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13.2. Epicyclic orbits

In the limit of quasi-circular orbits, which can be quantified as

Equation 17 (13.17)

we have Omegar(E, J) rightarrow kappa(r0), Omegatheta(E, J) rightarrow Omega(r0) and the radial periodic motion can be approximated by a harmonic oscillator with the epicyclic frequency defined by:

Equation 18 (13.18)

(We also have Jr ~ [E - E0(r0)] / kappa(r0), which is analogous to vperp2 / B for a gyrating charged particle in a magnetic field.) Thus if we separate out the motion of the guiding center by writing r(t) = r0 + r1(t) and theta dot(t) = Omega(r0) + theta
dot1(t), the linearized equation for the conservation of angular momentum leads to:

Equation 19 (13.19)

thus the epicycles are ellipses characterized by aspect ratio 2Omega / kappa (therefore they are usually elongated in the direction of the motion), with the star running in the opposite direction with respect to the guiding center (i.e., the motion in the epicycle is clockwise if the motion on the circular orbit at r0 is counter-clockwise).

Note that from Eq. (18) the condition for the stability of circular orbits (kappa2 > 0) formally coincides with the classical Rayleigh's criterion for the stability of a rotating fluid (4). A few important special cases should be noted. A pure harmonic potential (i.e., the mean field potential associated with a homogeneous sphere) implies solid body rotation, in the sense that Omega2 = 4pi G rho/3 = constant; in this case we have kappa = 2Omega, and orbits are closed in the form of ellipses centered at r = 0. A point mass generates a Keplerian potential; from the third law of planetary motion we see that in this case kappa = Omega , and thus orbits are closed in the form of ellipses with one focus at r = 0. For galaxy disks, since they are often characterized by a flat rotation curve, the typical relation should be kappa approx 21/2 Omega, and orbits are generally not closed. Some simple cases of orbits with Omega and kappa in rational ratio are shown in Fig. 13.3.

Figure 13.3

Figure 13.3. Quasi-circular orbits when the ratio of angular to radial frequency is rational (3/2, upper left; 2/3 lower left; 4, upper right; 1/4, lower right). (In a frame rotating with angular velocity Omegap the relevant angular frequency is the frequency in the inertial frame reduced by the value of Omegap)

In Chapter 14 we will show that the velocity distribution for a relatively cool disk, because of the epicyclic constraints, has an anisotropic pressure tensor for which the radial pressure exceeds the tangential pressure by the ratio 4Omega2 / kappa2.

For some purposes (e.g. for some detailed stellar dynamical studies of density waves where an integration along the unperturbed orbits is performed), it is of interest to have a full description of the epicyclic expansion, beyond the lowest order harmonic oscillator obtained by approximating the potential Phieff with a parabola in r0. Such a systematic expansion (5) is obtained by introducing an appropriate phase variable lambda. In order to do this, it is first convenient to consider the transformation (E, J) rightarrow (a, r0), where the dimensionless epicyclic energy a is given by

Equation 20 (13.20)

Thus the radial momentum can be expressed as a function

Equation 21 (13.21)

Now the phase variable lambda is introduced by replacing the radial velocity coordinate pr with

Equation 22 (13.22)

The complete epicyclic expansion is thus obtained by Taylor expansion of Eq. (21) around r = r0, which inserted in Eq. (22) gives

Equation 23 (13.23)

From here one gets the expression for dr / dlambda as well. Then from

Equation 24 (13.24)

Equation 25 (13.25)

one gets the desired expressions for lambda = lambda(t) and theta = theta(t), which completes the derivation. The first terms of the full expansion can be summarized by noting

Equation 26 (13.26)

Equation 27 (13.27)

with B3 = - A3 + 2A2 - 1 and

Equation 28 (13.28)

Here one can easily check that for the harmonic oscillator A2 = 1/2 and A3 = B3 = 0, while for the Keplerian case A2 = 0, A3 = 0, B3 = -1. Note that for the isochrone potential (see Chapter 21) A1 = 1 and all the other An vanish.

Many of these results find application in the study of the dynamics of galaxies. In addition, they are also of interest in some simple problems of celestial mechanics, where the potential Phi is often close to being Keplerian. For example, the potential of the Earth, in space (r > rT), because of its flattening at the poles, is approximately given by

Equation 29 (13.29)

where we have retained only the quadrupole term in the general solution to the Laplace equation (here at r = rT the quantity pi/2 - vartheta represents the geographical latitude); for the Earth one has J20 approx 10-3. We recall the expression for the Legendre polynomial P20(x) = (3x2 - 1)/2. The epicyclic theory easily allows us to study the precession of the perigee of a satellite on the equatorial plane, where vartheta = pi/2. The precession rate is proportional to the difference between kappa and Omega.

4 See, for example, Chandrasekhar, S. (1961), Hydrodynamic and Hydromagnetic Stability, Oxford University Press, Oxford; reprinted by Dover Back.

5 Shu, F.H. (1969), Astrophys. J., 158, 505; Mark, J.W-K. (1976), Astrophys. J., 203, 81. The analysis of Mark removes an undesired secular term present in the original derivation Back.

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