this process.

The ability of psychotherapeutic drugs to alter mood provides another line of evidence against the presence of the soul. If you produce a chemical imbalance in the brain, such as by depleting dopamine, noradrenaline and serotonin with tetrabenazine, you can induce depression in some people.

Correspondingly, many depressed people can be helped by drugs that increase the function of these neurotransmitters in the brain.

The brain is where thinking takes place, love and hatred reside, sensations become perceptions, personality is formed, memories and beliefs are held, and where decisions are made. As D.K. Johnson said: “There is nothing left for the soul to do.”

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Where Is the Rest of the Universe?

Where Is the Rest of the Universe?

7:04:00 PM   Amazing, Education, Nature, News, Science,

Most of the universe is made up of "stuff" that is invisible, possibly intangible and interacts with other things only via the force of gravity. Oh, yes, and physicists don't know what the stuff is or why it makes up so much of the universe — some four-fifths of its mass.

They call it dark matter.

So where is this mysterious stuff that makes up such a huge chunk of our universe, and when will scientists find it?

First, though, how do we know it's even out there?

Dark matter was first hypothesized in the 1930s when Swiss astronomer Fritz Zwicky realized that his measurements of the masses of galaxy clusters showed some of the mass in the universe was "missing." Whatever was making galaxies heavier, it didn't give off any light, nor did it interact with anything else except via gravity.

Vera Rubin, in the 1970s, found that the rotation of galaxies was not following the predictions of Newton's laws of motion; the stars in galaxies (notably Andromeda) all seemed to be orbiting the center at the same speed, rather than those farther out moving more slowly as theories of gravity said they should. Clearly, something was adding mass to the outer parts of galaxies, something nobody could see. [Does the Universe Have an Edge?]

Other pieces of evidence came from gravitational lensing, which happens when the gravity of a large object bends the light waves around that object. Per Albert Einstein's theory of general relativity, gravity bends space (like a sumo wrestler might deform the mat he is standing on), so light rays bend around massive objects even though light itself is massless. Observations indicated that there wasn't enough visible mass to bend the light by as much as it was bending around certain galaxy clusters — in other words, the galaxies were more massive than they should be.

Then there is the cosmic microwave background (CMB), the "echo" of the Big Bang, and supernovas. "What the CMB tells you is that the universe is spatially flat," said Jason Kumar, a professor of physics at the University of Hawaii. "Spatially flat" means that if you were to draw two lines across the universe, they would never meet, even if those lines were a billion light-years across. In a steeply curved universe, those lines would meet at some point in space.

Researchers then calculated how much matter the universe must have in order to be flat and produce the amount of normal matter (also called baryons) observed in the universe.

 "I ask myself, 'Is the amount of matter I have equal to baryonic matter, and it's not," Kumar said.  

There's now little dispute among cosmologists and astronomers that dark matter exists. Yet it seems unaffected by light, and it isn't charged like electrons or protons are. So far it has eluded direct detection.

 "That's kind of the mystery," Kumar said. There are may ways scientists have tried to "see" dark matter – either via its interactions with normal matter or looking for particles that dark matter might become. "Those experiments are going to keep getting better, and don't seem to have any hiccups so far as they go to better detectors."

What we know it isn't

A number of theories have come and gone as to what dark matter is. One of the first was logical enough: The matter was hidden within massive astrophysical compact halo objects, or MACHOs, such as neutron stars, black holes, brown dwarfs and rogue planets. They don't emit light (or they emit only very little), so they are effectively invisible to telescopes. [The 9 Biggest Unsolved Mysteries in Physics]

Yet surveys of galaxies looking for small distortions in the light of background stars produced by a MACHO passing by — called microlensing events — couldn't account for the amount of dark matter around galaxies, or even a significant portion of it. "MACHOs seem as ruled out as ever," said Dan Hooper, an associate scientist at the Fermi National Accelerator Laboratory in Illinois.

Dark matter doesn't appear to be clouds of gas that telescopes wouldn't see, either. Diffuse gas would absorb light from the galaxies that are farther away, and on top of that, ordinary gas would re-emit radiation at longer wavelengths – there'd be a massive radiation of infrared light in the sky. Since that doesn't occur we can rule that out as well, Kumar said.

What it might be

Weakly interacting massive particles, or WIMPs, are some of the strongest contenders to explain dark matter. WIMPs are heavy particles — about 10 to 100 times heavier than a proton — that were produced during the Big Bang, though just small quantities are left today. These particles interact with normal matter via gravity or the weak nuclear force. More massive WIMPs would move more slowly through space, and therefore be "cold" dark-matter candidates, while lighter ones would move faster, and be "warm" dark matter. [Wacky Physics: The Coolest Little Particles in Nature]

One way to find them is in "direct detection" experiments, such as the Large Underground Xenon (LUX) experiment, which is a container of liquid xenon in a South Dakota mine. If a xenon nucleus seems to "bounce" with no explanation, that would be a candidate for getting hit with a dark-matter particle. The magnitude of the bounce would give an idea of the new particle's mass. But Hooper said that LUX hasn't seen anything yet.

