Objects moving around the Sun, whether planets or smaller bodies, such as asteroids and comets, usually have elliptical orbits (see Ellipses and Other Conic Sections and Kepler's First Law). These orbits are the result of two competing tendencies, described by Newton's First and Second Laws of Motion.
Newton's First Law of Motion, the Law of Inertia, states that a body in motion tends to maintain its motion in the absence of a force. If it is not moving, it remains "at rest"; but if, like a body orbiting the Sun, it is headed in some direction at some speed (referred to by physicists as a velocity, which includes both its speed and its direction), it will maintain that motion unless and until a force is exerted on it. So in the absence of a force, the body would move in a straight line at constant speed, which would eventually carry it out of the Solar System, into interstellar space.
However, all bodies near the Sun (or any other reasonably massive body) experience a force, which we call the force of gravity, which pulls them toward the object creating that force. In the absence of anymotion, this force would inexorably pull them toward the Sun, faster and faster, until they ran into it.
Orbital motion is the result of combining these two tendencies. If there were no force acting on an object, it would move in a straight line at constant speed; but there is a force, which tries to pull it into the Sun. If the force trying to pull it into the Sun was the only thing which had to be considered, the object would fall into the Sun; but there is another thing which must be taken into account, namely the object's original motion. The combination of these two factors can be expressed with a diagram which shows each tendency, and the result:
In the image above the object is moving in one direction, while being pulled in another. The result is a change in the direction of motion, which is in the direction of the force being exerted on the object. The amount of the change is given by Newton's Second Law, the Force Law, which is
F (force being exerted) times m (mass being acted on) = a (the resulting acceleration)
In other words, the force of the Sun on the moving body changes its motion, by creating an acceleration in the direction of the force. The change of motion which results from the force is given by the acceleration imparted to the object, multiplied by the length of time the force is exerted
dv (the change of motion) = a times t (the length of time the force is exerted), or
dv = F / m times t
What this equation means is that the velocity can be changed by an amount dv either by using a large force for a short period of time, or a small force for a long period of time, or some combination of the two. For planets in small orbits, which have a strong gravitational force acting on them, a large change in motion can take place in a short period of time, but planets in large orbits can also have a large change of motion; it just takes longer for the weaker force of gravity at large distances from the Sun to accomplish the same thing.
Regardless of the details of the situation, the fact that the Sun exerts a gravitational force on the moving body means that after a while, it will be moving in a different direction from its original direction, and will have a motion which has "fallen" toward the Sun in three ways:
(1) The new direction of motion is more toward the Sun than the original motion.
(2) The body is closer to the Sun than it would have been, if it had followed its original path.
(3) The body will be moving slower than before if it is moving away from the Sun, and faster than before if it is moving toward the Sun, which represents a motion more toward the Sun than before, or at least not as much away from the Sun as before.
Now, if all the Sun did was give the body a little tug to the side at one moment in time, what is shown in the first diagram is all that would happen. The object would move a little to the side relative to its original motion, and that would be it. But of course the Sun continues to pull on the object at every moment in time. So in addition to the change of motion shown in the first diagram, there is an additional change of motion, later on.
As shown in the image above, the Sun's pull causes the object to move a little to the side of its original motion, changing both its position and direction of motion (and, unless it is moving perpendicular to the direction to the Sun, its speed as well). A little while later, the new motion of the object (and its new position) represent a new "original" motion. This motion is changed, just as the (truly) original motion was, by being pulled toward the Sun, resulting in an even more different motion relative to the original motion. So over a period of time, the original motion will move to the side more and more, changing what would have been a straight-line motion into a gradually curving motion. (It might not be obvious that it is a curving motion in the diagram, because each change of motion is shown by a straight line; but since the Sun is pulling on the body at every moment in time, the actual motion is an infinite number of infinitely small sideways motions, which represent a smoothly curving path, instead of a series of straight-line steps to the side.)
In other words, as a result of the continual pull of the Sun, a body moving through space near the Sun must follow a gradually curving path, which moves it further and further from its original motion. Now, what can we say about the nature of the curve the object follows? Unfortunately, with just the simple diagrams shown here, not much; but if you work through the math using calculus or (as Newton did) what we would now call differential geometry, the result turns out to be dependent upon the nature of the force which is exerted by the Sun. If that force was like that of a spring -- stronger when the spring is stretched further, weaker when it is stretched less -- then you would end up with one kind of curve. If the force had another kind of variation with distance from the Sun, you would end up with a different kind of curve. In fact, the orbital motion could be most anything, depending upon the equation which describes how the force exerted by the Sun changes as the object gets closer to the Sun, or farther away.
