Future Tense: 3D Or Not 3D

Future Tense: 3D Or Not 3D

12:56:00 AM   Science,

Despite the skepticism, of the commentariat, I believe that 3D television is inevitable. Within a few years, all television sets and all Blu-Ray players and all gaming consoles will be 3D capable. And because the capability is there, studios and networks will continue to use it as an audience magnet for important events.
But…
I also believe that the current implementation of 3D technology—using polarized or active shutter glasses is only an intermediate stage.
The ideal 3D system—the one we really want—will be some form of wall-sized holographic projection that does not require any special glasses at all. And while that technology was seemingly impossible only a few short years ago, I think we’re on the threshold of a breakthrough. I’ll explain.

Image courtesy dj-design.com

The history of photography, from the first pinhole cameras to the present has been about focusing light onto a flat image-sensitive surface. Whether that surface was film or an electronic sensor, the key element was the lens and its ability to focus light onto a flat plane. All of photography is about the relationship of the lens to the recording surface.
Current 3D imagery is still about the lens—two lenses. Stereoscopy is a carefully constructed illusion, produced by two images photographed about 2.25 inches apart. When it’s well done, the illusion is wonderful. But it’s still an illusion, and as some critics have pointed out, the illusion sometimes contradicts itself. The eyes are focusing on a flat plane that is often at a contradictory distance from the illusory depth of the stereoscopic image. (There are other issues as well. The lenses used, the depth of field, and the amount of separation also complicate the creation of the illusion.)
Holography, on the other hand, is a lensless process. It’s not about capturing an image on a flat plane, it’s about capturing a wave-form map of the light passing through a plane. With laser holography, you shoot a laser at a beam-splitter. Half the beam reflects off the object, the other half is the reference beam. When the two beams are reintegrated, you get an interference pattern. A laser hologram is the recording of that interference pattern onto a light-sensitive surface, a large piece of film. When light is aimed through the interference pattern, the original wave forms are reconstructed and the viewer perceives that as a true three-dimensional image.
A hologram presents the same wave forms to the eyes as the original image. It’s as if you’re looking through a window. You can move your head, tilt your head, stand on your head. It doesn’t matter. You’re not focusing on the static plane of the film or the screen. You’re focusing on what appears to be the actual object.
Unfortunately, laser-holography does not lend itself easily to capturing color or moving images. It can be done. I’ve seen color holograms and they’re so realistic that the image defies comparison with any other form of three-dimensional reproduction—with or without glasses. But projecting such a holographic image so that it can be viewed by an audience is another technological hurdle.
As I said above, I think we’re on the threshold of solving both parts of the problem—recording the image and projecting it. Up till now, the creation of a hologram required a carefully calibrated beam-splitter and laser. Not practical for a point-and-shoot situation at the park or a wedding or a vacation.
But the laser may no longer be necessary. We finally have the processing power to change the entire nature of photography. We can synthesize a holographic interference pattern from a digital model of an object. All we need is the digital model of the scene, real or constructed, and the computer can create a color hologram.
How do we get that digital model in real time without the laser? Multiple simultaneous photographs. Let me speculate. Imagine a camera the size of a small saucer. The rear is an electronic viewfinder. The front is an array of lenses. Two might be sufficient, but I suspect a dozen, set in a circle, might be even more efficient. The lenses need only have image-sensors of 2 megapixels each. But for sharpness and detail, you could put a 20-megapixel lens in the center. The little sensors record placement information. The one big sensor records texture, color, and detail. When you take a picture, the camera records 44 megapixels of information. 20 megapixels of actual photograph and another 24 megapixels of difference information. A multi-core processor in the camera constructs a 3D model of the scene and maps the 20-megapixel data onto it. Data can be interpolated as needed to fill in things missed by the central sensor, but picked up by the lesser quality edge sensors. Perhaps the professional models would have three 20-megapixel sensors spaced equidistantly around the edge of the saucer to capture even more detail. Or some other arrangement.
The point is that we don’t need a laser. If we have multiple views of the same scene, we have enough information for the software to construct a digital model and extrapolate an interference pattern for a color hologram viewable in white light. If we can record images at 30fps, we’ll have holographic video. (BTW, all those multiple sensors can give you real-time High Dynamic Range video too. And probably a lot of other nifty exposure effects, like synthesizing an amazingly high ISO.) Existing 3D movies might also be translatable into a holographic format for viewing without glasses.