Another way to see the WIMPs might be particle accelerators. Inside accelerators, atomic nuclei smash into each at near the speed of light, and in the process that energy of the collision gets turned into other particles, some new to science. So far, though, particle accelerators haven't detected anything that looks like a dark matter candidate, either.

Results from both direct detection and particle accelerators, however, have placed limits on the size and mass of this hypothetical dark-matter particle, Kumar said. The sensitivity of LUX is down to 200 MeV, or about a fifth of a proton's mass, and it could theoretically see particles as heavy as 1 TeV, which is comparable to some types of quarks. Since LUX saw nothing so far, that could well mean that dark matter isn't in that range.

Kumar said that it's possible that WIMPs are really heavy, and because they are so massive there just aren't that many of them, meaning the chance they'll hit a xenon atom is small. 

Another possibility: axions. These subatomic particles could be detected indirectly by the kinds of radiation they emit as they annihilate or as they decay into other kinds of particles, or appear in particle accelerators. Yet there hasn't been any direct evidence of axions, either.

Since detecting heavy, slow-moving "cold" particles, like WIMPs or axions, hasn't yielded results yet, some scientists are looking at the possibility of lighter, faster-moving particles, which they call "warm" dark matter. There has been a renewed interest in such a dark-matter model after scientists found evidence of an unknown particle, using the Chandra X-ray Observatory, in the Perseus cluster, a group of galaxies about 250 million light-years from Earth. The known ions in that cluster produce certain X-ray emission lines, and in 2014, scientists saw a new "line" that may correspond to an unknown lightweight particle, Nicola Menci, an astrophysicist at Italy's National Institute for Astrophysics (INAF), told Live Science in an email.

If dark-matter particles are light, scientists are going to have a difficult time detecting them directly, said Tracy Slatyer, a physicist at MIT. Slatyer has proposed new kinds of particles that might make up dark matter.

"Dark matter with [a] mass below about 1 GeV is really hard to detect with conventional direct detection experiments, because they work by looking for the unexplained recoils of atomic nuclei… but when the dark matter is much lighter than the atomic nucleus, the recoil energy is very small," Slatyer said. Protons — a hydrogen nucleus — can't be any lighter than about 938 MeV, so a particle that weighs in the keV range would be 1,000 times lighter. "Think bouncing a ping-pong ball off a bowling ball; the bowling ball doesn't move very much," she said.

Slatyer said there is a lot of research being done on how to find dark-matter particles if current methods fail. Using "superfluid" liquid helium, semiconductors and even breaking of chemical bonds in crystals, are among the ideas being floated.

Kumar said one reason dark matter is so mysterious is precisely that physicists feel that they understand, to a point, how the Big Bang nucleosynthesis ― the origins of matter ― worked. The Standard Model, which predicted the Higgs boson, has been very successful so far, so unless they've all been really wrong about something fundamental it's odd that nobody has managed to detect a dark-matter particle yet.

If, for example, dark-matter particles are very different from what many current models predict, it's possible that particle accelerators wouldn't have seen it. Accelerators like the Large Hadron Collider are better at seeing things that interact with the strong nuclear force, which decay into other particles.

"If that's the way your dark matter works it's a great machine to find it," he said. "But if there isn't a heavier particle like that it's harder." 

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How Hot Is Lightning?

How Hot Is Lightning?

6:58:00 PM   Amazing, Education, Nature, Science,

Lightning is one of the most destructive forces in nature. But for all the folklore and legends amassed over human history on lightning, we know surprisingly little about the inner workings of this powerful phenomenon, including something as simple as how the current that flows through a thunder-inducing flash is related to the temperature of the strike.

Lightening

"The basic physics of lightning, such as lightning initiation and lightning propagation, is not fully understood at this point," said Robert Moore, a lightning researcher from University of Florida in Gainesville.

"We know the basics, but not the details. So when anybody makes headway, it is major news."

Lightning causes more than $5 billion in damages every year in the U.S., as well as more fatalities than hurricanes.

"A direct hit from a lightning strike can melt a power cable or start a forest fire, where the amount of heat from the lightning plays a major role," said Xiangchao Li, a scientist from China who specializes in lightning research. Li and his team discovered a mathematical relationship between the current intensity and the temperature inside lightning. Their result was published last month in the journal Scientific Reports.

Although there are approximately 100,000 lightning strikes on Earth every single day, the randomness of the occurrences makes it difficult for scientists to study them in an effective or systematic way. So until Thor, the Norse god of lightning as well as other meteorological events, joins a lightning research team, scientists are left to their own devices.

