| Sir Isaac Newton and the Unification of Physics & Astronomy |
| Sir Isaac Newton (1642-1727) was by many standards the most important figure in the development of modern science. Many would credit he and Einstein with being the most original thinkers in that development. |
| | Newtonian Gravitation & 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. |
| | 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.) |
| | 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. |
| | 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! |
| | 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. |
| | 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. |
| | 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. |
| | Another Biography of Newton |
| | The Universal Law of Gravitation |
| | 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. |
| | 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. |
| | 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. |
| | 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. |
| | 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. |
| | 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. |
| | 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. |
| | 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. |
| | 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 Second Law of Motion: |
| | 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 First Law of Motion: |
| | Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. |
| | Newton's Third Law of Motion: |
| | For every action there is an equal and opposite reaction. |
| | his 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!). |
| | The Great Synthesis of Newton |
| | Kepler had proposed three Laws of Planetary motion based on the systematics that he found in Brahe's data. These Laws were supposed to apply only to the motions of the planets; they said nothing about any other motion in the Universe. Further, they were purely empirical: they worked, but no one knew a fundamental reason WHY they should work. |
| | Newton changed all of that. First, he demonstrated that the motion of objects on the Earth could be described by three new Laws of motion, and then he went on to show that Kepler's three Laws of Planetary Motion were but special cases of Newton's three Laws if a force of a particular kind (what we now know to be the gravitational force) were postulated to exist between all objects in the Universe having mass. In fact, Newton went even further: he showed that Kepler's Laws of planetary motion were only approximately correct, and supplied the quantitative corrections that with careful observations proved to be valid. |
| | Vectors: Velocities, Accelerations, and Forces |
| | In order to understand the discoveries of Newton, we must have an understanding of three basic quantities: (1) velocity, (2) acceleration, and (3) force. In this section we define the first two, and in the next we shall introduce forces. These three quantities have a common feature: they are what mathematicians call vectors. |
| | Velocity and Acceleration |
| | Let us now give a precise definition of velocity and acceleration. They are vectors, so we must give a magnitude and a direction for them. The velocity v and the acceleration a are defined in the following illustration, |
| | This illustration also demonstrates graphically that velocity (and therefore acceleration) is a vector: the direction of the rock's velocity is certainly of critical interest to the person standing under the rock in the two illustrations! |
| | Examples of Vector Quantities |
| | However, consider a velocity. If we say that a car is going 70 km/hour, we have not completely specified its motion, because we have not specified the direction that it is going. Thus, velocity is an example of a vector quantity. A vector generally requires more than one number to specify it; in this example we could give the magnitude of the velocity (70km/hour), a compass heading to specify the direction (say 30 degrees from North), and an number giving the vertical angle with respect to the Earth's surface (zero degrees except in chase scenes from action movies!). The adjacent figure shows a typical coordinate system for specifying a vector in terms of a length r and two angles, theta and phi. |
| | Graphical Representation of Vectors |
| | Vectors are often distinguished from scalar quantities either by placing a small arrow over the quantity, or by writing the quantitity in a bold font. It is also common to indicate a vector by drawing an arrow whose length is proportional to the magnitude of the vector, and whose direction specifies the orientation of the vector. |
| | In the adjacent image we show graphical representations for three vectors. Vectors A and C have the same magnitude but different directions. Vector B has the same orientation as vector A, but has a magnitude that is twice as large. Each of these represents a different vector, because for two vectors to be equivalent they must have both the same magnitudes and the same orientations. |
| | Examples of Scalar Quantities |
| | Vectors are quantities that require not only a magnitude, but a direction to specify them completely. Let us illustrate by first citing some examples of quantities that are not vectors. The number of gallons of gasoline in the fuel tank of your car is an example of a quantitity that can be specified by a single number---it makes no sense to talk about a "direction" associated with the amount of gasoline in a tank. Such quantities, which can be specified by giving a single number (in appropriate units), are called scalars. Other examples of scalar quantities include the temperature, your weight, or the population of a country; these are scalars because they are completely defined by a single number (with appropriate units). |
| | Uniform Circular Motion is Accelerated Motion |
| | Notice that velocity, which is a vector, is changed if either its magnitude or its direction is changed. Thus, acceleration occurs when either the magnitude or direction of the velocity (or both) are altered. |
| | In particular, notice from the adjacent image that circular motion (even at uniform angular velocity) implies a continual acceleration, because the direction of the velocity (indicated by the direction of the arrow) is continuously changing, even if its magnitude (indicated by the length of the arrow) is constant. This point, that motion on a curved path is accelerated motion, will prove crucial to our subsequent understanding of motion in gravitational fields |
| | How Many Accelerators Does Your Car Have? |
| | Be aware that in popular speech acceleration is assumed to be an increase in the magnitude of the velocity. As we have just seen, acceleration also occurs when the direction of the velocity is changed, even if the magnitude is constant; furthermore, in physics a decrease in the velocity is just as much an acceleration as an increase. Thus, your car actually has at least 3 accelerators: (1) the foot pedal called the "accelerator", that changes the magnitude of the velocity, (2) the brake, which also changes the magnitude of the velocity, and (3) the steering wheel, which changes the direction of the velocity! |
| | The Accomplishments of Newton |
| | Newton's accomplishments were of astonishingly broad scope. For example, as a sidelight to his fundamental contributions in physics and astronomy, he (in parallel with Liebnitz) invented the mathematical discipline of calculus, so if you have to take both physics and calculus courses, you have Newton to blame! No survey course such as this one can possibly do justice to what Newton accomplished. The poet Alexander Pope was moved to pen the lines |
| | and a study of Newton's discoveries suggests that Pope was indulging only slightly in hyperbole. We shall concentrate on three developments of most direct relevance to our discussion: (1) Newton's Three Laws of Motion, (2) the Theory of Universal Gravitation, and (3) the demonstration that Kepler's Laws follow from the Law of Gravitation. |
| | 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 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. |
| | 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. |
| | 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.) |
| | 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. |
| | 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 |
| | 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. |
| | The Sun is so massive compared with every other object in the Solar System, |
| | 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.) |
| | 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. |
| | 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. |
| | 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. |
| |