IV) Gravity in the Universe: Key Ideas

In this chapter we will try to derive/justify the equations that fully describe gravity throughout the Universe. We will then show applications of these equations in the Universe at large. Our route from the simple F=mg governing gravity here on Earth to the Universal Law rests on three main arguments: gravity is a vector force, the Moon orbits under the influence of gravity and gravity acts between all objects.

A) Gravity Is a Vector Quantity

The discussion of gravity in the previous section has ignored the vector nature of forces. We can build bridges, fly airplanes, throw fastballs, just about anything it seems, with this understanding. Is this a complete description of gravity? Is there anything else to learn? One clue that the simple F=mg description of gravity is incomplete comes from a fact that we've known for a very long time: the Earth is round!

Think about our treatment so far. We mentioned that the force is downward directed, but we haven't gone into more detail. In fact we pretty much brushed aside the fact that forces are vector quantities. By vector quantity we mean that forces have both a magnitude and directional aspect. You can have a changing vector if the direction changes, even if the magnitude stays constant. If you think about it, this is a crucial point. At each location on the Earth's surface, gravity must be directed pretty much towards the center of the Earth. In any event, the force appears to be central. That is, it pulls objects towards the center and not in any sideways direction.

This suggests a provisional modification of the law of gravity along the lines of:

Where is just a vector that points towards the center of the Earth but has a length of one. Notice that we now have reformulated the force law with vectors. Of course we really should have done this before, but we were sweeping the idea of "down" under the rug. Now that the "down" issue is out in the open we must write the force of gravity as a vector.

This formulation already has interesting consequences. Imagine again standing on a tower on a flat earth with the original simple law of gravity. Throw a rock sideways out the window. Eventually the rock will fall to the ground. Now throw the rock harder. It will hit the ground farther from the tower. On a spherical Earth, however, the ground curves away under the falling rock. If the direction of the force doesn't change, then a rock thrown hard enough will miss the Earth entirely and drop off into space....On the other hand if the direction of the force changes, as suggested by the provisional law, then path of the rock will keep curving back towards the Earth. If the amount the rock's path curves is balanced by the curvature of the Earth, then the rock will never hit the ground; The secret to orbiting is to fall but miss the ground!

B) Gravity Makes the Moon Go Around: The Inverse Square Law

Let's see how this works in detail. We can put this is a somewhat more quantitative manner by considering centripetal acceleration. If no force acts on an object in orbit it will continue to move with its initial velocity in a straight line. This is the "natural" motion of an object. Imagine, however, you come across an object moving in a circle. Its velocity is changing with time: it is accelerating! This is where the vector nature of the velocity is important. The speed, or magnitude, is not changing, but the direction is. Changing velocity means acceleration. This means a force must be acting on it. How large is this force? If the object is moving in uniform circular motion then one can use the standard formula,

The gravitational force that is producing it must have this magnitude and direction.

Well, we know a very good example of something moving in a circle in space... The Moon. Perhaps gravity is keeping the Moon in orbit! Let us calculate what must be the magnitude of the force keeping the moon in orbit to see if this could be true. We start by knowing that the moon takes a month to circle the Earth at a distance of 384,400 km. That means it is moving with a speed of about 1023 m/s. If we plug this into the standard formula for acceleration in circular motion we have

This suggests that the force of gravity gets weaker far away from the Earth according to the rule

But the Earth isn't anything special. It's just another planet. It seems reasonable that the other planets and the Sun ought to have gravity also. Each object would exert a force that would depend on its properties. So we really ought to have a force law that looks like:

Where K is different for each object exerting a gravitational force. The only task left is to figure out what K is for each situation.

C) Gravitational Forces Exist Between All Objects

A thought experiment will help us here. Imagine dropping a rock. It accelerates towards the Earth, moving faster and faster:

But wait a second... this sounds like it is violating conservation of momentum! Remember Newton's third law? What is going on here? The solution to this paradox is that the Earth is falling towards the rock also! The momentum of the rock and the Earth are equal but opposite, so the end result is that momentum conservation is not violated. The force is acting between the Earth and the rock. Remember, all objects exert a gravitational force on all other objects.

The important point to note is that there is a complete symmetry here between the Earth and the rock: we can interchange roles. In our derivation of the force of gravity acting on the rock we ended up with a force that was proportional to the mass of the rock.