Luckily such a device does exist. Known as an impulse current generator system, the device can create artificial lightning with currents up to tens of thousands of amperes. For perspective, a household or automotive fuse is usually rated well below a hundred amperes, and an electric current of just a few amperes can easily kill you. A natural lightning strike typically carries around 20-30,000 amperes of current. Certainly there are other factors such as size and setting of natural lightning that cannot be replicated in a laboratory, but just in terms of sheer current output, the lightning generated by the device can really give Thor a run for his money.

By using their artificial lightning system, Li and his team were able to dial up lightning strikes at will, with currents between 5,000 to 50,000 amperes. This resulted in artificial lightning strikes with temperatures as high as 17,000 F, twice as hot as the surface of the Sun.

This creates a new problem — at such high temperatures, a normal thermometer would explode. And even if it didn't, it wouldn't react quickly enough to register the temperature of the lightning strike. Fortunately, there is "light" in "lightning." Li and his team were able to record the lightning's temperature within a millisecond by measuring the intensity of the light at various wavelengths.

After striking lightning at the same place over and over again, they concluded that the relationship between the current and temperature of lightning is a highly logarithmic one, meaning that the temperature difference between lightning strikes with 1,000 and 10,000 amperes is similar to those with 10,000 and 100,000 amperes. This result provides solid evidence for previous theoretical predictions that lacked the support of data.

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Albert Einstein and Theory of Relativity, Newton vs. Einstein

Albert Einstein and Theory of Relativity, Newton vs. Einstein

5:32:00 PM   Education,

Newton's theory of gravitation was soon accepted without question, and it remained unquestioned until the beginning of this century. Then Albert Einstein shook the foundations of physics with the introduction of his Special Theory of Relativity in 1905, and his General Theory of Relativity in 1915 (Here is an example of a thought experiment in special relativity). The first showed that Newton's Three Laws of Motion were only approximately correct, breaking down when velocities approached that of light. The second showed that Newton's Law of Gravitation was also only approximately correct, breaking down in the presence of very strong gravitational fields.

Newton vs. Einstein: Albert's Turn to Kick Butt

We shall consider Relativity in more detail later. Here, we only summarize the differences between Newton's theory of gravitation and the theory of gravitation implied by the General Theory of Relativity. They make essentially identical predictions as long as the strength of the gravitational field is weak, which is our usual experience. However, there are three crucial predictions where the two theories diverge, and thus can be tested with careful experiments.

1. The orientation of Mercury's orbit is found to precess in space over time, as indicated in the adjacent figure (the magnitude of the effect is greatly exaggerated in this figure). This is commonly called the "precession of the perihelion", because it causes the position of the perihelion to move. Only part of this can be accounted for by perturbations in Newton's theory. There is an extra 43 seconds of arc per century in this precession that is predicted by the Theory of General Relativity and observed to occur (a second of arc is 1/3600 of an angular degree). This effect is extremely small, but the measurements are very precise and can detect such small effects very well.
2. Einstein's theory predicts that the direction of light propagation should be changed in a gravitational field, contrary to the Newtonian predictions. Precise observations indicate that Einstein is right, both about the effect and its magnitude. A striking consequence is gravitational lensing.
3. The General Theory of Relativity predicts that light coming from a strong gravitational field should have its wavelength shifted to larger values (what astronomers call a "red shift"), again contary to Newton's theory. Once again, detailed observations indicate such a red shift, and that its magnitude is correctly given by Einstein's theory.
4. The electromagnetic field can have waves in it that carry energy and that we call light. Likewise, the gravitational field can have waves that carry energy and are called gravitational waves. These may be thought of as ripples in the curvature of spacetime that travel at the speed of light. Just as accelerating charges can emit electromagnetic waves, accelerating masses can emit gravitational waves. However gravitational waves are difficult to detect because they are very weak and no conclusive evidence has yet been reported for their direct observation. They have been observed indirectly in the binary pulsar. Because the arrival time of pulses from the pulsar can be measured very precisely, it can be determined that the period of the binary system is gradually decreasing. It is found that the rate of period change (about 75 millionths of a second each year) is what would be expected for energy being lost to gravitational radiation, as predicted by the Theory of General Relativity.