As it happens, if we were to write down various random equations to represent the force of gravity, most of the curves produced by such forces would be chaotic -- that is, as the object gradually curves around the Sun, it would eventually end up back on the same side of the Sun that it started out on, but with a completely different position and motion from its original position and motion. In such cases, every "orbit" around the Sun would be different from the one before, and eventually the object would be ejected from the Solar System. There are, in fact, only two equations which the force of gravity can have, which provide "stable" orbits. One is the spring force law mentioned above. If that were the way gravity works, orbits would be ellipses, with the Sun at the center of the orbits. But that isn't the way gravity works. Instead, it follows an "inverse-square" law in which the further out a planet is, the weaker the force of gravity it experiences. The equation whichs gives the strength of the gravitational force of the Sun is (as discovered by Newton)
F = (a constant) times M (the Sun's mass) times m (the planet's mass) divided by r-squared,
where r is the distance between the Sun and planet.
Substituting the Force Law on the left side of the equation, m times a = (a constant) times M times m / r-squared, Dividing both sides by the mass of the planet, a = (a constant) times M / r-squared
So, given this force law for gravity, what is the nature of the orbital path which an object can follow? As discussed in Ellipses and Other Conic Sections, the orbit must be a conic section, but for practical reasons, only elliptical orbits can be stable for any great length of time, so all orbits are elliptical (as discussed in Kepler's First Law).
At this point, many people will ask (having read the page about Ellipses and Other Conic Sections), "But a circle is a kind of ellipse; so why can't the orbits be circles? Why do they have to be non-circular ellipses?" The reason is that there are an infinite number of elliptical shapes, ranging from perfectly circular to infinitely stretched out (a so-called "degenerate ellipse"). Any one of them can represent some elliptical orbit, but only one particular shape happens to be a circle; and it requires a very exact situation for that to occur. Namely, to have a circular orbit, the object must be moving exactly perpendicular to the direction to the Sun, at exactly the "circular" orbital velocity. Any faster or slower, and the object will have an elliptical orbit. Even the smallest movement inward or outward, and the object will have an elliptical orbit.
As an example, suppose that an object does have an exactly circular orbit, so that it is moving at exactly the right speed, and in exactly the right direction, to follow a perfect circle path around the Sun. This means that each second (or even each microsecond), as the planet moves forward, the pull of the Sun pulls it inward, away from its straight-line path, by exactly the amount required to stay on a perfectly circular path. But suppose that for some reason, the object goes just a little faster or a little slower than the "circular velocity". What happens then?
To see what happens then, suppose we imagine dropping a ball, and watching it fall. Since we are "dropping" the ball, which implies that it has no motion before we do so, the only thing to be concerned about is the downward force of gravity; so the ball will fall straight down, hitting the ground soon after we release it. But suppose, instead of dropping the ball, we throw it exactly sideways (horizontally), with some small speed.
As shown in the diagram above, the ball thrown to the side will have two motions -- the downward fall caused by gravity, which would have occurred if it had just been dropped, and the sideways motion due to its being thrown horizontally to the side, which would have not involved falling at all, if we could have somehow turned off the force of gravity. As a result of those two motions, the thrown ball lands off to the side at a distance equal to its speed times the length of time it took to fall, but it still falls to the ground, just the same as the ball that was simply dropped. This is because we can treat the two motions of the thrown ball -- the original motion of the ball, carried out over the time it takes it to fall, and the falling motion, identical to the motion it would have had if just dropped -- as completely independent. There is nothing in the law which describes the force of gravity which is in any way dependent upon the motion of the object; so the force of gravity and its effect (the falling of the ball) are the same whether the ball is stationary, or moving to the side (or, for that matter, moving in any other direction).
Of course, the actual motion of the ball is not two separate motions to the side and down, as shown above, but a continually curving motion, just like orbital motion (in fact, technically the ball is in orbit around the Earth when dropped or thrown to the side; it is just in an orbit which runs into the surface of the Earth instead of going around it, like a suborbital ballistic missile). This is shown below, along with similar motions which result from throwing the ball faster and faster to the side.
(Note: "Actual" motions is in quotes because the curves drawn are only crude approximations)
In the diagram above we see that whether a ball is dropped, or thrown to the side at various speeds, its overall motion is the same; it falls exactly the same amount, and moves to the side by an amount equal to its speed multiplied by the time it took to the fall (this is even true for the dropped ball; it has a sideways speed of zero, so it goes zero distance to the side). There is a difference in the curves followed by the different balls, of course. The faster a ball is thrown to the side, the more shallow the curve it follows, since its sideways speed is larger in comparison to the downward speed it has, as a result of gravity. But that doesn't effect how far it falls; just how far off to the side it lands.
