## jueves, 25 de octubre de 2012

﻿  Sir Isaac Newton: The
Universal Law of Gravitation        twitter:         @germza

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.

Newtonian Gravitation and the
Laws of Kepler

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.

Gravitational Perturbations
and the Prediction of New Planets

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.

Albert Einstein and
the Theory of Relativity

 Albert Einstein1879-1955
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 Online

Celestial Coordinate
System

It is useful to impose on the celestial sphere a coordinate system that is analogous to the latitude-longitude system employed for the surface of the Earth. For a more extensive discussion, see Astronomy without a Telescope.

## Right Ascension and Declination

This coordinate system is illustrated in the following figure (for which you should imagine the earth to be a point at the center of the sphere).
 The celestial coordinate system

In the celestial coordinate system the North and South Celestial Poles are determined by projecting the rotation axis of the Earth to intersect the celestial sphere, which in turn defines a Celestial Equator. The celestial equivalent of latitude is called declination and is measured in degrees North (positive numbers) or South (negative numbers) of the Celestial Equator. The celestial equivalent of longitude is called right ascension. Right ascension can be measured in degrees, but for historical reasons it is more common to measure it in time (hours, minutes, seconds): the sky turns 360 degrees in 24 hours and therefore it must turn 15 degrees every hour; thus, 1 hour of right ascension is equivalent to 15 degrees of (apparent) sky rotation.

## Equinoxes and Solstices

The zero point for celestial longitude (that is, for right ascension) is the Vernal Equinox, which is that intersection of the ecliptic and the celestial equator near where the Sun is located in the Northern Hemisphere Spring. The other intersection of the Celestial Equator and the Ecliptic is termed the Autumnal Equinox. When the Sun is at one of the equinoxes the lengths of day and night are equivalent (equinox derives from a root meaning "equal night"). The time of the Vernal Equinox is typically about March 21 and of the Autumnal Equinox about September 22.
The point on the ecliptic where the Sun is most north of the celestial equator is termed the Summer Solstice and the point where it is most south of the celestial equator is termed the Winter Solstice. In the Northern Hemisphere the hours of daylight are longest when the Sun is near the Summer Solstice (around June 22) and shortest when the Sun is near the Winter Solstice (around December 22). The opposite is true in the Southern Hemisphere. The term solstice derives from a root that means to "stand still"; at the solstices the Sun reaches its most northern or most southern position in the sky and begins to move back toward the celestial equator. Thus, it "stands still" with respect to its apparent North-South drift on the celestial sphere at that time. Traditionally, Northern Hemisphere Spring and Fall begin at the times of the corresponding equinoxes, while Northern Hemisphere Winter and Summer begin at the corresponding solstices. In the Southern Hemisphere, the seasons are reversed (e.g., Southern Hemisphere Spring begins at the time of the Autumnal Equinox).

## Coordinates on the Celestial Sphere

The right ascension (R.A.) and declination (dec) of an object on the celestial sphere specify its position uniquely, just as the latitude and longitude of an object on the Earth's surface define a unique location. Thus, for example, the star Sirius has celestial coordinates 6 hr 45 min R.A. and -16 degrees 43 minutes declination, as illustrated in the following figure.
 Right Ascension and Declination for Sirius

This tells us that when the vernal equinox is on our celestial meridian, it will be 6 hours and 45 minutes before Sirius crosses our celestial meridian, and also that Sirius is a little more than 16 degrees South of the Celestial Equator.

## Keeping your Perspective

Do not become confused because the perspectives in the celestial sphere diagram and the sky segment diagram containing Sirius are different. In the celestial sphere diagram one is imagining an outside view of the celestial sphere (from a vantage point beyond the most distant stars that we see with the naked eye). In the diagram showing the position of Sirius in the sky the view is instead the actual sky as viewed from the Earth (that is, from the center of the sphere in the first diagram).
Thus, the directions get reversed: moving to the right from the vernal equinox in the first diagram will look like moving to the left as viewed from its center, which is the perspective of the second diagram (that is, the actual view of the sky from Earth). That direction, by convention, is chosen to be the positive direction for right ascension.

