The Universal Law of Gravitation (KNOWLEDGE PAGE

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 "acceleration 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.

Newton knew that the force which caused the apple's acceleration (gravity) must be dependent upon the mass of the apple. And since the force acting to cause the apple's downward acceleration also causes the earth's upward acceleration (Newton's third law), that force must also depend upon the mass of the earth. So for Newton, the force of gravity acting between the earth and any other object is directly proportional to the mass of the earth, directly proportional to the mass of the object, and inversely proportional to the square of the distance which separates the centers of the earth and the object. 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 numerical value of G is very small, which is basically the reason for the force of gravity to be the weakest force of the nature. For the value of "G", refer to your text book..

But Newton's law of universal gravitation extends gravity beyond earth. Newton's law of universal gravitation is about the universality of gravity. Newton's place in the Gravity Hall of Fame is not due to his discovery of gravity, but rather due to his discovery that gravitation is universal. ALL objects attract each other with a force of gravitational attraction. Gravity is universal. This force of gravitational attraction is directly dependent upon the masses of both objects and inversely proportional to the square of the distance which separates their centers.

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 as described by 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.

Using Equations as a Guide to Thinking

The inverse square law proposed by Newton suggests that the force of gravity acting between any two objects is inversely proportional to the square of the separation distance between the object's centers. Altering the separation distance (r) results in an alteration in the force of gravity acting between the objects. Since the two quantities are inversely proportional, an increase in one quantity results in a decrease in the value of the other quantity. That is, an increase in the separation distance causes a decrease in the force of gravity and a decrease in the separation distance causes an increase in the force of gravity.

Furthermore, the factor by which the force of gravity is changed is the square of the factor by which the separation distance is changed. So if the separation distance is doubled (increased by a factor of 2), then the force of gravity is decreased by a factor of four (2 raised to the second power). And if the separation distance (r) is tripled (increased by a factor of 3), then the force of gravity is decreased by a factor of nine (3 raised to the second power). Thinking of the force-distance relationship in this way involves using a mathematical relationship as a guide to thinking about how an alteration in one variable effects the other variable. Equations can be more than merely recipes for algebraic problem-solving; they can be "guides to thinking."

The proportionalities expressed by Newton's universal law of gravitation is represented graphically by the following illustration. Observe how the force of gravity is directly proportional to the product of the two masses and inversely proportional to the square of the distance of separation. In the above figure, the figure on the left hand side indicates the effect of "mass" if the diatnce between the two objects remains fixed at a given value "d". The right hand figure shows the effect of changing the distance while keeping the mass constant, and the last part of it shows the effect of changing both the distance and the mass.

Check your understanding of the inverse square law as a guide to thinking by answering the following questions below.

1 . Suppose that two objects attract each other with a force of 16 units (like 16 N or 16 lb). If the distance between the two objects is doubled, what is the new force of attraction between the two objects?

Answer: If the distance is increased by a factor of 2, then distance squared will increase by a factor of 4. Thus, the inverse square law implies that the force will be "1/4" of the original 16 units. Therefore, the force of gravity becomes 4 units.

2. Suppose the distance in question 1 is tripled. What happens to the forces between the two objects?

Answer: Again using inverse square law, we get distance squared to go up by a factor of 9. The force decreases by a factor of 9 and becomes 1.78 units.

3. If you wanted to make a profit in buying gold by weight at one altitude and selling it at another altitude for the same price per weight, should you buy or sell at the higher altitude location? What kind of scale must you use for this work?

Answer: To profit, buy at a high altitude and sell at a low altitude. Explanation is left to the student.

4. What would happen to your weight if the mass of the Earth somehow increased by 10%?

Answer: Your weight is nothing but force of gravity between the earth and you (as an object with a mass m). As shown in the above graph, changing one of the masses results in change in force of gravity. In this case, if the earth' mass goes up by 10%, then the force of gravity on you, or your weight, will increase by the same amount, that is 10%.

5. The planet Jupiter is more than 300 times as massive as Earth, so it might seem that a body on the surface of Jupiter would weigh 300 times as much as on Earth. But it so happens a body would scarcely weigh three times as much on the surface of Jupiter as it would on the surface of the Earth. Explain why this is so.

Answer: The effect of greater mass of Jupiter is partly off set by its larger radius which is about 10 times the radius of the earth. This means the object is times farther from the center of the Jupiter compared to the earth. Inverse of the distance brings a factor of 100 to the denominator and as a result, the force increases by a factor of 300 due to the mass, but decreases by a factor of 100 due to the distance squared. The net effect is that the force increases 3 times.

