3.4: Planetary Problems
( \newcommand{\kernel}{\mathrm{null}\,}\)
Leaving a more detailed study of oscillations for Chapter 5, let us now discuss the so-called planetary systems 10 whose description, somewhat surprisingly, may be also reduced to an effectively 1D problem. Consider two particles that interact via a conservative, central force F21=−F12=nrF(r), where r and nr are, respectively, the magnitude and direction of the distance vector r≡r1−r2 connecting the two particles (Figure 3).

Generally, two particles moving without constraints in 3D space, have 3+3=6 degrees of freedom, which may be described, e.g., by their Cartesian coordinates {x1,y1,z1,x2,y2,z2} However, for this particular form of interaction, the following series of tricks allows the number of essential degrees of freedom to be reduced to just one.
First, the central, conservative force of particle interaction may be described by a timeindependent potential energy U(r), such that F(r)=−∂U(r)/∂r.11 Hence the Lagrangian of the system is L≡T−U(r)=m12˙r21+m22˙r22−U(r). Let us perform the transfer from the initial six scalar coordinates of the particles to the following six generalized coordinates: three Cartesian components of the distance vector r≡r1−r2, and three scalar components of the following vector: R≡m1r1+m2r2M, with M≡m1+m2, which defines the position of the center of mass of the system, with the total mass M. Solving the system of two linear equations (31) and (32) for r1 and r2, we get r1=R+m2Mr,r2=R−m1Mr. Plugging these relations into Eq. (30), we see that it is reduced it to L=M2˙R2+m2˙r2−U(r), where m is the so-called reduced mass: m≡m1m2M, so that 1m≡1m1+1m2. Note that according to Eq. (35), the reduced mass is lower than that of the lightest component of the two-body system. If one of m1,2 is much less than its counterpart (like it is in most star-planet or planetsatellite systems), then with a good precision m≈min[m1,m2].
Since the Lagrangian function (34) depends only on R rather than R itself, according to our discussion in Sec. 2.4, all Cartesian components of R are cyclic coordinates, and the corresponding generalized momenta are conserved: Pj≡∂L∂˙Rj≡M˙Rj=const,j=1,2,3. Physically, this is just the conservation law for the full momentum P≡MR of our system, due to the absence of external forces. Actually, in the axiomatics used in Sec. 1.3 this law is postulated - see Eq. (1.10) - but now we may attribute the momentum P to a certain geometric point, with the center-of-mass radius vector R. In particular, since according to Eq. (36) the center moves with a constant velocity in the inertial reference frame used to write Eq. (30), we may consider a new inertial frame with the origin at point R. In this new frame, R≡0, so that the vector r (and hence the scalar r ) remain the same as in the old frame (because the frame transfer vector adds equally to r1 and r2, and cancels in r=r1−r2 ), and the Lagrangian (34) is now reduced to L=m2˙r2−U(r) Thus our initial problem has been reduced to just three degrees of freedom - three scalar components of the vector r. Moreover, Eq. (37) shows that dynamics of the vector r of our initial, twoparticle system is identical to that of the radius vector of a single particle with the effective mass m, moving in the central potential field U(r).
Two more degrees of freedom may be excluded from the planetary problem by noticing that according to Eq. (1.35), the angular momentum L=r×p of our effective single particle of mass m is also conserved, both in magnitude and direction. Since the direction of L is, by its definition, perpendicular to both of r and v=p/m, this means that the particle’s motion is confined to the plane whose orientation is determined by the initial directions of the vectors r and v. Hence we can completely describe particle’s position by just two coordinates in that plane, for example by the distance r to the origin, and the polar angle φ. In these coordinates, Eq. (37) takes the form identical to Eq. (2.49): L=m2(˙r2+r2˙φ2)−U(r). Moreover, the latter coordinate, polar angle φ, may be also eliminated by using the conservation of angular momentum’s magnitude, in the form of Eq. ( 2.50):12 Lz=mr2˙φ= const.