The Seasons

There is a popular misconception that the seasons on the Earth are caused by varying distances of the Earth from the Sun on its elliptical orbit. This is not correct. One way to see that this reasoning may be in error is to note that the seasons are out of phase in the Northern and Southern hemispheres: when it is Summer in the North it is Winter in the South.

## Seasons in the Northern Hemisphere

The primary cause of the seasons is the 23.5 degree of the Earth's rotation axis with respect to the plane of the ecliptic, as illustrated in the adjacent image (Source). This means that as the Earth goes around its orbit the Northern hemisphere is at various times oriented more toward and more away from the Sun, and likewise for the Southern hemisphere, as illustrated in the following figure.

 The Seasons in the Northern Hemisphere

Thus, we experience Summer in the Northern Hemisphere when the Earth is on that part of its orbit where the N. Hemisphere is oriented more toward the Sun and therefore the Sun rises higher in the sky and is above the horizon longer, and the rays of the Sun strike the ground more directly. Likewise, in the N. Hemisphere Winter the hemisphere is oriented away from the Sun, the Sun only rises low in the sky, is above the horizon for a shorter period, and the rays of the Sun strike the ground more obliquely. In fact, as the diagram indicates, the Earth is actually closer to the Sun in the N. Hemisphere Winter than in the Summer (as usual, we greatly exaggerate the eccentricity of the elliptical orbit in this diagram). The Earth is at its closest approach to the Sun (perihelion) on about January 4 of each year, which is the dead of the N. Hemisphere Winter. (The time for perihelion, aphelion, and the solstices for any year 1992-2000 is available in this compilation.) For a more extensive introduction to how variations in the amount of solar energy reaching the Earth's surface influence climate, see this discussion of solar databases for global change models.

## Another Fallacy to Avoid

Incidentally, one should be precise in terminology. A common student answer for the cause of the seasons is that "the Earth tips toward the Sun in the Summer, . . .". This conveys the impression that the Earth moves around its orbit and at certain times of the year the rotation axis suddenly tips one way or another and thus we have seasons. As the preceding diagram makes clear, the rotation axis of the Earth remains pointed in the same direction (except for small effects from precession) as it moves around its orbit. It is the relative location of the Sun with respect to this constant tilt angle that causes the seasons, not some elaborate square dance of the Earth bowing to its partner as it moves around its orbit!

## Southern Hemisphere Seasons

As is clear from the preceding diagram, the seasons in the Southern Hemisphere are determined from the same reasoning, except that they are out of phase with the N. Hemisphere seasons because when the N. Hemisphere is oriented toward the Sun the S. Hemisphere is oriented away, and vice versa:
 The Seasons in the Southern Hemisphere

## The Lag of the Seasons

The preceding reasoning for the causes of the seasons is idealized. In reality, we know that the seasons "lag": for example, the hottest temperatures in the Summer usually occur a month or so after the time of maximum insolation (the time when maximum solar energy is deposited during a day at a point on the surface of the Earth). This is because the Earth and its atmosphere store heat (the oceans are particularly effective heat sinks). Thus, a detailed description of the seasons is quite complicated since it must take into account complex local variations in the storage of solar energy. However, the basic reason for the seasons is simple, as described above.

