This note started out in response to a thread in the Usenet group
sci.astro.amateur asking why planetary orbits are elliptical. There are
places you can go to get the mathematical derivation of those orbits,
the poster observed, but nowhere you can get an ordinary explanation of
just *why* the orbit is elliptical.

It turns out that the poster really wanted to know why the orbits weren't circular—he wasn't concerned about whether the particular out-of-round shape was an ellipse or something else—but consider the first question anyway. (I'm like that.) Can we say, succinctly and intuitively, how an elliptical orbit comes out from the so-called inverse square law?

My first reaction is to say no, and here's why. Scientists often prefer mathematical derivations to those in ordinary English, for two reasons: one, because they are more precise; and two, because they are more compact. It's possible, if you are very well versed in the language of mathematics, to turn a mathematical derivation into an ordinary English one that is just as precise, but the price for preserving that precision is an extraordinary lack of concision and intuitiveness.

As it turns out, even the mathematical derivation of the orbit is
not a gem of brevity. Newton, in his *Principia Mathematica*,
first derived the inverse square law of gravity from the properties of
an elliptical orbit, and he needed a whole book of structure to build up
underneath it. Even if we ignore the setup as merely preliminary, just
the proof of the inverse square law itself fills up the better part of a
page with extremely dense geometrical arguments that assume the reader
is intimately familiar with the properties of ellipses. Today, even a
physicist would be hard put to follow the arguments. To be fair,
physicists today would use a different language to derive the orbital
equations, but that derivation too is fairly lengthy.

Well, if the mathematical arguments are long, imagine what an ordinary English explanation would look like! So what are we doing here? I still think that a full explanation of why planetary orbits are elliptical is too involved to do in English; it's best to stick with the mathematics. But just perhaps, we can get enough of a flavor of the argument in ordinary English to satisfy our intuition. I warn you, however: This is still tricky going. The fact that it isn't simple and straightforward to demonstrate is an indication that Newton uncovered a relationship of great depth.

Just to be clear, here's what's really involved. Back in the early 17th century, Kepler announced that planetary orbits obeyed two simple laws. The first law states that planets move in elliptical orbits, with the Sun at one focus.

What does that mean, exactly—what is an ellipse? You can think of an ellipse as a squashed circle. You can draw one easily: Hammer two nails into a wooden board, near the center. Then, take a length of string long enough to wrap around the nails with room to spare, and tie it in a loop. Next, loop the string around the nails. If you stick a pencil inside the string loop and pull it taut, you can move the pencil around in a gently curving path. This path is an ellipse, and the two nails are at the ellipse's two foci (singular, focus). Figure 1 shows what this looks like.

Figure 1. Drawing an ellipse with two nails, a board, and a loop of string.

Two foci? Yes, an ellipse has two of them. The Sun, for instance,
is at one of the foci of the Earth's elliptical orbit. People often
wonder what's at the other focus, expecting something unusual to be
there—an anti-Earth, perhaps—but in fact, there is nothing at all of
interest at the other focus. It is sometimes called the *empty
focus*.

Kepler's second law states that in general, planets do not move in their orbits at a uniform speed. Rather, they move faster when closer to the Sun, and slower when further from the Sun. We can imagine them sweeping out little wedge-shaped slices of the orbital disc in each unit of time. The slices near the Sun are short and squat, while those far from the Sun are long and thin. Kepler's second law states that although the shapes are different, the areas are the same: Planets sweep out equal areas in equal times. Figure 2 illustrates this law.

Figure 2. The two wedges have equal areas and are traced out by the planet in equal times. The Sun is at

What Newton showed was that given a planet that obeyed these two
laws, the Sun must exert a force—that is, gravity—that drops off with
increasing distance, in inverse proportion to the square of the distance
(hence, "inverse square"). That is to say, if one planet is *k*
times as far as another identical planet, the Sun's force is stronger on
the second, closer one, by a factor of *k* squared.

Now that you know what I'm trying to show, let me reiterate that I'm
not going to show it in any kind of rigorous detail. My emphasis will
be on imparting a *visceral* intuition for why an orbit that
adheres to Kepler's laws follows an inverse square law of gravity,
without going too deeply into the mathematics.

To begin with, let's consider one interesting property of an ellipse. If you build a barrier in an elliptical shape, and you roll a tennis ball from one of the foci, in any direction whatsoever, it will bounce off the barrier and roll to the other focus.

This property applies to anything that bounces elastically, like a ball. For instance, if you put a light source at one focus of an ellipsoidal mirror, all the light will bounce off the mirror and converge at the other focus. (In fact, that's one reason why these points are called the foci.) An auditory analogue of this phenomenon can be found in some buildings, such as the U.S. Capitol Building, where you can whisper at one point and be heard at another point a long way away, even with people in the way. The reason is the ellipsoidal ceiling: the two points are the ceiling's foci.

