# 15. Keplerian Orbits

*Michael Fowler*

## Preliminary: Polar Equations for Conic Section Curves

As we shall find, Newton's equations for particle motion in an inverse-square central force give orbits that are conic section curves. Properties of these curves are fully discussed in the accompanying "Math for Orbits" lecture, here for convenience we give the relevant polar equations for the various possibilities.

For an ellipse, with eccentricity and semilatus rectum (perpendicular distance from focus to curve)

Recall the eccentricity is defined by the distance from the center of the ellipse to the focus being whereis the semi-major axis, and

For a parabola,

For a hyperbolic orbit with an attractive inverse square force, the polar equation with origin at the center of attraction is

where (Of course, the physical path of the planet (say) is only one branch of the hyperbola.)

The origin is at the center of attraction (the Sun), geometrically this is one focus of the hyperbola, and for this attractive case it's the focus "inside" the curve.

For a hyperbolic orbit with a *repulsive* inverse
square force (such as Rutherford scattering), the origin is the focus "outside"
the curve, and to the right (in the usual representation):

with angular range

## Summary

We'll begin by stating Kepler's laws, then apply Newton's Second Law to motion in a central force field. Writing the equations vectorially leads easily to the conservation laws for angular momentum and energy.

Next, we use Bernoulli's change of variable to prove that the inverse-square law gives conic section orbits.

A further vectorial investigation of the equations,
following Hamilton, leads naturally to an unsuspected *third *conserved
quantity, after energy and angular momentum, the Runge Lenz vector.

Finally, we discuss the rather surprising behavior of the momentum vector as a function of time.

## Kepler's Statement of his Three Laws

**1**. The planets all move in elliptical orbits with the
Sun at one focus.

**2**. As a planet moves in its orbit, the line from the
center of the Sun to the center of the planet sweeps out equal areas in equal
times, so if the area *SAB* (with curved side *AB*) equals the area *SCD*,
the planet takes the same time to move from *A* to *B* as it does
from *C* to *D*.

For my Flashlet illustrating this law, click here.

For the JavaScript version, click here.

**3**. The time it takes a planet to make one complete
orbit around the sun(one planet year) is
related to the length of the semimajor axis of the ellipse :

In other words, if a table is made of the length of year for each planet in the Solar System, and the length of the semimajor axis of the ellipse , and is computed for each planet, the numbers are all the same.

These laws of Kepler's are precise (apart from tiny
relativistic corrections, undetectable until centuries later) but they are only
*descriptive* -- Kepler did not understand why the planets should behave in
this way. Newton's great achievement was to prove that all this complicated
behavior followed from one simple law of attraction.

## Dynamics of Motion in a Central Potential: Deriving Kepler's Laws

### Conserved Quantities

The equation of motion is:

.

Here we use the hat ^ to denote a unit vector, so gives the magnitude (and sign) of the force. For Kepler's problem, .

(Strictly
speaking, we should be using the *reduced* mass for planetary motion, for our
Solar System, that is a small correction. It can be put in at the end if
needed.)

Let's see how using vector methods we can easily find constants of motion: first, angular momentum -- just act on the equation of motion with

Since , we have , which immediately integrates to

,

a constant, the angular momentum, and note that so the motion will always stay in a plane, withperpendicular to the plane.

This establishes that motion in a purely **central**
force obeys **a conservation law: that of angular momentum.**

(As we've
discussed earlier in the course, *conserved quantities in dynamical systems
are always related to some underlying symmetry* of the Hamiltonian. The
conservation of angular momentum comes from the spherical symmetry of the
system: the attraction depends only on distance, not angle. In quantum
mechanics, the angular momentum operator is a rotation operator: the three
components of the angular momentum vector are conserved, are constants of the
motion, because the Hamiltonian is invariant under rotation. That is, the
angular momentum operators commute with the Hamiltonian. The classical analogy
is that they have zero Poisson brackets with the Hamiltonian.)

To get back to Kepler's statement of his Laws, notice that when the planet moves through an incremental distance it "sweeps out" an area , so the rate of sweeping out area is Kepler's Second Law is just conservation of angular momentum!

Second, conservation of energy: this time, we act on the equation of motion with :

This immediately integrates to

Another conservation law coming from a simple integral: **conservation
of energy**. What symmetry does *that* correspond to? The answer is the
invariance of the Hamiltonian under *time*: the central force is time
invariant, and we're assuming there are time-dependent potential terms, (such
as from another star passing close by).

### Standard Calculus Derivation of Kepler's First Law

The first mathematical proof that an elliptic orbit about a focus meant an inverse-square attraction was given by Newton, using Euclidean geometry (even though he invented calculus!). The proof is notoriously difficult to follow. Bernoulli found a fairly straightforward calculus proof in polar coordinates by changing the variable to

The first task is to express in polar, meaning coordinates.

The simplest way to find the expression for acceleration is to parameterize the planar motion as a complex number: position , velocity , notice this means since theensures the term is in the positivedirection, and differentiating again gives

For a central force, the only acceleration is in the direction, so which integrates to give

the constancy of angular momentum.

Equating the radial components,

which isn't ready to integrate yet, because varies too. But since the angular momentum is constant, we can eliminate from the equation, giving:

This doesn't look too promising, but Bernoulli came up with two clever tricks. The first was to change from the variable to its inverse, . The other was to use the constancy of angular momentum to change the variable to .

Putting these together:

so

Therefore

and similarly

Going from to in the equation of motion

we get

or

This equation is easy to solve! The solution is

where is a constant of integration, determined by the initial conditions.