The Modern Theory of Gravitation

And there is stands to the present day. Our best current theory of gravitation is the General Theory of Relativity. However, only if velocities are comparable to that of light, or gravitational fields are much larger than those encountered on the Earth, do the Relativity theory and Newton's theories differ in their predictions. Under most conditions Newton's three laws and his theory of gravitation are adequate. We shall return to this issue in our subsequent discussion of cosmology. For a more comprehensive introduction to both Special and General Relativity, see the links at Relativity on the WWW, and The Light Cone (An Illuminating Introduction to Relativity), and Albert Einstein OnlineIn this section we deal with general properties of the celestial sphere: constellations, the naming of stars, and a general coordinate system for the celestial sphere that is analogous to the latitude-longitude system on the surface of the Earth and that allows us to specify precisely a location on the celestial sphere. We shall also consider some general aspects of timekeeping and calendars, because historically, the regular apparent motions of the heavens provided many of the ideas and much of the terminology that we use in timekeeping. For further discussion of many of the topics in this section, see Astronomy without a Telescope.

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Newton,'s Three Laws Of Motion

Newton,'s Three Laws Of Motion

5:28:00 PM   Education,

Newton's Three Laws
of Motion

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Let us begin our explanation of how Newton changed our understanding of the Universe by enumerating his Three Laws of Motion.

Newton's First Law of Motion:

I. Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.

This we recognize as essentially Galileo's concept of inertia, and this is often termed simply the "Law of Inertia".

Newton's Second Law of Motion:

II. The relationship between an object's mass m, its acceleration a, and the applied force F is F = ma. Acceleration and force are vectors (as indicated by their symbols being displayed in slant bold font); in this law the direction of the force vector is the same as the direction of the acceleration vector.

This is the most powerful of Newton's three Laws, because it allows quantitative calculations of dynamics: how do velocities change when forces are applied. Notice the fundamental difference between Newton's 2nd Law and the dynamics of Aristotle: according to Newton, a force causes only a change in velocity (an acceleration); it does not maintain the velocity as Aristotle held.
This is sometimes summarized by saying that under Newton, F = ma, but under Aristotle F = mv, where v is the velocity. Thus, according to Aristotle there is only a velocity if there is a force, but according to Newton an object with a certain velocity maintains that velocity unless a force acts on it to cause an acceleration (that is, a change in the velocity). As we have noted earlier in conjunction with the discussion of Galileo, Aristotle's view seems to be more in accord with common sense, but that is because of a failure to appreciate the role played by frictional forces. Once account is taken of all forces acting in a given situation it is the dynamics of Galileo and Newton, not of Aristotle, that are found to be in accord with the observations.

Newton's Third Law of Motion:

III. For every action there is an equal and opposite reaction.

This law is exemplified by what happens if we step off a boat onto the bank of a lake: as we move in the direction of the shore, the boat tends to move in the opposite direction (leaving us facedown in the water, if we aren't careful!). 
There is a popular story that Newton was sitting under an apple tree, an apple fell on his head, and he suddenly thought of the Universal Law of Gravitation. As in all such legends, this is almost certainly not true in its details, but the story contains elements of what actually happened.

What Really Happened with the Apple?

Probably the more correct version of the story is that Newton, upon observing an apple fall from a tree, began to think along the following lines: The apple is accelerated, since its velocity changes from zero as it is hanging on the tree and moves toward the ground. Thus, by Newton's 2nd Law there must be a force that acts on the apple to cause this acceleration. Let's call this force "gravity", and the associated acceleration the "accleration due to gravity". Then imagine the apple tree is twice as high. Again, we expect the apple to be accelerated toward the ground, so this suggests that this force that we call gravity reaches to the top of the tallest apple tree.

Sir Isaac's Most Excellent Idea

Now came Newton's truly brilliant insight: if the force of gravity reaches to the top of the highest tree, might it not reach even further; in particular, might it not reach all the way to the orbit of the Moon! Then, the orbit of the Moon about the Earth could be a consequence of the gravitational force, because the acceleration due to gravity could change the velocity of the Moon in just such a way that it followed an orbit around the earth. This can be illustrated with the thought experiment shown in the following figure. Suppose we fire a cannon horizontally from a high mountain; the projectile will eventually fall to earth, as indicated by the shortest trajectory in the figure, because of the gravitational force directed toward the center of the Earth and the associated acceleration. (Remember that an acceleration is a change in velocity and that velocity is a vector, so it has both a magnitude and a direction. Thus, an acceleration occurs if either or both the magnitude and the direction of the velocity change.)

But as we increase the muzzle velocity for our imaginary cannon, the projectile will travel further and further before returning to earth. Finally, Newton reasoned that if the cannon projected the cannon ball with exactly the right velocity, the projectile would travel completely around the Earth, always falling in the gravitational field but never reaching the Earth, which is curving away at the same rate that the projectile falls. That is, the cannon ball would have been put into orbit around the Earth. Newton concluded that the orbit of the Moon was of exactly the same nature: the Moon continuously "fell" in its path around the Earth because of the acceleration due to gravity, thus producing its orbit. By such reasoning, Newton came to the conclusion that any two objects in the Universe exert gravitational attraction on each other, with the force having a universal form:

The constant of proportionality G is known as the universal gravitational constant. It is termed a "universal constant" because it is thought to be the same at all places and all times, and thus universally characterizes the intrinsic strength of the gravitational force.