Planetary and Satellite Motion

After reading this section, it is recommended to check the following movie of Kepler's laws.

http://www.archive.org/details/kepler_full_cc (movie length is about 7 minutes)

Kepler's Three Laws

Newton's law of gravitation was preceded by three important discoveries about planetary motion by the German astronomer Johannes Kepler.

Kepler's three laws of planetary motion can be described as follows:

• Law of Orbits Kepler's First Law is illustrated in the image shown above. The Sun is not at the center of the ellipse, but is instead at one focus (generally there is nothing at the other focus of the ellipse). The planet then follows the ellipse in its orbit, which means that the Earth-Sun distance is constantly changing as the planet earth goes around its orbit. For purpose of illustration we have shown the orbit as rather eccentric; remember that the actual orbits are much less eccentric than this.

• Law of Areas Kepler's second law is illustrated in the preceding figure. The line joining the Sun and planet sweeps out equal areas in equal times, so the planet moves faster when it is nearer the Sun. Thus, a planet executes elliptical motion with constantly changing angular speed as it moves about its orbit. The point of nearest approach of the planet to the Sun is termed perihelion; the point of greatest separation is termed aphelion. Hence, by Kepler's second law, the planet moves fastest when it is near perihelion and slowest when it is near aphelion.

• Law of Periods In this equation P represents the period of revolution for a planet (in some other references the period is denoted as "T") and R represents the length of its semi-major axis. The subscripts "1" and "2" distinguish quantities for planet 1 and 2 respectively. The periods for the two planets are assumed to be in the same time units and the lengths of the semi-major axes for the two planets are assumed to be in the same distance units. Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit. Thus, we find that Mercury, the innermost planet, takes only 88 days to orbit the Sun but the outermost planet (Pluto) requires 248 years to do the same.

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 . 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. 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.

On June 21, the summer solstice, the top of the axis (the North Pole) is pointed directly toward the sun. Areas north of the equator experience longer days and shorter nights.
On December 21, the winter solstice, the top of earth's axis is pointed directly away from the sun. Areas north of the equator experience shorter days and longer nights.
Halfway in between the summer and winter solstices are the equinoxes. At these times the earth's axis is pointing neither toward nor away from the sun. On both equinoxes, all locations on earth receive exactly 12 hours of daylight and 12 hours of night.

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:

It is strongly recommended that you also visit the following web address for more details. (If the links do not take you to the referenced web page, please copy the link and paste it to a new page)

Ocean Tides

Lunar Tides

The tides at a given place in the Earth's oceans occur about an hour later each day. Since the Moon passes overhead about an hour later each day, it was long suspected that the Moon was associated with tides. Newton's Law of Gravitation provided a quantitative understanding of that association.

Differential Forces

Consider a water molecule in the ocean. It is attracted gravitationally by the Earth, but it also experiences a much smaller gravitational attraction from the Moon (much smaller because the Moon is much further away and much less massive than the Earth). But this gravitational attraction of the Moon is not limited to the water molecules; in fact, the Moon exerts a gravitational force on every object on and in the Earth. Tides occur because the Earth is a body of finite extent and these forces are not uniform: some parts of the Earth are closer to the Moon than other parts, and since the gravitational force drops off as the inverse square distance, those parts experience a larger gravitational tug from the Moon than parts that are further away. In this situation, which is illustrated schematically in the adjacent figure, we say that differential forces act on the body (the Earth in this example). The effect of differential forces on a body is to distort the body. The body of the Earth is rather rigid, so such distortion effects are small (but finite). However, the fluid in the Earth's oceans is much more easily deformed and this leads to significant tidal effects.

We may illustrate the basic idea with a simple model of a planet completely covered by an ocean of uniform depth, with negligible friction between the ocean and the underlying planet, as illustrated in the adjacent figure. The gravitational attraction of the Moon produces two tidal bulges on opposite sides of the Earth.