Now in the diagram above, the ground is shown as a flat horizontal surface. Real ground may not be quite that, due to dips and hollows, bumps and peaks; but to a certain extent, we think of the surface of the Earth as being flat on the grand scale, despite any local variations. But of course, the surface of the Earth is not flat. It is curved, because the Earth is a sphere. In fact, if you were to go about 5 miles horizontally in any direction, the apparently horizontal surface of the Earth beneath your feet would be about 16 feet below the horizontal plane defined by your original position, and slightly tilted downward as well (by about 5 arcmins, or a twelfth of a degree). We can show that by putting part of a circular arc beneath the horizontal surface in the last diagram, to show the actual curvature of the Earth:
Now, consider what the motion of the falling balls looks like to observers at different places along their path. For the ball dropped straight downward, we see it fall straight down, and hit the surface of the Earth. But for the balls thrown to the side, although the balls fall just as much as the ball dropped straight down, they do not hit the surface of the Earth in the time shown. They do fall to the original horizontal plane, extended into space; but since the Earth's surface curves out from under their paths, they are still above the surface of the Earth, at the time that the dropped ball hits the surface. This does not mean that the balls won't hit the Earth; but it does mean that their motions, as observed by someone running along with them, look a little different than we might expect. The faster a ball is thrown to the side, the further it goes around the Earth while it is falling. As a result, the surface of the Earth "falls out" from beneath it by a greater amount, and the falling ball therefore appears to fall less than it really did. (I might note that this is the cause of the vertical Coriolis Effect, which makes things seem to weigh less at the Equator than at the Poles, even though they actually weigh more at the Equator than at the Poles, due to the extra mass of the equatorial bulge.)
Now, in the example above, the amount that the balls move to the side is relatively small, and the amount that the Earth's surface drops out from under a horizontal plane is small in comparison to the amount that the balls fall. But suppose that we throw a ball to the side at a very fast speed -- in fact, at 5 miles per second, or 18,000 miles per hour, so that in one second, the ball would be five miles away from its starting point, at a place where the curvature of the Earth causes its surface to lie 16 feet below the horizontal plane at the starting point. What would we see in that case?
The Earth's force of gravity accelerates (changes the motion) of a ball in a downward direction, at a rate of 32 feet per second, each second. This means that over a period of one second, the ball has an average downward speed of 16 feet per second (its initial speed of zero, averaged with the downward speed of 32 feet per second, a second later; assuming, of course, that it didn't hit the ground in the meantime). As a result of this, any ball, whether dropped, thrown horizontally or in any other direction, will fall 16 feet relative to its original straight-line motion, during the first second it falls (presuming, again, that it doesn't run into anything). And in particular, a ball thrown at 5 miles per second to the side will fall 16 feet downward in one second, despite the fact that it is also moving to the side at a much faster speed. However, in that second it will travel 5 miles around the curve of the Earth, and as discussed a few paragraphs above, 5 miles in any direction from your starting point, a "horizontal" Earth is actually 16 feet lower than the horizontal plane we started on. So although the ball falls 16 feet relative to its original horizon, the ground 5 miles away is also 16 feet below the original horizon, and as a result the ball would be at exactly the same height it started out with. Of course, in addition to falling downward, the ball will have a downward motion relative to its original horizontal motion; but just as the surface of the Earth is tilted downward by 5 arcmin at the 5-mile distant location, so is the motion of the ball. In other words, someone moving along with a ball moving to the side at 5 miles per second would see it apparently not falling at all, but simply moving in a perfectly straight-line path, always the same height above the surface of the Earth, and always moving exactly parallel to the surface of the Earth.
This does not mean that the ball is not falling. It is falling, just as much as any other ball. However, it is "falling" around the Earth in an orbit parallel to the surface of the Earth -- that is, in a perfectly circular orbit, parallel to the "circular" (or more accurately spherical) surface of the Earth. We call this situation an "orbit", and the velocity required to achieve it an "orbital velocity", or more accurately, the "circular orbital velocity".
To summarize the discussion thus far, if we drop a ball, we see it fall straight to the surface of the Earth; but if we throw it horizontally to the side, we see it move sideways and fall, at the same time. For slow speeds, and small distances moved to the side during the fall of the ball, there is no obvious difference between the way the dropped ball falls, and the way a thrown ball falls. But if we throw things to the side with very high speeds, then the curvature of the Earth makes it appear as though they are not falling as fast as they are really falling; and if we throw something sideways at the "circular orbital velocity" of 5 miles per second, it will not appear to fall at all, but to just go around the Earth in a circular orbit, never getting any higher or lower, because its motion is parallel to the curved surface of the Earth.
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