Turning this around we must conclude that the force on the Earth is proportional to the mass of the Earth.

But the force on the Earth and rock are of the same magnitude (otherwise momentum wouldn't be conserved!) so both must be proportional to both the mass of the rock and the mass of the Earth. We thus obtain

Since we had already determined that the force of gravity is proportional to the inverse square of the distance we can write

where G is a undetermined constant and the negative sign just means the force is together, not apart. This finally gives us the full Newtonian theory of gravity:

where G (Newton's Constant) is, as previously mentioned, equal to 6.67x10^-11m³s^-2kg^-1, M is the mass of one object, m is the mass of the other object, R (radius) is the distance between the objects and F is the resulting gravitational force pulling the objects together (hence the negative sign). This Universal Law of Gravitation applies between all bodies in the universe regardless of how they are composed or what their size. Of course, between most objects (say two tennis balls), the force of attraction is very tiny and not noticeable, but it is always there.

D) A Closer Look At Gravity

Let's look at the law of gravity more closely. The first thing one notices is that the force exerted by an object is directly proportional to its mass. Double the mass of an object and it will tug on other objects with twice the strength. Triple the mass and the force is three times stronger. This doesn't seem very remarkable until one thinks that the Earth is about 100,000,000,000,000,000,000,000 times more massive than the average student and it take the whole Earth to give us a sense of weight... The gravitational force between two students can be approximated

This is about a millionth of the force a feather exerts brushing against your face. The gravitational attraction between day-to-day objects is very, very tiny!

The force of gravity decreases in a very particular way as the distance between two objects increases. If the force is 1 Newton at a distance of 1 meter then at a distance of 2 meters the force would be 1/4th of a Newton. At 3 meters, 1/9th of a Newton and at 4 meters 1/16th of a Newton. In other words, the force is proportional to the inverse of the square of the distance. Notice that the distance is measured between the centers of the two objects. Intensity of light also drops off as one moves away from a light source proportional to the inverse of the square of the distance.

An example of both of these points is that your weight on different planets depends on both the mass and size of the planet. Take Jupiter as an example. Jupiter is 318 times more massive than the Earth, but also about 11 times larger.

So a person would weigh only about 2 1/2 times as much on Jupiter, not 318 times as much (as one might conclude if mass alone determined force).

Finally, notice that the force is negative. By convention, this means that the force of gravity between two objects is attractive, and never repulsive. This convention comes from the vector nature of the forces (F) and displacements (R) discussed in the next section. The answer for the direction of the Force depends on the direction of R. The direction of R is somewhat arbitrary, toward the person or toward the Earth in the example used above. The sign on the Force is positive if it is in the same direction as R and negative if it is in the opposite direction of R. Scientists have adopted a convention for the direction of R that creates a force in the opposite direction (negative) as R when attractive and the same direction (positive) as R when repulsive. The force also acts in the direction along the line connecting the two objects. There is never a sideways part of the force. That is to say: gravity is a "together" force.

A beautiful example of this "together" quality of gravity is the formation of the Sun and Solar System. The Solar System formed out of a huge cloud of gas and dust that was many times larger than it is presently. Slowly, over time, the cloud contracted under the influence of gravity to form what we currently know as our Solar System.

This also is why our planets are round like spheres. When there is enough mass, the rigidity of the object is insufficient to hold its shape against the force of gravity. All the matter wants to be as close to the center as possible. The shape that allows this is the sphere. To see this, imagine any other shape, say a cube. The cube's corners are farther away from the center than the faces. Under intense gravity, the corners will be flattened out and you end up with a sphere.

E) Gravitational Force and Gravitational Potential Energy

Let's consider the role of energy again now that we have the force law for gravity in general. When we first considered how energy and gravity were tied together, we calculated the energy it took to lift an object high above the floor. The analogous question in the case of gravity in general is "How much energy does it take to separate two objects?" Actually, as stated, this question doesn't make a lot of sense because you'd need to know how far apart the objects are to start with and how far they are to be separated. We will assume that the distance the objects start at can be denoted r, and they are to be separated completely. That is, to infinity.