 Without getting too much into the technical details, there are two bulges because of the differential gravitational forces. The liquid at point A is closer to the Moon and experiences a larger gravitational force than the Earth at point B or the ocean at point C. Because it experiences a larger attraction, it is pulled away from the Earth, toward the Moon, thus producing the bulge on the right side. Loosely, we may think of the bulge on the left side as arising because the Earth is pulled away from the water on that side because the gravitational force exerted by the Moon at point B is larger than that exerted at point C. Then, as our idealized Earth rotates under these bulges, a given point on the surface will experience two high and two low tides for each rotation of the planet. Spring Tides and Neap Tides

Another complication of a realistic model is that not only the Moon, but other objects in the Solar System, influence the Earth's tides. For most their tidal forces are negligible on Earth, but the differential gravitational force of the Sun does influence our tides to some degree (the effect of the Sun on Earth tides is less than half that of the Moon). For example, particularly large tides are experienced in the Earth's oceans when the Sun and the Moon are lined up with the Earth at new and full phases of the Moon. These are called spring tides (the name is not associated with the season of Spring). The amount of enhancement in Earth's tides is about the same whether the Sun and Moon are lined up on opposite sides of the Earth (full Lunar phase) or on the same side (new Lunar phase). Conversely, when the Moon is at first quarter or last quarter phase (meaning that it is located at right angles to the Earth-Sun line), the Sun and Moon interfere with each other in producing tidal bulges and tides are generally weaker; these are called neap tides. The figure shown above illustrates spring and neap tides.

Moon Tides
How The Moon Affects Ocean Tides...

 The word "tides" is a generic term used to define the alternating rise and fall in sea level with respect to the land, produced by the gravitational attraction of the moon and the sun. To a much smaller extent, tides also occur in large lakes, the atmosphere, and within the solid crust of the earth, acted upon by these same gravitational forces of the moon and sun. What are Lunar Tides Tides are created because the Earth and the moon are attracted to each other, just like magnets are attracted to each other. The moon tries to pull at anything on the Earth to bring it closer. But, the Earth is able to hold onto everything except the water. Since the water is always moving, the Earth cannot hold onto it, and the moon is able to pull at it. Each day, there are two high tides and two low tides. The ocean is constantly moving from high tide to low tide, and then back to high tide. There is about 12 hours and 25 minutes between the two high tides. Tides are the periodic rise and falling of large bodies of water. Winds and currents move the surface water causing waves. The gravitational attraction of the moon causes the oceans to bulge out in the direction of the moon. Another bulge occurs on the opposite side, since the Earth is also being pulled toward the moon (and away from the water on the far side). Ocean levels fluctuate daily as the sun, moon and earth interact. As the moon travels around the earth and as they, together, travel around the sun, the combined gravitational forces cause the world's oceans to rise and fall. Since the earth is rotating while this is happening, two tides occur each day. What are the different types of Tides When the sun and moon are aligned, there are exceptionally strong gravitational forces, causing very high and very low tides which are called spring tides, though they have nothing to do with the season. When the sun and moon are not aligned, the gravitational forces cancel each other out, and the tides are not as dramatically high and low. These are called neap tides.
 Spring Tides When the moon is full or new, the gravitational pull of the moon and sun are combined. At these times, the high tides are very high and the low tides are very low. This is known as a spring high tide. Spring tides are especially strong tides (they do not have anything to do with the season Spring). They occur when the Earth, the Sun, and the Moon are in a line. The gravitational forces of the Moon and the Sun both contribute to the tides. Spring tides occur during the full moon and the new moon.  Neap Tides During the moon's quarter phases the sun and moon work at right angles, causing the bulges to cancel each other. The result is a smaller difference between high and low tides and is known as a neap tide. Neap tides are especially weak tides. They occur when the gravitational forces of the Moon and the Sun are perpendicular to one another (with respect to the Earth). Neap tides occur during quarter moons. The Proxigean Spring Tide is a rare, unusually high tide. This very high tide occurs when the moon is both unusually close to the Earth (at its closest perigee, called the proxigee) and in the New Moon phase (when the Moon is between the Sun and the Earth). The proxigean spring tide occurs at most once every 1.5 years.

 High Tide / Low Tide Examples A view of the tides at Halls Harbour on Nova Scotia's Bay of Fundy. This is a time lapse of the tidal rise and fall over a period of six and a half hours. During the next six hours of ebb the fishermen unload their boats on the dock. That's a high tide every 12 and 1/2 hours! There are two high tides every 25 hours.