## Simulating the Apparent Motion of the Sun

One can use the Starry Night program for Windows and the Macintosh to simulate the appearance of the sky at any time, from any chosen vantage point in the Solar System. Thus, by choosing different points on the surface of the Earth at different times of the year, this program can be used to show the motion of the Sun through the sky and illustrate clearly the preceding points about the causes for the seasons. Here is an extreme example:
In the N. Hemisphere Summer at latitudes above the Arctic Circle (23.5 degrees away from the N. Pole) the Sun stays above the horizon for the entire day (midnight sun). The adjacent image illustrates the midnight sun. This GOES-8 weather satellite visible light image is taken from a vantage point high above the western hemisphere, with the North at the top. Even though the local time for the longitude line under the satellite is near midnight, the Northernmost portion of the globe is illuminated by sunlight (the lighted portion actually extends below the arctic circle in this image because of sunlight scattering in the atmosphere). This simulation of the midnight sun was made using the Starry Night program with a "fisheye lens" perspective to show a wide (180 degree) region of the sky from a vantage point at the North Pole on July 4, 1996. As the movie illustrates, the Sun moves more or less parallel to the horizon and never goes below it during the course of a day at these latitudes at this time of the year. Conversely, in the N. Hemisphere Winter the Sun never comes above the horizon for the entire day at this latitude. This is an extreme example of the difference in insolation in Winter and Summer for the N. Hemisphere that is responsible for the seasons. The exact time of sunrise and sunset (and similar data for moonrise and moonset) may be calculated for any date and 22,000 named cities in the United States, or by specifying the latitude and longitude of any location worldwide, using this program. For locations in the United States, a table of corresponding information for an entire year (past, present, or future) may be calculated using this program

Precession of the
Earth's Rotation Axis

The Earth's rotation axis is not fixed in space. Like a rotating toy top, the direction of the rotation axis executes a slow precession with a period of 26,000 years (see following figure).

## Pole Stars are Transient

Thus, Polaris will not always be the Pole Star or North Star. The Earth's rotation axis happens to be pointing almost exactly at Polaris now, but in 13,000 years the precession of the rotation axis will mean that the bright star Vega in the constellation Lyra will be approximately at the North Celestial Pole, while in 26,000 more years Polaris will once again be the Pole Star.

## Precession of the Equinoxes

Since the rotation axis is precessing in space, the orientation of the Celestial Equator also precesses with the same period. This means that the position of the equinoxes is changing slowly with respect to the background stars. This precession of the equinoxes means that the right ascension and declination of objects changes very slowly over a 26,000 year period. This effect is negligibly small for casual observing, but is an important correction for precise observations.

## The Dawning of the Age of Aquarius (Almost)

Because of the precession of the equinoxes, the vernal equinox moves through all the constellations of the Zodiac over the 26,000 year precession period. Presently the vernal equinox is in the constellation Pisces and is slowly approaching Aquarius.
 The Vernal Equinox

This is the origin of the "Age of Aquarius" celebrated in the musical Hair: a period when according to astrological mysticism and related hokum there will be unusual harmony and understanding in the world. We could certainly use a dose of harmony and understanding in this old world; unfortunately, it is unlikely to come because of something as irrelevant as the position of the vernal equinox with respect to the constellations of the Zodiac.

Orbit and
Phases of the Moon

The orbit of the Moon is very nearly circular (eccentricity ~ 0.05) with a mean separation from the Earth of about 384,000 km, which is about 60 Earth radii. The plane of the orbit is tilted about 5 degrees with respect to the ecliptic plane.

## Revolution in Orbit

The Moon appears to move completely around the celestial sphere once in about 27.3 days as observed from the Earth. This is called a sidereal month, and reflects the corresponding orbital period of 27.3 days The moon takes 29.5 days to return to the same point on the celestial sphere as referenced to the Sun because of the motion of the Earth around the Sun; this is called a synodic month (Lunar phases as observed from the Earth are correlated with the synodic month). There are effects that cause small fluctuations around this value that we will not discuss. Since the Moon must move Eastward among the constellations enough to go completely around the sky (360 degrees) in 27.3 days, it must move Eastward by 13.2 degrees each day (in contrast, remember that the Sun only appears to move Eastward by about 1 degree per day). Thus, with respect to the background constellations the Moon will be about 13.2 degrees further East each day. Since the celestial sphere appears to turn 1 degree about every 4 minutes, the Moon crosses our celestial meridian about 13.2 x 4 = 52.8 minutes later each day.