The Center of Mass for a Binary System

If you think about it a moment, it may seem a little strange that in Kepler's Laws the Sun is fixed at a point in space and the planet revolves around it. Why is the Sun privileged? Kepler had rather mystical ideas about the Sun, endowing it with almost god-like qualities that justified its special place. However Newton, largely as a corollary of his 3rd Law, demonstrated that the situation actually was more symmetrical than Kepler imagined and that the Sun does not occupy a privileged postion; in the process he modified Kepler's 3rd Law. Consider the diagram shown to the right. We may define a point called the center of mass between two objects through the equations

where R is the total separation between the centers of the two objects. The center of mass is familiar to anyone who has ever played on a see-saw. The fulcrum point at which the see-saw will exactly balance two people sitting on either end is the center of mass for the two persons sitting on the see-saw. Here is a Center of Mass Calculator that will help you make and visualize calculations concerning the center of mass. (Caution: this applet is written under Java 1.1, which is only supported by the most recent browsers. It should work on Windows systems under Netscape 4.06 or the most recent version of Internet Explorer 4.0, but may not yet work on Mac or Unix systems or earlier Windows browsers.)

Newton's Modification of Kepler's Third Law

Because for every action there is an equal and opposite reaction, Newton realized that in the planet-Sun system the planet does not orbit around a stationary Sun. Instead, Newton proposed that both the planet and the Sun orbited around the common center of mass for the planet-Sun system. He then modified Kepler's 3rd Law to read,

where P is the planetary orbital period and the other quantities have the meanings described above, with the Sun as one mass and the planet as the other mass. (As in the earlier discussion of Kepler's 3rd Law, this form of the equation assumes that masses are measured in solar masses, times in Earth years, and distances in astronomical units.) Notice the symmetry of this equation: since the masses are added on the left side and the distances are added on the right side, it doesn't matter whether the Sun is labeled with 1 and the planet with 2, or vice-versa. One obtains the same result in either case. Now notice what happens in Newton's new equation if one of the masses (either 1 or 2; remember the symmetry) is very large compared with the other. In particular, suppose the Sun is labeled as mass 1, and its mass is much larger than the mass for any of the planets. Then the sum of the two masses is always approximately equal to the mass of the Sun, and if we take ratios of Kepler's 3rd Law for two different planets the masses cancel from the ratio and we are left with the original form of Kepler's 3rd Law:

Thus Kepler's 3rd Law is approximately valid because the Sun is much more massive than any of the planets and therefore Newton's correction is small. The data Kepler had access to were not good enough to show this small effect. However, detailed observations made after Kepler show that Newton's modified form of Kepler's 3rd Law is in better accord with the data than Kepler's original form.

Two Limiting Cases

We can gain further insight by considering the position of the center of mass in two limits. First consider the example just addressed, where one mass is much larger than the other. Then, we see that the center of mass for the system essentially concides with the center of the massive object:

This is the situation in the Solar System: the Sun is so massive compared with any of the planets that the center of mass for a Sun-planet pair is always very near the center of the Sun. Thus, for all practical purposes the Sun IS almost (but not quite) motionless at the center of mass for the system, as Kepler originally thought. However, now consider the other limiting case where the two masses are equal to each other. Then it is easy to see that the center of mass lies equidistant from the two masses and if they are gravitationally bound to each other, each mass orbits the common center of mass for the system lying midway between them:

This situation occurs commonly with binary stars (two stars bound gravitationally to each other so that they revolve around their common center of mass). In many binary star systems the masses of the two stars are similar and Newton's correction to Kepler's 3rd Law is very large. Here is a Java applet that implements Newton's modified form of Kepler's 3rd law for two objects (planets or stars) revolving around their common center of mass. By making one mass much larger than the other in this interactive animation you can illustrate the ideas discussed above and recover Kepler's original form of his 3rd Law where a less massive object appears to revolve around a massive object fixed at one focus of an ellipse.
These limiting cases for the location of the center of mass are perhaps familiar from our afore-mentioned playground experience. If persons of equal weight are on a see-saw, the fulcrum must be placed in the middle to balance, but if one person weighs much more than the other person, the fulcrum must be placed close to the heavier person to achieve balance.
Here is a Kepler's Laws Calculator that allows you to make simple calculations for periods, separations, and masses for Keplers' laws as modified by Newton (see subsequent section) to include the effect of the center of mass. (Caution: this applet is written under Java 1.1, which is only supported by the most recent browsers. It should work on Windows systems under Netscape 4.06 or the most recent version of Internet Explorer 4.0, but may not yet work on Mac or Unix systems or earlier Windows browsers.)