A Few Facts About Lunar Tides

 The gravitational force of the moon is one ten-millionth that of earth, but when you combine other forces such as the earth's centrifugal force created by its spin, you get tides.   The sun's gravitational force on the earth is only 46 percent that of the moon. Making the moon the single most important factor for the creation of tides.   The sun's gravity also produces tides. But since the forces are smaller, as compared to the moon, the effects are greatly decreased.   Tides are not caused by the direct pull of the moon's gravity. The moon is pulling upwards on the water while the earth is pulling downward. Slight advantage to the moon and thus we have tides.   Whenever the Moon, Earth and Sun are aligned, the gravitational pull of the sun adds to that of the moon causing maximum tides.   Spring tides happen when the sun and moon are on the same side of the earth (New Moon) or when the sun and moon are on opposite sides of the earth (Full Moon).   When the Moon is at first quarter or last quarter phase (meaning that it is located at right angles to the Earth-Sun line), the Sun and Moon interfere with each other in producing tidal bulges and tides are generally weaker; these are called neap tides.   Spring tides and neap tide levels are about 20% higher or lower than average.   Offshore, in the deep ocean, the difference in tides is usually less than 1.6 feet   The surf grows when it approaches a beach, and the tide increases. In bays and estuaries, this effect is amplified. (In the Bay of Fundy, tides have a range of 44.6 ft.)   The highest tides in the world are at the Bay of Fundy in Nova Scotia, Canada.   Because the earth rotates on its axis the moon completes one orbit in our sky every 25 hours (Not to be confused with moon's 27 day orbit around the earth), we get two tidal peaks as well as two tidal troughs. These events are separated by about 12 hours.   Since the moon moves around the Earth, it is not always in the same place at the same time each day. So, each day, the times for high and low tides change by 50 minutes.   The type of gravitational force that causes tides is know as "Tractive" force.