## Lunar Phases

The Moon appears to go through a complete set of phases as viewed from the Earth because of its motion around the Earth, as illustrated in the following figure.
 Phases of the Moon

In this figure, the various positions of the Moon on its orbit are shown (the motion of the Moon on its orbit is assumed to be counter-clockwise). The outer set of figures shows the corresponding phase as viewed from Earth, and the common names for the phases.

Here is an animation of actual lunar phases, and here is a Java applet illustrating the orbit of the moon around the Earth and the corresponding phases of the Moon as viewed from Earth. Notice that you can set this applet to a top view, an Earth view, or both on a split screen, and that you can start and stop the animation with a button. Also, note that in this applet the position of the Sun is shown to the left, whereas in the above figure the view is such that the position of the Sun is to the right.

## Perigee and Apogee

The largest separation between the Earth and Moon on its orbit is called apogee and the smallest separation is called perigee. Here is an online Lunar Perigee and Apogee Calculator that will allow you to determine the date, time, and distance of lunar perigees and apogees for a given year (Credit: John Walker).

## Rotational Period and Tidal Locking

The Moon has a rotational period of 27.3 days that (except for small fluctuations) exactly coincides with its (sidereal) period for revolution about the Earth. As we will see later, this is no coincidence; it is a consequence of tidal coupling between the Earth and Moon. Because of this tidal locking of the periods for revolution and rotation, the Moon always keeps essentially the same face turned toward the Earth (small fluctuation mean that over a period of time we can actually see about 55% of the Lunar surface from the Earth).

Solar
Eclipses

One consequence of the Moon's orbit about the Earth is that the Moon can shadow the Sun's light as viewed from the Earth, or the Moon can pass through the shadow cast by the Earth. The former is called a solar eclipse and the later is called a lunar eclipse. The small tilt of the Moon's orbit with respect to the plane of the ecliptic and the small eccentricity of the lunar orbit make such eclipses much less common than they would be otherwise, but partial or total eclipses are actually rather frequent.

## Frequency of Eclipses

For example there will be 18 solar eclipses from 1996-2020 for which the eclipse will be total on some part of the Earth's surface. The common perception that eclipses are infrequent is because the observation of a total eclipse from a given point on the surface of the Earth is not a common occurrence. For example, it will be two decades before the next total solar eclipse visible in North America occurs.
The next total solar eclipse will be on August 11, 1999, with the path of totality crossing the North Atlantic, Europe, the Middle East, and India. In this section we consider solar eclipses and in the next we discuss lunar eclipses.

## Geometry of Solar Eclipses

The geometry associated with solar eclipses is illustrated in the following figure (which, like most figures in this and the next section, is illustrative and not to scale).
 Geometry of solar eclipses

The shadow cast by the Moon can be divided by geometry into the completely shadowed umbra and the partially shadowed penumbra.

## Types of Solar Eclipses

The preceding figure allows three general classes of solar eclipses (as observed from any particular point on the Earth) to be defined:
1. Total Solar Eclipses occur when the umbra of the Moon's shadow touches a region on the surface of the Earth.
2. Partial Solar Eclipses occur when the penumbra of the Moon's shadow passes over a region on the Earth's surface.
3. Annular Solar Eclipses occur when a region on the Earth's surface is in line with the umbra, but the distances are such that the tip of the umbra does not reach the Earth's surface.
As illustrated in the figure, in a total eclipse the surface of the Sun is completely blocked by the Moon, in a partial eclipse it is only partially blocked, and in an annular eclipse the eclipse is partial, but such that the apparent diameter of the Moon can be seen completely against the (larger) apparent diameter of the Sun.
A given solar eclipse may be all three of the above for different observers. For example, in the path of totality (the track of the umbra on the Earth's surface) the eclipse will be total, in a band on either side of the path of totality the shadow cast by the penumbra leads to a partial eclipse, and in some eclipses the path of totality extends into a path associated with an annular eclipse because for that part of the path the umbra does not reach the Earth's surface.

