Many disk galaxies contain a bar: a linear structure crossing the disk. Here we analyze bars in terms of the orbits which occur in the potential of a rotating bar, and discuss how such structures may form and evolve.
Bars may be described as strong or weak, depending on the amplitude of the nonaxisymmetric part of the potential. Orbits in strong bars are most easily analyzed in a frame of reference which rotates with the bar, while those in weak bars can be studied by the epicyclic approximation, which illustrates the role of resonances.
The equation of motion for a particle in a frame of reference r = (x,y,z) rotating at angular velocity Omega_b about the z axis is
2 d r d | 0 1 0| (1) --- = - -- Phi_eff + 2 Omega_b |-1 0 0| v , 2 dr | 0 0 0| dtwhere v is the velocity in the rotating frame and the effective potential is
1 2 2 (2) Phi_eff = Phi(r) - - Omega_b R . 2The term proportional to Omega_b in Eq. (1) represents the Coriolis force, which comes about through conservation of angular momentum; thus a particle moving outward with velocity v_x along the x axis accelerates at -2 Omega_b v_x in the y direction. The term proportional to Omega_b^2 in Eq. (2) represents the outward-directed centrifugal pseudoforce present in a rotating frame of reference. Note that when Omega_b = 0 we recover the equations of motion for a non-rotating frame.
The rotating-frame analog of the total energy is the Jacobi integral,
1 2 (3) E_J = - |v| + Phi_eff . 2This quantity is conserved by motion in a rotating frame of reference.
As an example of a strong bar, BT87 use the logarithmic potential
1 2 2 2 2 2 (4) Phi(x,y) = - v_0 ln(R_c + x + y / q ) , 2where v_0 is the asymptotic circular speed, R_c is the core radius, and q is the axial ratio of the potential; q < 1 for a bar elongated along the x axis. A contour plot of Phi_eff for this potential (BT87, Fig. 3-13) displays five places where
d d (5) -- Phi_eff = -- Phi_eff = 0 . dx dyThese equilibrium positions (in the rotating frame of reference) are called the Lagrange points. One, conventionally known as the L3 point, occupies the origin and occurs even in a non-rotating potential. Points L1 and L2 fall along the x axis, while L4 and L5 lie along the y axis; these are the places where the centrifugal pseudoforce balances gravity. L3 is a minimum of Phi_eff and hence is always stable, while L1 and L2 are saddle points and thus are always unstable. The points L4 and L5, though they mark maxima of Phi_eff, are stable for a logarithmic barred potential.
Motion in the vicinity of a Lagrange point (x_L,y_L) may be studied by expanding the effective potential in powers of x-x_L and y-y_L (BT87, Ch. 3.2.2). The key results are outlined here (see also BT87, Ch. 4.6.3). In the special case of a non-rotating potential with a finite core radius, a star near the L3 point executes independent and generally incommensurate harmonic motions in the x and y directions. For the case of a rotating potential the motion may likewise be decomposed into the sum of two periodic motions: one a retrograde motion about an epicycle, and the other a prograde motion of the guiding center. Because two motions are involved, it follows that orbits near the L3 point must have another integral of motion in addition to E_J. Similar results are obtained at the L4 and L5 points when these are stable.
Numerical integration of Eq. (1) provides a way to study orbits which do not stay close to a Lagrange point (e.g. Contopoulos & Papayannopoulos 1980). Just as in the earlier discussion of orbits in triaxial systems, here too each closed, stable orbit parents an orbit family. Close to the core of a barred potential the only important orbit families are the prograde x_1 family, which is aligned with the bar, and the retrograde x_4 family, which is nearly circular. For slightly smaller values of -E_J two new types of closed orbits may arise (BT87, Fig. 3-17): the stable x_2 orbits and the unstable x_3 orbits. Both are elongated perpendicular to the bar, but only the x_2 orbits, which are rounder than x_3 orbits of the same E_J, can parent an orbit family. At yet-smaller values of -E_J these perpendicular orbits disappear, and finally the x_1 orbits likewise vanish when -E_J is small enough for the star to reach the L1 and L2 points. At comparable values of -E_J one may also find closed orbits circling the L4 and L5 points.
To understand the effects of resonances, consider the driven harmonic oscillator,
2 d x i Omega_1 t (6) --- + k x = alpha e , 2 dtwhere k is the spring constant, alpha is the amplitude of the driving force, and Omega_1 is the driving frequency. The solution has the form
i Omega_0 t i Omega_1 t (7) x(t) = e + A e ,where by direct substitution it follows that
2 (8) Omega_0 = k ,and
2 2 (9) A = alpha / (Omega_0 - Omega_1) .Thus if the driving force is zero the system oscillates at its natural frequency Omega_0, while if |alpha| > 0 there is also an oscillation at the driving frequency Omega_1. As Omega_1 -> Omega_0 the amplitude of the driven response diverges. Notice also that the sign of the response changes on moving through a resonance -- there is a phase shift of pi radians.
Now consider the effect of a bar with pattern speed Omega_b on a star moving in a circular orbit with angular speed Omega. Relative to the star, the angular speed of the bar is Omega - Omega_b, and because the bar is bisymmetric the star feels a perturbation at twice this angular speed. This perturbation is in resonance with the star's epicyclic frequency kappa if
(10) 2 |Omega - Omega_b| = kappa .Because Omega and kappa depend on radius, there are specific radii in the disk where Eq. (10) is satisfied. At the outer Lindblad resonance (OLR),
(11) Omega_b = Omega + kappa/2 .Depending on the rotation curve and on the value of Omega_b, there may be zero, one, or two inner Lindblad resonances (ILRs) where
(12) Omega_b = Omega - kappa/2 .Finally, there is the corotation resonance (CR) where
(13) Omega_b = Omega .
To link these results with the above discussion of orbits in strong bars, note that these resonances mark transitions between orbital families. If the pattern speed of the bar is higher than the peak value of Omega - kappa/2 then no LIRs exist and the x_1 family extends all the way from the origin to the CR. The x_2 family, on the other hand, occurs only at those radii where Omega_b < Omega - kappa/2. Because only the x_1 family is elongated with the bar, we may guess that bars in disk galaxies have pattern speeds greater than the maximum value of Omega - kappa/2 (BT87, Ch. 6.5.1(a)).
Last modified: March 20, 1995