Weight and the Gravitational Force

We have seen that in the Universal Law of Gravitation the crucial quantity is mass. In popular language mass and weight are often used to mean the same thing; in reality they are related but quite different things. What we commonly call weight is really just the gravitational force exerted on an object of a certain mass. We can illustrate by choosing the Earth as one of the two masses in the previous illustration of the Law of Gravitation:

Thus, the weight of an object of mass m at the surface of the Earth is obtained by multiplying the mass m by the acceleration due to gravity, g, at the surface of the Earth. The acceleration due to gravity is approximately the product of the universal gravitational constant G and the mass of the Earth M, divided by the radius of the Earth, r, squared. (We assume the Earth to be spherical and neglect the radius of the object relative to the radius of the Earth in this discussion.) The measured gravitational acceleration at the Earth's surface is found to be about 980 cm/second/second.

Mass and Weight

Mass is a measure of how much material is in an object, but weight is a measure of the gravitational force exerted on that material in a gravitational field; thus, mass and weight are proportional to each other, with the acceleration due to gravity as the proportionality constant. It follows that mass is constant for an object (actually this is not quite true, but we will save that surprise for our later discussion of the Relativity Theory), but weight depends on the location of the object. For example, if we transported the preceding object of mass m to the surface of the Moon, the gravitational acceleration would change because the radius and mass of the Moon both differ from those of the Earth. Thus, our object has mass m both on the surface of the Earth and on the surface of the Moon, but it will weigh much less on the surface of the Moon because the gravitational acceleration there is a factor of 6 less than at the surface of the Earth.

We now come to the great synthesis of dynamics and astronomy accomplished by Newton: the Laws of Kepler for planetary motion may be derived from Newton's Law of Gravitation. Furthermore, Newton's Laws provide corrections to Kepler's Laws that turn out to be observable, and Newton's Law of Gravitation will be found to describe the motions of all objects in the heavens, not just the planets.

Acceleration in Keplerian Orbits

Kepler's Laws are illustrated in the adjacent animation. The red arrow indicates the instantaneous velocity vector at each point on the orbit (as always, we greatly exaggerate the eccentricty of the ellipse for purposes of illustration). Since the velocity is a vector, the direction of the velocity vector is indicated by the direction of the arrow and the magnitude of the velocity is indicated by the length of the arrow. Notice that (because of Kepler's 2nd Law) the velocity vector is constantly changing both its magnitude and its direction as it moves around the elliptical orbit (if the orbit were circular, the magnitude of the velocity would remain constant but the direction would change continuously). Since either a change in the magnitude or the direction of the velocity vector constitutes an acceleration, there is a continuous acceleration as the planet moves about its orbit (whether circular or elliptical), and therefore by Newton's 2nd Law there is a force that acts at every point on the orbit. Furthermore, the force is not constant in magnitude, since the change in velocity (acceleration) is larger when the planet is near the Sun on the elliptical orbit.

Newton's Laws and Kepler's Laws

Since this is a survey course, we shall not cover all the mathematics, but we now outline how Kepler's Laws are implied by those of Newton, and use Newton's Laws to supply corrections to Kepler's Laws.

1. Since the planets move on ellipses (Kepler's 1st Law), they are continually accelerating, as we have noted above. As we have also noted above, this implies a force acting continuously on the planets.
2. Because the planet-Sun line sweeps out equal areas in equal times (Kepler's 2nd Law), it is possible to show that the force must be directed toward the Sun from the planet.
3. From Kepler's 1st Law the orbit is an ellipse with the Sun at one focus; from Newton's laws it can be shown that this means that the magnitude of the force must vary as one over the square of the distance between the planet and the Sun.
4. Kepler's 3rd Law and Newton's 3rd Law imply that the force must be proportional to the product of the masses for the planet and the Sun.

Thus, Kepler's laws and Newton's laws taken together imply that the force that holds the planets in their orbits by continuously changing the planet's velocity so that it follows an elliptical path is (1) directed toward the Sun from the planet, (2) is proportional to the product of masses for the Sun and planet, and (3) is inversely proportional to the square of the planet-Sun separation. This is precisely the form of the gravitational force, with the universal gravitational constant G as the constant of proportionality. Thus, Newton's laws of motion, with a gravitational force used in the 2nd Law, imply Kepler's Laws, and the planets obey the same laws of motion as objects on the surface of the Earth!

Conic Sections and Gravitational Orbits

The ellipse is not the only possible orbit in a gravitational field. According to Newton's analysis, the possible orbits in a gravitational field can take the shape of the figures that are known as conic sections (so called because they may be obtained by slicing sections from a cone, as illustrated in the following figure).