Presenting the synthesized interference pattern on a video panel might require greater resolution than our video displays are currently capable of. 1900x1080 might be insufficient for the kind of clarity and crispness we’ve gotten used to. We might have to go up to 4K resolution. Or more. But screens of that resolution have already been demonstrated in the lab and could become consumer products in the foreseeable future.
If and when we can put an interference pattern on a video panel, one that will allow white light shone through it to construct the digitally-synthesized waveform, we will have true 3D television. If we can find a way to recreate that waveform on a theater-sized screen, we’ll have 3D movies that don’t require glasses.
There are still going to be technical issues. If the photographic window is the size of a saucer, then how do you expand that window to the size of a home theater, or for that matter an Imax installation? Part of the image processing will have to address that issue. You don’t want Beverly Hills Chihuahua to turn into Bel Air St. Bernard.
But I think that problem could be solvable too. It’ll take some experimentation, some engineering, a lot of testing and tweaking, but if we’re construct a digital model of the scene when we photograph it, we can also reconstruct that digital model for whatever size screen it’s going to be presented on, whether it’s your laptop, your home theater, or the local multiplex.
(The bandwidth problems could be horrendous, of course. It all depends on how much information actually needs to be transmitted and where the bulk of the processing is done—maybe most of it can be done in the receiver. I’m sure that Scotty will figure it out just before the last commercial.)
Some of what I’ve described here has already been demonstrated in the lab. Adobe has shown some very sophisticated technology for mapping detailed textures onto 3D surfaces, other companies have produced computer generated interference patterns for credit cards and driver’s licenses. Those are the two front-end issues. The last piece of the puzzle is the presentation window.
If it all comes together, I suspect we could see the first experimental units before the decade is over. And if I’m right, then the current generation of 3D televisions will be obsolete right on schedule. Ten years.
What do you think?

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2nd Generation Intel® Core™ i7 Processor Specifications