## Total Solar Eclipses

A total solar eclipse requires the umbra of the Moon's shadow to touch the surface of the Earth. Because of the relative sizes of the Moon and Sun and their relative distances from Earth, the path of totality is usually very narrow (hundreds of kilometers across). The following figure illustrates the path of totality produced by the umbra of the Moon's shadow. (We do not show the penumbra, which will produce a partial eclipse in a much larger region on either side of the path of totality; we also illustrate in this figure the umbra of the Earth's shadow, which will be responsible for total lunar eclipses to be discussed in the next section.)
 Solar eclipse (not to scale)

As noted above, the images that we show in discussing eclipses are illustrative but not drawn to scale. The true relative sizes of the Sun and Earth and Moon, and their distances, are very different than in the above figure.

## Animations of Solar Eclipses

Here are three animations that illustrate observations in a solar eclipse. The first demonstrates generally the case of a total solar eclipse; the next two are simulated views of two recent solar eclipses from unusual vantage points, one from the Moon and one from the Sun (these last two were constructed using the program Starry Night).
In these last two simulations, the shadow cast on the Earth is the penumbra, which can cover a region thousands of kilometers in diameter. If the eclipse is total, the path of totality traced out by the umbra is much narrower.

## Appearance of a Total Solar Eclipse

If you are in the path of totality the eclipse begins with a partial phase in which the Moon gradually covers more and more of the Sun. This typically lasts for about an hour until the Moon completely covers the Sun and the total eclipse begins. The duration of totality can be as short as a few seconds, or as long as about 8 minutes, depending on the details.
As totality approaches the sky becomes dark and a twilight that can only be described as eerie begins to descend. Just before totality waves of shadow rushing rapidly from horizon to horizon may be visible. In the final instants before totality light shining through valleys in the Moon's surface gives the impression of beads on the periphery of the Moon (a phenomenon called Bailey's Beads). The last flash of light from the surface of the Sun as it disappears from view behind the Moon gives the appearance of a diamond ring and is called, appropriately, the diamond ring effect (image at right). As totality begins , the solar corona (extended outer atmosphere of the Sun) blazes into view. The corona is a million times fainter than the surface of the Sun; thus only when the eclipse is total can it be seen; if even a tiny fraction of the solar surface is still visible it drowns out the light of the corona. At this point the sky is sufficiently dark that planets and brighter stars are visible, and if the Sun is active one can typically see solar prominences and flares around the limb of the Moon, even without a telescope (see image at left). The period of totality ends when the motion of the Moon begins to uncover the surface of the Sun, and the eclipse proceeds through partial phases for approximately an hour until the Sun is once again completely uncovered. Here is a movie of the 1994 total solar eclipse (3.1 MB MPEG; Source; here is a QuickTime version, but note that it is 15 MB in length). A partial solar eclipse is interesting; a total solar eclipse is awe-inspiring in the literal meaning of the phrase. If you have an opportunity to observe a total solar eclipse, don't miss it! It is an experience that you will never forget.

## Patterns of Eclipses

Because solar eclipses are the result of periodic motion of the Moon about the Earth, there are regularities in the timing of eclipses that give cycles of related eclipses. These cycles were known and used to predict eclipses long before there was a detailed scientific understanding of what causes eclipses. For example, the ancient Babylonians understood one such set of cycles called the Saros, and were able to predict eclipses based on this knowledge. Here is a link to a discussion of such cycles and regularities in eclipse patterns.

## Solar Eclipse Resources

Here are some resources for those interested in keeping track of eclipses.