For the ellipse (and its special case, the circle), the plane intersects opposite "edges" of the cone. For the parabola the plane is parallel to one edge of the cone; for the hyperbola the plane is not parallel to an edge but it does not intersect opposite "edges" of the cone. (Remember that these cones extend forever downward; we have shown them with bottoms because we are only displaying a portion of the cone.)

Examples of Gravitational Orbits

We see examples of all these possible orbitals in gravitational fields. In each case, the determining factor influencing the nature of the orbit is the relative speed of the object in its orbit.

• The orbits of some of the planets (e.g., Venus) are ellipses of such small eccentricity that they are essentially circles, and we can put artificial satellites into orbit around the Earth with circular orbits if we choose.
• The orbits of the planets generally are ellipses.
• Some comets have parabolic orbits; this means that they pass the Sun once and then leave the Solar System, never to return. Other comets have elliptical orbits and thus orbit the Sun with specific periods.
• The gravitational interaction between two passing stars generally results in hyperbolic trajectories for the two stars.

Thus, Kepler's elliptical orbitals are but one example of the possible orbits in a gravitational field. Only ellipses (and their special case, the circle) lead to bound orbits; the others are associated with one-time gravitational encounters. Here is a set of Java applets, taken from the Famous Curves Applet Index that illustrate the geometrical properties of these gravitational orbits:

• Java applet illustrating properties of a circle
• Java applet illustrating properties of an ellipse
• Java applet illustrating properties of an hyperbola
• Java applet illustrating properties of a parabola

For a given central force, increasing the velocity causes the orbit to change from a circle to an ellipse to a parabola to a hyperbola, with the changes occurring at certain critical velocities. For example, if the speed of the Earth (which is in a nearly circular gravitational orbit) were increased by about a factor of 1.4, the orbit would change into a parabola and the Earth would leave the Solar System.

Computing the orbit of the Earth as an ellipse around the center of mass for the Earth-Sun system assumes that they are the only two masses in the Universe. In reality, the Universal Law of Gravitation implies that the Earth interacts gravitationally not only with the Sun, but with every other mass in the Universe: the Moon, the other planets, asteroids and comets, the distant stars.

The Two-Body Approximation

However, from the form of the gravitational force

we see that the interactions are largest when two situations are fulfilled: (1) the product of the masses of the two objects is large, which maximizes the numerator of the expression for the strength of the gravitational force, and (2) the objects are near each other, which minimizes the denominator of the force equation. The two-body approximation that the orbits of the planets are determined only by the gravitational interaction between the Sun and the planet is possible because

1. The Sun is so massive compared with every other object in the Solar System,
2. Objects outside the Solar System such as stars are so distant that the distance squared factor in the denominator renders their gravitational interactions with the planets negligible.

For example, the Sun is about 300,000 times more massive than the Earth, and about 1000 times more massive than the largest planet. Thus, the product of the mass of a planet and the mass of the Sun is always much larger than the product of the masses of any two planets, and it is a good initial approximation to neglect all interactions except that of the planet and the Sun. EXERCISE: use the Astrophysical Calculator to compare the products of the masses of the Sun and Mars with the product of the masses of Mars and the Earth. The answer is the ratio of the masses of the Sun and the Earth, which is about 300,000, as noted above.

Gravitational Perturbations

However, the small deviations from this ideal picture have consequences if careful measurements are made. These small deviations from the simplified picture are called perturbations. They can be calculated systematically using Newton's laws of motion and gravitation from the positions of the known masses in the Solar System. If we account carefully for all known gravitational perturbations on the motion of observed planets and the motion of the planet still deviates from the prediction, there are two options:

1. Newton's Law of Gravitation requires modification,
2. There is a previously undetected mass that is perturbing the orbits of the observed planets.

We shall see that the history of astronomy following the introduction of the Law of Gravitation by Newton gives examples of both.

The Discovery of the New Planet Neptune

In 1846, the planet Neptune was discovered after its existence was predicted because of discrepancies between calculations and data for the planet Uranus. Astronomers found the new planet almost exactly at the position predicted by the calculations of Leverrier (Adams had also calculated the position independently). We illustrate the situation schematically in the adjacent diagram. The dominant interaction between Uranus and the Sun is indicated with the heavy line, but some perturbations associated with other masses are indicated by thin lines. By using Newton's laws to calculate the perturbations on the orbit of Uranus by an hypothesized new planet, Leverrier and Adams were able to predict where the planet had to be in order to cause the observed deviations in the position of Uranus. Once astronomers took this calculation seriously, they found the new planet within hours of turning their telescopes on the region of the sky implicated by the calculations. This precise prediction of the new planet and its location was striking confirmation of the power of Newton's theory of gravitation. (Although in truth it must be said that both Leverrier and Adams made an incorrect assumption in their calculations concerning the radius of the new planet's orbit. Fortunately, the error largely cancelled out of the calculations and had little effect on their final results.)

