In n space dimensions, many orbits can formally be decomposed into n independent periodic motions. Such regular orbits may be represented as winding paths on an n-dimensional torus. On the other hand, irregular or stochastic orbits defy any such representation; such orbits may wander anywhere permitted by conservation of energy.
Constants of motion are functions of phase-space coordinates and time which are constant along orbits:
(1) C(r(t), v(t), t) = constant ,where r(t) and v(t) = dr/dt are a solution to the equations of motion. The function C(r,v,t) must be constant along every orbit, with a value which depends on the orbit. In a phase-space of 2n dimensions there are always 2n independent constants of motion. For example, the 2n initial conditions (r_0,v_0) of an orbit can serve as constants of motion; given phase-space coordinates (r,v) at time t, integrate the orbit backwards to t = 0 and read off the initial (r_0,v_0).
Integrals of motion are functions of phase-space coordinates alone which are constant along orbits:
(2) I(r(t), v(t)) = constant .An integral of motion can't depend on time; thus all integrals are constants of motion, but not all constants are integrals. Integrals are important because they constrain the shapes of orbits. In a phase-space of 2n dimensions, the condition I(r,v) = const. defines a hypersurface of 2n-1 dimensions; thus if there are N independent integrals of motion then each orbit is confined to a sub-space of 2n-N dimensions. Regular orbits are those which have N = n (isolating) integrals.
Consider the motion of a star in a spherically-symmetric potential, Phi = Phi(|r|). The orbit of the star remains in a plane perpendicular to the angular momentum vector, and it's natural to adopt a polar coordinate system; call the coordinates R = |r| and phi. The system has n = 2 degrees of freedom, so the phase space has 4 dimensions.
The equations of motion can be derived by starting with the lagrangian,
1 2 2 2 (3) L(R,phi,Rdot,phidot) = - (Rdot + R phidot ) - Phi(R) , 2where Rdot = dR/dt and phidot = d(phi)/dt. Differentiating with respect to Rdot and phidot yields the momenta conjugate to R and phi,
dL (4) ------- = Rdot = v_R , d(Rdot) dL 2 (5) --------- = R phidot = R v_phi = J ; d(phidot)here v_R and v_phi are velocities in the radial and azimuthal directions. The hamiltonian may now be expressed as a function of the coordinates and conjugate momenta:
1 2 2 2 (6) H(R,phi,v_R,J) = - (v_R + J / R ) + Phi(R) . 2Then the equations of motion are
dR dH (7) -- = ------ = v_r , dt d(v_r) d(phi) dH J (8) ------ = -- = --- , dt dJ R^2 d(v_r) dH d(Phi) J^2 (9) ------ = - -- = - ------ + --- , dt dR dR R^3 dJ dH (10) -- = - ------ = 0 . dt d(phi)Here dJ/dt = 0 because the conjugate coordinate phi does not appear in H; phi is called a cyclic coordinate.
The system has two integrals of motion. One, of course, is the total energy E, numerically equal to the value of H. The other is the angular momentum J. These quantities are given by
2 2 (11) E = - (v_R + v_phi ) + Phi(R) , (12) J = R v_phi .Each of these integrals defines a hypersurface in phase space, and the orbit must remain in the intersection of these hypersurfaces. This can be visualized by ignoring the phi coordinate and drawing surfaces of constant E and J in the three-dimensional space (R,v_R,v_phi). Surfaces of constant E are figures of revolution about the R axis, while surfaces of constant J are hyperbolas in the (r,v_phi) plane. The intersection of these surfaces is a closed curve, and an orbiting star travels around this curve.
For an orbit of a given J, the system may be reduced to one degree of freedom by defining the effective potential,
J^2 (13) Psi(R) = Phi(R) + ----- ; 2 R^2the corresponding equations of motion are then just
dR (14) -- = v_R , dt d(v_R) d(Psi) (15) ------ = - ------ . dt dRBecause Psi(R) diverges as R -> 0, the star is energetically prohibited from coming too close to the origin, and shuttles back and forth between turning points R_min and R_max.
In addition to its periodic radial motion described by Eqs. 14 & 15, a star also executes a periodic azimuthal motion as it orbits the center of the potential. If the radial and azimuthal periods are incommensurate, as is usually the case, the resulting orbit never returns to its starting point in phase space; in coordinate space such an orbit is a rosette (BT87, Fig. 3-1). The Keplerian potential is a very special case in which the radial and azimuthal periods of all bound orbits are equal. The only other potential in which all orbits are closed is the harmonic potential generated by a uniform sphere; here the radial period is half the azimuthal one and all bound orbits are ellipses centered on the bottom of the potential well. Thus in the Keplerian case all stars advance in azimuth by Delta phi = 2 pi between successive pericenters, while in the harmonic case they advance by Delta phi = pi. Galaxies typically have mass distributions intermediate between these extreme cases, so most orbits in spherical galaxies are rosettes advancing by pi < Delta phi < 2 pi between pericenters.
This combination of radial and azimuthal motions can be represented as a path on a 2-torus; that is, on a rectangle with its opposing edges glued together. Associate the azimuthal direction with the long edge of the rectangle, and the closed loop formed by the intersection of E = const. and J = const. with the short edge of the rectangle; then glue opposing edges together to render both coordinates periodic and produce what is known as an invariant torus. Stellar orbits wind solenoid-like on the surface of the torus, typically making between one and two turns through the hole of the torus for each turn made around its circumference.