# Masa crítica

Una esfera (simulada) de plutonio rodeada por bloques de un reflector de neutrones carburo de tungsteno. Una re-creación de un accidente de criticidad ocurrido en 1945 para medir la radiación producida cuando se agregaba un bloque adicional de reflector, tornando a la masa supercrítica.
En física, la masa crítica es la cantidad mínima de material necesaria para que se mantenga una reacción nuclear en cadena. La masa crítica de una sustancia fisible depende de sus propiedades físicas (en particular su densidad) y nucleares (su enriquecimiento y sección eficaz de fisión), su geometría (su forma) y su pureza, además de si está rodeada o no por un reflector de neutrones. Al rodear a un material fisible por un reflector de neutrones la masa crítica resulta menor. En el caso de una esfera rodeada por un reflector de neutrones, la masa crítica es de unos cincuenta y dos kilogramos para el uranio 235 y de diez kilogramos para el plutonio 239.
"Crítico" se refiere a un estado de equilibrio dinámico en la reacción de fisión en cadena; en él no existe aumento de la potencia, temperatura y densidad de neutrones en el tiempo. "Subcrítico" se refiere a la incapacidad de mantener o sostener en el tiempo una reacción nuclear en cadena; al introducir una cierta cantidad de neutrones en un conjunto subcrítico, la población de neutrones disminuirá a lo largo del tiempo (por fenómenos de absorción en el material o por fuga). "Supercrítico" se refiere a un sistema en el que la cantidad de procesos de fisión por unidad de tiempo aumenta hasta el punto en que algún mecanismo de realimentación intrínseco hace que el reactor alcance un punto de equilibrio dinámico (se ponga crítico) a una mayor temperatura o potencia o se destruya (en cuyo caso se desarma el conjunto crítico).
Es posible que un conjunto alcance el estado de crítico a potencias muy próximas a cero. Si fuera posible hacer un experimento en el que se agregue una cantidad exacta de material fisible a una masa levemente subcrítica, se podría crear un conjunto con una masa exactamente crítica, y en ese caso la reacción en cadena de fisión mantendría exactamente una generación de neutrones (ya que el consumo del combustible producido por el mismo proceso de fisión tornaría al conjunto nuevamente subcrítico).
Si fuera posible hacer un experimento en el que se agregara una cantidad exacta de material fisible a una masa levemente subcrítica, se estaría creando un conjunto con una masa apenas supercrítica, y en ese caso la temperatura del conjunto aumentaría hasta un valor máximo para luego de un cierto tiempo disminuir nuevamente a la temperatura del entorno (ya que el consumo del material fisible en la reacción en cadena haría que el conjunto se volviera subcrítico).

## [editar]Masa crítica de una esfera desnuda

Superior: Una esfera de material fisible es demasiado pequeña para permitir que la reacción en cadena se automantenga, debido a que los neutrones generados por la fisión pueden escapar fácilmente del sistema . Centro: Al incrementar la masa de la esfera hasta alcanzar la masa crítica, la reacción nuclear se automantiene. Inferior: Al recubrir la esfera original por un reflector de neutrones, se aumenta la eficiencia de la reacción y se permite que el sistema posea una reacción autosostenida.
La esfera es la forma que posee menor masa crítica. Es posible reducir la masa crítica si se rodea la esfera con un material reflector de neutrones, o algún otro material.
En el caso de una esfera desnuda (sin reflector de neutrones) la masa crítica es de más de 50 kg para el uranio-235 y 10 kg para el plutonio-239.
La siguiente tabla presenta la masa crítica de esferas desnudas de algunos isótopos con vidas medias de más de 100 años.
IsótopoMasa CríticaReferencia
protactinio-231750±180 kg
uranio-23315 kg[1]
uranio-23552 kg[2]
neptunio-2367 kg[3]
neptunio-23760 kg
plutonio-2389.04–10.07 kg
plutonio-23910 kg[4]
plutonio-24040 kg[5]
plutonio-242100 kg[6]
americio-24155–77 kg
americio-242m9–14 kg
americio-243180–280 kg
curio-2437.34–10 kg[7]
curio-244(13.5)–30 kg[8]
curio-2459.41–12.3 kg[9]
curio-24639–70.1 kg[10]
curio-2476.94–7.06 kg[11]
californio-2496 kg[12]
californio-2515 kg[13]