The Accidental Discovery of Pluto

Later, similar calculations on supposed perturbations of the orbits of Uranus and Neptune suggested the presence of yet another planet beyond the orbit of Neptune. Eventually, in 1930, a new planet Pluto was discovered, but we now know that the calculations in this case were also in error because of an incorrect assumption about the mass of the new planet. It is now believed that the supposed deviations in the orbits of Neptune and Uranus were errors in measurement because the actual properties of Pluto would not have accounted for the supposed perturbations. Thus, the discovery of Pluto was a kind of accident.

Effects Beyond Newtonian Perturbations

The power of Newton's theory became apparent as detailed calculations accounted more and more precisely for the orbits of the planets. Any deviations from the expected behavior soon became viewed as evidence for unseen masses in the Solar System. However, later observations of anomalies in the orbit of Mercury could not be accounted for by the gravitational perturbation of a new planet (the hypothetical new planet, which turned out not to exist, was called Vulcan). As we discuss in the next section, early in this century this forced the replacement of Newton's Law of Gravitation with Einstein's Theory of General Relativity.

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NETWORK THEOREMS

NETWORK THEOREMS

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Superposition Theorem: Two Loop Problem To apply the superposition theorem to calculate the current through resistor R1 in the two loop circuit shown, the individual current supplied by each battery is calculated with the other battery replaced by a short circuit. For R1 =Ω, R2 = Ω, R3 = Ω, and voltages V1 = V and V2 = V, the calculated currents are = A, = A with a resultant current in R1 of = A. Note: To avoid dealing with so many short circuits, any resistor with value zero will default to 1 when a voltage is changed. It can be changed back to a zero value if you wish to explore the effects of short circuits. Ohms and amperes are the default units, but if you put in resistor values in kilohms, then the currents will be milliamperes. Other approaches to two-loop circuits

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Superposition Theorem The total current in any part of a linear circuit equals the algebraic sum of the currents produced by each source separately. To evaluate the separate currents to be combined, replace all other voltage sources by short circuits and all other current sources by open circuits. Application to two-loop problem	Index DC Circuits
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Thevenin's Theorem Any combination of batteries and resistances with two terminals can be replaced by a single voltage source e and a single series resistor r. The value of e is the open circuit voltage at the terminals, and the value of r is e divided by the current with the terminals short circuited. Thevenin voltage Thevenin resistance Numerical example Norton equivalent to AC Version Application in Digital to Analog Converter

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Thevenin Voltage The Thevenin voltage e used in Thevenin's Theorem is an ideal voltage source equal to the open circuit voltage at the terminals. In the example below, the resistance R2 does not affect this voltage and the resistances R1 and R3 form a voltage divider, giving Thevenin resistance Numerical example Norton equivalent	Index DC Circuits
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Thevenin/Norton Resistance The Thevenin resistance r used in Thevenin's Theorem is the resistance measured at terminals AB with all voltage sources replaced by short circuits and all current sources replaced by open circuits. It can also be calculated by dividing the open circuit voltage by the short circuit current at AB, but the previous method is usually preferable and gives The same resistance is used in the Norton equivalent. Thevenin voltage Numerical example Norton equivalent

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Thevenin Example Replacing a network by its Thevenin equivalent can simplify the analysis of a complex circuit. In this example, the Thevenin voltage is just the output of the voltage divider formed by R1 and R3. The Thevenin resistance is the resistance looking back from AB with V1 replaced by a short circuit. For R1 =Ω, R2 = Ω, R3 =Ω, and voltage V1 = V the Thevenin voltage is V since R1 and R3 form a simple voltage divider. The Thevenin resistance is = Ω. Apply to two-loop problem AC example Norton equivalent

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Norton's Theorem Any collection of batteries and resistances with two terminals is electrically equivalent to an ideal current source i in parallel with a single resistor r. The value of r is the same as that in the Thevenin equivalent and the current i can be found by dividing the open circuit voltage by r. Norton current Norton resistance Numerical example Thevenin equivalent	Index DC Circuits
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Norton Current The value i for the current used in Norton's Theorem is found by determining the open circuit voltage at the terminals AB and dividing it by the Norton resistance r. Norton resistance Numerical example Thevenin equivalent	Index DC Circuits
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Norton Example Replacing a network by its Norton equivalent can simplify the analysis of a complex circuit. In this example, the Norton current is obtained from the open circuit voltage (the Thevenin voltage) divided by the resistance r. This resistance is the same as the Thevenin resistance, the resistance looking back from AB with V1 replaced by a short circuit. For R1 = Ω, R2 = Ω, R3 = Ω, and voltage V1 = V the open circuit voltage is = V since R1 and R3 form a simple voltage divider. The Norton resistance is = Ω. and the resulting Norton current is = A Apply to two-loop problem Thevenin equivalent

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