When we found the potential energy in the simple case of gravity (in the Gravitational Force and Energy section) we calculated how quickly the falling object was moving when it reached the bottom. This was easy because we could assume constant acceleration. In the general case, however, the force changes depending on how far apart two objects are. As the objects get closer, the net force on those objects gets greater. This means the acceleration won't be constant. Calculating the potential energy converted into kinetic energy becomes much harder. It takes the application of a bit of calculus, which gives the following result:

This is the amount of energy it takes to move two objects from infinity to a distance r apart. The negative sign tells you that you get energy when you allow two objects to fall together. At infinity, the potential energy is zero. Let us consider two objects very far apart and not moving at all. Then the potential energy is zero, and the kinetic energy is zero,

As the objects slowly fall together they will move more and more quickly. But energy is conserved, so we still have

Thus we have

Turning this around gives the amount of kinetic energy required to move an object from a distance r out to infinity. This is the escape velocity. Throw an object up on the surface of the Earth. It rises and then falls. Throw it harder. It rises farther and then falls again. Keep throwing harder. At some point it will rise so high that the force of gravity will be appreciably smaller at the top of the trajectory. If you throw fast enough (give it enough kinetic energy) the object will leave the Earth entirely. We can solve for this velocity from the above equation,

How much is this on the surface of the Earth? Using M_earth=6x10²⁴ kg and R_earth=6.4x10⁶ m we get

This is very, very fast! Which explains why such powerful rockets are required to send people to the Moon!

F) Gravity Predicts the Orbits of the Planets: Kepler's Third Law

Almost everything in the Universe orbits around something else. Moons orbit around planets, planets around stars, stars around galaxies, and galaxies in clusters. As we saw, the Moon orbits around the Earth because it keeps falling towards the Earth but missing it (see description in the Historical Perspectives section). The important thing is that the object in orbit has a bit of sideways motion, so it doesn't just smack into the other object, but misses it instead. If two objects are released from at rest, then they will fall towards each other and collide.

One of Newton's greatest triumphs was to use the Law of Gravitation to derive the motions of the planets around the Sun. We will look at a simpler version of this derivation where we assume all the orbits are exact circles. Newton's ability to mathematically connect to Kepler's Third Law demonstrates one of the hallmarks of good science. Truly grand theories should simplify our understanding of nature. Before Newton, Kepler's Third law was considered a basic law of nature that applied only to planets. Newton showed that scientists could set aside Kepler's idea as a basic law and consider it to be a consequence of his own more fundamental and universally applicable Law of Gravitation. Newton's is a law that applies to all objects, not just planets.

Consider a planet orbiting the Sun. How long will it take to move once around in its orbit? We assume that the mass of the Sun is much larger than that of any of the planets. Just as the Earth doesn't move much under the gravitational influence of a dropped rock, the Sun doesn't move much under the influence of the planets. Thus we can take it to be stationary. If a planet is at a distance R_p then the force on the planet is

Setting this equal to the force required to keep the planet in circular motion (from back in our earlier discussion),

and solving for the velocity, we get the following

Notice again that the expression does not involve the mass of the planet. Any object, planet, asteroid, comet, or grain of dust will have the same orbital velocity at the same distance. Notice also that the velocity is proportional to the inverse of the square root of the distance. It drops off only slowly with distance. At the Earth's orbit, the velocity is 30 km/s, at the distance of Jupiter 13 km/s and even at far off Pluto's distance the Sun still whips the planets around at a healthy 4.7 km/s (Mach 15!).

Once we have the velocity we can obtain the orbital period quite easily. The distance traveled once around the Sun is

so the time it takes to go this distance (the orbital period) is

Squaring both sides we obtain

which is Kepler's third law! But this formula could have been derived with any central object: it's a general rule. Just plug the mass of the central object in for the mass of the Sun and go! Let's try it out for the Earth and the Moon. The moon is about 384,000 kilometers from the Earth, and the mass of the Earth is about 6 x 10²⁴ kilograms. Putting it all together we get T=2,363,000 seconds. This is 27.35 days, which is just right! Doing the calculation again for satellites in low Earth orbit (M_earth= 6 x 10 ²⁴ kg, R=6400 km (distance from the center!)) we get about 88 minutes which again is just right (the shuttle takes about 90 minutes to orbit Earth.