2nd Generation Intel® Core™ i7 Processor Specifications

12:47:00 AM   Amazing Fcacs,

Processor Number	Cache	Clock Speed	# of Cores/ # of Threads	Max TDP	Memory Types	Graphics
Intel® Core™ i7-2720QM Processor (6M Cache, 2.20 GHz)	6.0 MB	2.20 GHz	4 / 8	45	DDR3-1066/1333/1600	Processor Graphics
Intel® Core™ i7-2600 Processor (8M Cache, 3.40 GHz)	8.0 MB	3.40 GHz	4 / 8	95	DDR3-1066/1333	Intel® HD Graphics 2000
Intel® Core™ i7-2600K Processor (8M Cache, 3.40 GHz)	8.0 MB	3.40 GHz	4 / 8	95	DDR3-1066/1333	Intel® HD Graphics 3000
Intel® Core™ i7-2600S Processor (8M Cache, 2.80 GHz)	8.0 MB	2.80 GHz	4 / 8	65	DDR3-1066/1333	Intel® HD Graphics 2000
Intel® Core™ i7-2630QM Processor (6M Cache, 2.00 GHz)	6.0 MB	2.00 GHz	4 / 8	45	DDR3-1066/1333	Processor Graphics
Intel® Core™ i7-2820QM Processor (8M Cache, 2.30 GHz)	8.0 MB	2.30 GHz	4 / 8	45	DDR3-1066/1333/1600	Processor Graphics
Intel® Core™ i7-2620M Processor (4M Cache, 2.70 GHz)	4.0 MB	2.70 GHz	2 / 4	35	DDR3-1066/1333	Processor Graphics
Intel® Core™ i7-2635QM Processor (6M Cache, 2.00 GHz)	6.0 MB	2.00 GHz	4 / 8	45	DDR3-1066/1333	Processor Graphics
Intel® Core™ i7-2710QE Processor (6M Cache, 2.10 GHz)	6.0 MB	2.10 GHz	4 / 8	45	DDR3-1066/1333/1600	Intel® HD Graphics 3000
Intel® Core™ i7-2629M Processor (4M Cache, 2.10 GHz)	4.0 MB	2.10 GHz	2 / 4	25	DDR3-1066/1333	Processor Graphics
Intel® Core™ i7-2649M Processor (4M Cache, 2.30 GHz)	4.0 MB	2.30 GHz	2 / 4	25	DDR3-1066/1333	Processor Graphics
Intel® Core™ i7-2657M Processor (4M Cache, 1.60 GHz)	4.0 MB	1.60 GHz	2 / 4	17	DDR3-1066/1333	Processor Graphics
Intel® Core™ i7-2617M Processor (4M Cache, 1.50 GHz)	4.0 MB	1.50 GHz	2 / 4	17	DDR3-1066/1333	Processor Graphics
Intel® Core™ i7-2677M Processor (4M Cache, 1.80 GHz)	4.0 MB	1.80 GHz	2 / 4	17	DDR3-1066/1333	Processor Graphics
Intel® Core™ i7-2637M Processor (4M Cache, 1.70 GHz)	4.0 MB	1.70 GHz	2 / 4	17	DDR3-1066/1333	Processor Graphics
Intel® Core™ i7-2655LE Processor (4M Cache, 2.20 GHz)	4.0 MB	2.20 GHz	2 / 4	25	DDR3-1066/1333	Intel® HD Graphics 3000
Intel® Core™ i7-2715QE Processor (6M Cache, 2.10 GHz)	6.0 MB	2.10 GHz	4 / 8	45	DDR3-1066/1333/1600	Intel® HD Graphics 3000
Intel® Core™ i7-2610UE Processor (4M Cache, 1.50 GHz)	4.0 MB	1.50 GHz	2 / 4	17	DDR3-1066/1333	Intel® HD Graphics 3000

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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

---

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

5:21:00 PM   Education,

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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6 Famously Terrible Movies That Were Almost Awesome

6 Famously Terrible Movies That Were Almost Awesome

2:27:00 PM   Amazing Fcacs,

At some point in the last six months, you've sat in a theater watching a terrible movie that took $200 million to make and said to yourself, "Why the hell did they ever think this would be good?"
The reason is that lots of bad movies looked like good movies in the early stages. It takes a whole bunch of people to make a movie, and often the finished product is only as good as its shittiest element. That messy process is how we wound up with ...

#6. X-Men: The Last Stand

Via Chillnmasti.blogspot.com
X-Men: The Last Stand was the unforgivable follow-up to X2. We say "unforgivable" because the second X-Men movie was widely regarded as one of the greatest superhero flicks of all time, if just for one scene.

You know, the stabbing scene.

Yet, despite coming hard on the heels of that success and boasting a larger budget than most nations' militaries, The Last Stand somehow managed to spit in the face of just about everything the X-Men franchise had going for it. Everyone's favorite characters from the first two films were either killed off, depowered, reduced to cheap gags or, in Nightcrawler's case, completely left out.

"Audiences just can't root for a blue protagonist."

If you followed the debacle that was the third X-Men movie, you probably blame Twentieth Century Fox or the departure of director Bryan Singer. But the truth is that all the film's problems can be traced to one equally bad movie that steered both franchises into a kryptonite iceberg faster than a speeding bullet.

Via Superman.wikia.com
Nice work, dickface.