A direct corollary of this conservation is the so-called 2nd Kepler law: 13 the radius vector r sweeps equal areas A in equal times. Indeed, in the linear approximation in dA<<A, the area differential dA is equal to the area of a narrow right triangle with the base being the arc differential rdφ, and the height equal to r-see Figure 4. As a result, according to Eq. (39), the time derivative of the area, dAdt=r(rdφ)/2dt≡12r2˙φ=Lz2m, remains constant. Since the factor Lz/2m is constant, integration of this equation over an arbitrary (not necessarily small!) time interval Δt proves the 2nd Kepler law: A∝Δt.

Now note that since ∂L/∂t=0, the Hamiltonian function H is also conserved, and since, according to Eq. (38), the kinetic energy of the system is a quadratic-homogeneous function of the generalized velocities ˙r and ˙φ, we have H=E, so that the system’s energy E, E=m2˙r2+m2r2˙φ2+U(r), is also a first integral of motion. 14 But according to Eq. (39), the second term on the right-hand side of Eq. (41) may be represented as m2r2˙φ2=L2z2mr2, so that the energy (41) may be expressed as that of a 1D particle moving along axis r, E=m2˙r2+Uef (r), in the following effective potential: Uef (r)≡U(r)+L2z2mr2 (The physical sense of the second term is similar to that of the first term in the Uef spelled out in Eq. (6), and will be discussed again in Sec. 4.6 below.) So the planetary motion problem has been reduced to the dynamics of an effectively-1D system. 15
Now we may proceed just like we did in Sec. 3, with due respect to the very specific effective potential (44) which, in particular, diverges at r→0 - besides the very special case of an exactly radial motion, Lz=0. In particular, we may solve Eq. (43) for dr/dt to get dt=(m2)1/2dr[E−Uef(r)]1/2 This equation allows us not only to get a direct relationship between time t and distance r, similar to Eq. (26), t=±(m2)1/2∫dr[E−Uef(r)]1/2=±(m2)1/2∫dr[E−U(r)−L2z/2mr2]1/2, but also do a similar calculation of the angle φ. Indeed, integrating Eq. (39), φ≡∫˙φdt=Lzm∫dtr2, and plugging dt from Eq. (45), we get an explicit expression for the particle trajectory φ(r) : φ=±Lz(2m)1/2∫drr2[E−Uef(r)]1/2=±Lz(2m)1/2∫drr2[E−U(r)−L2z/2mr2]1/2. Note that according to Eq. (39), the derivative dφ/dt does not change sign at the reflection from any classical turning point r≠0, so that, in contrast to Eq. (46), the sign on the right-hand side of Eq. (48) is uniquely determined by the initial conditions and cannot change during the motion.
Let us use these results, valid for any interaction law U(r), for the planetary motion’s classification. (Following a good tradition, in what follows I will select the arbitrary constant in the potential energy in the way to provide U→0, and hence Uef →0 at r→∞.) The following cases should be distinguished.
If U(r)<0, i.e. the particle interaction is attractive (as it always is in the case of gravity), and the divergence of the attractive potential at r→0 is faster than 1/r2, then Uef(r)→−∞ at r→0, so that at appropriate initial conditions the particle may drop on the center even if Lz≠0 - the event called the capture. On the other hand, with U(r) either converging or diverging slower than 1/r2, at r→0, the effective energy profile Uef(r) has the shape shown schematically in Figure 5. This is true, in particular, for the very important case U(r)=−αr, with α>0 which describes, in particular, the Coulomb (electrostatic) interaction of two particles with electric charges of opposite signs, and the Newton gravity law (1.15). This particular case will be analyzed below, but now let us return to the analysis of an arbitrary attractive potential U(r)<0 leading to the effective potential shown in Figure 5 , when the angular-momentum term in Eq. (44) dominates at small distances r.

Figure 3.5. Effective potential profile of an attractive central field, and two types of motion in it.