Now in general, planets and stars, etc. don't orbit in perfect circles, but rather in ellipses where the distance changes first larger then smaller and back. It turns out that a more detailed analysis nevertheless duplicates Kepler's law where the radius is replaced by the semi-major axis, which is half the length of the long axis of the ellipse.

G) Gravity Lets Us Detect the Invisible Stuff (stars, planets, galaxies, structure in the Universe, black holes, etc.)

From an astronomical perspective, one of the most important aspects of gravity is that it gives us a tool to determine the mass of objects forever beyond our reach. Consider a star. It is very difficult to determine the mass of a star. Even if one knows the distance, and hence the absolute brightness of the star, it requires complicated (and possibly incorrect!) computer models of the structure of the star to work back to determine the mass of the star. This is a very unsatisfactory state of affairs. But now consider a binary star system. The two stars orbit each other in accordance with Newton's law of gravitation. The force of gravity depends on the mass of the stars! By making careful observations of the orbits of the binary stars, one can determine the mass directly without modeling. So, Newton's Law of Gravitation is at the very foundation of what we know about stars.

In a similar way, the existence of planets can be inferred even if the planets themselves cannot be seen. Imagine looking at the Solar System from many light years away. Even Jupiter, the largest planet, would be at least a billion times fainter than the Sun, and it would be lost in the glare. It would be hopeless to try to look for planets directly. So how do astronomers detect extra-solar planets? The key lies with gravity. As a planet orbits a star, it exerts a gravitational force on the star. The star responds to this force by moving slightly. As the planet orbits, the star wobbles slightly. Although we cannot see the planet directly, we can see this wobble!

But remember the gravitational force is reciprocal. If the Earth attracts the Moon, the Moon also attracts the Earth. So the Earth should also move. Of course the Earth is much more massive than the Moon, so it doesn't move as much. The movement of the Earth-Moon system is like a parent swinging a child around. The child (Moon) moves in a large circle, while the parent (Earth) moves in a small circle. This is the principle through which planets around other stars have been found. We think of planets orbiting stars; in reality, planets and their stars orbit the "center of mass" of the stellar system, so orbiting planets cause their stars to wobble slightly. This predicted wobble has been detected by scientists.

On a larger scale, gravity allows us to probe the contents of galaxies. By observing the orbits of the stars in galaxies, we can determine the masses of the galaxies. If we compare these masses to the number of stars, we discover something very interesting: There is much more mass in galaxies than can be accounted for by the visible stars and gas. Ninety percent of the mass in galaxies is dark.

H) Gravity Is the Weakest of the Four Fundamental Forces in Nature

There are four fundamental forces in the Universe. They are: the electromagnetic force, the "weak" force, the "strong" force, and gravity. The electromagnetic force is perhaps the most common force. It is what holds atoms together, drives chemical reactions, and keeps objects from floating through each other. You have, perhaps, heard that most of matter is empty space. So why don't objects pass right through each other? The electromagnetic force is the reason. All the chemical reactions are also consequences of the electromagnetic force (taste, touch, smell). The very light we see with our eyes is a consequence of the electromagnetic force. Light is, after all, electromagnetic radiation.

The "weak" and "strong" forces are both only important in the atomic nucleus. Among other things, the weak force is responsible for radioactive decay. The strong force is what keeps nuclei and nucleons together. Without it, all matter would disintegrate in the tiniest fraction of a second. When huge "atom smashers" (more correctly particle accelerators) are used to probe the constituents of matter, they are probing the strong and weak forces.

The strong force is, reasonably enough, the strongest of the four forces, next comes the electromagnetic force, followed by the weak force. Gravity is by far the weakest force. The electromagnetic force between an electron and a proton is about 10,000,000,000,000,000,000,000,000,000,000,000,000,000 (10⁴⁰) (ten thousand trillion, trillion, trillion) times stronger than the gravitational force!

I) Gravity Is Nevertheless the Main Mover and Shaper in the Universe

Everywhere that we have looked, from the motions of the planets in our Solar System to the slow dance of binary stars in distant galaxies, Gravity seems to be the same. The same rules apply in our small corner of the Universe as apply in all other parts of the Universe. Distant parts of the Universe may be strange and unfamiliar, but some things stay the same. Gravitation is Universal!