Singer left The Last Stand in July 2004 to direct Superman Returns, but also brought with him two of X2's screenwriters, his longtime editor/composer John Ottman and actor James "Cyclops" Marsden. Fox had no choice but to step in and make sure their dismembered picture beat Superman to the finish line over James Marsden's dead body.

Which it did.

The Awesome Movie We Missed Out On:
This:

Via Goodcomics.comicbookresources.com

If you're not a comics geek, that is an issue from one of the greatest story lines ever: "The Dark Phoenix Saga." And, sure enough, this was precisely where Bryan Singer wanted the third movie to pick up from, as it was exactly where X2 left off.
According to X2 screenwriter Michael Dougherty, The Last Stand was supposed to come off as a goddamn monster movie: "The idea was that you open up with Alkali Lake but it's completely barren and dried up and there are these odd reports of strange phenomena going on around the world accompanied by bright lights in the sky." You know, such as ships getting torn in two ...

Via Comicbookmovie.com

... and San Francisco getting fucking destroyed:

Via Comicbookmovie.com

Via Comicbookmovie.com

Instead of being shoved into the corner like Jennifer Grey, the second coming of Jean Grey/Phoenix was supposed to be heralded as if "a very god-like force had entered their reality ... causing disruptions around the world." In nerd terms, instead of X-Men: The Last Stand, you would have gotten X-Men: The Best Movie Ever, with a three-way war for control of Phoenix between the X-Men, Magneto's Brotherhood and the same Hellfire Club that would eventually be recycled into X-Men: First Class.
And someone traded that film for Superman Returns.

At long last, Superman fans can watch his son play piano.

#5. Alien: Resurrection

Via Dethorchid.com
Coming off the disastrous Alien 3, the development of Alien: Resurrection actually started off in the most promising way possible. The studio asked the up-and-coming screenwriter Joss Whedon to resurrect the franchise from the dreary and slow third film. Whedon's then-credits included Buffy the Vampire Slayer and Speed, and he would shortly thereafter get nominated for an Academy Award for Toy Story.
However, the decision to kill Ripley off in Alien 3 and the prospects for a short-lived Alien vs. Predator franchise severely tied Whedon's hands for the project, particularly when it came to all things Ripley. The studio suggested that they just clone her back in. Whedon did what they said, at which point it was suggested that he add some more clones.

Above: The difference between "horror" and "horrible."

Things kind of went downhill from there.
The Awesome Movie We Missed Out On:
First of all, let's go back one movie in the series. Check out the original, hilariously misleading teaser trailer for Alien 3.

Yep, it promised that the aliens would be showing up on Earth. A few dozen drafts of the screenplay later, it wound up taking place on a remote asteroid prison with a lonely, dog-shaped alien.
So Whedon's original draft of the next film (which you can find on the Internet) included the climax that so many of us anticipated for the Alien series: an "epic final battle on Earth, for Earth."
Leading up to that was a huge soldiers vs. aliens battle on board a massive ship that is slowly crashing toward Earth. Then a new, much more terrifying alien is born and starts wreaking havoc ... and no, it's not the weird-ass human/alien hybrid thing that made it into the film.

Which promptly started tonguing Ripley.

Here's how Whedon described it in the screenplay:
"An alien, to be sure, but nothing we've seen so far, its forelegs arch out of its back like spiders legs, its back legs set on enormous haunches, thick and powerful. Its head is long, eyeless, like the others, but along its white expanse red veins, coming out of the skin and running like thick black hairs to the back. It has retracted pincers at the side of head that come out when its tongue does. It's much bigger than the others, nearly the size of the queen herself.
And it's bone white."

So pretty much not that.

Then we see it in action:
"It LEAPS up to the ceiling in a second, quick and effortless as a monstrous flea. [It] leaps again and lands on the screaming soldier ... pincers SWING out and pin either side of his head. His eyes go wide as:
Its tongue SHOOTS into his throat. We watch it drain the blood from his body. We can see it, see its stomach swell, red tinged, as his body goes blue and slack."