La masa crítica del uranio depende del grado en que este presente (enriquecido) en el isótopo uranio-235: para un enriquecimiento del 20 % de U-235 la masa crítica es de más de 400 kg; para el 15 % de U-235, la masa crítica excede los 600 kg.
La masa crítica es inversamente proporcional al cuadrado de la densidad: si por ejemplo la densidad se incrementa en un 1% la masa crítica se reducirá en un 2%, entonces el volumen será menor en un 3% y el diámetro se reducirá en 1%. La probabilidad de que un neutrón por cm recorrido colisione con un núcleo es proporcional a la densidad, o sea aumentaría en nuestro ejemplo en un 1% , lo cual compensa el hecho que la distancia recorrida antes de que el neutrón salga del sistema es un 1% menor. Esto es algo que debe tenerse en cuenta al realizar cálculos más precisos de las masas críticas de los isótopos del plutonio que los valores indicativos que se muestran en la tabla previa, porque el plutonio tiene un gran número de fases cristalinas cuyas densidades son sumamente distintas entre si.
Es importante notar que no todos los neutrones contribuyen a la reacción en cadena. Algunos escapan al sistema, y otros pueden sufrir capturas radioactivas.
Sea q la probabilidad que un dado neutrón origine una fisión en un núcleo. Considerando en forma simplificada sólo a los neutrones prontos o instantáneos, y si se llama ν al número de neutrones prontos generados en la fisión nuclear de un átomo. Por ejemplo, en el caso del uranio-235 ν = 2.7. Por lo tanto la criticidad tendrá lugar cuando νq = 1. La dependencia con la geometría, la masa y la densidad se manifiesta a través del factor q.
Si se considera la sección eficaz total de interacción $\sigma$ (usualmente medida en barn), entonces el camino libre medio de un neutrón pronto se expresa como $\ell^{-1} = n \sigma$ donde $n$ es la densidad de número atómico. La mayoría de las interacciones son choques con cambio de dirección y energía del neutrón, de forma tal que un neutrón recorrerá una trayectoria aleatoria hasta que escape del medio en el que se encuentra o produzca una reacción de fisión. En la medida en que los otros mecanismos de desaparición de neutrones no sean significativos, entonces el radio de la esfera de masa crítica se puede calcular en forma aproximada como el producto del camino libre medio $\ell$ y la raíz cuadrada de uno más la cantidad de eventos de colisión por evento de fisión (lo que llamamos $s$), dado que la distancia neta recorrida en un recorrido aleatorio es proporcional a la raíz cuadrada de la cantidad de pasos que se dan:
$R_c \simeq \ell \sqrt{s} \simeq \frac{\sqrt{s}}{n \sigma}$
Donde, es necesario recordar nuevamente, que esto es solo un estimación gruesa.
La criticidad se puede expresar en función de la masa total M, la masa nuclear m, la densidad ρ, y un factor f que incluye los efectos geométricos y de otro tipo como:
$1 = \frac{f \sigma}{m \sqrt{s}} \rho^{2/3} M^{1/3}$
En donde queda confirmado la dependencia de la masa crítica con la inversa del cuadrado de la densidad tal como se mencionó previamente.
Alternativamente, esto se puede expresar en forma más sucinta en función de la densidad superficial de núcleos Σ:
$1 = \frac{f' \sigma}{m \sqrt{s}} \Sigma$
Donde se reescribió el factor f como f' para tener en cuenta que los dos valores pueden diferir dependiendo de efectos geométricos y de como se defina Σ. Por ejemplo, para una esfera sólida desnuda de Pu-239 la criticidad se obtiene a 320 kg/m², independientemente de la densidad, y para U-235 a 550 kg/m². O sea que la criticidad depende de que un neutrón típico "vea" una cierta cantidad de núcleos alrededor de él de manera que la densidad superficial de núcleos este por encima de un cierto valor umbral.
Esto se aplica en las armas nucleares del tipo de implosión, donde una masa esférica de material físil que es significativamente menos que la masa crítica, se vuelve supercrítica mediante un rápido aumento de $\rho$ (y en consecuencia también de $\Sigma$), ver la próxima sección. En efecto, los programas de armas nucleares más sofisticados permiten fabricar dispositivos funcionales con menos material que el requerido en los dispositivos más primitivos.
Más allá de las matemáticas, este resultado puede ser explicado mediante una simple analogía física. Si se considera el humo que expele un motor diésel por el caño de escape, inicialmente el humo parece negro, pero luego gradualmente es posible ver a través de él sin inconvenientes. Esto no se debe a que se haya modificado la sección eficaz de scattering de todas las partículas de hollín, pero en cambio a que el hollín se ha dispersado en el aire. Si se considera un cubo transparente cuyo lado es $L$, lleno con hollín, entonces la profundidad óptica de este medio es inversamente proporcional al cuadrado de $L$, y por lo tanto proporcional a la densidad superficial de las partículas de hollín: es posible ver a través de este cubo imaginario solo con hacer mayor el cubo (sin variar la cantidad total de hollín contenido en él).
Varias incertezas contribuyen a la determinación de un valor preciso de las masas críticas, incluyendo (1) conocimiento preciso de las secciones eficaces, (2) cálculo de los efectos geométricos. Este último problema es el que dio un fuerte impulso al desarrollo del método Montecarlo en física computacional por Nicholas Metropolis y Stanislaw Ulam. En efecto, aún para una esfera homogénea sólida, el cálculo exacto no es para nada trivial. Finalmente es de notar que el cálculo podría ser realizado si se supusiera una aproximación de un medio continuo para el transporte de neutrones, de forma tal que el problema se reduce a un problema de difusión. Sin embargo, como las dimensiones típicas del problema no son significativamente más largas que el camino libre medio, dicha aproximación tiene muy poca utilidad práctica.
Finalmente, debe notarse que para ciertas geometrías ideales, la masa crítica desde un punto de vista formal podría ser infinita, y entonces se utilizan otros parámetros para describir la criticidad. Por ejemplo, si se considera el caso de una placa infinita de material fisionable. Para todo espesor finito, esto se corresponde con una masa infinita. Sin embargo, la criticidad es solo alcanzable cuando el espesor de la placa excede un valor crítico.