This proves that *Kepler's First Law follows from the
inverse-square nature of the force*, because (see beginning of lecture) the
equation above is *exactly* the standard equation
of an ellipse of semi major axisand
eccentricity , with the origin at one focus:

Comparing the two equations, we can find the geometry of the ellipse in terms of the angular momentum, the gravitational attraction, and the initial conditions. The angular momentum is

## A Vectorial Approach: Hamilton's Equation and the Runge Lenz Vector

(*Mainly following Milne, *Vectorial Mechanics*, p
235 on.)*

Laplace and Hamilton developed a rather different approach
to this inverse-square orbit problem, best expressed vectorially, and made a
surprising discovery: even though conservation of angular momentum and of
energy were enough to determine the motion completely, *for the special case
of an inverse-square central force, something else was conserved*. So the
system has another symmetry!

Hamilton's approach (actually vectorized by Gibbs) was to apply the operator to the equation of motion :

Now

so

**This is known as Hamilton's equation. **In fact, it's
pretty easy to understand on looking it over: has
magnitude and direction perpendicular to ,
, etc.

It isn't very useful, though -- *except* in one case, the
inverse-square: (so )

Then it becomes tractable:

and -- surprise -- this **integrates immediately** to

whereis a vector constant of integration, that is to say we find

is **constant** throughout the motion!

This is unexpected: we found the usual conserved quantities,
energy and angular momentum, and indeed they were sufficient for us to find the
orbit. But for the special case of the inverse-square law, something else is
conserved. It's called the **Runge Lenz** vector (sometimes Laplace Runge
Lenz, and in fact Runge and Lenz don't really deserve the fame -- they just
rehashed Gibbs' work in a textbook).

From our earlier discussion, this conserved vector must correspond to a symmetry. Finding the orbit gives some insight into what's special about the inverse-square law.

### Deriving the Orbital Equation from the Runge-Lenz Vector

The Runge Lenz vector gives a very quick derivation of the elliptic orbit, without Bernoulli's unobvious tricks in the standard derivation presented above.

First, taking the dot product of with the angular momentum , we find , meaning that the constant vector lies in the plane of the orbit.

Next take the dot product of with , and since , we find , or

where and is the angle between the planet's orbital position and the Runge Lenz vector .

This is the standard equation
for an ellipse, with the semi-latus rectum
(the perpendicular distance from a focus to the ellipse), *e* the
eccentricity.

E**vidently points along the major axis**.

The point is that the direction of the major axis remains
the same: the elliptical orbit repeats indefinitely. If the force law is
changed slightly from inverse-square, the orbit **precesses**: the whole
elliptical orbit rotates around the central focus, the Runge Lenz vector is no
longer a conserved quantity. Strictly speaking, of course, the orbit isn't
quite elliptical even for once around in this case. The most famous example,
historically, was an extended analysis of the precession of Mercury's orbit, most
of which precession arises from gravitational pulls from other planets, but
when all this was taken into account, there was left over precession that led
to a lengthy search for a planet closer to the Sun (it didn't exist), but the
discrepancy was finally, and precisely, accounted for by Einstein's theory of
general relativity.

### Variation of the Momentum Vector in the Orbit (Hodograph)

It's interesting and instructive to track how the *momentum*
vector changes as time progresses, this is easy from the Runge Lenz equation.
(Hamilton did this.)

From , we have

That is,

Staring at this expression, we see that

** goes in a circle
of radius about a point distancefrom the momentum plane origin. **

Of course, is not moving in this circle at a uniform rate (except for a planet in a circular orbit), its angular progression around its circle matches the angular progression of the planet in its elliptical orbit (because its location on the circle is always perpendicular to thedirection from the circle center).

An orbit plotted in momentum space is called a **hodograph**.

### Orbital Energy as a Function of Orbital Parameters Using Runge-Lenz

We'll prove that the total energy, and the time for a complete
orbit, *only depend on the length of the major axis *of the ellipse. So a
circular orbit and a very thin one going out to twice the circular radius take
the same time, and have the same total energy per unit mass.

Take and square both sides, giving

Dividing both sides by ,

Putting in the values found above, we find

So the total energy, kinetic plus potential, depends only on the length of the major axis of the ellipse.

Now for the time in orbit: we've shown area is swept out at a rate , so one orbit takes time , and , so

This is Kepler's famous Third Law: , easily proved for circular orbits, not so easy for ellipses.

### Important Hint!

Always remember that for Kepler problems with a given
massive Sun, both **the time in orbit and the total orbital energy/unit mass only
depend on the length of the major axis**, they are independent of the length
of the minor axis. This can be very useful in solving problems.

### The Runge-Lenz Vector in Quantum Mechanics

This is fully discussed in advanced quantum mechanics texts,
we just want to mention that, just as spherical symmetry ensures that the total
angular momentum and its components commute with the Hamiltonian, and as a
consequence there are degenerate energy levels connected by the raising
operator, an analogous operator can be constructed for the Runge-Lenz vector,
connecting states having the same energy. Furthermore, this raising operator,
although it commutes with the Hamiltonian, does *not* commute with the
total angular momentum, meaning that states with different total angular
momentum can have the same energy. This is the degeneracy in the hydrogen atom
energy levels that led to the simple Bohr atom correctly predicting all the
energy levels (apart from fine structure, etc.). It's also worth mentioning
that these two vectors, angular momentum and Runge-Lenz, both sets of rotation
operators in three dimensional spaces, combine to give a complete set of
operators in a four dimensional space, and the inverse-square problem can be
formulated as the mechanics of a free particle on the surface of a sphere in
four-dimensional space.