According to the analysis in Sec. 3, such potential profile, with a minimum at some distance r0, may sustain two types of motion, depending on the energy E (which is determined by initial conditions):
(i) If E>0, there is only one classical turning point where E=Uef , so that the distance r either grows with time from the very beginning or (if the initial value of ˙r was negative) first decreases and then, after the reflection from the increasing potential Uef, starts to grow indefinitely. The latter case, of course, describes the scattering of the effective particle by the attractive center. 16
(ii) On the opposite, if the energy is within the range Uef(r0)≤E<0, the system moves periodically between two classical turning points rmin and rmax− see Figure 5. These oscillations of the distance r correspond to the bound orbital motion of our effective particle about the attracting center.
Let us start with the discussion of the bound motion, with the energy within the range (50). If the energy has its minimal possible value, E=Uef(r0)≡min[Uef(r)], the distance cannot change, r=r0= const, so that the orbit is circular, with the radius r0 satisfying the condition dUef/dr=0. Using Eq. (44), we see that the condition for r0 may be written as L2zmr30=dUdr|r=r0. Since at circular motion, the velocity v is perpendicular to the radius vector r,Lz is just mr0v, the lefthand side of Eq. (52) equals mv2/r0, while its right-hand side is just the magnitude of the attractive force, so that this equality expresses the well-known 2nd Newton law for the circular motion. Plugging this result into Eq. (47), we get a linear law of angle change, φ=ωt+ const, with the angular velocity ω=Lzmr20=vr0, and hence the rotation period τφ≡2π/ω obeys the elementary relation τφ=2πr0v. Now let the energy be above its minimum value (but still negative). Using Eq. (46) just as in Sec. 3, we see that the distance r now oscillates with the period τr=(2m)1/2∫rmaxrmindr[E−U(r)−L2z/2mr2]1/2. This period is not necessarily equal to another period, Tφ, that corresponds to the 2π-change of the angle. Indeed, according to Eq. (48), the change of the angle φ between two sequential points of the nearest approach, |Δφ|=2Lz(2m)1/2∫rmaxrmindrr2[E−U(r)−L2z/2mr2]1/2 is generally different from 2π. Hence, the general trajectory of the bound motion has a spiral shape see, e.g., an illustration in Figure 6.

The situation is special, however, for a very important particular case, namely that of the Coulomb potential described by Eq. (49). Indeed, plugging this potential into Eq. (48), we get φ=±Lz(2m)1/2∫drr2(E+α/r−L2z/2mr2)1/2. This is a table integral, 17 giving φ=±cos−1L2z/mαr−1(1+2EL2z/mα2)1/2+ const. The reciprocal function, r(φ), is 2π-periodic: r=p1+ecos(φ+const), so that at E<0, the orbit is a closed line, 18 characterized by the following parameters: 19 p≡L2zmα,e≡(1+2EL2zmα2)1/2 The physical meaning of these parameters is very simple. Indeed, the general Eq. (52), in the Coulomb potential for which dU/dr=α/r2, shows that p is just the circular orbit radius 20 for the given Lz:r0=L2z/mα≡p, so that min[Uef(r)]≡Uef(r0)=−α2m2L2z. Using this equality together with the second of Eqs. (60), we see that the parameter e (called the eccentricity) may be represented just as e={1−Emin[Uef(r)]}1/2. Analytical geometry tells us that Eq. (59), with e<1, is one of the canonical representations of an ellipse, with one of its two focuses located at the origin. The fact that planets have such trajectories is known as the 1st Kepler law. Figure 7 shows the relations between the dimensions of the ellipse and the parameters p and e.21

In particular, the major semi-axis a and the minor semi-axis b are simply related to p and e and hence, via Eqs. (60), to the motion integrals E and Lz :
a=p1−e2=α2|E|,b=p(1−e2)1/2=Lz(2m|E|)1/2. As was mentioned above, at E→min[Uef(r)] the orbit is almost circular, with r(φ)≅r0≈p. On the contrary, as E is increased to approach zero (its maximum value for the closed orbit), then e→1, so that the aphelion point rmax=p/(1−e) tends to infinity, i.e. the orbit becomes extremely extended − see the red lines in Figure 8 .