In a geometrical sense, what this property represents is that if you
draw a line perpendicular to the ellipse at any point *P*, the
lines drawn from *P* make equal angles to that perpendicular.
(OK, we have to use *some* mathematics.) This is shown in Figure
3.

Figure 3. A ball (or light, or sound) from the focus at

Now, suppose that *P* is a planet in an elliptical orbit,
with its two foci at *T* and *S*. And suppose that this
planet traces out a small section of its orbit in a certain period of
time. If the Sun is at *T*, it sweeps out one wedge. But if the
Sun is at *S*, it sweeps out a different wedge. (See Figure 4.)

Figure 4. The planet sweeps out a different wedge, depending on which focus the Sun is at.

In moving along that small section of orbit, the planet changes its
direction slightly. Since we're talking about the same stretch of
orbit, it undergoes the same change in direction, whether the Sun is at
*T* or it's at *S*. This change in direction can only
come from the gravity of the Sun.

This sounds like we've just shown that the force of the Sun on a
planet is the same for different distances, since *TP* is clearly
longer than *SP*. But remember that the planet doesn't move at
the same speed in the two different cases. If *TP* is twice as
long as *SP*, as in Figure 4, then the shaded wedge extending
from *T* contains twice the area of the shaded wedge extending
from *S*; if it were three times as long, the larger wedge would
have three times the area of the smaller wedge; and so on. Those ratios
are only the same because of the equal-angle property we saw in Figure
3.

In short, if the Sun is at *T*, it takes twice as long to go
along that section of orbit as it does if the Sun is at *S*.

This is critical! The distance covered is the same in either case,
so the planet moves one-half as fast with the Sun at *T* as it
does with the Sun at *S*. That means that the change in motion
is only one-half as much, too. For instance, if you go through half of
a circular orbit at 10 meters per second, the total change in motion is
20 meters per second, since you go from 10 meters per second in one
direction to 10 meters per second in exactly the opposite direction.
But if you do it at only 5 meters per second, the total change in motion
is only 10 meters per second.

To be sure, the planet changes speed as well as direction, but it
does so in proportion to the angle that *TP* or *SP* makes
with the perpendicular, *VP*. Since both angles are the same,
the proportions are the same, so the total change in motion is still
only half as much with the Sun at *T* as it is with the Sun at
*S*.

What's more, that change in motion also takes twice as long when the
Sun is at *T*. If you combine those two effects—half the change
in motion, in twice the amount of time—you can see that the rate of
change in motion, the *acceleration*, is one-fourth as much if
the Sun is at *T* as it is if the Sun is at *S*. And
since the force exerted by the Sun must be proportional to the
acceleration, it follows that the force is one-fourth, too.

Of course, there's only one Sun, and it can only be at one of the
foci; it can't be continually moving back and forth between the foci.
But the planet does move. And as Figure 5 shows, there is a spot in its
orbit, labelled *P'*, where the force exerted when the Sun is at
*S* is the same as the force exerted when the Sun is at
*T* and the planet is at *P*. So the planet at
*P'* feels one-fourth the force from the Sun at *S* that
the planet at *P* does.

Figure 5. A single planet at

The fact that we ended up with a ratio of one-fourth comes from the
fact that *TP* is twice as long as *SP*. If *TP*
were some other factor *k* times as long as *SP*, the
planet would undergo one *k*th the change in motion with the Sun
at *T*, in *k* times as long a stretch of time, as it
would with the Sun at *S*—meaning that the force would be
smaller by a factor of *k* squared. So—aha!—here's our first
glimpse of understanding the inverse square law of gravity.

We have to be careful, though, even when we're not quite being
rigorous. We haven't shown that all points on the orbit are related by
an inverse square law. We've only shown that that relationship exists
between two symmetrically placed points *P* and *P'* on
the orbit. We haven't said a thing about the relationship between any
other kind of pairs of points. Nevertheless, it's an important first
step, one that's worth setting apart and putting in bold:

Finding 1. An inverse square relationship exists between any two symmetrically placed points on the planet's orbit.

Next, let's try to relate another pair of points on the orbit. For
this step, we'll fix the Sun at *S*. Suppose we extend the wedge
from *P*, through the Sun at *S*, to a new point on the
orbit, *Q*. These two wedges aren't the same size or shape, but
they do have one thing in common: They make equal angles at *S*.
I'll also draw lines connecting those wedges back to *T*.
(See Figure 6.) Note that since *P*, *S*, and *Q*
are all in a straight line, the wedges actually trace out a triangle,
*TPQ*.

Figure 6. The wedges swept out by planets at two opposite points in the orbit,

In this diagram, *TP* is still twice as long as *SP*.
Notice that *SP* is longer than *SQ*—let's say, by a
ratio of 4 to 3. (It could be any ratio, but picking a particular value
will make the discussion easier.) That means that in some arbitrary
units, *TP* has length 8, *SP* has length 4, *SQ*
has length 3, and *TQ* has length 9; remember that because of the
way an ellipse is constructed, *TP+SP* has to be equal to
*TQ+SQ*. Figure 7, below, is just the same as Figure 6, but with
all the lengths filled in. Pay close attention now; this is the
toughest part of the entire explanation!