## [editar]Diseño de explosivos

Si dos pedazos de material subcrítico no se juntan lo suficientemente rápido, puede ocurrir una pre-detonación nuclear, en cuyo caso una explosión muy pequeña separará las partes.
Una bomba atómica debe almacenarse en una configuración subcrítica hasta el momento en que se la desee detonar. En el caso de una bomba de uranio, basta con mantener el combustible en forma de piezas separadas, la dimensión de cada una de ellas es menor que el tamaño crítico tanto porque sean muy pequeñas o porque sus formas previenen alcanzar la criticidad. Para producir la detonación, las partes de uranio se juntan rápidamente. En Little Boy, esto se realizó disparando una pequeña parte de uranio desde un cañón tipo arma de fuego hacia un agujero correspondiente ubicado en un pedazo mayor de uranio, un diseño conocido como bomba de fisión de tipo revolver.
También sería posible construir un explosivo de Pu-239 con una pureza teórica del 100%. Pero en la realidad esto no es práctico porque el Pu-239 "de calidad armamento" esta contaminado con pequeñas cantidades de Pu-240, el cual posee una fuerte tendencia hacia la fisión espontánea. Por esta razón, en un arma del tipo revolver se produciría una reacción nuclear antes de que las masas de plutonio estuvieran en la posición adecuada para producir una explosión de magnitud. Aún teniendo en cuenta la impureza en Pu-240, se podría en principio construir un arma tipo revolver. Sin embargo no sería un arma demasiado práctica, ya que debería ser muy larga para acelerar la masa de plutonio a muy altas velocidades para compensar los efectos mencionados anteriormente.
Por ello es que se recurre a otro método. El plutonio se coloca en forma de una esfera subcrítica (o con otra forma), la cual puede o no ser hueca. La detonación se produce haciendo explotar una carga de un explosivo convencional que rodea la esfera, aumentando así su densidad (y haciendo colapsar la cavidad interna si es que la hubiera) para producir una configuración que es supercrítica. A este método se lo llama arma de implosión.

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