 FAQs About Lunar Tides From - "The Astronomy Cafe" Why are there no ocean tides at the equator? "Tides are a very complex phenomenon. For any particular location, their height and fluctuation in time depends to varying degrees on the location of the Sun and the Moon, and to the details of the shape of the beach, coastline, coastline depth and prevailing ocean currents. The tidal bulge of the Moon follows along the path on the earth's surface which intersects with the orbital plane of the Moon. This plane is tilted about 23 degrees with respect to the equatorial plane of the earth. The result is that near the equator, the difference between high tide and low tide is actually rather small, compared to other latitudes. To see this, draw a circle inscribed in an ellipse, with the major axis of the ellipse rotated by 23 degrees with respect to the circle's horizontal diameter. Now measure the height of the elliptical contour just above the 'equator' of the circle. You will see that it is quite small compared to other positions on earth, particularly at latitudes of 23 degrees or so. Even larger differences can occur depending on the shape of a bay or inlet or continental shelf." - Dr. Odenwald's ASK THE ASTRONOMER Why are ocean tides so different everywhere? "Because they depend on many factors including the geometry of your local coastline, and exactly where the Sun and Moon are located. Also, like the surface of a vibrating drum, the world oceans have vibratory modes that get stimulated in changing ways from minute to minute. Finally, there are storms at sea and elsewhere which move large quantities of water. Detailed forecasts are available for high and low tides in all sea ports." - Dr. Odenwald's ASK THE ASTRONOMER  Why aren't the Atlantic and Pacific coast tides the same? "The nature of tides on the Earth's oceans is very complex. The oceans are, of course, being periodically 'forced' by a number of tidal sources including the Moon and the Sun, but this forcing has a number of different periods and harmonics. The two dominant periods are sue to the Sun and Moon, these are referred to as the S1 and M2 'modes' which have roughly 12 hour periods because they raise TWO water tides on the ocean diametrically opposite each other. But, for a variety of reasons, any given port will not have two high and two low tides each day; also called 'semi-diurnal tides'. A careful monitoring of the tides at any port for several years will show that in addition to the major modes, there are as many as 300 minor or 'harmonic' modes as well. The World Ocean is a complex dynamical system. The natural velocity of a water disturbance depends on the depth and salinity of the water at each point it passes. When bodies of land circumscribe bodies of water, they produce a collection of resonating systems that favor water oscillations with certain frequencies over others. From among the 300+ harmonics that can be measured, every port and coastal location has its own unique signature depending on its latitude, longitude, water depth and salinity. The result is that the 'two high two low' tide rule can be strongly modified so that the time between successive high tides can be greater than or less that 12 hours in many cases. The result is that for some locations, there can be days when only one high tide occurs. Looking at the Atlantic and Pacific Coast tide tables for 1995, the data for the various 'Standard Ports' showed that virtually all days had two high tides and two low tides in San Diego, San Francisco, New York and Charleston. There were, however a few days every few months when only a single high tide occurred." - Dr. Odenwald's ASK THE ASTRONOMER  What is a Proxigean Spring Tide? "The Moon follows an elliptical path around the Earth which has a perigee distance of 356,400 kilometers, which is about 92.7 percent of its mean distance. Because tidal forces vary as the third power of distance, this little 8 percent change translates into 25 percent increase in the tide- producing ability of the Moon upon the Earth. If the lunar perigee occurs when the Moon is between the Sun and the Earth, it produces unusually high Spring high tides. When it occurs on the opposite side from the Earth that where the Sun is located ( during full moon) it produces unusually low, Neap Tides. The High, High Tide is called the Proxigean Spring Tide and it occurs not more than once every 1.5 years. Some occurrences are more favorable that others. A very interesting book "Tidal Dynamics" by Fergus J. Wood, published in 1986 by Reidel Publishing Company, talks at great length about these tides, and their environmental consequences. Because of the gravitational nature of the interaction between the Earth, the Moon, and the water on the Earth, there is a curious amplification event called 'evection' that occurs when the Moon is at its closest 'perigee' distance called its 'proxigee'. The Moon draws even closer to the Earth than its ordinary perigee distance. Because of the complex dynamics of the Earth's oceans, their inertia, friction with the ocean floor, internal viscosity and the distribution of the continents, the maximum tides do not always coincide with the optimal times of proxigee. Still, these tides can produce enormous damage when all factors come together optimally. There are many recorded instances of unusually high storm or coastal flooding during the proxigean times. On January 9, 1974 the Los Angeles Times reported 'Giant Waves Pound Southland Coast".  During the last 400 years, there have been 39 instances or 'Extreme Proxigean Spring Tides' where the tide-producing severity has been near the theoretical maximum. The last one of these was on March 7 1995 at 22:00 hours Greenwich Civil Time during a lunar Full Moon. There were, in fact cases of extreme tidal flooding recorded during these particular spring tides which occur once every 31 years." - Dr. Odenwald's ASK THE ASTRONOMER  If the Moon were to escape, what would happen to the Earth's oceans? "What happens is that the lunar water tides on the Earth go away, but the solar water tides still occur, but with about 1/3 or so the amplitude. There are still daily high and low tides, but they would be noticeably smaller. There would be no 'Spring' or 'Neap' tides, however."