Points on the surface of the torus may be parameterized by a pair of angles (theta_1,theta_2); moreover, by stretching the torus appropriately, the motion of a star can be described by a pair of linear relations:
theta_1(t) = theta_1(0) + omega_1 t , (16) theta_2(t) = theta_2(0) + omega_2 t ,where the omega_i are integrals of motion. Together, the angles (theta_1,theta_2) and their conjugate angular frequencies or actions (omega_1,omega_2) define a coordinate system for the four-dimensional phase space; in these action-angle variables the equations of motion take their simplest possible form, governed by the hamiltonian
1 2 2 (17) H(theta_1,theta_2,omega_1,omega_2) = - (omega_1 + omega_2 ) . 2
In describing axisymmetric galaxy models it's natural to use cylindrical coordinates (R,phi,z), where R and phi are polar coordinates in the equatorial plane, and z is the coordinate perpendicular to that plane. In these coordinates, the potential has the form Phi = Phi(R,z). The equations of motion are identical to Eqs. 7-10, with additional expressions for z and v_z:
dz (18) -- = v_z , dt d(v_z) d(Phi) (19) ------ = - ------ . dt dz
Once again, there are two classical integrals of motion:
2 2 2 (20) E = - (v_R + v_phi + v_z ) + Phi(R,z) , (21) J_z = R v_phi .
Just as for spherical potentials, it's possible to define an effective potential
J_z^2 (22) Psi(R,z) = Phi(R,z) + ----- . 2 R^2Instead of governing motion along a line as in the spherical case, the effective potential now governs the star's motion in the meridional plane, which rotates about the z axis with angular velocity omega = J_z/R^2. The radial motion is described by Eqs. 14-15, while the vertical motion is described by Eqs. 18-19 with Phi replaced by Psi.
On the meridional plane, which has coordinates (R,z), the effective potential Psi has a minimum at R > 0, z = 0 and a steep angular momentum barrier as R -> 0 (BT87, Fig. 3-2). If only the energy E constrains the motion of a star on this plane, one might expect a star to travel everywhere within some closed contour of constant Psi. But in many cases this is not observed; instead, stars launched from rest at different points along a contour of constant Psi follow distinct trajectories. This implies the existence of a third integral besides the classic integrals given by Eqs. 20-21. No general expression for a third integral exists, but it can be evaluated by following a given orbit until its velocity on the meridional plane vanishes, and noting where on the zero-velocity contour the star lies. The result is a proper integral of motion, rather than just a constant, because this procedure makes no reference to the current time t.
The existence of a third integral implies that a star's orbit is a combination of three periodic motions: radial, azimuthal, and vertical. Thus the orbit can be represented as a path on an invariant 3-torus, with action-angle variables (theta_i,omega_i), where i = 1, 2, 3.
Some axisymmetric potentials also have orbits which wander everywhere energetically permitted on the meridional plane. For such orbits, the description in terms of motion on a invariant 3-torus breaks down. These are examples of irregular or stochastic orbits; in an axisymmetric potential, they respect only the two classical integrals, E and J_z.
Non-axisymmetric potentials, with Phi = Phi(x,y) or Phi(x,y,z) in Cartesian coordinates (x,y,z), admit an even richer variety of orbits. The only classical integral of motion in such a potential is the energy,
1 2 (23) E = - |v| + Phi(r) . 2Some potentials nonetheless permit other integrals of motion, and in such potentials regular orbits may be mapped onto invariant tori. But not all regular orbits can be continuously deformed into one another; consequently, orbits can be grouped in to topologically distinct orbit families. Each regular orbit family will require a different invariant torus. Some sense of the variety of possible orbits in non-axisymmetric galaxies is available by examining orbits in separable and scale-free potentials.
In a separable potential all orbits are regular and the mapping to the invariant tori can be constructed analytically; all integrals of motion are known. Separable potentials are rather special, mathematically speaking, and it's highly unlikely that real galaxies have such potentials. However, numerical experiments show that non-axisymmetric galaxy models with finite cores or shallow cusps usually generate potentials with many key features of separable potentials.
The orbits in a separable potential may be classified into distinct families, each associated with a set of closed and stable orbits. In two dimensions, for example, there are two types of closed, stable orbits; one type (i) oscillates back and forth along the major axis, and the other type (ii) loops around the center. Because these orbits are stable, other orbits which start nearby will remain nearby at later times. The families associated with types (i) and (ii) are known as box and loop orbits, respectively (BT87, Ch. 3.3.1).
In three dimensions, a separable potential permits four distinct orbit families:
In scale-free models all properties have either a power-law or a logarithmic dependence on radius. In particular, scale-free models with density profiles proportional to r^-2 have logarithmic potentials and flat rotation curves. While real galaxies are not entirely scale-free, such steep power-law density distributions are reasonable approximations to the central regions of some elliptical galaxies and to the halos of galaxies in general.
If the density falls as r^-2 or faster, then box orbits are replaced by minor orbital families called boxlets (Gerhard & Binney 1985, Miralda-Escude & Schwarzschild 1989). Each boxlet family is associated with a closed and stable orbit arising from a resonance between the motions in the x and y directions. Moreover, some irregular orbits also exist; these have no integrals of motion apart from the energy E. In principle, such an orbit can wander everywhere on the phase-space hypersurface of constant E, but in actuality such orbits show a complicated and often fractal-like structure.
Last modified: February 8, 1997