"OK, that all sounds good, but we're going to just give you this instead."

So then the ship crashes on Earth, and Ripley and Call (Winona Ryder's character) are forced to do battle with this horrific new alien before it can go on a rampage through Earth's population.
As you know if you've seen the movie, virtually all of this was cut, for budget reasons. As for what was left, as Whedon puts it:
"I don't remember writing, 'A withered, granny-lookin' Pumkinhead-kinda-thing makes out with Ripley.' Pretty sure that stage direction never existed in any of my drafts."

Not that Whedon has ever had an issue ratcheting up the creep factor.

#4. Wild Wild West

Via Impawards.com
One of the worst, most unnecessary movies of all time, Wild Wild West stars Will Smith and Kevin Kline as racist jokes in cowboy outfits. It managed to score five Razzie Awards for being the shittiest piece of shit ever filmed (more than Howard the Duck, Catwoman or The Love Guru could pull off) despite its $140 million price tag. Will Smith later publicly apologized to actor Robert Conrad from the original TV series for taking such a steaming dump on the source material.

Via Amazon
"Why does this box set smell like asshole?"

The Awesome Movie We Missed Out On:
Originally, the movie was supposed to be directed by Richard Donner, who incidentally directed three episodes of the original TV series. A pre-meltdown Mel Gibson was slated to star, and their version of the film was written by Shane Black, whose credits at the time included the often brutally violent first two Lethal Weapon movies and The Last Boy Scout. In short, Wild Wild West was damn near a clever, witty, bloody action comedy set in the Old West. But at some point Donner and Gibson left the project to make a different movie with a different writer -- Maverick.

The world will never forgive them.

So Smith was left to make the gurgling hamster fart that became Wild Wild West, which he considers to be the biggest mistake of his career if for no other reason than the role he turned down to make it: Neo from The Matrix.

Via NY Daily News
He got in 12,932 little fights, and his mom got scared ...

OK, so that might very well have ruined The Matrix (we're pretty sure the script would have gone through several rewrites to Will Smithify the role of befuddled geek Thomas Anderson). So maybe getting a shitty Wild Wild West was worth getting The Matrix in exchange.

#3. Cool World

Via Movi.ca
The Brad Pitt/Kim Basinger animated/live action feature Cool World was an utter disaster, scoring a solid 3 percent on Rotten Tomatoes. That's worse than Batman & Robin, Showgirls and Stop! or My Mom Will Shoot, which incidentally came out that same year.

Via Amazon
Seriously, take that in for a couple of minutes.

It was supposed to be the world's first animated horror film (and it is pretty damn horrifying, just not in that way), but according to director Ralph Bakshi, the only horror was his experience working on the film with Paramount.
The Awesome Movie We Missed Out On:
It was supposed to be a "hard-R, gritty, sexy, noir horror/thriller." The original storyboards for Ralph Bakshi's script look like a mix between Who Framed Roger Rabbit and Sin City.

Via Animationarchive.org

Via Animationarchive.org

Oh, and the female lead up there wasn't supposed to be Kim Basinger -- Bakshi wanted Drew Barrymore.

Getty
Which would have worked out great since she's already a cartoon.

Everything seemed to be moving forward for that version of the film until Bakshi found a "surprise" waiting for him while on location. As Bakshi puts it, "Frank Mancuso, Jr. [the film's producer] had the script rewritten in secret. I had a huge fight with the guy and punched Mancuso, Jr. in the mouth."

Getty
He punched his hairline into a comb-over.

Also not helping were some even-less-welcome screenwriting suggestions from Basinger, who, according to Bakshi, decided she wanted it to be a PG movie (it wound up PG-13). The result, in Bakshi's own words, "was a total disaster."

Via Animationarchive.org
Judging from the storyboards, we now want to fight the staff, too.