- Dr. Odenwald's ASK THE ASTRONOMER  Why does the Moon produce TWO water tides on the Earth and not just one? "It is intuitively easy to understand why the gravitational pull of the Moon should produce a water tide on the Earth in the part of the ocean closest to the moon along the line connecting the center of the Moon with the center of the Earth. But in fact not one but TWO water tides are produced under which the Earth rotates every day to produce about two high tides and two low tides every day. How come? It is not the gravitational force that is doing it, but the change in the gravitational force across the body of the Earth. If you were to plot the pattern of the Moon's 'tidal' gravitational force added to the Earth's own gravitational force, at the Earth's surface, you would be able to resolve the force vectors at different latitudes and longitudes into a radial component directed towards the Earth's center, and a component tangential to the Earth's surface. On the side nearest the moon, the 'differential' gravitational force is directed toward the Moon showing that for particles on the Earth's surface, they are being tugged slightly towards the Moon because the force of the Moon is slightly stronger at the Earth's surface than at the Earth's center which is an additional 6300 kilometers from the Moon. On the far side of the Earth, the Moon is tugging on the center of the Earth slightly stronger than it is on the far surface, so the resultant force vector is directed away from the Earth's center. The net result of this is that the Earth gets deformed into a slightly squashed, ellipsoidal shape due to these tidal forces. This happens because if we resolve the tidal forces at each point on the Earth into a local vertical and horizontal component, the horizontal components are not zero, and are directed towards the two points along the line connecting the Earth and the Moon's centers. These horizontal forces cause rock and water to feel a gravitational force which results in the flow of rock and water into the 'tidal bulges'. There will be exactly two of these bulges. At exactly the positions of the tidal bulges where the Moon is at the zenith and at the nadir positions, there are no horizontal tidal forces and the flow stops. The water gets piled up, and the only effect is to slightly lower the weight of the water along the vertical direction. Another way of thinking about this is that the gravitational force of the Moon causes the Earth to accelerate slightly towards the Moon causing the water to get pulled towards the Moon faster than the solid rock on the side nearest the Moon. On the far side, the solid Earth 'leaves behind' some of the water which is not as strongly accelerated towards the Moon as the Earth is. This produces the bulge on the 'back side' of the Earth."- Dr. Odenwald's ASK THE ASTRONOMER  What Causes Tides? "There are several kinds of tides. The ones that break upon a beach every 10 seconds to a minute are caused by sea level disturbances out in the ocean produced by such things as storms. Also, the various circulation currents of sea water can have velocity components directed towards the land which will bring water up onto the beach. As this water travels towards the beach from deep water to shallow water, its amplitude will increase until it finally 'breaks' as a full-fledged breaker, suitable for surfing etc. Now, underlying this minute to minute activity is a slower water wave which causes an alternating pattern of high-tide, low-tide, high-tide, low-tide in most places on the Earth that are directly on the ocean. This roughly 6 hour cycle is caused by the gravitational tugging of the Moon upon the Earth. This 'tidal' pull causes the shape of the solid Earth to be not perfectly round by something like a few dozen yards over its entire 27,000 mile circumference. The Earth gets distorted a small bit, but because it is solid rock its a small effect. The water in the oceans, however, gets distorted into a roughly ellipsoidal ( football-like) shape with a much larger amplitude. The orientation of this shape changes from minute to minute as the Moon orbits the Earth, which is why the high and low tide times change all the time. The Moon causes these tides by deforming the oceans, and as the Earth rotates under this ocean bulge, it causes a high tide to propagate onto beaches. Because there are two bulges, we get two high tides, and also two low tides each day. The Sun also causes tides on the Earth because even though it is so far away, it is very massive. These solar tides are about half as strong as the ones produced by the Moon, and they cause the so-called Spring tides and the Neap Tides. When the bulge of ocean water raised by the Moon is added the a similar tidal bulge raised by the Sun, you get a higher, high tide called the Spring Tide. When the solar low tide is added to the lunar low tide, you get the Neap Tide. There may be even weaker tides caused by the gravitational influences of the planets Mars and Venus, but they are probably lost in the daily noise of individual tides."- Dr. Odenwald's ASK THE ASTRONOMER  When the Earth, Moon and Sun are aligned for Spring Tides, are they highest at Full or New Moon? "Spring tides are about the same height whether at New or Full Moon, because the tidal bulge occurs on both sides of the Earth...the side toward the Moon ( or sun) and the side away from the Moon (or Sun). They will not be equally high because the distance between the Earth and Sun, and the Earth and Moon both vary and so will their tide producing effectiveness. The highest Spring tides occur when the Moon is at its closest to the Earth...the so-called Perigee Tide."- Dr. Odenwald's ASK THE ASTRONOMER Kepler’s laws of planetary motion, in astronomy and classical physics, laws describing the motions of the planets in the solar system. They were derived by the German astronomer Johannes Kepler, whose analysis of the observations of the 16th-century Danish astronomer Tycho Brahe enabled him to announce his first two laws in the year 1609 and a third law nearly a decade later, in 1618. Kepler himself never numbered these laws or specially distinguished them from his other discoveries. Kepler’s three laws of planetary motion can be stated as follows: (1) All planets move about the Sun in elliptical orbits, having the Sun as one of the foci. (2) A radius vector joining any planet to the Sun sweeps out equal areas in equal lengths of time. (3) The squares of the sidereal periods (of revolution) of the planets are directly proportional to the cubes of their mean distances from the Sun. Knowledge of these laws, especially the second (the law of areas), proved crucial to Sir Isaac Newton in 1684–85, when he formulated his famous law of gravitation between Earth and the Moon and between the Sun and the planets, postulated by him to have validity for all objects anywhere in the universe. Newton showed that the motion of bodies subject to central gravitational force need not always follow the elliptical orbits specified by the first law of Kepler but can take paths defined by other, open conic curves; the motion can be in parabolic or hyperbolic orbits, depending on the total energy of the body. Thus, an object of sufficient energy—e.g., a comet—can enter the solar system and leave again without returning. From Kepler’s second law, it may be observed further that the angular momentum of any planet about an axis through the Sun and perpendicular to the orbital plane is also unchanging. The usefulness of Kepler’s laws extends to the motions of natural and artificial satellites as well as to unpowered spacecraft in orbit in stellar systems or near planets. As formulated by Kepler, the laws do not, of course, take into account the gravitational interactions (as perturbing effects) of the various planets on each other. The general problem of accurately predicting the motions of more than two bodies under their mutual attractions is quite complicated; analytical solutions of the three-body problem are unobtainable except for some special cases. It may be noted that Kepler’s laws apply not only to gravitational but also to all other inverse-square-law forces and, if due allowance is made for relativistic and quantum effects, to the electromagnetic forces within the atom.