The above relations enable, in particular, a ready calculation of the rotation period T≡Tr=τφ. (In the case of a closed trajectory, τr and τφ coincide.) Indeed, it is well known that the ellipse area A= πab. But according to the 2nd Kepler law (40), dA/dt=Lz/2m= const. Hence τ=AdA/dt=πabLz/2m. Using Eqs. (60) and (63), this important result may be represented in several other forms: τ=πp2(1−e2)3/2(Lz/2m)=πα(m2|E|3)1/2=2πa3/2(mα)1/2. Since for the Newtonian gravity (1.15), α=Gm1m2=GmM, at m1<<m2 (i.e. m<<M this constant is proportional to m, and the last form of Eq. (64b) yields the 3rd Kepler law: periods of motion of different planets in the same central field, say that of our Sun, scale as τ∝a3/2. Note that in contrast to the 2nd Kepler law (which is valid for any central field), the 1st and the 3rd Kepler laws are potentialspecific.
Now reviewing the above derivation of Eqs. (59)-(60), we see that they are also valid in the case of E≥0 - see the top horizontal line in Figure 5 and its discussion above, if we limit the results to the physically meaningful range r≥0. This means that if the energy is exactly zero, Eq. (59) (with e=1 ) is still valid for all values of φ (except for one special point φ=π where r becomes infinite) and describes a parabolic (i.e. open) trajectory - see the magenta lines in Figure 8.
Moreover, if E>0, Eq. (59) is still valid within a certain sector of angles φ, Δφ=2cos−11e≡2cos−1(1+2EL2zmα2)−1/2<π, for E>0, and describes an open, hyperbolic trajectory (see the blue lines in Figure 8). As was mentioned earlier, such trajectories are typical, in particular, for particle scattering.
10 This name is very conditional, because this group of problems includes, for example, charged particle scattering (see Sec. 3.7 below).
11 See, e.g., MA Eq. (10.8) with ∂/∂θ=∂/∂φ=0.
12 Here index z stands for the coordinate perpendicular to the motion plane. Since other components of the angular momentum equal zero, the index is not really necessary, but I will still use it - just to make a clear distinction between the angular momentum Lz and the Lagrangian function L.
13 This is one of the three laws deduced, from the extremely detailed astronomical data collected by Tycho Brahe (1546-1601), by Johannes Kepler in the early 17th century. In turn, the three Kepler laws have become the main basis for Newton’s discovery, a few decades later, of the gravity law (1.15). That relentless march of physics...
14 One may argue that this fact should have been evident from the very beginning because the effective particle of mass m moves in a potential field U(r), which conserves energy.
15 Note that this reduction has been done in a way different from that used for our testbed problem (shown in Figure 2.1) in Sec. 2 above. (The reader is encouraged to analyze this difference.) To emphasize this fact, I will keep writing E instead of H here, though for the planetary problem we are discussing now, these two notions coincide.
16 In the opposite case when the interaction is repulsive, U(r)>0, the addition of the positive angular energy term only increases the trend, and the scattering scenario is the only one possible.
17 See, e.g., MA Eq. (6.3a).
18 It may be proved that for the power-law interaction, U∝rv, the orbits are closed curves only if v=−1 (our current case of the Coulomb potential) or if v=+2 (the 3D harmonic oscillator) - the so-called Bertrand theorem.
19 Let me hope that the difference between the parameter p and the particle momentum’s magnitude is absolutely clear from the context, so that using the same (traditional) notation for both notions cannot lead to confusion.
20 Mathematicians prefer a more solemn terminology: the parameter 2p is called the latus rectum of the ellipse.
21 In this figure, the constant participating in Eqs. (58)-(59) is assumed to be zero. A different choice of the constant corresponds just to a different origin of φ, i.e. a constant turn of the ellipse about the origin.