#2. Robin Hood

Via Filmdocket.com
When most of us saw the ads for the Russell Crowe Robin Hood movie last year, the only reaction was, "What's the point?" This is a character that has been brought to the screen almost 100 times, according to IMDB, and the ads made it look like the same old shit. A blandly heroic Robin Hood shoots arrows at the evil soldiers who oppress the people while he presumably robs from the rich and gives to the poor. The only saving grace of Ridley Scott's contribution was that its artistic liberties did not include crazy shit like Morgan Freeman inventing the telescope in 1194.

History!

So why did they even bother making it? Is this the best story to spend $200 million bringing to the screen?
Well, it got made because the original script was apparently freaking amazing. It turns out the production of Scott's Robin Hood is ultimately the story of two films: the movie you saw ...

Or more likely didn't see.

... and a wholly abandoned project called Nottingham.
The Awesome Movie We Missed Out On:
Nottingham would've been the single most original Robin Hood movie in history. The original script (written by Ethan Reiff and Cyrus Voris) so excited studios when it turned up in 2007 that it became the subject of a fierce bidding war (Universal wound up paying freaking $1.5 million for it). It was a totally different take on the story -- the Sheriff of Nottingham is the main character and protagonist. Shit gets real once the sheriff investigates a string of grisly murders in the area. The sheriff then pursues an assholish Robin Hood (Russell Crowe) for the crimes only to find out that Robin, while a dickhead, was actually being framed.

It was more than "Robin Hood has a bow."

The film was to climax with an epic siege of the city of Nottingham between Prince John and King Richard, all while the Sheriff tries to discover the identity of the real killer using 12th century detective techniques (Reiff is a history buff and researched how actual killings in that era were investigated).

Cue sunglasses and an inappropriate scream.

Then, director Ridley Scott came on board and said, "What is this shit? We're making a Robin Hood movie! Get all that standard Robin Hood stuff back in there. That's what everybody wants to see." The movie was renamed Robin Hood and lots more scenes with people shooting bows and arrows were added. A few rewrites later, very little of the original screenplay remained. Today, Nottingham is a cautionary tale for every young, aspiring screenwriter out there. It doesn't matter what you write: the director and the star will decide what makes it onto the screen.

Did they mention he had a bow? Because he totally did.

#1. The Godfather: Part III

Via Geneticwriters.wordpress.com
It's the Rocky V of the Godfather trilogy, and for more reasons than just the poorly acted, painfully unwelcome supporting characters forced onto the audience at gunpoint.

She must have known someone in the film crew. But who?

All talk of Sofia Coppola's horrible acting aside, perhaps the biggest blow to The Godfather: Part III was the notable absence of Robert Duvall's consigliere Tom Hagen from the whole damn affair. The short story is that Robert Duvall's price to reprise his role was too high, but the more accurate story is that Sonny and Fredo's deaths had pretty much put Tom Hagen on equal footing with Michael for the male lead, a fact which the corresponding actors' proposed salaries didn't reflect. Duvall later said, "If they paid Pacino twice what they paid me, that's fine, but not three or four times, which is what they did."

Also, we're pretty sure Duvall looked this awesome when he said that.

Add it all up, and The Godfather: Part III was released to a chorus of boos, to the point that it's a massive understatement to say it's the worst of the Godfather films.
The Awesome Movie We Missed Out On:
The Godfather: Part III was supposed to be such an ostentatious Greek tragedy that Coppola originally wanted to title the movie The Death of Michael Corleone.

Probably at the hands of Al Pacino.

The film would have chronicled the fall of one of the greatest cinematic characters of all time, centering on a civil war between Michael Corleone and the last moral fiber left in the family, good ol' Tom Hagen. How do we know this? Because this is precisely the spectacular ending that the first two films had been hinting toward all along. Coppola would later lament in his DVD commentary that Duvall's absence "was a profound loss ... to this movie," adding that the film seemed "incomplete" without the crucial inclusion of Tom Hagen.

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