Still, gravity is very much weaker than the other forces, so why don't the other forces dominate the Universe? There are two reasons. The first deals with the range of the forces. Gravity and electromagnetism have infinite range. These forces weaken with distance, but not very quickly. By contrast, the weak and strong forces are limited to distances small compared to atoms. At distances farther than a hundred thousandth of a nanometer, the weak and strong forces are negligible. But, electromagnetism is infinite in range and also much stronger than gravity. Why isn't it more important? The reason is that electric charge comes in two types, positive and negative. Like charges repel, and opposite charges attract. So imagine a lonely positive charge, perhaps a proton, sitting in the Universe. It will exert a tremendous force on any loose negative charges nearby, and they will move towards it. But as soon as the first negative charge reaches it, then taken together the proton and negative charge have a net charge of zero, so they stop attracting other particles! Any free charge quickly cancels out with an opposite charge, so most of the Universe is neutral and hence exerts no electromagnetic forces. This leaves gravity, which is always attractive and never cancels itself out.

J) Gravity Will Determine the Fate of the Universe

How will the Universe end? We know now that the Universe began 13.7 billion years ago, when the Universe was very, very small and the density tended towards infinity. Since then, the Universe has been expanding. If the Universe continues to expand, it will eventually freeze as the temperature drops to absolute zero and the stars burn out. On the other hand, if the expansion stops, the Universe will re-collapse, and the world will end in fire as we return to a state much like the initial Big Bang.

But what controls whether the expansion stops or not? Gravity! As the Universe expands, the galaxies are getting farther and farther away from each other. But remember, gravity is a "together" force. It tries to pull galaxies together. So the expansion is slowed down by gravity. The same is true if I throw a ball into the air. The away-from-the-Earth motion is slowed by gravity, and eventually the ball will come to a stop and then reverse its course and return down... That is, unless I throw the ball very, very hard indeed, and it reaches escape velocity! Then it will continue to travel farther and farther away.

Getting back to the Universe, if the initial expansion is fast enough, then the Universe is "open," and it will expand forever. If the Universe is expanding too slowly, then it will slow more and finally stop and then re-collapse. Another way of saying this is to consider the mass of the Universe. If the Universe is too massive, then it has a lot of gravity, and it will re-collapse. If it is light, then the Universe's expansion will be able to overcome its gravity, and it will continue to expand.

K) Gravitational Waves

In Newton's Universal Law of Gravitation, the gravitational force is determined by the masses and positions of objects. Let us imagine we change the position of an object. Now the gravitational forces on other objects will be slightly different. In Newtonian Theory, this change takes place instantaneously. But Special Relativity forbids this! Nothing can go faster than light. At best, the change in the gravitational force can propagate outwards at the speed of light. A propagating change? This is just a wave!

But gravitational waves are more than simply the change in the force of gravity from some distant object. In Einstein's GR, gravity is not really a force, it is a warping of space and time. So the change that is propagating out from a perturbed system is a distortion of spacetime! It's like a ripple in spacetime itself. As the ripple goes by, objects are stretched and stressed (very gently). But stretching objects takes energy. Where does this energy come from? It can't just appear because energy is conserved! (Energy is neither created nor destroyed - it may change form, (motion to heat or potential to kinetic) but it can't simply disappear or appear.) The energy comes from the original object. The gravitational wave takes its energy from the source of the wave. So accelerating objects lose energy.

Now for most objects in our experience, this is hardly important. The masses involved are small and the velocities far less than that of light, and so the gravitational radiation is miniscule, almost too tiny to comprehend. Is gravitational radiation important under any circumstances? Yes, but it requires high speeds and large masses. A neutron star is the dead remnant of a massive star that has burned up all its nuclear fuel. It is a little bit more massive than the Sun, but much, much smaller, being only about 10 kilometers across. Usually neutron stars are born in isolation, but occasionally two are produced together. If they are close enough together, then their orbital velocity is very high, and gravitational waves are produced copiously (large mass and large speed). But this means that the system loses energy. But losing energy means that the neutron stars fall closer together. This makes the orbit faster, which in turn increases the gravitational waves! A vicious cycle sets in with the neutron stars spiraling in faster and faster. Eventually, they merge together entirely with a tremendous collision. Some scientists believe that gamma-ray bursts, the most powerful explosions in the Universe, are powered by these collisions of neutron stars in distant galaxies.