Figure 7. The same figure as in Figure 6, with the lengths filled in.

When the planet is at *P*, its distance from the Sun is 4,
and when it's at *Q*, its distance is 3. To ensure that the
planet sweeps out equal areas in equal times (Kepler's second law
again), it must be moving faster at *Q* than at *P*, by a
factor of 4 to 3. OK so far?

Now consider the thin wedges emanating in both directions from the
Sun at *S*. They obviously cover the same angle. If we imagine
the planet's orbit to be a kind of gigantic mirror, the section of
mirror at *Q* is able to get that diverging wedge to converge
again at *T* in a distance of 9, whereas the section at
*P* is able to get it to converge at *T* in a distance of
only 8. So the planet changes direction more at *P* than it does
at *Q*, by a factor of 9 to 8.

Put those together, and you have the planet changing its motion more
at *Q* than at *P*, by a factor of 4 x 8 = 32 to
3 x 9 = 27.

*Except!*

The "bounce" of the lines at *P* is shallower than it is at
*Q*. This has two effects: First, it means that more of the
Sun's gravitational effect is diverted into changing the planet's
*speed* (as opposed to its direction) at *P* than it is at
*Q*; and secondly, it means that the planet is moving faster at
*P* than you might expect from Kepler's second law (because more
of its motion is directed toward or away from the Sun and therefore has
less impact on the area the planet sweeps). These two effects are
weakest when the planet is at the ends of the elliptical orbit, and
strongest when it's halfway in between. In fact, combined, they are
proportional to the product of the planet's distance from the Sun, at
*S*, and its distance from the empty focus, at *T*.

As a result, we have to apply corrections to the changes in motion
at *P* and *Q*, equal to 4 x 8 = 32 and
3 x 9 = 27, respectively. That means that the actual,
corrected changes of motion at *P* and at *Q* are related
by a factor of 32 x 27 to 27 x 32; or, in other
words, they're exactly equal!

But remember that the planets don't take equal times to sweep out
the wedges at *P* and *Q*. It takes longer to sweep out
the wedge at *P*, because that wedge covers more area, by a
ratio of 4 to 3, squared, or 16 to 9. Since the change in motion is the
same, the rate of change in motion—the acceleration, again—must be
greater at *Q* by that same 16 to 9 ratio. Of course, that's
just what the inverse square law says.

There's nothing in what I wrote that depends specifically on the particular lengths I chose, so the same reasoning can be applied to any two opposite points on the orbit:

Finding 2. An inverse square relationship exists between any two points on the planet's orbit that are opposite the Sun from one another.

At last, we're ready to put everything together. What we're going to show is that any point on the orbit is related by an inverse square law to a special point, called the perihelion point. That point is the one that's closest to the Sun ("perihelion" comes from two Greek words meaning "around" and "Sun"), and it lies on one "end" of the ellipse.

Here's how we're going to do it. We start out at a point *P*
on the orbit. We draw a line from *P* through *S* to a
point *Q*; Finding 2 tells us that *P* and *Q* are
related by the inverse square law.

Then, we move from *Q* across to a symmetrically placed point
*Q'*; here, it is Finding 1 that tells us that the inverse square
law relates *Q'* to *Q*, and therefore back to *P*.
Then we draw a line from *Q'* through *S* to *R*,
then across to *R'*, etc. We keep applying Findings 1 and 2
alternately to give us a whole slew of points, all related by the
inverse square law. Figure 8 illustrates the basic idea.

Figure 8. Successive applications of Findings 1 and 2 get us closer and closer to the perihelion point.

As we collect more and more points under the unbrella of the inverse
square law, we get closer and closer to the perihelion point. In fact,
the limit of this sequence is always the perihelion point. (Actually,
there are two limits; the other limit is the aphelion point, the point
furthest from the Sun, and directly opposite the perihelion point.
Either one will serve our purposes.) Since any arbitrary point on the
orbit can be related to the perihelion point by the inverse square law,
*all* points on the orbit are so related:

Finding 3. An inverse square relationship exists amongst all the points on the planet's orbit.

And that's what we wanted to see all along.

Of course, Newton went on to show much more than just this. Using his methods, he was able to use another law of Kepler to show that the inverse square law applied equally to all the planets, to apply this universal law to determine the perturbative effects of one planet upon another, to explain the precession of the equinoxes, etc.—all of which are beyond the scope of this article. But for those who might have been mystified by the appearance of the inverse square relationship out of the ellipse, I hope this article has given you at least an inkling of it.

Copyright (c) 2006, 2010 Brian Tung